Algebra without an actual infinity is proposed for applied mathematics.The first four chapters present an allnumber mat
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Table of contents :
Contents
Preface
Finite Precision for Numbers
Chapter 1  Numbers
1.2 GeometricVectors
1.3 Quaternions
1.4 Translation Back to Geometry
1.5 SingularLabelNumbers
1.6 Exercises  Numbers
Chapter 2  Particles
2.2 Inertial Reference Frame
2.3 The UnspecifiedSpeedParameter
2.4 CompoundLabelNumbers and Components
2.5 Adding Hyperbolic Angles
2.6 Energy, Time Dilation, Length Contraction
2.7 SpaceLike and TimeLike Invariants
2.8 Electric Current Density
2.9 Motion Faster than Light
2.10 AntiMatter
2.11 Distributed Material Theory
2.12 Exercises  Particles
Chapter 3  Fields
3.2 AllNumber Notation
3.3 Gauges and SuperPotentials
3.4 Lorentz Transformation
3.5 BiotSavart Law
3.6 Electric EnergyMomentum of an Electron
3.7 Maxwell's Wave Equation
3.8 Forces Using GeometricVector Notation
3.9 Force Density Invariant
3.10 Area and Volume Differential Operators
3.11 Exercises  Fields
Chapter 4  Waves
4.2 Development of the Dirac Equation
4.3 Solutions to the Dirac Equation
4.4 Particle Properties
4.5 Two Alternative Arrangements
4.6 Lorentz Transformation of a Dirac Spinor
4.7 Exercises  Waves
Chapter 5  Proposed Theory
5.2 Cantor's Theory of Infinite Sets
5.3 Algebra Field for LocalReal Numbers
5.4 Lorentz Transformation with NonFinite #s
5.5 Dirac Equation Development
5.6 Force Density Using the ComplexConjugate
5.7 Spin of a Photon
5.8 Exercises  Proposed Theory
Appendix A  Octonions and Sedonions
Appendix B  Spooky Action at a Distance
Appendix C  Discovering an Abstraction
The Storybook
Glossary
Index
Back Cover
SPECIAL ALGEBRA FOR
SPECIAL RELATIVITY
SECOND EDITION
2 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
The other book by Paul C Daiber: ALIEN INVASION MATH STORY
SPECIAL ALGEBRA FOR
SPECIAL RELATIVITY Proposed Theory of NonFinite Numbers
Paul C Daiber
SECOND EDITION
4 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Copyright © 2020 by Paul C Daiber All rights reserved. This book or any portion thereof may not be reproduced or used in any manner whatsoever without the express written permission of the publisher except for the use of brief quotations in a book review, scholarly journal or other critical document. Daiber, Paul C, 1960 – Special Algebra for Special Relativity p. cm. Includes index. Paperback ISBN 9798698633808 1. Special Relativity, Electricity, Waves, Algebra, Mathematics, Infinity, Math, Abstraction I. Title.
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For My Wife Sue
Table of Contents
Finite Precision for Numbers ........................................................................ix Chapter 1 – Numbers .................................................................................. 1 1.1 Process from Descartes ......................................................................... 1 1.2 GeometricVectors ................................................................................ 2 1.3 Quaternions ......................................................................................... 4 1.4 Translation Back to Geometry.............................................................. 19 1.5 SingularLabelNumbers ...................................................................... 20 1.6 Exercises ............................................................................................ 21 Chapter 2 – Particles ................................................................................. 33 2.1 HypercomplexPlane ........................................................................... 33 2.2 Inertial Reference Frames ................................................................... 36 2.3 The UnspecifiedSpeedParameter ....................................................... 38 2.4 CompoundLabelNumbers and Components ........................................ 39 2.5 Adding HyperbolicAngles.................................................................... 42 2.6 Energy, Time Dilation, Length Contraction ........................................... 46 2.7 SpaceLike and TimeLike Invariants .................................................... 48 2.8 Electric Current Density ....................................................................... 52 2.9 Motion Faster than Light ..................................................................... 55 2.10 AntiMatter ........................................................................................ 64 2.11 Distributed Material Theory ............................................................... 72
2.12 Exercises .......................................................................................... 83 Chapter 3 – Fields ..................................................................................... 89 3.1 GeometricVector Notation .................................................................. 89 3.2 AllNumber Notation ........................................................................... 94 3.3 Gauges and SuperPotentials ............................................................. 105 3.4 Lorentz Transformation ..................................................................... 108 3.5 BiotSavart Law ................................................................................ 116 3.6 Electric EnergyMomentum of an Electron .......................................... 119 3.7 Maxwell’s Wave Equation .................................................................. 126 3.8 Forces Using GeometricVector Notation............................................. 132 3.9 Force Density Invariant ..................................................................... 133 3.10 Area and Volume Differential Operators ............................................ 143 3.11 Exercises ........................................................................................ 151 Chapter 4 – Waves .................................................................................. 159 4.1 Differential Operator ......................................................................... 159 4.2 Development of the Dirac Equation .................................................... 162 4.3 Solutions to the Dirac Equation .......................................................... 166 4.4 Particle Properties ............................................................................. 169 4.5 Two Alternative Arrangements ........................................................... 173 4.6 Lorentz Transformation of a Dirac Spinor ........................................... 175 4.7 Exercises .......................................................................................... 181 Chapter 5 – Proposed Theory ................................................................... 187 5.1 LocalReal Numbers .......................................................................... 187 5.2 Cantor’s Theory of Infinite Sets.......................................................... 196 5.3 Algebra Field for LocalReal Numbers ................................................. 204 5.4 Lorentz Transformation with NonFinite Numbers ............................... 209 5.5 Dirac Equation Development.............................................................. 229
5.6 Force Density Using the ComplexConjugate ...................................... 237 5.7 Spin of a Photon ............................................................................... 245 5.8 Exercises .......................................................................................... 248 Appendix A – Octonions and Sedonions .................................................... 257 Appendix B – Spooky Action at a Distance ................................................ 277 Appendix C – Discovering an Abstraction .................................................. 289 The Storybook ........................................................................................ 295 Glossary ................................................................................................. 296 Index ..................................................................................................... 307 Back Cover ............................................................................................. 310
Preface The first four chapters of Special Algebra for Special Relativity present an allnumber mathematical structure for Special Relativity. The fifth chapter restricts a measurable quantity to finite precision by limiting placevalue digits to a maximum count before and after the decimal point. For example, each side of a unit square has small magnitude finite imprecision added to it. The finite imprecision adder is a nonfinite type of number because it isn’t knowable. Finite imprecision larger than a measurable quantity is the division reciprocal of small magnitude imprecision. In Special Relativity large magnitude imprecision is added to timespace hyperbolic angle “” (that relates to speed by “v = c*tanh”) using a Lorentz Transformation. Large magnitude imprecision models electromagnetism by uniting Maxwell’s Equations with the Dirac Equation. Precision improves with time to cause measurable dynamics. Electromagnetic field force density components are calculated using the same process by which electric current density components are calculated. Included are energy density and Poynting Vector components. Uniting those three empirically derived electromagnetic phenomena into one mathematical model is new and that success suggests quantities in our geometric world actually do have finite imprecision and suggests finite imprecision numbers should also apply to more modern theories of physics.
ix SYNOPSIS – FINITE PRECISION FOR NUMBERS
Finite Precision for Numbers Real numbers are replaced by rational numbers for which the count of known or knowable placevalue digits before and after the decimal point is limited to Aristotle’s potential infinity (the finite, temporary, largest natural number). Outside that limit the rational number is unknown and unknowable. The small magnitude unknown portion is finite because Aristotle’s potential infinity is finite. The large magnitude unknown portion is the division reciprocal of the small portion. In Special Relativity, energy and momentum of an electron is found by a Lorentz Transformation in which the trivially small unknown portion is added to rational timespace hyperbolic angle “” (alpha, that relates to speed by “v = c*tanh”). Simultaneously, energy and momentum of a photon is found by adding the large magnitude unknown portion to “”. Both together in one expression means an electron and photon are one in the same particle. Similarly, the Dirac Equation is Lorentz Transformed with the large magnitude unknown portion to result in Maxwell’s Equations. The calculation that finds measurable electric current density components results in electromagnetic force density components. Included in force density are empirically derived electromagnetic energy density and Poynting Vector components. Uniting those electromagnetic phenomena into one derived mathematical model is new and it suggests continuum quantities in our physical world actually do have finite precision. This new construction of numbers can be applied to more modern theories of physics.
The algebra field for rational numbers requires • • •
Any sum or product of two rational numbers be a rational number, as well as negatives and reciprocals (except division by zero) Zero and one be included as identity elements Commutative and associative properties for addition and multiplication as well as the distributive property of multiplication over addition
x SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Irrational numbers “2” and “log23” are included in the rational numbers’ algebra field because none of the above criteria are violated. To prove “2” is irrational, set “2 = p/q” from which “p2 = 2*q2”, from which “p” must be even and “q” must be even, and that observation is inconsistent with the ability of either “p” or “q” to be odd because both can be divided by “2” until one is odd. Inconsistent observations mean no ratio “p/q” of natural numbers “{1, 2, 3, …}” possibly equals “2”. To prove “log23 = p/q” is irrational, derive “2^(log23) = 2^(p/q)” and then “2p = 3q” and observe no natural numbers “{1, 2, 3, …}” for “p” and “q” apply because even number “2p” cannot equal odd number “3q”. Irrational numbers with rational numbers form real numbers, and the algebra field as defined for rational numbers applies to real numbers. Cantor defined real numbers in the late 1800’s by stating real numbers had quantity “2^N0” over any finite or infinite (N0) interval. “N0” (called “aleph null”) was forced to be positive actual infinity through his Continuum Hypothesis: No set has a quantity between “N0” and “2^N0”. Attempt to prove “2” and “log23” are included in Cantor’s set of real numbers. If “2” equals a ratio of two infinity numbers, both are “2^N0” and so, maybe, both are even and cannot be odd, and therefore “2” appears excluded. If “log23” equals a ratio of two infinity numbers, then “2^(2^N0)” equates to “3^(2^N0)”, but there is no algebra to calculate “3^(2^N0)” because the later developed algebra field theory applies only to finite numbers. Perhaps, we are mistakenly forcing “N0” to be finite. Cantor’s general approach of organizing numbers into sets was structured into Axiomatic Set Theory. Axiom of Infinity addressed only Aristotle’s finite potential infinity, and not positive actual infinity. To insert actual infinity into Axiomatic Set Theory, Cantor’s Continuum Hypothesis was added as another axiom, effectively. Also, Axiomatic Set Theory had no axiom that addressed reciprocal of zero, other than excluding division by zero from rational numbers. Propose a reciprocalofzero axiom to specify calculations not possible: No operation that includes reciprocal of integer zero can result in a finite number. Per the proposed axiom, “0/0” and “1/0  1/0” are not permitted operations. Also “1/0 + 1/0” is not permitted because the positive or negative feature of “0” is unspecified. “1/0 + 7 = 1/0” and “(1/0)*7 = 1/0” are accepted. Also “2^(1/0) = 3^(1/0) = 0 or 1/0”. “1/0” is the absolute maximum magnitude of numbers and is both or either positive and negative.
xi SYNOPSIS – FINITE PRECISION FOR NUMBERS “1/0” is an irrational number and applies to proofs of irrationality. For “2”, “1/0” is both even and odd because “1/0 = 1/0 + 1”. For “log23”, “2^(1/0) = 3^(1/0)”. “p/q = 0/0” is not permitted, therefore, “q*2 = p” and “q*log23 = p” apply instead, as “(1/0)*2 = 1/0” and “(1/0)*log23 = 1/0”, to state “1/0” is the smallest number (other than “0”) multiplied by an irrational number for which the result does not have contribution after the decimal point. “1/0” is larger than positive “N0”, “0 < N0 < 1/0”, because if “N0 < 2^N0” then “N0 2*N0”, “N0 1/0”, and “2^N0 1/0”. Another property of “N0” is no contribution after the decimal point, analogous to a natural number, because “N0” is the quantity of members in a set. Given those two properties, neither “(2^N0)*2” nor “(2*N0)*log23” can equal “2^N0”, and that proves irrational numbers are not included in Cantor’s set of real numbers. The proof required the proposed reciprocalofzero axiom so that an irrational number had a quantity “1/0” nonpattern placevalue digits after the decimal point. It supposed “1/0 + N0 = 1/0”, “0*N0 = 0”.
Algebra for calculating with “1/0” was set up to remove actual infinity “N0” from numbers. “1/0” is not rational, and because continuum quantities in our physical world are rational numbers to which imprecision is added, “1/0” does not exist as a physical quantity. Assume finite potential infinity “Lmax” is the quantity of placevalue digits before and after a decimal point in a rational number that can be nonzero. Use base two (all digits “0”’s and “1”’s). For example, if adjacent unit lengths on a square are “f” and “g”, then, idealistically, “f = g = 1”, but when the square is constructed in the physical world “f” and “g” each have finite imprecision added and therefore “f g”. Added imprecision is a localzero “CminC0”. To construct “CminC0”, have quantity “Lmax” zeros after the decimal point. Starting at “Lmax + 1”, placevalue digits are all “d” for which “d = b  b”. “b” can be either “0” or “1” with that value unknowable until an observation is made, analogous to Schrödinger’s Cat. Each “b” is independent. For “Lmax = 3”, “CminC0 = 0.000dddd……”. Note that “CminC0 < 2^Lmax”.
xii SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Six dots “……” represents quantity “1/0”. For example, “……” represents the quantity of zeros after the decimal point for an integer. To create an integer, truncate the number at the decimal point and discard in one bulk operation, and that contrasts to three dots “…” which represent one at a time in a count, as for finite natural numbers. Another example of six dots is the quantity of nonpattern placevalue digits for an irrational number. Also, for example, “1 = 0.9999……” to contrast with “1 0.9999…” because “0.9999…” has a localzero added to it.
Probability Distribution for “Lmax = 0” for localzero “CminC0 = 0.dddd……” left and localinfinity “CmaxC = 1/CminC0” right. Higher values of “Lmax” make the spike taller and make the two arms further apart.
As time increases, potential infinity “Lmax” increases in value, and each “d” in sequence becomes “1”, “0”, “0”, or “1”. If each “d” of a localzero is randomly replaced with “1”, “0”, “0”, or “1”, then, because there is quantity “1/0” of “d”’s (per six dots “……”), not all will be zero. Therefore, a division reciprocal of a localzero exists, called a localinfinity, “CmaxC = 1/CminC0”. Localinfinity “CmaxC” is both positive and negative. Also, magnitude of a localinfinity is larger than “2^Lmax” and so is larger than what is possible for a rational number. A localreal number is a rational number (called a truncated number) plus either “CminC0” or “CmaxC” with the selection unknowable. Localreal numbers replace real numbers.
xiii SYNOPSIS – FINITE PRECISION FOR NUMBERS
Mechanical energymomentum components and timespace location components for an electron are given below, respectively. mB*c*coshM + q*mB*c*sinhM
;
c*tB*coshM + q*c*tB*sinhM
“mB” is electron rest mass. “tB” is time on a clock mounted on the electron. “c” is speedoflight. Electron speed “vM” relates to hyperbolic angle “M” by “vM = c*tanhM”. “q” is a 2x2 Pauli Spin Matrix conforming to “q2 = 1” and indicates direction in space. Reference frame “B” is stationary with respect to the particle. Reference frame “M” is moving, for example, the interior of a bus in which the electron moves along the floor toward the front with speed “vM”. Reference frame “S” is stationary, for example, the roadside where observer (you or me) is located. Speed of the bus relative to the roadside is “vS/M = c*tanhS/M”. Hyperbolic angle “M” is a rational truncated number. Use Lorentz Transformation hyperbolic angle “S/M = (1  q)*”. Imprecision term “” (xi) is either a localinfinity “CmaxC” or else a localzero “CminC0”. The “1  q” factor keeps components finite and applies because rest mass is a truncated number, as is time measured on the electron’s clock. Lorentz Transformation “S = M + S/M” has First Case for “” maxC “C ” and Second Case for “” “CminC0”. The Second Case is trivial because “CminC0” is zero, so that “S/M” is zero such that the electron when observed in “S” is the same as when observed in “M”. In contrast, the First Case (with “C” positive, “+”) has the electron observed as a photon because energy component “mB*c*exp(M)/2” equals momentum component “mB*c*exp(M)/2”, and time component “c*tB*exp(M)/2” equals space component “c*tB*exp(M)/2”. mB*c*coshS + q*mB*c*sinhS mB*c*exp(M)/2 + q*mB*c*exp(M)/2
First Case “C” positive
c*tB*coshS + q*c*tB*sinhS c*tB*exp(M)/2 + q*c*tB*exp(M)/2
First Case “C” positive
xiv SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Electron dynamics are modeled by the Dirac Equation. The Dirac Equation is developed below with a First Case and a Second Case. Begin with a relationship of momentum components per the Pythagorean Theorem, and split it into two equations to form a matrix equation. (mB*c*(1 + coshS)*PPS)*(mB*c*(1 + coshS)*QQS) = (q*mB*c*sinhS*QQS)*(q*mB*c*sinhS*PPS) (mB*c*(1 + coshS))
(q*mB*c*sinhS)
PPS *
( q*mB*c*sinhS) (mB*c*(1 + coshS))
0 =
QQS
0
For the First Case, rest mass “mB” terms are zero because the “q” in “S/M = (1  q)*” creates zero factor “exp(CmaxC)” (for “C” “+”). And, infinity factor “exp(CmaxC)” cancels that zero factor so that “mB*c*coshS = mB*c*exp(M)/2” and “mB*c*sinhS = mB*c*exp(M)/2”. To account for electric charge, the rightside is given nonzero value “a”. (mB*c*exp(M)/2) (q*mB*c*exp(M)/2)
PPS *
(q*mB*c*exp(M)/2) (mB*c*exp(M)/2)
a =
QQS
q*a
Mechanical equals total minus electrical. mB*c*cosh(S) = i*ħ*tS  QB*VtS q*mB*c*sinh(S) = qx*(i*ħ*xS  QB*VxS) + qy*(i*ħ*yS  QB*VyS) + qz*(i*ħ*zS  QB*VzS) “tS = /ctS”, “xS = /xS”, “yS = /yS”, and “zS = /zS”. “V” components form the external voltage invariant. “ħ” is Planck’s constant. For the First Case, zero electron rest mass “mB” coincides to zero electron electric charge “QB”, as can be derived from the classic radius of an electron calculation, and the above operators reduce to: mB*c*exp(M)/2 = i*ħ*tS q*mB*c*exp(M)/2 = qx*(i*ħ*xS) + qy*(i*ħ*yS) + qz*(i*ħ*zS)
xv SYNOPSIS – FINITE PRECISION FOR NUMBERS Pauli Spin Matrices substitute for “qx”, “qy”, and “qz”. 0 1 qx =>
0 i qy =>
1 0
i 0
1000 0100 /ct
1
+ /x
0
1
+ /y 0 i00 i 0 0 0 1_FirstCase 2_FirstCase
1_FirstCase 2_FirstCase
*
= 3_FirstCase 4_FirstCase
3_FirstCase 4_FirstCase
1 000 0 1 0 0
0 1 000 i 0 0 i 0
0100 1000
001 0 0 0 0 1
1 0 1 =>
0001 0010
0010 0001
+ /z
0
qz =>
“PPS” and “QQS = q*PPS” were replaced by the Dirac Spinor () (psi). Similarly, “a” and “q*a” were replaced by spinor () (phi). Electric field components are “E”. Magnetic field “B”. Electric current density components are “J”, with adjusted measurement units. 1_FirstCase = 3_FirstCase = Ez + i*c*Bz 2_FirstCase = 4_FirstCase = (Ex + i*c*Bx)  i*(Ey + i*c*By)
1_FirstCase = 3_FirstCase = Jt  Jz 1_FirstCase = 3_FirstCase = Jx + i*Jy 1000 0100 /ct
0001 0010 + /x
0010 0001
+ /z
+ /y 0100 1000
001 0 0 0 0 1
000 i 0 0 i 0 0 i00 i 0 0 0
(Ez + i*c*Bz) (Ex + i*c*Bx)  i*(Ey + i*c*By) *
1 000 0 1 0 0
(Jt  Jz) (Jx + i*Jy) =
(Ez + i*c*Bz) (Ex + i*c*Bx)  i*(Ey + i*c*By)
(Jt  Jz) (Jx + i*Jy)
xvi SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Rotate the three Pauli Spin Matrices, each through “qx”, “qy” and “qz”. Second Portion: 1000 0100
0001 0010
/ct
+ /y 0010 0001
000 i 0 0 i 0 + /z
0100 1000
001 0 0 0 0 1
0 i00 i 0 0 0
(Ex + i*c*Bx) (Ey + i*c*By)  i*(Ez + i*c*Bz)
+ /x
* 1 000 0 1 0 0
(Jt  Jx) (Jy + i*Jz) =
(Ex + i*c*Bx) (Ey + i*c*By)  i*(Ez + i*c*Bz)
(Jt  Jx) (Jy + i*Jz)
Third Portion: 1000 0100 /ct
0001 0010 + /z
0010 0001
+ /y
+ /x 0100 1000
001 0 0 0 0 1
000 i 0 0 i 0 0 i00 i 0 0 0
(Ey + i*c*By) (Ez + i*c*Bz)  i*(Ex + i*c*Bx) *
1 000 0 1 0 0
(Jt  Jy) (Jz + i*Jx) =
(Ey + i*c*By) (Ez + i*c*Bz)  i*(Ex + i*c*Bx)
(Jt  Jy) (Jz + i*Jx)
Second Case Dirac Equation: 1000 0100 mB*c* 0 0 1 0 0 0 0 1
1000 0100 + (i*ħ*/ct  QB*Vt)*
0001 0010 + (i*ħ*/x  QB*Vx)*
0010 0001 000 i 0 0 i 0
+ (i*ħ*/y  QB*Vy)* 0 i00 i 0 0 0
0100 1000 001 0 0 0 0 1
1 2
0 0
+ (i*ħ*/z  QB*Vz)* * = 1 000 3 0 0 1 0 0 4 0
xvii SYNOPSIS – FINITE PRECISION FOR NUMBERS Dirac Spinor is “1”, “2”, “3”, “4”. Electric charge is “QB”. Voltage invariant components are “V”. Electron rest mass is “mB”. First Case and Second Case, together, as one mathematical model, pertain to a combined photon/electron particle. Per the model, the electromagnetic field is developed from “CmaxC” with “C” “+”, and the electron matterwave Dirac Spinor is developed from “CminC0”. As time progresses, “M” becomes more precise to change observations of the projected photon. To conceptually justify this new model, refer to the classic radius of the electron, for which matter mass “mB” is equated to electric field energy “E/c2”. The Dirac Spinor is postprocessed into a prediction for a measured electric current density invariant. By the same equations, the First Case is postprocessed into a prediction for a measured electromagnetic field force density invariant, and it includes the empirically discovered energy density and Poynting Vector. Because separate empirically discovered models of physics are united, the First Case Dirac Equation form of Maxwell’s Equations appears to be fundamental to electromagnetic field theory. The discovery of something more fundamental justifies a claim that the new number system with finite precision is valid in applied mathematics and is ready for use in more modern theories. Also note that one particle is at two places, to violate a preconceived notion for geometric space that that isn’t possible. And note material (fermion electrons) is one in the same as force (boson photons), to unite what appear to be opposites. Take this radical notion further by supposing perceived reality results from numbers, alone from objects, interacting by becoming more precise with respect to each other, to form patterns we see as the Dirac Equation and other mathematical models of physics. Per this theory, the universe is fundamentally numbers and not physical stuff. For further information, please read Special Algebra for Special Relativity. What didn’t fit into a textbook was placed in Alien Invasion Math Story. Second Editions read better.
xviii SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
1 CHAPTER 1  NUMBERS
Chapter 1 – Numbers 1.1 Process from Descartes Formal mathematics in the written record began about twentyfive hundred years ago in ancient Greece, when Pythagoras presented his proof for the Pythagorean Theorem. Along with that proof came the fresh idea that numbers could be separated from items being counted. About four hundred years ago in La Géométrie, Descartes solved geometric problems using an allnumber algebra and plotted solutions on a (Des)Cartesian grid. He had separated algebra from geometry. The three steps that form the “Process from Descartes” are: 1) Mathematically model a physically real phenomenon with geometry 2) Translate the mathematical model into more abstract allnumber algebra to do the analysis 3) Translate the allnumber algebraic finishedcalculation back into geometry as a finalresult The Process from Descartes, what he called “analytic geometry”, explicitly removed analysis from geometry and measurements but did so temporarily. The Process from Descartes is the basis of applied mathematics in this book. This book will show that physical material and force results from numbers becoming more precise relative to each other. Time dependent precision improvement causes number interactions that we model as mathematical dynamic physics. And that means step 3 is only preparation for a measurement and is not a reversion away from numbers to a more fundamental concept of what the physical universe is. Numbers are fundamental, and not material or force. Pythagoras said, “All is number,” and yes, he got it right.
2 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
1.2 GeometricVectors GeometricUnitVectors and Components. Newton’s Second Law is our example: Force “F” equals mass “m” times acceleration “a”. “F” and “a” are geometricvectors, as indicated by bold font. F = m*a ;
Fx*ix + Fy*iy + Fz*iz = m*ax*ix + m*ay*iy + m*az*iz
Fx = m*ax ; Fy = m*ay ; Fz = m*az
(Component equations)
Figure 1. Tabletop Coordinate System, Location P is 3r = x*ix + y*iy + z*iz
Figure 2. Righthand coordinate system, and Lefthand system
3 CHAPTER 1  NUMBERS Three perpendicular geometricunitvectors “ix”, “iy”, and “iz” identify directions along which length components “x”, “y” and “z” are measured by counting measurement units (inches or centimeters) on measuring tapes that have their zero points at the origin. Per tradition, a righthand coordinate system for “x, y, z” is always selected: Righthand fingers curl through positive “ix” and then positive “iy”, and the thumb points to positive “iz”. Push slippery ice of mass “m” with force “Fx”, and the ice moves with acceleration “ax” per “Fx = m*ax”. The push was parallel with the “x” direction and not necessarily along the front edge of the table. DotProduct Operations: ix•ix = 1 ; ix•iy = 0 ; ix•iz = 0 iy•ix = 0 ; iy•iy = 1 ; iy•iz = 0 iz•ix = 0 ; iz•iy = 0 ; iz•iz = 1
CrossProduct Operations: ixxix = 0 ; ixxiy = iz ; ixxiz = iy iyxix = iz ; iyxiy = 0 ; iyxiz = ix izxix = iy ; izxiy = ix ; izxiz = 0
Time GeometricUnitVector. The time component in timespace is “c*t”. Time “t” is measured by a clock in seconds relative to a time origin by counting a repeated ticking internal to the clock mechanism. “c 300,000,000” meters/second is the speedoflight and is a measurement unit conversion factor from time units to length units. “it” is the time geometricunitvector. Location “P” in timespace: 4r
= c*t*it + x*ix + y*iy + z*iz
“4” presubscript indicates the quantity of terms in the expression. Each term consists of a component factor (“c*t”, “x”, “y”, or “z”) multiplied by a geometric direction factor (“it”, “ix”, “iy”, or “iz”). We could speculate “ixxit” should include an “i” factor (“i2 = 1”) because, perhaps, “it•it = 1”, but we don’t go there because geometricvectors are only used to set up a problem. EngineeringCalculationAlgebra. Lack of division inverse to dot and crossproducts illustrates that physical properties of geometric vectors took priority over mathematical properties. Priority to physical properties is appropriate in engineering because an engineer applies known theory
4 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY in a simple and efficient calculation. In the time of Maxwell (late 1800’s) and Minkowski (early 1900’s) (slightly over a hundred years ago) geometric vectors had just been introduced by Heaviside and was, perhaps, the best algebra available.
1.3 Quaternions An allnumber algebra using quaternions fits Special Relativity so well it appears quaternions are inherently the correct algebra. An allnumber algebra has only mathematical properties, no physical properties. It might seem counterintuitive, but algebra separated away from the physical, geometric world better models subtle symmetries. TheoryDevelopmentAlgebra is the name for the new intention of the allnumber algebra, which is to expand known theories into new theories, as opposed to using known theories in engineering. The difference between engineeringcalculationalgebra and theorydevelopmentalgebra evolved (in the writing of this book) from the difference in Pythagorean mathematics between logistica (routine calculations) and arithmetica (theory). Quaternions and LabelNumbers. Three quaternions “jx”, “jy”, and “jz” (with the number “1” as the fourth) are in Hamilton’s paper dated 1843. Each quaternion squares to negative one. Quaternions “jx”, “jy”, and “jz”, complex number factor “i”, integer one “1”, and unit magnitude constructions of these special numbers, are given categorical name (in this book) “labelnumbers”. Translation to allnumber. Place labelnumbers in front of components. “4r = c*t*it + x*ix + y*iy + z*iz” becomes i*4r = i*c*t + jx*x + jy*y + jz*z qx = jx/i ; qy = jy/i ; qz = jz/i ;
or
4r
= 1*c*t + qx*x + qy*y + qz*z
qx2 = qy2 = qz2 = +1
5 CHAPTER 1  NUMBERS Matrix Isomorphs of Quaternions. An “isomorph” is a functional equal with respect to an algebra. Products of labelnumbers are calculated using matrix isomorphs because matrix isomorphs provide internal mathematical structure for the calculation. A traditional association of matrix isomorphs to quaternion labelnumbers is given below. 0 i jx =>
0 1
i
jy => i 0
0 i qy =>
1 0
i
0
0
i
i =>
1 0
0 1 qx =>
0
jz => 0
i
1
0
qz => i 0
1 0 1 =>
0
1
0 1
(“qy” is the negative of its corresponding Pauli Spin Matrix.) The “1” and “i” terms in the matrices also have isomorphs. 1
0
0 1
1 =>
i => 0 1
1 0
“jx”, “jy”, “jz” can be rotated to “jy”, “jz”, “jx” or to “jz”, “jx”, “jy”, but with matrices not rotated. Multiply matrices per the below equation. It applies only if matrix terms (“a, b, c, d, e, f, g, h”) commute (and so cannot be quaternions). a
b
c
d
e
f
g
h
*
a*e + b*g
a*f + b*h
c*e + d*g
c*f + c*h
=
Labelnumbers have the following product equations. jx*jy = jy*jx = jz jy*jz = jz*jy = jx jz*jx = jx*jz = jy
; ; ;
i*jx = jx*i i*jy = jy*i i*jz = jz*i
jx2 = jy2 = jz2 = i2 = 1
;
qx2 = qy2 = qz2 = 12 = +1
6 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY qx = jx/i ; qy = jy/i ; qz = jz/i qx*jy = jy*qx = qy*jx = jx*qy = qz ; qy*jz = jz*qy = qz*jy = jy*qz = qx ; qz*jx = jx*qz = qx*jz = jz*qx = qy ;
qx*qy = qy*qx = i*qz = jz qy*qz = qz*qy = i*qx = jx qz*qx = qx*qz = i*qy = jy
Quaternion Division Operation. To divide one matrix by another, alter the numerator to create a matrix divided by itself. All possible factorings and regroupings end with the same result. Alternatively, use a reciprocal. qy/jz = (jz*qx)/jz = (qx*jz)/jz = qx*(jz/jz) = qx ; qy/jz = qy*(jz) = qx Comparing Quaternions to GeometricUnitVectors. Geometricunitvectors had no “i” factor in a crossproduct result (“ixxiy = iz”). In contrast, quaternions did have an “i” factor (“qx*qy = i*qz”). An engineer does not want an “i” factor in a calculation that predicts a measurement because every measurement is a real number. In contrast, the “i” is wanted for theory development. Exponential Function with Quaternions. Argument “s” can have a labelnumber included as a factor, for example “s = qx*α” (alpha). In contrast, a geometricunitvector cannot be in the argument of an exponential function because there is no algebra by which to calculate. exp(s) = 1 + s + s2/2 + s3/6 + s4/24 + s5/120 + s6/6! + … exp(qx*α) = 1 + qx*α + (qx*α)2/2 + (qx*α)3/6 + (qx*α)4/24 + … = 1 + qx*α + α2/2 + qx*α3/6 + α4/24 + … = (1 + α2/2 + α4/24 + …) + qx*(α + α3/6 + …) = cosh(α) + qx*sinh(α) Complexconjugate operator “*i” swaps “i” for “i” and reverses factors. i*i = i
1*i = 1
7 CHAPTER 1  NUMBERS Quaternion hypercomplexconjugate operator “*j” swaps “jx”, “jy”, “jz” for “jx”, “jy”, “jz”, respectively, and reverses factors. i*j = i
; 1*j = 1
jx*j = jx ; jy*j = jy ; jz*j = jz ;
qx*j = qx ; qy*j = qy ; qz*j = qz
qx = (qx)*j = (jy*qz)*j = (qz)*j*(jy)*j = (qz)*(jy) = qz*jy = qx Both an allnumber expression and its conjugate relate to the same geometricvector expression. The “*j” symbol is removed after “4r*j” is used as a factor in a multiplication operation, and not before. 4r
*j
= (1*c*t + qx*x + qy*y + qz*z)*j = c*t*1*j + x*qx*j + y*qy*j + z*qz*j
Division reciprocal can substitute for negative. i*i = i = 1/i ;
jx*j = jx = 1/jx ; jy*j = jy = 1/jy ; jz*j = jz = 1/jz
An Algebra that Includes Quaternions. Product equations above must be organized into a mathematical group. The group’s criteria are used in a proof so that the proof is not ambiguous or illogical. Criteria for a group: • • • • •
Closure Identity element Commutative property Associative property Inverse operation
Two groups combine to form an algebra field with: •
Distributive property
8 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Number Sets. Every natural number, integer, and rational number is finite. Natural numbers “N” used for counting start with one and have one added repeatedly. Include zero and negatives to form integers “Z”. N = {1, 2, 3, …}
;
Z = {…, 3, 2, 1, 0, 1, 2, 3, …}
Rational numbers “Q” are each a ratio of two integers: “q Q” if “q = m/n” with “m, n Z” and “n ≠ 0”. A rational number “q” has multiple selections of “m” and “n”, for example “1.5 = 3/2 = 6/4”. Integers are a subset of rational numbers: “Z Q” because the denominator “n” may equal “1”. An integer lacks a decimal point. Descartes assigned the name “real” to real numbers to distinguish them from complex or imaginary numbers. Today, real numbers “R” are rational numbers united with numbers proven to be irrational. “Q R” Algebra Field for Real Numbers. Integers “Z”, rational numbers “Q”, and real numbers “R” each form a group with respect to addition. The example group “{R, +}” uses real numbers “R”. Criteria are: •
.1. Addition Closure Property. Each element “a” formed by addition is an element of “R”: “a = b + c”. “a, b, c R”
•
.2. Addition Identity Property. An addition identity element “e” is an element of the set “R”, for which “a = a + e”. “a, e R” (“e” is integer zero “0”)
•
.3. Addition Commutative Property. The order of elements in an addition operation has no effect on the result of the operation, so that “a + b = b + a”. “a, b R”
•
.4. Addition Associative Property. The order of addition operations has no effect on the result of the operations, so that “(a + b) + c = a + (b + c)”. “a, b, c R”
•
.5. Addition Inverse Property. Each element “a” formed by the addition inverse operation, “a = b” (such that “a + b = e”), is an element of “R”. “a, b, e R”
9 CHAPTER 1  NUMBERS Rational numbers “Q” and real numbers “R” each form a group with respect to multiplication. “{R, *}” Criteria: •
.6. Multiplication Closure Property. Each element “a” formed by multiplication “a = b*c” is an element of “R”. “a, b, c R”
•
.7. Multiplication Identity Property. A multiplication identity element “f” is an element of the set “R”, for which “a = a*f”. “a, f R” (“f” is integer one “1”)
•
.8. Multiplication Commutative Property. The order of elements in a multiplication operation has no effect on the result of the operation, so that “a*b = b*a”. “a, b R”
•
.9. Multiplication Associative Property. The order of multiplication operations has no effect on the result of the operations, so that “(a*b)*c = a*(b*c)”. “a, b, c R”
•
.10. Multiplication Inverse Property. Each element “a” formed by the multiplication inverse operation, “a = 1/b” (such that “a*b = f”), is an element of “R”. “a, b, f R” with the exception “b 0”
The two groups above are brought together to form an algebra field “{R, +, *}” (or “{Q, +, *}”) by use of the distributive property. •
.11. Distributive Property of Multiplication over Addition. “a*(b + c) = a*b + a*c” and “(a + b)*c = a*c + b*c”. “a, b, c R” Use algebra field criteria to prove properties.
•
.12. Property of One Addition Identity Element “e”: If “e1 + a = a” and if also there is “e2 + a = a” then from “a = a” we have “e1 + a = e2 + a”, and after subtracting “a” we have “e = e1 = e2”.
•
.13. Property of One Multiplication Identity Element “f”: If “f1*a = a” and if “f2*a = a”, then from “a = a” we have “f1*a = f2*a”, and, after dividing out “a” (for “a 0”), we have “f = f1 = f2”.
10 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •
.14. Property of “0 = 0”: The addition identity element “0” is its own addition inverse, so that “0 = 0”.
•
.15. Property of “1 = 1/1” and “1 = 1/(1)”: The multiplication identity element “1” is its own multiplication inverse, “1 = 1/1”. And, the addition inverse of the multiplication identity element is its own multiplication inverse, “1 = 1/(1)”.
•
.16. Property of “b*0 = 0”: Any number “b” multiplied by the additive identity element zero “0” equals the additive identity element zero “0” as the product. If “a = a + 0”, then “0 = a  a” and “b*0 = b*a  b*a = 0”.
•
.17. Inverse AntiCommutative Properties: The addition inverse anticommutative property is “(a  b) = b  a”. The multiplication inverse anticommutative property is “1/(a/b) = b/a”.
Exponent operation properties for rational numbers and integers are listed below. The exponent operation “^” is repeated multiplication (for example, “4^5 = 4*4*4*4*4”). Similarly, the multiplication operation “*” is repeated addition “+” (for example, “4*5 = 4 + 4 + 4 + 4 + 4”). •
.18. Exponent Closure Property: Each element “a” formed by the exponent operation “a = b^c” is an element of “Q”. “a, b Q”, “c Z”, and “b ≠ 0” if “c < 0”.
•
.19. Exponent Identity Property: An exponent identity element “g” is an element of set “Z” for which “a = a^g”. “a Q”, “g Z” (“g” is integer one “1”)
•
.20. No Exponent Inverse Property: Element “c” formed by the root inverse exponent operation, “c = a^(1/b)”, is not an element of “Q” if “b ≠ 1” or “b ≠ 1”. “a Q”, “b Z”
•
.21. Base Inverse Property: Element “b” formed by the logarithm inverse exponent operation, “logac = b”, is an element of “Z” only if “c = a^b”. “a, c Q”
11 CHAPTER 1  NUMBERS •
.22. No Exponent Associative Property: The order of exponent operations cannot be altered, so that “(a^b)^c” does not necessarily equal “a^(b^c)”. “a Q”, “b, c Z”
•
.23. No Exponent Commutative Property: The order of elements in an exponent operation has an effect on the result of the operation so that “a^b ≠ b^a”. “a, b Z”, “a ≠ b”
•
.24. Distributive Property for Exponent Operation. “a^(b + c) = (a^b)*(a^c)”. “a Q”, “b, c Z”
Complex Numbers. “{1, i}” in a group with multiplication, “{{1, i}, *}”: •
Closure: No holes in the multiplication table
•
Identity: Identity element is integer one.
•
Commutative Property: Applies without exception
•
Associative Property: Applies without exception
•
Inverse: The ratio of any two numbers is in the set of numbers 1/1 = 1 ; i/1 = i ; 1/i = 1*(i) = i ; i/i = 1
“{1, i}” combines with real numbers “R” to form “complex numbers”, “C”, that conform to the criteria of an algebra with the same properties as for real numbers “R”, “{C, +, *}”. Conjugate operation “*i” does not affect “{C, +, *}”. Through an algebraic manipulation “i” is placed in the numerator. 1/(5 + i*7) = (5  i*7)/((5  i*7)*(5 + i*7)) = (5  i*7)/(25 + 49) = 5/74  i*7/74 “a + i*b” with “a, b R” is called the “summationform”. 2x2 matrix isomorphs apply.
12 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1 0 1 =>
0 1 ;
0
1
i =>
a b ;
a + i*b =>
1 0
b a
*
+1
1
+i
i
+1
(+1)*(+1) = +1
(+1)*(1) = 1
(+1)*(+i) = +i
(+1)*(i) = i
1
(1)*(+1) = 1
(1)*(1) = +1
(1)*(+i) = i
(1)*(i) = +i
i
(i)*(+1) = i
(i)*(1) = +i
(i)*(+i) = +1
(i)*(i) = 1
+i
(+i)*(+1) = +i
(+i)*(1) = i
(+i)*(+i) = 1
(+i)*(i) = +1
Table 1. Multiplication Table for Complex Number Factor. “+1” is along the major diagonal by using a conjugate in the left column (first factor).
QuaternionHypercomplex Numbers. “{{1, jx, jy, jz}, *}” criteria: •
Closure: No holes in the multiplication table
•
Identity: Identity element is integer one
•
Commutative Property: Applies with “1” or “1” as a factor or as the product. The anticommutative property applies when two different (non“1”) quaternions are factors.
•
Associative Property: Applies without exception
•
Inverse: The ratio of any two numbers is in the set of numbers
jx/jy = (jy*jz)/jy = (jz*jy)/jy = jz*(jy/jy) = jz*1 = jz ;
jx/jy = jx*(jy) = jz
“{1, jx, jy, jz}” with “R” forms “quaternionhypercomplex numbers”, “QH”, “{QH, +, *(anticommute)}”. Conjugate operation “*j” does not affect the algebra field.
13 CHAPTER 1  NUMBERS *
+1
1
+jx
jx
+1
(+1)*(+1) = +1
(+1)*(1) = 1
(+1)*(+jx) = +jx
(+1)*(jx) = jx
1
(1)*(+1) = 1
(1)*(1) = +1
(1)*(+jx) = jx
(1)*(jx) = +jx
jx
(jx)*(+1) = jx
(jx)*(1) = +jx
(jx)*(+jx) = +1
(jx)*(jx) = 1
+jx
(+jx)*(+1) = +jx
(+jx)*(1) = jx
(+jx)*(+jx) = 1
(+jx)*(jx) = +1
jy
(jy)*(+1) = jy
(jy)*(1) = +jy
(jy)*(+jx) = +jz
(jy)*(jx) = jz
+jy
(+jy)*(+1) = +jy
(+jy)*(1) = jy
(+jy)*(+jx) = jz
(+jy)*(jx) = +jz
jz
(jz)*(+1) = jz
(jz)*(1) = +jz
(jz)*(+jx) = jy
(jz)*(jx) = +jy
+jz
(+jz)*(+1) = +jz
(+jz)*(1) = jz
(+jz)*(+jx) = +jy
(+jz)*(jx) = jy
*
+jy
jy
+jz
jz
+1
(+1)*(+jy) = +jy
(+1)*(jy) = jy
(+1)*(+jz) = +jz
(+1)*(jz) = jz
1
(1)*(+jy) = jy
(1)*(jy) = +jy
(1)*(+jz) = jz
(1)*(jz) = +jz
jx
(jx)*(+jy) = jz
(jx)*(jy) = +jz
(jx)*(+jz) = +jy
(jx)*(jz) = jy
+jx
(+jx)*(+jy) = +jz
(+jx)*(jy) = jz
(+jx)*(+jz) = jy
(+jx)*(jz) = +jy
jy
(jy)*(+jy) = +1
(jy)*(jy) = 1
(jy)*(+jz) = jx
(jy)*(jz) = +jx
+jy
(+jy)*(+jy) = 1
(+jy)*(jy) = +1
(+jy)*(+jz) = +jx
(+jy)*(jz) = jx
jz
(jz)*(+jy) = +jx
(jz)*(jy) = jx
(jz)*(+jz) = +1
(jz)*(jz) = 1
+jz
(+jz)*(+jy) = jx
(+jz)*(jy) = +jx
(+jz)*(+jz) = 1
(+jz)*(jz) = +1
Table 2. Multiplication Table for Quaternions. “+1” is along the major diagonal using the conjugate in the left column.
Quaternions can be placed exclusively in the numerator. 1/(3 + jx*5 + jy*7) = (3  jx*5  jy*7)/((3  jx*5  jy*7)*(3 + jx*5 + jy*7)) = (3  jx*5  jy*7)/(9 + 25 + 49  35*(jx*jy + jy*jx)) = (3  jx*5  jy*7)/83 = 3/83  jx*5/83  jy*7/83 Any quaternionhypercomplex number can be written in summationform “a + jx*b + jy*c + jz*d” with “a, b, c, d R”. Each of three rotation options for 2x2 matrix isomorph substitutions apply.
14 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY (a + i*d) (c + i*b)*i a + jx*b + jy*c + jz*d =>
(c + i*b) (a + i*d)*i
QuaternionComplexHypercomplex Numbers. “{{1, i, jx, qx}, *}” criteria: •
Closure: No holes in the multiplication table
•
Identity: The identity element is integer one
•
Commutative Property: Applies without exception
•
Associative Property: Applies without exception
•
Inverse: The ratio of any two numbers is in the set of numbers
“{1, i, jx, qx}” combines with “{1, jx, jy, jz}” to form a 16x16 multiplication table for “{1, i, jx, jy, jz, qx, qy, qz}”. Inverse example: qx/jy = (jx/i)/jy = i*(jx/jy) = i*(jz) = jz/i = qz “{1, i, jx, jy, jz, qx, qy, qz}” combines with “R” to form “quaternioncomplexhypercomplex numbers”, “{QCH, +, *(anticommute)}”. Conjugates apply. Labelnumbers can be placed exclusively in the numerator. 1/(3 + i*5 + qy*7) = (3 + i*5  qy*7)/((3 + i*5  qy*7)*(3 + i*5 + qy*7)) = (3 + i*5  qy*7)/(16 + i*30  49) = (3 + i*5  qy*7)/(65 + i*30) = (3 + i*5  qy*7)*(65  i*30)/((65  i*30)*(65 + i*30)) = (45  i*415 + qy*455 + jy*210)/5125 Summationform “ar + i*ai + qx*br + jx*bi + qy*cr + jy*ci + qz*dr + jz*di” with “ar, ai, br, bi, cr, ci, dr, di R” has three 2x2 matrix isomorphs. ((ar + dr) + i*(ai + di))
((ci  br) + i*(cr + bi))*i
((ci + br) + i*(cr + bi)) ((ar  dr) + i*(ai + di))*i
15 CHAPTER 1  NUMBERS *
+1
1
+i
i
+1
(+1)*(+1) = +1
(+1)*(1) = 1
(+1)*(+i) = +i
(+1)*(i) = i
1
(1)*(+1) = 1
(1)*(1) = +1
(1)*(+i) = i
(1)*(i) = +i
i
(i)*(+1) = i
(i)*(1) = +i
(i)*(+i) = +1
(i)*(i) = 1
+i
(+i)*(+1) = +i
(+i)*(1) = i
(+i)*(+i) = 1
(+i)*(i) = +1
jx
(jx)*(+1) = jx
(jx)*(1) = +jx
(jx)*(+i) = +qx
(jx)*(i) = qx
+jx
(+jx)*(+1) = +jx
(+jx)*(1) = jx
(+jx)*(+i) = qx
(+jx)*(i) = +qx
+qx
(+qx)*(1) = +qx
(+qx)*(1) = qx
(+qx)*(+i) = +jx
(+qx)*(i) = jx
qx
(qx)*(+1) = qx
(qx)*(1) = +qx
(qx)*(+i) = jx
(qx)*(i) = +jx
*
+jx
jx
+qx
qx
+1
(+1)*(+jx) = +jx
(+1)*(jx) = jx
(1)*(+qx) = +qx
(+1)*(qx) = qx
1
(1)*(+jx) = jx
(1)*(jx) = +jx
(1)*(+qx) = qx
(1)*(qx) = +qx
i
(i)*(+jx) = +qx
(i)*(jx) = qx
(i)*(+qx) = jx
(i)*(qx) = +jx
+i
(+i)*(+jx) = qx
(+i)*(jx) = +qx
(+i)*(+qx) = +jx
(+i)*(qx) = jx
jx
(jx)*(+jx) = +1
(jx)*(jx) = 1
(jx)*(+qx) = i
(jx)*(qx) = +i
+jx
(+jx)*(+jx) = 1
(+jx)*(jx) = +1
(+jx)*(+qx) = +i
(+jx)*(qx) = i
+qx
(+qx)*(+jx) = +i
(+qx)*(jx) = i
(+qx)*(+qx) = 1
(+qx)*(qx) = 1
qx
(qx)*(+jx) = i
(qx)*(jx) = +i
(qx)*(+qx) = 1
(qx)*(qx) = +1
Table 3. Multiplication Table for a Quaternion and the Complex Number Factor. Conjugates for the first factor place “+1” on the major diagonal.
DotProduct and CrossProduct. Analogous to geometric vectors, allnumber expressions, for example “jz*qz = i” and “jy*qz = qx”, can be separated using dotproduct “•” and crossproduct “x”. jz*qz = jz•qz + jzxqz = i + 0 = i 1•1 = 1 jx•1 = 0 jy•1 = 0 jz•1 = 0
1•jx = 0 jx•jx = 1 jy•jx = 0 jz•jx = 0
1•jy = 0 jx•jy = 0 jy•jy = 1 jz•jy = 0
;
jy*qz = jy•qz + jyxqz = 0 + qx = qx 1•jz = 0 jx•jz = 0 jy•jz = 0 jz•jz = 1
16 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1x1 = 0 jxx1 = jx jyx1 = jy jzx1 = jz
1xjx = jx jxxjx = 0 jyxjx = jz jzxjx = +jy
1xjy = jy jxxjy = +jz jyxjy = 0 jzxjy = jx
1xjz = jz jxxjz = jy jyxjz = +jx jzxjz = 0
“4rC” and “4rD” are factors in “6A”. “3A” is “area”. 4rC
= 1rC + 3rC ;
1rC 1rC
= 1*c*tC
*j
;
3rC
= qx*xC + qy*yC + qz*zC
= c*tC*1*j ;
3rC
*j
= xC*qx*j + yC*qy*j + zC*qz*j
*4rD = 4rC*j•4rD + 4rC*jx4rD
4rC
*j
1rC
*j
3rC
*j
4rC
*j
•1rD = c*tC*(1*j*1)*c*tD = c*tC*c*tD •3rD = xC*qx*j*qx*xD + yC*qy*j*qy*yD + zC*qz*j*qz*zD = (xC*xD + yC*yD + zC*zD) •4rD = 1rC*j•1rD + 3rC*j•3rD = c*tC*c*tD  (xC*xD + yC*yD + zC*zD) 6A
= 4rC*jx4rD = 1rC*j*3rD + 3rC*j*1rD + 3rC*jx3rD
3B
= 1rC*j*3rD + 3rC*j*1rD = (c*tC*qx*xD + c*tC*qy*yD + c*tC*qz*zD) + (qx*j*xC*c*tD + qy*j*yC*c*tD + qz*j*zC*c*tD)
= (c*tC*xD  c*tD*xC)*qx + (c*tC*yD  c*tD*yC)*qy + (c*tC*zD  c*tD*zC)*qz = Bx*qx + By*qy + Bz*qz 3A
= 3rC*jx3rD = xC*qx*j*qy*yD + xC*qx*j*qz*zD + yC*qy*j*qx*xD + yC*qy*j*qz*zD + zC*qz*j*qx*xD + zC*qz*j*qy*yD = (yC*zD  zC*yD)*jx + (zC*xD  xC*zD)*jy + (xC*yD  yC*xD)*jz = Ax*jx + Ay*jy + Az*jz
17 CHAPTER 1  NUMBERS 6A
= 3B + 3A = Bx*qx + By*qy + Bz*qz + Ax*jx + Ay*jy + Az*jz = (Bx+ i*Ax)*qx + (By+ i*Ay)*qy + (Bz+ i*Az)*qz
Division operation as an inverse applies only to a complete multiplication operation “*” and not to a dotproduct or crossproduct (or triplevectorproduct or remnantproduct) alone. TripleVectorProduct and RemnantProduct. A fourterm summationform number “4rB” multiplied by a sixterm summationform number “6A” may be split between a triplevectorproduct “■” and a remnantproduct “♦”. The triplevectorproduct “■” has a mathematically imaginary result. The remnantproduct “♦” has a mathematically real result (as given by the “r” (real) and “i” (imaginary) subscripts in the summationform in quaternioncomplexhypercomplex algebra). 4rB*6A
= 4rB*(4rC*jx4rD) = 4rB■(4rC*jx4rD) + 4rB♦(4rC*jx4rD)
TripleVectorProduct. 4rB■6A
= 1rB*3A + 3rBx3B + 3rB•3A
4rB■(4rC
*j
x4rD) = 1rB*(3rC*jx3rD) + 3rBx(1rC*j*3rD) + 3rBx(3rC*j*1rD) + 3rB•(3rC*jx3rD)
= c*tB*((yC*zD  zC*yD)*jx + (zC*xD  xC*zD)*jy + (xC*yD  yC*xD)*jz)  ((yB*Bz  zB*By)*jx + (zB*Bx  xB*Bz)*jy + (xB*By  yB*Bx)*jz) + xB*(yC*zD  zC*yD)*qx*jx + yB*(zC*xD  xC*zD)*qy*jy + zB*(xC*yD  yC*xD)*qz*jz = c*tB*yC*zD*jx  c*tB*zC*yD*jx + c*tB*zC*xD*jy  c*tB*xC*zD*jy + c*tB*xC*yD*jz  c*tB*yC*xD*jz  ((yB*c*tC*zD  yB*c*tD*zC  zB*c*tC*yD + zB*c*tD*yC)*jx + (zB*c*tC*xD  zB*c*tD*xC  xB*c*tC*zD + xB*c*tD*zC)*jy + (xB*c*tC*yD  xB*c*tD*yC  yB*c*tC*xD  yB*c*tD*xC)*jz) + xB*yC*zD*i  xB*zC*yD*i + yB*zC*xD*i  yB*xC*zD*i + zB*xC*yD*i  zB*yC*xD*i
18 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = i*(xB*yC*zD + yB*zC*xD + zB*xC*yD  zB*yC*xD  xB*zC*yD  yB*xC*zD) + jx*(c*tB*yC*zD + zB*c*tC*yD + yB*zC*c*tD  c*tB*zC*yD  yB*c*tC*zD  zB*yC*c*tD) + jy*(c*tB*zC*xD + xB*c*tC*zD + zB*xC*c*tD  c*tB*xC*zD  zB*c*tC*xD  xB*zC*c*tD) + jz*(c*tB*xC*yD + yB*c*tC*xD + xB*yC*c*tD  c*tB*yC*xD  xB*c*tC*yD  yB*xC*c*tD) Notice there is no term with two “x” dimension factors, etc. Notice “i*4rB■6A 4rB■(i*6A)”. Contravariant vector “4kA” is perpendicular (orthogonal or normal) to each covariant factor “4rB”, “4rC”, and “4rD”. “World volume” “1w = (4rA*j•(4rB■(4rC*jx4rD)))” is in the denominator so that “4kA” is real and is per length, as opposed to length cubed measurement units of imaginary volume, “4VA = 4rB■(4rC*jx4rD)”. 4k A
= 4rB■(4rC*jx4rD)/(4rA*j•(4rB■(4rC*jx4rD))) •4rB = 4kA*j•4rC = 4kA*j•4rD = 0
4k A
*j
4r A
= 4kB■(4kC*jx4kD)/(4kA*j•(4kB■(4kC*jx4kD)))
Remnantproduct is the sum of terms not in the triplevectorproduct. 4rB♦(4rC 4rB♦6A
*j
x4rD) = 4rB*(4rC*jx4rD)  4rB■(4rC*jx4rD)
= 1rB*3B + 3rBx3A + 3rB•3B
4rB♦(3rC
*j
x3rD) = 1rB*(1rC*j*3rD) + 1rB*(3rC*j*1rD) + 3rBx(3rC*jx3rD) + 3rB•(1rC*j*3rD) + 3rB•(3rC*j*1rD)
= c*tB*(c*tC*xD  c*tD*xC)*qx + c*tB*(c*tC*yD  c*tD*yC)*qy + c*tB*(c*tC*zD  c*tD*zC)*qz + (yB*Az  zB*Ay)*qx + (zB*Ax  xB*Az)*qy + (xB*Ay  yB*Ax)*qz + xB*(c*tC*xD  c*tD*xC)*qx*qx + yB*(c*tC*yD  c*tD*yC)*qy*qy + zB*(c*tC*zD  c*tD*zC)*qz*qz
19 CHAPTER 1  NUMBERS = (c*tB*c*tC*xD  c*tB*c*tD*xC)*qx + (c*tB*c*tC*yD  c*tB*c*tD*yC)*qy + (c*tB*c*tC*zD  c*tB*c*tD*zC)*qz + (yB*(xC*yD  yC*xD)  zB*(zC*xD  xC*zD))*qx + (zB*(yC*zD  zC*yD)  xB*(xC*yD  yC*xD))*qy + (xB*(zC*xD  xC*zD)  yB*(yC*zD  zC*yD))*qz + xB*c*tC*xD  xB*c*tD*xC + yB*c*tC*yD  yB*c*tD*yC + zB*c*tC*zD  zB*c*tD*zC = 1*(xB*c*tC*xD + yB*c*tC*yD + zB*c*tC*zD  xB*c*tD*xC  yB*c*tD*yC  zB*c*tD*zC) + qx*(c*tB*c*tC*xD + yB*xC*yD + zB*xC*zD  c*tB*c*tD*xC  yB*yC*xD  zB*zC*xD) + qy*(c*tB*c*tC*yD + zB*yC*zD + xB*yC*xD  c*tB*c*tD*yC  zB*zC*yD  xB*xC*yD) + qz*(c*tB*c*tC*zD + xB*zC*xD + yB*zC*yD  c*tB*c*tD*zC  xB*xC*zD  yB*yC*zD) Triplevectorproduct and remnantproduct each result in a fourterm summationform number, one imaginary and the other real. There is a pattern to the count of terms in the products: One (for a real number), four (for timespace location), six (for area), four (for volume), and one (for world volume). This pattern of numbers – 1,4,6,4,1 – is a row of Pascal’s Triangle.
1.4 Translation Back to Geometry The translation for step three has the following equivalences. “1” and “i” become “it” “qx” and “jx” become “ix”
; ;
“qy” and “jy” become “iy” “qz” and “jz” become “iz”
Axial Vectors. “Polar vectors” have “1”, “qx”, “qy”, and/or “qz” with real components. Examples are timespace location, contravariant location, energymomentum, frequencywavenumber, and electric field. “Axial vectors” have “jx”, “jy”, and/or “jz” such that their components are imaginary. Examples are area (only the imaginary
20 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY portion), torque, and magnetic field. There are other imaginary allnumber expressions, but we do not consider these to be axial vectors: volume fourterm summationform, and world volume. Axial geometricvectors are called “pseudovectors” because direction of an axial geometricvector changes if the coordinate system is lefthand rather than righthand. In allnumber algebra, “qx”, “qy”, and “qz” may be thought of as direction indicated by the thumb and “jx”, “jy”, and “jz” may be thought of as direction of the curl of fingers in a plane, with no thumb applied. Step three translation forces the thumb to be applied.
1.5 SingularLabelNumbers 1 = 1 + q1
alpha (includes “i  jx”) beta gamma
β1 = q2 + i*q3 = q2*1 1 = 1 + β1 = 2*1
(1, 2, 3) are substituted by (x, y, z), (x, z, y), (z, x, y), etc. with handedness retained. Therefore, (x, z, y) and (x, y, z) are not valid.
x =>
1
1
i
i
i
i
; βx => 1
1
; x =>
1+i
1+i
1i
1i
The above 2x2 matrix isomorphs are singular matrices because the determinant is zero. For terms that commute, the determinant is calculated by multiplying upper left by lower right and subtracting from it upper right by lower left: x = (1+i)*(1i)  (1i)*(1+i) = 0 A singularlabelnumber multiplied by its quaternion hypercomplexconjugate equals zero:
1*j*1 = (1  q1)*(1 + q1) = 0
;
β1*j*β1 = (q2  i*q3)*(q2 + i*q3) = 0
1*j*1 = (1*j + β1*j)*(1 + β1) = 1*j*1 + 1*j*β1 + β1*j*1 + β1*j*β1 = 0 + (1  q1)*((q2)*(1 + q1)) + ((1  q1)*(q2))*(1 + q1) + 0 = 0
21 CHAPTER 1  NUMBERS Squares differ between the three varieties.
12 = (1 + q1)*(1 + q1) ; = 1 + 2*q1 + q1*q1 = 2*1
β12 = (q2 + i*q3)*(q2 + i*q3) = q2*q2 + q2*i*q3 + i*q3*q2 + i*q3*i*q3 =0
12 = (1 + β1)*(1 + β1) = 12 + 1*β1 + β1*1 + β12 = 2*1 + q2*1*j*1 + q2*1*1 + 0 = 2*1 + q2*0 + 2*β1 + 0 = 2*1 A singularlabelnumber in the denominator is a division by zero and not allowed, and that gives us “singularity theorems”. Examples: ((1 + qx)/2)*((3  qx)/2) = (1 + qx)/2 ((1 qx)/2)*exp(qx*α) = ((1 qx)/2)*exp(α) An exponential function with a singular label number argument has a result of unit magnitude, for example, “exp((i + jx)*/2) = qx”. Singularlabelnumbers should be thought of as part of a group rather than as individual anomalies.
1.6 Exercises 1) Write geometricunitvector representations for timespace “4r” and energymomentum “4p”: Use “E/c” for the energy component (time component), and “px”, “py”, and “pz” for the three momentum components (three space components). Write allnumber representations for timespace location “4r” and “4p”. Write hypercomplexconjugates of “4r” and “4p”. 2) Write matrix multiplication operations for the below. jx2 = jy2 = jz2 = i2 = 1
;
qx2 = qy2 = qz2 = 12 = +1
22 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY jx*jy = jy*jx = jz jy*jz = jz*jy = jx jz*jx = jx*jz = jy
; ; ;
i*jx = jx*i i*jy = jy*i i*jz = jz*i
qx = jx/i qy = jy/i qz = jz/i
qx*jy = jy*qx = qy*jx = jx*qy = qz ; qx*qy = qy*qx = i*qz = jz qy*jz = jz*qy = qz*jy = jy*qz = qx ; qy*qz = qz*qy = i*qx = jx qz*jx = jx*qz = qx*jz = jz*qx = qy ; qz*qx = qx*qz = i*qy = jy
x*j*x = (1  qx)*(1 + qx) = 0 ; x2 = (1 + qx)*(1 + qx) = 2*x y*j*y = (1  qy)*(1 + qy) = 0 ; y2 = (1 + qy)*(1 + qy) = 2*y z*j*z = (1  qz)*(1 + qz) = 0 ; z2 = (1 + qz)*(1 + qz) = 2*z βx*j*βx = (qy  i*qz)*(qy + i*qz) = 0 ; βx2 = (qy + i*qz)*(qy + i*qz) = 0 βy*j*βy = (qz  i*qx)*(qz + i*qx) = 0 ; βy2 = (qz + i*qx)*(qz + i*qx) = 0 βz*j*βz = (qx  i*qy)*(qx + i*qy) = 0 ; βz2 = (qx + i*qy)*(qx + i*qy) = 0 x*j*x = (x*j + βx*j)*(x + βx) ; x2 = (x + βx)*(x + βx) = 2*x y*j*y = (y*j + βy*j)*(y + βy) ; y2 = (y + βy)*(y + βy) = 2*y z*j*z = (z*j + βz*j)*(z + βz) ; z2 = (z + βz)*(z + βz) = 2*z 3) Write dot product and cross product multiplication tables using “q”’s rather than “j”’s. And, again for “q*j” on the left. 4) Translate the below summationform allnumber expressions into summationform expressions that have geometricunitvectors as factors. Explain in words what the geometricunitvector translations represent in our physical world. 4rC 3rC 6A
= c*tC + qx*xC + qy*yC + qz*zC = c*tC + xC*qx*j + yC*qy*j + zC*qz*j
*j
= Bx*qx + By*qy + Bz*qz + Ax*jx + Ay*jy + Az*jz
4VA
= i*VAt + jx*VAx + jy*VAy + jz*VAz
World Volume = i*w
23 CHAPTER 1  NUMBERS 5) For the below set of timespace location geometricvectors “4rA”, “4rB”, “4rC” and “4rD”, find the corresponding set of contravariant geometricvectors “4kA”, “4kB”, “4kC” and “4kD”. Check with the dot product. And, then, as another check, repeat the process but use the derived contravariant geometricvectors “4kA”, “4kB”, “4kC” and “4kD” substituting for the original (covariant) geometricvectors, to find “4rA”, “4rB”, “4rC” and “4rD”. First Exercise: 4rA = 1*ix 4rB = 1*iy
4rC
= 3*it + 5*iz
4rD
= 2*it + 7*iz
Second Exercise: 4rA = 1*ix 4rB = 1*iy
4rC
= 2*it + 5*iz
4rD
= 3*it + 7*iz
Third Exercise: 4rA = 2*it + 1*ix
4rB
= 3*it + 1*iy
4rC
= 5*iz
4rD
= 7*iz
6) Prove the two timespace location geometricvectors “4rC” and “4rD” are each perpendicular to the crossproduct of “4rC” and “4rD”. Use the triplevectorproduct set equal to zero. 7) A row in Pascal’s Triangle is 1 3 3 1. Using threedimensional geometricunitvectors, relate the first “1” to a real number, the first “3” to the polar vector for location, the second “3” to the axial vector for area formed by the crossproduct, and the second “1” to the volume formed by the dotproduct. Next, for the row 1 2 1, model space using complex numbers: For two locations on the complexplane modeled by “2rA” and “2rB”, take the complexconjugate of one of them, and then multiply them using the dot product, and relate the result to the 1 2 1 model of terms in vector expressions for the complexplane. 8) .A. Prove “a*b = ((a2 + b2)  (a2  b2))/4” for rational numbers by stating which criteria are used in each step of the proof. .B. Prove “(a  b) = b  a” and “1/(a/b) = b/a” for rational numbers, also by referencing criteria. .C. Prove quadratic equation solution “x = b/(2*a) ((b*b  4*a*c))/(2*a)” is the solution for “a*x2 + b*x + c = 0”. For the quadratic equation solution, what limitations must be placed onto “a”, “b”, “c”, and “x” for integers, rational
24 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY numbers, real numbers, and for complex numbers to apply. .D. Prove the distributive property of the exponential operation over multiplication with reference to an algebra and criteria. Explain why this distributive property does not apply to “exp(qx*x + qy*y)”. 9) Prove the below singularity theorem. Why cannot “1 qx” be divided by both sides to equate “exp(qx*α)” to “exp(α)”? (1 qx)*exp(qx*α) = (1 qx)*exp(α) Select Solutions 1)
4p
4r
= c*t*it + x*ix + y*iy + z*iz ; 4r = 1*c*t + qx*x + qy*y + qz*z *j *j *j *j *j 4r = c*t*1 + x*qx + y*qy + z*qz = c*t*1  x*qx  y*qy  z*qz
= (E/c)*it + px*ix + py*iy + pz*iz ; 4p = 1*(E/c) + qx*px + qy*py + qz*pz *j *j *j *j *j 4p = (E/c)*1 + px*qx + py*qy + pz*qz = (E/c)*1  px*qx  py*qy  pz*qz 2) 0 i
jx2 = 1 =>
0
i
* i 0
i 0
0 i
0
1
1
0
jx*jy = jz =>
* i 0
0*0 + i*i
0*i + i*0
=
1 0 =
i*0 + 0*i i*i + 0*0
0 1
0*0 + i*1 0*1 + i*0 =
i 0 =
i*0 + 0*1 i*1 + 0*0
0 i
i*jx = jx*i i 0
0 i *
0 i
= i 0
i*0 + 0*i i*i + 0*0 0*i + i*0 0*0 + i*i 0 i i 0 = = * 0*0 + i*i 0*i + i*0 i*i + 0*0 i*0 + 0*i i 0 0 i
25 CHAPTER 1  NUMBERS qx = jx/i 0 i
i 0 /
= (
i 0 0 i
jx/i = (jx*i)*(i/i) = (jx*i) = qx
0 i i 0 i 0 i 0 0 i i 0 0*i + i*0 0*0 + i*i 0 1 * )*( / )= * == i 0 0 i 0 i 0 i i 0 0 i i*i + 0*0 i*0 + 0*i 1 0
0 1
0 1
qx2 = +1 =>
0*0 + 1*1 0*1 + 1*0
*
=
1 0
1 0
0 1
1*0 + 0*1 1*1 + 0*0
0 1
qx*jy = qz => 0 1
= 1*0 + 0*1 1*1 + 0*0
0 i
0*0 + 1*i
* 1 0
1 0
= 1 0
qx*qy = jz =>
0 1
0*0 + 1*1 0*1 + 1*0
* 1 0
1 0 =
0*i + 1*0
i 0
= i 0
0 1 =
1*0 + 0*i
1*i + 0*0
0 i
x*j*x = (1  qx)*(1 + qx) = 0 1 1
1 1 *
1 1
1*1 + 1*1
1*1 + 1*1
= 1 1
0 0 =
1*1 + 1*1
1*1 + 1*1
0 0
βx*j*βx = (qy  i*qz)*(qy + i*qz) = 0 i*1
i
i*1 i
i i
* i i*1
= i i*1
i i *
i i
i*i + i*i i*i + i*i =
i i
0 0 =
i*i + i*i
i*i + i*i
0 0
x*j*x = (x*j + βx*j)*(x + βx) = 0 1i 1i * 1+i 1+i
(12i2)(12i2)
1+i 1+i
(12i2)(12i2)
= 1i 1i
0 0 =
(12i2)+(12i2) (12i2)+(12i2)
0 0
26 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY x2 = (1 + qx)*(1 + qx) = 2*x 1 1
1 1 *
1 1
1*1+1*1
1*1+1*1
= 1 1
2 2 =
1*1+1*1
1*1+1*1
2 2
βx2 = (qy + i*qz)*(qy + i*qz) = 0 i i
i 2i2
i i *
i i
i 2i2
= i i
0 0 =
(i 2i2) (i 2i2)
0 0
x2 = (x + βx)*(x + βx) = 2*x 1+i 1+i 1+i 1+i (12+2*i+i2)+(12i2) (12+2*i+i2)+(12i2) 2+2*i 2+2*i * = = 1i 1i 1i 1i (12i2)+(122*i+i2) (12i2)+(122*i+i2) 22*i 2+2*i 3) Solution: 1•1 = 1 qx•1 = 0 qy•1 = 0 qz•1 = 0
1•qx = 0 qx•qx = +1 qy•qx = 0 qz•qx = 0
1•qy = 0 1•qz = 0 qx•qy = 0 qx•qz = 0 qy•qy = +1 qy•qz = 0 qz•qy = 0 qz•qz = +1
1x1 = 0 qxx1 = qx qyx1 = qy qzx1 = qz
1xqx = qx qxxqx = 0 qyxqx = +jz qzxqx = jy
1xqy = qy qxxqy = jz qyxqy = 0 qzxqy = +jx
1*j•1 = 1 qx*j•1 = 0 qy*j•1 = 0 qz*j•1 = 0
1*j•qx = 0 qx*j•qx = 1 qy*j•qx = 0 qz*j•qx = 0
1*j•qy = 0 1*j•qz = 0 j qx* •qy = 0 qx*j•qz = 0 qy*j•qy = 1 qy*j•qz = 0 qz*j•qy = 0 qz*j•qz = 1
1*jx1 = 0 qx*jx1 = qx qy*jx1 = qy qz*jx1 = qz
1*jxqx = qx qx*jxqx = 0 qy*jxqx = jz qz*jxqx = +jy
1xqz = qz qxxqz = +jy qyxqz = jx qzxqz = 0
1*jxqy = qy qx*jxqy = +jz qy*jxqy = 0 qz*jxqy = jx
1*jxqz = qz qx*jxqz = jy qy*jxqz = +jx qz*jxqz = 0
27 CHAPTER 1  NUMBERS
4) Solutions: 4rC
4VA
= c*tC*it + xC*ix + yC*iy + zC*iz
;
3B
= Bx*ix + By*iy + Bz*iz A = Ax*ix + Ay*iy + Az*iz 3
= VAt*it + VAx*ix + VAy*iy + VAz*iz ; World Volume = w 5) Select Solution, First Exercise: 4rA = 1*ix 4rB = 1*iy 4rC = 3*it + 5*iz 4rA 4rA
= qx*1 *j
= 1*qx*j
4rB
= qy*1
4r B
*j
4rC
= 1*qy*j
= 3 + qz*5
4r C
*j
4 rD 4rD
= 3*1*j + 5*qz*j
= 2*it + 7*iz
= 2 + qz*7 4rD
*j
= 2*1*j + 7*qz*j
x3rA = (7*qz*j)x(qx*1) = 7*qz*j*qx*1 = jy*7 *j *j 1rD *3rA = (2*1 )x(qx*1) = 2*1 *qx*1 = qx*2 *j *j 3rD *1rA = (7*qz )x(0) = 0 *j *j *j *j *j 4rD x4rA = (2*1 + 7*qz )x(qx*1) = 2*1 *qx*1 + 7*qz *qx*1 = qx*2 + jy*7 3rD
*j *j
x3rB = (1*qx*j)x(qy*1) = 1*qx*j*qy = jz*1 1rA *3rB = (0)x(qy*1) = 0 *j *j 3rA *1rB = (1*qx )x(0) = 0 *j *j *j 4rA x4rB = (1*qx )x(qy*1) = 1*qx *qy = jz*1 3rA
*j *j
x3rC = (1*qy*j)x(qz*5) = 1*qy*j*qz*5 = jx*5 1rB *3rC = (0)x(qz*5) = 0 *j *j *j 3rB *1rC = (1*qy )x(3) = 1*qy *3 = qy*(3) j j * * *j *j 4rB x4rC = (1*qy )x(3 + qz*5) = 1*qy *3 + 1*qy *qz*5 = qy*(3) + jx*5 3rB
*j *j
x3rD = (5*qz*j)x(qz*7) = 0 *j *j 1rC *3rD = (3*1 )x(qz*7) = 3*1 *qz*7 = qz*21 *j *j *j 3rC *1rD = (5*qz )x(2) = 5*qz *2 = qz*(10) *j *j *j *j *j 4rC x4rD = (3*1 + 5*qz )x(2 + qz*7) = 3*1 *qz*7 + 5*qz *2 = qz*(21  10) = qz*11 3rC
*j *j
28 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY *j
x4rA) = 1rC*(3rD*jx3rA) + 3rCx(1rD*j*3rA) + 3rCx(3rD*j*1rA) + 3rC•(3rD*jx3rA) = (3)*(jy*7) + (qz*5)x(qx*2) + (qz*5)x(0) + (qz*5)•(jy*7) = jy*21 + jy*(10) + 0 + 0 = jy*11
4rC■(4rD
*j
x4rB) = 1rD*(3rA*jx3rB) + 3rDx(1rA*j*3rB) + 3rDx(3rA*j*1rB) + 3rD•(3rA*jx3rB) = (2)*(jz*1) + (qz*7)x(0) + (qz*7)x(0) + (qz*7)•(jz*1) = jz*2 + i*7
4rD■(4rA
*j
x4rC) = 1rA*(3rB*jx3rC) + 3rAx(1rB*j*3rC) + 3rAx(3rB*j*1rC) + 3rA•(3rB*jx3rC) = (0)*(jx*5) + (qx*1)x(0) + (qx*1)x(qy*(3)) + (qx*1)•(jx*5) = jz*3 + i*5
4rA■(4rB
*j
x4rD) = 1rB*(3rC*jx3rD) + 3rBx(1rC*j*3rD) + 3rBx(3rC*j*1rD) + 3rB•(3rC*jx3rD) = (0)*(0) + (qy*1)x(qz*21) + (qy*1)x(qz*(10))) + (qy*1)•(0) = jx*(21+10) = jx*(11)
4rB■(4rC
1w
= (4rA*j•(4rB■(4rC*jx4rD))) = (1*qx*j)•(jx*(11)) = i*11
= 4rC■(4rD*jx4rA)/1w = (jy*11)/(i*11) = qy*1 *j 4kC = 4rD■(4rA x4rB)/1w = (jz*2 + i*7)/(i*11) = (7/11) + qz*(2/11) *j 4kD = 4rA■(4rB x4rC)/1w = (jz*3 + i*5)/(i*11) = (5/11) + qz*(3/11) *j 4kA = 4rB■(4rC x4rD)/1w = (jx*(11))/(i*11) = qx*(1) 4k B
•4rA = 0 4kC •4rA = 0 *j 4kD •4rA = 0 *j 4kA •4rB = 0 4k B
*j *j
•4rC = 0 4kC •4rB = 0 *j 4kD •4rB = 0 *j 4kA •4rC = 0 4kB
•4rD = 0 •4rD = (7/11)*2  (2/11)*7 = 0 *j 4kD •4rC = (5/11)*3  (3/11)*5 = 0 *j 4kA •4rD = 0
*j
4kB
*j
*j
4kC
*j
6) The triplevectorproduct being equal to zero indicates the geometricvector is perpendicular to the crossproduct of that geometricvector with another geometricvector. *j
x4rD) = 1rC*(3rC*jx3rD) + 3rCx(1rC*j*3rD) + 3rCx(3rC*j*1rD) + 3rC•(3rC*jx3rD)
4rC■(4rC
29 CHAPTER 1  NUMBERS = i*(xC*yC*zD + yC*zC*xD + zC*xC*yD  zC*yC*xD  xC*zC*yD  yC*xC*zD) + jx*(c*tC*yC*zD + zC*c*tC*yD + yC*zC*c*tD  c*tC*zC*yD  yC*c*tC*zD  zC*yC*c*tD) + jy*(c*tC*zC*xD + xC*c*tC*zD + zC*xC*c*tD  c*tC*xC*zD  zC*c*tC*xD  xC*zC*c*tD) + jz*(c*tC*xC*yD + yC*c*tC*xD + xC*yC*c*tD  c*tC*yC*xD  xC*c*tC*yD  yC*xC*c*tD) = i*(0) + jx*(0) + jy*(0) + jz*(0) =0 7) Solutions: The first “1” in “1 3 3 1” pertains to real numbers. The first “3” in “1 3 3 1” pertains to polar vectors. 3rB
= xB*ix + yB*iy + zB*iz = xC*ix + yC*iy + zD*iz 3rD = xD*ix + yD*iy + zD*iz 3rC
The second “3” in “1 3 3 1” pertains to axial vectors. 3A = 3rCx3rD = (xC*ix + yC*iy + zC*iz)x(xD*ix + yD*iy + zD*iz) = ((yC*zD*  zC*yD)*ix + (zC*xD*  xC*zD)*iy + (xC*yD*  yC*xD)*iz) = Ax*ix + Ay*iy + Az*iz
The second “1” in “1 3 3 1” pertains to the dotproduct of a crossproduct to form a scalar (nonvector) result. V = 3rB•(3rCx3rD) = (xB*ix + yB*iy + zB*iz)•(Ax*ix + Ay*iy + Az*iz) = xB*Ax + yB*Ay + zB*Az The first “1” in “1 2 1” pertains to real numbers. The “2” in “1 2 1” pertains to a complex number, including one complex number times another. = xA + i*yA ; C = 2rA*j*2rB = (xA + yA*i*i)*(xB + i*yB) *i = (xA*xB + yA*yB*(i*i*i)) + (i*xA*yB + i*i*yA*xB) 2rA = xA + yA*i = (xA*xB + yA*yB) + i*(xA*yB  yA*xB) 2rB = xB + i*yB = A + i*B 2rA
30 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The second “1” in “1 2 1” pertains to a scalar number that is the result of a dotproduct operation. 2rA
= xA + i*yA *i r 2 A = xA + yA*i 2rB = xB + i*yB
;
A = 2rA*j•2rB = (xA*xB + yA*yB*(i*i*i)) = (xA*xB + yA*yB)
8) Solution not given. 9) Solution:
(1 qx)*exp(qx*α) = (1 qx)*(cosh(α) + qx*sinh(α)) = (1 qx)*cosh(α) + qx*(1 qx)*sinh(α) = (1 qx)*cosh(α) qx*(1 qx)*sinh(α) = (1 qx)*cosh(α) + (1 qx)*sinh(α) = (1 qx)*(cosh(α) + sinh(α)) = (1 qx)*exp(α)
Division by “1 qx” is prohibited because a singularlabelnumber cannot be in the denominator, to avoid a division by zero that would lead to the nonsense result of equating “exp(qx*α)” to “exp(α)”.
31 CHAPTER 1  NUMBERS Further Thought. 1) By what criteria are some theories of pure mathematics segregated away from applied mathematics? 2) Labelnumbers “i”, “j”, and “k” in “{1, i, j, k, i*j, j*k, k*i, i*j*k}” square to negative one and commute. With negatives, prove this set forms a division algebra by reviewing each of the five criteria. 3) Write analogous equations to those below, using Pauli Spin Matrices (“PSMx = jx/i”, “PSMy = jy*i” and “PSMz = jz/i”). qx = jx/i ; qy = jy/i ; qz = jz/i qx*jy = jy*qx = qy*jx = jx*qy = qz qy*jz = jz*qy = qz*jy = jy*qz = qx qz*jx = jx*qz = qx*jz = jz*qx = qy
qx2 = qy2 = qz2 = 12 = +1
; ; ; ;
qx*qy = qy*qx = i*qz = jz qy*qz = qz*qy = i*qx = jx qz*qx = qx*qz = i*qy = jy
4) 2x2 matrix isomorphs for quaternions did not include intermediate combinations of the three 2x2 matrix isomorphs. Assign “a = exp(i*)” and “b = exp(i*)” in the below 2x2 matrix. Select “” (theta), “” (phi) and “” (delta) to create “1”, “jx”, “jy”, and “jz”. Try to select other values for “”, “” and “” retaining “jx = jy*jz = jz*jy”, “jy = jz*jx = jx*jz”, and “jz = jx*jy = jy*jx”. Try a different format for the 2x2 matrix, too. a*sin()
b*i*cos()
exp(i*)*sin() exp(i*)*cos() =
b*cos()
a*i*sin()
exp(i*)*cos()
exp(i*)*sin()
5) If organized as a group, singularlabelnumbers would have each of the five criteria of a mathematical group addressed. Try to define an algebra for singularlabelnumbers. 6) For a better understanding, push math to extremes, and further. Quaternions are pushed to an extreme in octonions, sedonions, and beyond. Read the Appendix on octonionsedonion algebra.
32 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
33 CHAPTER 2 – PARTICLES
Chapter 2 – Particles Einstein’s Special Theory of Relativity provides the method for transforming mathematical descriptions of physical entities, called invariants, from one constant speed observer vantage to a different constant speed observer vantage. Two examples of invariants are timespace location and energymomentum. A person seated in a moving bus has a speed vantage from which they measure time, spacelocation, energy, and momentum component values for a baseball thrown forward from the back of the bus. Because the bus is moving, a person standing on the roadside has a different vantage, and so measures different component values for the same baseball. The change in component values is quantified through the mathematics of Special Relativity by use of the Lorentz Transformation. A Lorentz Transformation is addition of timespace hyperbolic angles.
2.1 HypercomplexPlane In this chapter, Special Relativity is presented with the “x” component the only space component. Timespace location “2r = 1*c*t + qx*x” is visualized by plotting components “c*t” and “x” on a Cartesian grid called a timespace “hypercomplexplane”. The hypercomplexplane is an analogy to the “complexplane” used to illustrate a complex number.
Figure 3. ComplexPlane and Radial Coordinate System.
A complexplane is drawn on a sheet of paper using two space dimensions “x” and “y”. “” is theta.
34 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2z
= x + i*y = r*cos + i*r*sin = r*exp(i*)
1 = (cos2 + sin2) ; r = (x2 + y2) Rotate the complexplane coordinate system by rotating the sheet of paper. For the hypercomplexplane, substitute “jx*x” for “i*y” and “i*c*t” for “x” and find “i*2r = i*c*t + jx*x”. The rightside of “i*2r = i*c*t + jx*x” has two labelnumbers that square to negative one, “i2 = 1” and “jx2 = 1”. That makes it different from the right side of “2z = x + i*y”. Because of that difference, the hypercomplexplane cannot be rotated by rotating the sheet of paper. Divide “i*2r = i*c*t + jx*x” by “i” to get “2r = c*t + qx*x”, in which 2 “qx = +1”. “rhyperbolic” is the hyperbolicradius. “α” (alpha) is the timespace hyperbolicangle. 2r
= 1*c*t + qx*x = 1*rhyperbolic*coshα + qx*rhyperbolic*sinhα = rhyperbolic*exp(qx*α)
1 = (cosh2α  sinh2α) coshα = c*t/rhyperbolic
; rhyperbolic = ((c*t)2  x2) ;
sinhα = x/rhyperbolic
; tanhα = x/(c*t)
exp(qx*α) = coshα + qx*sinhα To illustrate a timespace hypercomplexplane on a spacespace sheet of paper, replace “qx” with “i” and call it “qxillustrated”. 2rillustrated
= 1*c*t + i*x = 1*rillustrated*cosαillustrated + i*rillustrated*sinαillustrated = rillustrated*exp(i*αillustrated)
1 = (cos2αillustrated + sin2αillustrated)
; rillustrated = ((c*t)2 + x2)
cosαillustrated = c*t/rillustrated ; sinαillustrated = x/rillustrated ; tanαillustrated = x/(c*t) exp(i*αillustrated) = cosαillustrated + i*sinαillustrated
35 CHAPTER 2 – PARTICLES The difference between circular trigonometric functions used in the “2rillustrated” illustration and hyperbolic trigonometric functions used in the “2r” timespace location expression causes distortion. Vertical and horizontal displacements are illustrated correctly. Lines at an angle are distorted by appearing longer than in timespace reality (because “rillustrated = ((c*t)2 + x2) > rhyperbolic = ((c*t)2  x2)”). Diagonal lines (for which “(c*t)2 = x2”) are completely distorted because any point illustrated on a diagonal has a zero hyperbolicradius in timespace reality.
Figure 4. The Illustrated HypercomplexPlane with the Hyperbolic Radial Coordinate System.
A light source at the origin emits one photon that travels along the positive “x” axis at the speedoflight. The plot of points “c*t” and “x” is called the worldline which, for the photon, is the diagonal from the origin to the upperright. The worldline of the light source is straight up the “c*t” axis, because “x = 0”, and the hyperbolicradius equals “c*t”. In contrast to the light source, the hyperbolic radius for the photon remains zero because “c*t = x”. Because “x = rhyperbolic*sinhα”, “α” is large (to the ultimate “1/0” if we allow division by zero).
36 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
2.2 Inertial Reference Frames Selected origins for “c*t” and “x” are valid for both a moving and a stationary inertial (constant speed) reference frame. The visualization is a big rectangular prism bus moving down a road toward the right. Inertial reference frame “M” is the interior of the moving bus. A person seated on the moving bus takes measurements of component values. These are written with subscript “M”. Inertial reference frame “S” is the stationary roadside. A person standing on the roadside also takes measurements. Those component values are written with subscript “S”. “(c*tM, xM) = (c*tS, xS) = (0, 0)”: When “c*tM = c*tS = 0”, the back of the bus (“xM = 0”) and the speed limit sign (“xS = 0”) identify the same point in space. “xM” is measured using measurement tape along the floor of the bus. “xS” is measured using measurement tape along the roadside. “tM” is measured using a clock mounted on the bus wall. “tS” is measured using a clock mounted on the roadside. Define a physical system (of components and labelnumbers) inside bus “M”, and then apply a “Lorentz Transformation” using the speed of the bus “S/M” to describe the same physical system observed from roadside “S”. A Lorentz Transformation is a hyperbolicangle rotation. Rest Frame “B” of the object in motion is a third inertial reference frame. The baseball’s own rest frame “B” has the ball stopped. Therefore, in “B”, there is advancement of time “tB”, but no change in space location (“xB = 0”). And there is rest mass energy per “EB = mB*c2”, but no momentum, “pxB = 0”. Speed “vM” could be written “vM/B” because “vB = 0”. The “Relative” in “Special Relativity”. “Relativity” is the ancient concept that all inertial frames have the same preference. In the movie Agora watch a reenactment of Hypatia of Alexandria (who lived around 390 AD / 390 CE) measure locations on a ship’s deck of weights dropped from the mast to verify there is no
37 CHAPTER 2 – PARTICLES preferred reference frame. Galileo (around 1600 AD / 1600 CE) described in one of his books the experiment of dropping weights from a ship’s mast. And Galileo created a visualization for the lack of a preferred reference frame by describing what a person would observe when isolated inside a moving ship. Einstein (in year 1905) added to relativity the requirement that the speedoflight be the same regardless of reference frame and that meant time and space measurements were specific to a reference frame. The change in space is easy to imagine. Place a measuring tape along the floor inside the bus and another measuring tape on the ground along the roadside and see the two coincide only when time equals zero (at a nonrelativistic speed). The change in time is not so easy to imagine, and that’s the trick. Geometric Problem Definition. On bus “M” we have measured or have calculated component values for timespace location “2r” and energymomentum “2p” for a moving baseball. 2r
;
= c*tM*itM + xM*ixM
2p
= (EM/c)*itM + pxM*ixM
Baseball “B” of rest mass “mB” was thrown at speed “vM” in the positive “+xM” direction inside the bus at “tM = 0”. “xM” can be calculated from “xM = vM*tM”. Total energy “EM” equals rest mass energy “EB = mB*c2” added to kinetic energy. Mechanical momentum “pxM” equals relativistic mass “EM/c2” times speed “vM”, “pxM = vM*(EM/c2)”. Geometricunitvectors “itM” and “ixM” are the same in energymomentum “2p” as they are in timespace “2r”. The bus moves at speed “vS/M”. Our objective is to use “M” components and bus speed “vS/M” to calculate the same four components relative to roadside “S”. 2r
= c*tS*itS + xS*ixS
;
2p
= (ES/c)*itS + pxS*ixS
38 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 5. Moving and Stationary Reference Frames.
2.3 The UnspecifiedSpeedParameter Begin the second step by replacing: itM with exp(qx*)
;
ixM with qx*exp(qx*)
If “itM” had been replaced with “1” and “ixM” with “qx”, then the replacement would have implied preference for moving coordinate system “M”. To avoid that preference, “exp(qx*)” uses “unspecifiedspeedparameter”, “” (sigma, or, in English, esse). If “” is included in a labelnumber, then we call that labelnumber a “compoundlabelnumber”. A compoundlabelnumber contrasts with a “simplelabelnumber”, such as “1” or “qx”. “” is unknown and unknowable. It is different from independent variable “x” because “x” is intended to be substituted with a selected single valued real number. In other words, although “x” is not known, “x” is knowable. In contrast, “” is unknowable. We are less likely to mistakenly assume a preferred reference frame if continuously reminded about the unspecifiedspeedparameter
39 CHAPTER 2 – PARTICLES “” by using compoundlabelnumbers. (Explicit use of “” ensures we do not violate “gauge invariance”.) 1M = exp(qx*) = cosh  qx*sinh qxM = qx*1M = qx*exp(qx*) = sinh + qx*cosh = exp(qx*)*qx = 1M*qx
2.4 CompoundLabelNumbers and Components Retain component values in the translation from a geometric representation to the allnumber (more abstract) representation. 2r
= c*tM*itM + xM*ixM
2r
= 1M*c*tM + qxM*xM = exp(qx*)*(1*c*tM + qx*xM) = exp(qx*)*(1*c*tB*cosh(αM) + qx*c*tB*sinh(αM)) = exp(qx*)*c*tB*exp(qx*αM)
2p
= (EM/c)*itM + pxM*ixM
2p
= 1M*(EM/c) + qxM*pxM = exp(qx*)*(1*EM/c + qx*pxM) = exp(qx*)*(1*mB*c*cosh(αM) + qx*mB*c*sinh(αM)) = exp(qx*)*mB*c*exp(qx*αM)
An advantage of allnumber expressions is use of “exp”. c*tB*exp(qx*αM) = c*tB*coshαM + qx*c*tB*sinhαM = c*tM + qx*xM c*tM = c*tB*coshαM
;
xM = c*tB*sinhαM
EM/c = mB*c*coshαM
;
pxM = mB*c*sinhαM
“αM” (alpha) relates to speed “vM” of the baseball on the bus. (vM/c) = tanhαM
αM = atanh(vM/c)
40 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 6. Hyperbolic tangent relationship between speed “v” and hyperbolicangle “α”.
For the above equations, a baseball was thrown forward from the back of the bus at time “tM = 0”, and the baseball has constant speed “vM” in the positive “xM” direction. Speedparameter “αM” is the hyperbolicangle for motion of the baseball inside the bus. “c*tB” and “mB*c” are each hyperbolicradius in their respective “c*tM”/”xM” or “EM/c”/”pxM” hypercomplexplanes and are calculated from the Pythagorean Theorem. 12 = cosh2αM  sinh2αM ; (c*tB)2 = (c*tM)2  xM2 ; (mB*c)2 = (EM/c)2  pxM2
Figure 7. Energy “E” and momentum “px” approach infinity as speed “v” approaches speedoflight “c” per “cosh” and “sinh” functions.
41 CHAPTER 2 – PARTICLES
Figure 8. Time and space in “M” for an object “B” at time “tM”.
Figure 9. Energy and momentum for an object “B” of mass “m” moving in “M”.
“xM = vM*tM” and “pxM = vM*EM/c2” are proven valid. xM = c*tB*sinhαM = c*tB*coshαM*tanhαM = c*tM*tanhαM = c*tM*vM/c = vM*tM pxM = mB*c*sinhαM = mB*c*coshαM*tanhαM = (EM/c)*tanhαM = (EM/c)*vM/c = vM*EM/c2
42 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY NonRelativistic Approximation “αM < 0.3” means “tanh” and “sinh” functions are not noticeable (unless extreme precision is needed). tanhαM sinhαM vM/c αM
αM kxM” because “cosh(αM) > sinh(αM)”, and that contrasts with “M/c < kxM” for “2k”. Spacelike yardstick “2s” partners with timelike “2r”. 2s
= 1M*qx*sB*exp(qx*αM) = qxM*sB*exp(qx*αM)
49 CHAPTER 2 – PARTICLES
Figure 12. Hypercomplexplane for frequency and wavenumber.
Figure 13. Timelike and spacelike.
Figure 14. World lines for location in “M” of many baseballs of different speeds “vM” at the same time “tB” showing timelike locations on the hyperbola.
50 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 15. Location in “M” of a yardstick.
Figure 16. Locations of right ends of many yardsticks, each at a different speed, to illustrate spacelike locations.
Lorentz Transformation from TimeLike to SpaceLike. αS/M = (i  jx)*/2 2r
= 1M*(c*tB)*exp(qx*αM)*1/1 = 1M*(c*tB)*exp(qx*αM)*(exp(qx*((i  jx)*/2))/exp(qx*((i  jx)*/2))) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*(i*/2  jx*/2))/exp(qx*(i*/2  jx*/2))) = 1M*(c*tB)*exp(qx*αM) *(exp(jx*/2)*exp(i*/2))/(exp(jx*/2)*exp(i*/2)) = 1M*(c*tB)*exp(qx*αM)*(jx*(i))/(jx*(i))
51 CHAPTER 2 – PARTICLES = 1M*(c*tB)*exp(qx*αM)*(qx/qx) = (1M/qx)*((qx*c*tB)*exp(qx*αM)) = 1S*(qx*c*tB)*exp(qx*αM) 1S = 1M/qx = 1M*qx = qxM
;
qxS = qxM/qx = 1M
c*tS + qx*xS = (qx*c*tB)*exp(qx*αM) c*tS = qx*c*tB*qx*sinh(αM) = xM ; qx*xS = qx*c*tB*cosh(αM) = qx*c*tM 2r
= 1S*c*tS + qxS*xS = (qxM)*(xM) + (1M)*(c*tM) = 1M*c*tM + qxM*xM = 2r
vS/c = tanhαS = tanh(αM + (i  jx)*/2) = tanh(αM + i*/2  jx*/2) = sinh(αM + i*/2  jx*/2)/cosh(αM + i*/2  jx*/2) = (sinh(αM)*cosh(i*/2  jx*/2) + cosh(αM)*sinh(i*/2  jx*/2)) /(cosh(αM)*cosh(i*/2  jx*/2) + sinh(αM)*sinh(i*/2  jx*/2)) = (sinh(αM)*cosh(i*/2)*cosh(jx*/2) + sinh(αM)*sinh(i*/2)*sinh(jx*/2) + cosh(αM)*sinh(i*/2)*cosh(jx*/2) + cosh(αM)*cosh(i*/2)*sinh(jx*/2)) /(cosh(αM)*cosh(i*/2)*cosh(jx*/2) + cosh(αM)*sinh(i*/2)*sinh(jx*/2) + sinh(αM)*sinh(i*/2)*cosh(jx*/2) + sinh(αM)*cosh(i*/2)*sinh(jx*/2)) = (sinh(αM)*cos(/2)*cos(/2) + qx*sinh(αM)*sin(/2)*sin(/2) + i*cosh(αM)*sin(/2)*cos(/2)  jx*cosh(αM)*cos(/2)*sin(/2)) /(cosh(αM)*cos(/2)*cos(/2) + qx*cosh(αM)*sin(/2)*sin(/2) + i*sinh(αM)*sin(/2)*cos(/2)  jx*sinh(αM)*cos(/2)*sin(/2)) = (qx*sinhαM)/(qx*coshαM) = tanhαM = vM/c
52 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY vS/M/c = tanhαS/M = tanh((i  jx)*/2) = tanh(i*/2  jx*/2) = sinh(i*/2  jx*/2)/cosh(i*/2  jx*/2) = (sinh(i*/2)*cosh(jx*/2) + cosh(i*/2)*sinh(jx*/2)) /(cosh(i*/2)*cosh(jx*/2) + sinh(i*/2)*sinh(jx*/2) = (i*sin(/2)*cos(/2)  jx*cos(/2)*sin(/2)) /(cos(/2)*cos(/2) + qx*sin(/2)*sin(/2)) = 0/qx = 0 The tangent with a sum of angles identity was not used because it is derived using a division by cosine, and cosine of half pi equals zero. Bus “M” moves at speed “vS/M/c = 0”. Observer “S” observes the baseball inside the bus per “c*tS = xM”, “xS = c*tM” and “vS = vM” by looking through the bus window and seeing ticks of a clock are nodes along a line and seeing nodes along a line are ticks of a clock. There are no examples in which time and space are swapped, as if a theory of physics prevents it from happening. “αS/M = (i  jx)*/2” includes a singularlabelnumber factor “i  jx” so that “αS/M = (i  jx)*/2” is a variation of the number zero. As such, “αS/M = (i  jx)*/2” cannot be in a denominator, which it is not. Also, “exp(qx*((i  jx)*/2)) = qx” must be unit magnitude, which it is, because “qx” is unit magnitude. A hypercomplex hyperbolicangle “αS/M” is called (in this book) an “exotic Lorentz Transformation”.
2.8 Electric Current Density Timelike electric current density is a static electric charge. Spacelike electric current density is electric current in a wire. TimeLike Electric Current Density. 2J
= 1M*(dQB/dxB)*exp(qx*αM) = 1M*xB*exp(qx*αM)
53 CHAPTER 2 – PARTICLES “xB = dQB/dxB” (Coulombs per centimeter) refers to rest frame “B” twice: Electric charge “QB” and rest space location “xB”.
Figure 17. Hypercomplexplane for charge density “JtB = xB = dQB/dxB” as the timelike hyperbolicradius.
SpaceLike Electric Current Density. Inside a wire: •
Stationary positive electric charges “2J+” with “xB+ > 0” (in the atomic nuclei), for which “vM+ = 0”, “αM+ = 0” and “exp(qx*αM+) = 1”
•
Moving negative electric charges “2J” with “xB < 0” (valence electrons), for which “vM/c = tanh(αM) > 0” and, for a physically real wire, “vM/c αM”.
The sum “2J = 2J+ + 2J” time component in “M” must be zero. 2J
= 2J+ + 2J= 1M*xB+*exp(qx*αM+) + 1M*xB*exp(qx*αM) = 1M*xB+ + 1M*xB*exp(qx*αM) = 1M*xB+ + 1M*xB*(coshαM + qx*sinhαM) = 1M*(xB+ + xB*coshαM) + qxM*xB*sinhαM= qxM*xB*sinhαM= qxM*(xB+/coshαM)*sinhαM= qxM*xB+*tanhαM= qxM*xB+*(vM/c)
54 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The requirement of a zero time component was satisfied by setting “xB+ + xB*coshαM = 0”, from which “xB = xB+/coshαM”. Regardless of “vM/c” being small, current “2J = qxM*xB+*(vM/c)” in a wire produces a very noticeable magnetic field around the wire. A wire with electric current can, as a piece of metal, move in the positive “x” direction with speed “vS”. With respect to “S”, the hyperbolicradius of the current “qx*(xB+*(vM/c))” is spacelike. Spacelike current density in “S” is not commonly useful. 2J
= 1S*qx*(xB+*(vM/c))*exp(qx*αS/M) = qxS*(xB+*(vM/c))*exp(qx*αS/M)
Figure 18. Hypercomplexplane for a spacelike current density.
SpaceLike Invariant Complement for EnergyMomentum. “nB” relates to rest mass “mB” as length “sB” relates to time “c*tB”. 2pspacelike
= 1M*qx*nB*c*exp(qx*αM) = qxM*nB*c*exp(qx*αM) = 1M*nB*c*sinh(αM) + qxM*nB*c*cosh(αM) = 1M*(EM/c) + qxM*pxM
“αS/M = (i  jx)*/2” has been used to create “2pspacelike” from “2p”, but although we can create “2pspacelike”, we cannot find a physical example. Can there be momentum without energy?
55 CHAPTER 2 – PARTICLES Angular Momentum Density Invariant. In twodimensional timespace, the axis of rotation for angular momentum is limited to being in the “x” direction. The visualization is a spinning ring. Use the righthandrule: Fingers curl with the direction of rotation and the righthand thumb points in the direction of angular momentum. If we look to a higher value of “x”, then a clockwise rotation (top to the right) has positive angular momentum. “HxB” is restframe angular momentum of a spinning ring. It is similar to restframe electric charge “QB” because its value does not change with respect to speed of an observer. Like “QB”, “HxB” is an invariant in itself: Factor “1 = cosh(αM)/cosh(αM)” has the numerator from relativistic mass and the denominator from time dilation. If “HxB = dHxB/dxB” is density of angular momentum along a particle’s length, then the (spacelike) invariant is 2H
= 1M*qx*HxB*exp(qx*αM) = 1M*HxB*sinh(αM) + qxM*HxB*cosh(αM) = 1M*HtM + qxM*HxM
Space component “HxM = HxB*cosh(αM)” is positive or negative depending on spin direction of “HxB”. Time component “HtM = HxB*sinh(αM)” is positive or negative depending on “sinh(αM)” and quantifies angular momentum that passes a point “xM”. Angular momentum density is similar to electric current density, with the difference being spacelike vs timelike, and that difference shows in their respective conservation laws: “((2)*jsn)x(2H) = 0” as opposed to “((2)*jsn)•(2J) = 0”. (“sn” will be explained later.)
2.9 Motion Faster than Light Inside bus “M” a baseball moves toward the front at speed “vM < c”, as observed by a person seated on the bus. A person standing on the roadside “S” observes the baseball moving at speed “vS > c”. Bus speed hyperbolicangle “αS/M = i*/2” is not real, “αS/M R”. vS/M/c = tanhS/M = tanh(i*/2) = i*tan(/2) = i*(1/0) = 1/0
56 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Bus speed “vS/M/c = 1/0” is instantaneous movement along “x” so that the bus is at all locations “x” simultaneously, but only for an instant. General form for “αS/M = i*/2” and “αM R”: 2r
= 1M*(c*tB)*exp(qx*αM) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*αS/M)/exp(qx*αS/M)) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*i*/2)/exp(qx*i*/2)) = 1M*(c*tB)*exp(qx*αM)*(exp(jx*/2)/(exp(jx*/2)) = 1M*(c*tB)*exp(qx*αM)*(jx/jx) = (1M/jx)*((jx*c*tB)*exp(qx*αM)) = (i*1M/qx)*((i*qx*c*tB)*exp(qx*αM))
1S = i*1M/qx = i*qxM
;
qxS = qx*i*qxM = i*1M
Hyperbolic radius “i*qx*c*tB” is spacelike and imaginary for “tB” real. c*tS + qx*xS = (i*qx*c*tB)*exp(qx*αM) c*tS = i*qx*c*tB*qx*sinh(αM) = i*xM
; c*tS = c*tM*coshαS/M + xM*sinhαS/M = c*tM*cosh(i*/2) + xM*sinh(i*/2) = c*tM*cos(/2) + i*xM*sin(/2) = i*xM
qx*xS = i*qx*c*tB*cosh(αM) = i*qx*c*tM
;
xS = c*tM*sinhαS/M + xM*coshαS/M = c*tM*sinh(i*/2) + xM*cosh(i*/2) = i*c*tM*sin(/2) + xM*cos(/2) = i*c*tM
“2r” is confirmed to be invariant. 2r
= 1S*c*tS + qxS*xS = (i*qxM)*(i*xM) + (i*1M)*(i*c*tM) = qxM*xM + 1M*c*tM = 1M*c*tM + qxM*xM = 2r
“c*tB” is real: c*tB = ((c*tS)2  xS2) = ((i*xM)2  (i*c*tM)2) = ((c*tM)2  xM2) = c*tB
57 CHAPTER 2 – PARTICLES Time and space (in “M”) is swapped for space and time (in “S”), respectively, and that means “M” is drawn on the “S” hypercomplexplane up from the “xS” axis toward the “c*tS” axis, to illustrate “αS = αM + i*/2” drawn down from the “c*tS” axis. αS = αM + αS/M = αM + i*/2 vS/c = tanhαS = tanh(αM + i*/2) = sinh(αM + i*/2)/cosh(αM + i*/2) = (sinh(αM)*cosh(i*/2) + cosh(αM)*sinh(i*/2))/ (cosh(αM)*cosh(i*/2) + sinh(αM)*sinh(i*/2)) = (sinh(αM)*cos(/2) + cosh(αM)*i*sin(/2))/ (cosh(αM)*cos(/2) + sinh(αM)*i*sin(/2)) = (i*coshαM)/(i*sinhαM) = cothαM For “αM R” “vS/c = cothαM > 1” so that “vS > c”. “c*tS = i*xM” and “xS = i*c*tM” mean “c*tS” and “xS” are imaginary for “c*tM” and “xM” real. The “S” hypercomplexplane requires “c*tS” and “xS” be real. To force “c*tS” and “xS” to be real make the hyperbolicradius (in “B”) imaginary. tB = i*B mB = i*B
(tau) ; B R
;
(nu) ; B R ;
i*qx*c*tB = i*qx*c*(i*B) = qx*c*B i*qx*mB*c = i*qx*(i*B)*c = qx*B*c
The imaginary hyperbolicradius causes “c*tM” and “xM” to be imaginary and not plotted on the “M” hypercomplex plane. Colliding Rods Visualization. Two parallel rods move perpendicular to their length to collide and bounce. All points along the contacting surface coincide everywhere in “xB” at one instant in time “tB”. Because we are observing a rod collision along “xB” at one instant “tB” and not one location “xB” for all time “tB”, we have “αB = i*/2” and not “αB = 0”.
58 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The two rods are on the floor of a bus, “αM = 0”. Because “vS/M, αS/M > 0”, the front of the bus is at an earlier time “tM” compared to the back, for each time “tS”, and that means the rod collision observed from “S” starts in the rear of bus and moves forward at speed “vS = c2/vS/M > c”. Spooky Action at a Distance. Einstein gave the name “spooky action at a distance” to the polarization of a pair of entangled photons determined simultaneously in two observations a macroscopic distance away. The information of the direction of polarization had to travel from one particle to its entangled partner particle instantaneously over the macroscopic distance. Perhaps this is another example of “α = i*/2”. Violation of CauseandEffect. A hyperlightspeed signal can arrive before it was emitted. The signal can be two rods colliding. Set “M > 0” (mu) and “αM = M + i*/2” so that “vM = c*coth(M)”. (If “M < 0” then the adder is “ i*/2”.) Bus “M” moves backward relative to roadside “S” with negative hyperbolicangle speedparameter “αS/M < 0”. M R ; M > 0 ; αM = M + i*/2 ; αS = αM + αS/M = i*/2 + (M + αS/M ) 2r
αS/M R ; αS/M < 0
= 1S*(c*tB)*exp(qx*αS) = 1S*(i*c*B)*exp(qx*αS) = 1S*(i*c*B)*exp(qx*(αM + αS/M)) = 1S*(i*c*B)*exp(qx*(i*/2 + M1 + αS/M)) = 1S*(i*c*B)*exp(qx*i*/2)*exp(qx*(M + αS/M)) = 1S*(i*c*B)*jx*exp(qx*(M + αS/M)) = qxS*(i*i*c*B)*exp(qx*(M + αS/M)) = qxS*(c*B)*exp(qx*(M + αS/M))
c*tS = c*B*sinh(M + αS/M)
xS = c*B*cosh(M + αS/M)
Spacelike equations for “c*tS” and “xS” depend on angle “M + αS/M”, which is positive for “αS/M = 0” and negative for “αS/M < M”.
59 CHAPTER 2 – PARTICLES •
For “αS/M = 0”, the colliding rods move forward inside the bus at speed “vMrods = c*tanh(M) < c” so that their contact point moves at speed “vS = c*coth(M) > c” and “c*tS > 0” and “xS > 0”.
•
In contrast, for “αS/M < M” the rods move backward at speed “c < vSrods = c*tanh(M + αS/M) < 0” so that their contact point moves backward at speed “vS = c*coth(M + αS/M) < c”. “c*tS < 0” with “xS > 0” means increasing values of “xS” occur for decreasing values of “c*tS” and that properly corresponds to negative direction movement because “S” time is only seen moving positively.
The contact point was seen moving forward on the bus but moving backward from the roadside so that past locations of the contact point on the bus are future locations of the contact point from the roadside. Time goes backward.
Figure 19. Illustration of a violation of causeandeffect for motion faster than the speedoflight. The observer sees time as moving up vertically. The arrow on the farright figure points right, but the “S” observer sees the activity as moving left, toward the source of the emission.
Visualization of the Violation of CauseandEffect. A solar system has two small planets in counterorbits. Reference frame “M” is Planet M. “S” is Planet S. At “c*tM = 0” and “c*tS = 0” the two planets are a few days passed each other with Planet M on the left and moving left. Therefore, “vS/M/c = tanhαS/M” with “αS/M < 0” and “vS/M < 0” (and “vM/S > 0”).
60 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Planet M sends a hyperlightspeed signal to Planet S at speed “vM/c = cothM” with “M > 0” and “vM > 0” / “vM > c”. Planet S received it at “c*tM = 7” and “xM = tM*vM = c*tM*coth(M)” (“xM > 0”). 2r
= 1M*(i*c*B)*exp(qx*αM) = 1M*(i*c*B)*exp(qx*(M + i*/2)) = 1M*(i*c*B)*(jx)*exp(qx*M) = 1M*(c*B)*(qx)*exp(qx*M)
c*tM = c*B*sinh(M)
;
xM = c*B*cosh(M) = (c*tM/sinh(M))*cosh(M) = c*tM*coth(M) = tM*vM
Note that “xM > 0” and “xM > c*tM” because “M R”. Relative to Planet S, the hyperlightspeed signal timespace location components “c*tS” and “xS” are found using the below component equations, as given previously. c*tS = c*B*sinh(M + αS/M)
;
xS = c*B*cosh(M + αS/M)
If “M = 0”, such that the signal arrived instantly (as would happen for colliding rods), then “c*tS < 0”, and the signal was received on Planet S prior to being emitted, per “c*tS = c*B*sinh(M + αS/M)”.
Figure 20. Instant signal sent from Planet M to Planet S and then another instant signal is sent back to Planet M. The second signal arrives prior to the first signal being emitted because “M = 0” and “S(second) = 0” with the two planets moving apart.
61 CHAPTER 2 – PARTICLES
Figure 21. Hyper light speed signal from Planet M to Planet S and then another signal back to Planet M. The second signal arrives prior to the first signal being emitted because “M  S(second) < αS/M” with the two planets moving apart.
But people on Planet S were unaware of the emission time of the signal because it occurred on Planet M, far away from them, and so are unaware the signal moved backward in time. To make everyone aware of the reverse passage of time, Planet S sent a second hyperlightspeed signal to Planet M the instant they received the signal from Planet M. For this second signal, “S(second) < 0” because the signal traveled from Planet S to the left towards Planet M. “” in “ i*/2” is because the second signal is on the left side of the “c*tS” axis on the hypercomplexplane. αS(second) = S(second)  i*/2 2r(second)
;
S(second) < 0
= 1S*(i*c*B(second))*exp(qx*αS(second)) = 1S*(i*c*B(second))*exp(qx*(S(second)  i*/2)) = 1S*(i*c*B(second))*(jx)*exp(qx*S(second)) = 1S*(c*B(second))*(qx)*exp(qx*S(second))
c*tS(second) = c*B(second)*sinh(S(second)) ; xS(second) = c*B(second)*cosh(S(second)) “S(second) < 0” in “c*tS(second)” and “xS(second)” component equations means “c*tS(second) > 0” for “c*B(second) > 0”. And “xS(second) < 0” for left motion. Relative to Planet S, the total (first signal plus second signal) time elapsed, and the total distance traveled, are given below.
62 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY c*tS(total) = c*tS + c*tS(second) = c*B*sinh(M + αS/M)  c*B(second)*sinh(S(second)) xS(total) = xS + xS(second) = c*B*cosh(M + αS/M)  c*B(second)*cosh(S(second)) The general form of the Lorentz Transformation is now applied to find the arrival time of this second signal on Planet M. 2r(second)
= 1S*(c*B(second))*(qx)*exp(qx*S(second)))*(exp(qx*αM/S)/exp(qx*αM/S)) c*tM(second) = c*B(second)*sinh(S(second) + αM/S) xM(second) = c*B(second)*cosh(S(second) + αM/S) c*tM(total) = c*tM + c*tM(second) = c*B*sinh(M)  c*B(second)*sinh(S(second) + αM/S) = c*B*sinh(M)  c*B(second)*sinh(S(second)  αS/M) xM(total) = xM + xM(second) = c*B*cosh(M)  c*B(second)*cosh(S(second)  αS/M) “xM(total) = 0” when “c*B*cosh(M) = c*B(second)*cosh(S(second)  αS/M)”. c*B(second) = c*B*cosh(M)/cosh(S(second) + αM/S) c*tM(total) = c*B*sinh(M)  c*B(second)*sinh(S(second)  αS/M) = c*B*(sinh(M)  cosh(M)*tanh(S(second)  αS/M)) For the second signal to be received by Planet M prior to the first signal being emitted, “c*tM(total) < 0”, and that requires “sinh(M) < cosh(M)*tanh(S(second)  αS/M)”, or “tanh(M) < tanh(S(second)  αS/M)” and that means “M < S(second)  αS/M”. It was specified that “M > 0”, “αS/M < 0” and “S(second) < 0”, therefore “M  S(second) > 0”. If both “M” and “S(second)” are small enough in magnitude (so that the signal speeds are very fast), then “M  S(second) < αS/M” to the result “c*tM(total) < 0”. If the first signal is a weapon, then the weapon is countered by a second weapon counterattack that goes backward in time to destroy the enemy prior to their initial attack. But then the attack is not initiated,
63 CHAPTER 2 – PARTICLES and, therefore, the counterattack is not initiated. To avoid such a strange condition, a rule of nature prevents controlled or prescribed information from being transmitted faster than the speedoflight, to not violate causeandeffect. The hypothetical violation of causeandeffect for hyperlightspeed signals is a classic feature of Special Relativity. Matterwaves. An electron’s matterwave moves at phase speed “vp”. vpM/c = M/kxM = (ħ*M)/(ħ*kxM) = (EM)/(pxM) = coshαM/sinhαM = c/vM Per the above equation (which assumes no potential energy for the electron) phase speed “vpM/c” is the reciprocal of group speed of the electron, “c/vM”. If group speed “vM = 0” (for a stopped electron), then phase speed “vpM” equals reciprocal of zero. If all of space cycled as a wave per “T = exp(i*M*tM)”, then information of the value of “T” can be thought of as being transmitted in both directions of “x” instantly because it goes to the extremes of “x” and back instantly. The wave carries no controlled information and so is not a signal. Controlled information is formed from the interference pattern of the waves. The interference pattern moves at the group speed. The group forms the probability function of particle location, and the group speed is slower than the speedoflight.
Figure 22. Wavelength node points for a matterwave, with “vpM > c”.
64 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
2.10 AntiMatter Antimatter was first proposed because of the alternative/opposite electron identified in the solution of the Dirac Equation. A few years later, the antimatter electron, called the positron, was discovered experimentally. Many years later, Feynman proposed that antimatter is matter that moves backward in time. (The below proposed use of Special Relativity as the basis of a theory for antimatter might be new with this book. A search didn’t find it anywhere.)
Figure 23. Back and front are swapped for antimatter. But more than that, antimatter is turned insideout, like a glove, such that the left glove appears to be a right glove. Momentum for matter is on the left figure and is to the right. Momentum for antimatter is on the right figure and is to the left. An observer made of matter feels the push of the momentum of antimatter as if it moves to the right, because the observed momentum is a reaction from the push that drove the antimatter backward in time and space.
An observer sits inside bus “M” and observes a baseball with speed “vM” pass their seat toward the front of the bus, “αM, vM > 0”. Bus speed is represented by “αS/M”. αS/M = (N + 1/2)*i*2* ;
NZ
“αS/M = (N + 1/2)*i*2*” is written simplified to “αS/M = i*”.
65 CHAPTER 2 – PARTICLES vS/M/c = tanhαS/M = tanh(i*) = i*tan() = i*0 = 0 “αS/M = i*” has no effect on speed. Total speed “vS” of the baseball inside the bus is found by adding hyperbolicangles. αS = αM + αS/M = αM i* vS/c = tanhαS = tanh(αM i*) = sinh(αM i*)/cosh(αM i*) = (sinh(αM)*cosh(i*) + cosh(αM)*sinh(i*)) /(cosh(αM)*cosh(i*) + sinh(αM)*sinh(i*)) = (sinh(αM)*cos() cosh(αM)*i*sin()) /(cosh(αM)*cos() sinh(αM)*i*sin()) = (sinhαM)/(coshαM) = tanhαM “vS = vM = c*tanhαM” even though “αS αM”. General form: 2r
= 1M*(c*tB)*exp(qx*αM) = 1M*(c*tB)*exp(qx*αM)*(exp(qx*i*)/exp(qx*i*)) = 1M*(c*tB)*exp(qx*αM)*(exp(jx*)/exp(jx*)) = 1M*(c*tB)*exp(qx*αM)*(1/1) = (1M)*((c*tB)*exp(qx*αM))
c*tS = c*tM*coshαS/M + xM*sinhαS/M = c*tM*cosh(i*) + xM*sinh(i*) = c*tM*cos() i*xM*sin() = c*tM*(1) i*xM*0 = c*tM c*tS = c*tB*coshαS = c*tB*cosh(αM + αS/M) = c*tB*cosh(αM + i*) = c*tB*(cosh(αM)*cosh(i*) + sinh(αM)*sinh(i*)) = c*tB*(cosh(αM)*cos() + i*sinh(αM)*sin()) = c*tB*cosh(αM) = c*tM
66 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY xS = c*tM*sinhαS/M + xM*coshαS/M = c*tM*sinh(i*) + xM*cosh(i*) = i*c*tM*sin() + xM*cos() = i*c*tM*0 + xM*(1) = xM xS = c*tB*sinhαS = c*tB*sinh(αM + αS/M) = c*tB*sinh(αM + i*) = c*tB*(sinh(αM)*cosh(i*) + cosh(αM)*sinh(i*)) = c*tB*(sinh(αM)*cos() + cosh(αM)*sin()) = c*tB*sinh(αM) = xM 1S = 1M*exp(qx*αS/M) = 1M*exp(∓qx*i*) = 1M*exp(∓jx*) = 1M*(1) = 1M qxS = qxM*exp(qx*αS/M) = qxM 2r
= 1S*c*tS + qxS*xS = (1M)*(c*tM) + (qxM)*(xM) = 1M*c*tM + qxM*xM = 2r
Observer “S” sees time “tS” go forward and, due to “c*tS = c*tM”, sees time going backward for what observer “M” sees going forward. To visualize this: Observer “S” watches time pass “tBmatter” on his clock and through the bus window sees a clock for “tBantimatter” made of antimatter and recognizes “tBantimatter = tBmatter”. “Bmatter” is the rest frame (“αBmatter = 0”) for matter. In contrast, “αBantimatter = i*” creates the negative. (It is oneinthesame with “αS/M = i*” for the bus, with the difference being how reference frames are defined.) For antimatter insert a negative. c*tBmatter = +((c*tS)2  xS2) c*tBantimatter = ((c*tS)2  xS2) tBantimatter*cosh(αBantimatter) = tBmatter*cosh(αBmatter) = tBmatter*cosh(αBmatter)
67 CHAPTER 2 – PARTICLES “c*tS = c*tM” is complemented by “xS = xM”. Measuring tape on the floor of the bus has increasing numbers that are negative of the tape on the roadside, and therefore the bus points backward. The clock held by the antimatter person “M” is observed by person “S” to be moving to smaller numbers and those numbers are reversed lefttoright so that the clock hand moves clockwise, just like the matter clock. Imagine the baseball is rolling on the floor toward the front of the bus (to the right) inside a little toy car frame. Person “M” seated in the bus sees headlights in front and taillights in back. Person “S” standing on the roadside sees the little car frame moving to the right, too, per “vS = vM”, but with taillights leading and the headlights following. Person “S” looks at the whole bus and sees the back faces positive “xS” (to the right) and the front faces negative “xS” and concludes “vS = vM” because of the double negative, time and space. Frequency. Person “M” seated in the bus hears clock ticks at frequency “M = B”. Person “S” on the roadside hears tick frequency “S = M” because “coshS = coshM”. The hyperbolicradius of the frequency invariant for antimatter is negative. Bmatter = Bantimatter
same as
c*tBmatter = c*tBantimatter
S = Bantimatter*coshS = Bantimatter*cosh(M + i*) = Bantimatter*cosh(i*) = Bantimatter*1 = Bantimatter = Bmatter = Bmatter*1 = Bmatter*cosh(0) = Bmatter*coshM Our observation of frequency “S” is positive, both for observing the antimatter baseball’s clock ticks and for observing the matter baseball’s clock ticks. Therefore “M = Bantimatter” is negative. Energy and Momentum. “ES/c = EM/c” and “pxS = pxM”. Person “S” is impacted by an antimatter baseball and it feels the same as if it were matter. The difference is that the matter baseball was received by person “S”, and the antimatter baseball was launched by person “S”. Person “S” felt the reaction force from launching it backward in time and space.
68 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY To help visualize matter and antimatter feeling the same, imagine the baseball recomposes itself into an electric field. The moving electric field induces a magnetic field and therefore has inertia same as mass. Antimatter has the electric field reversed because antimatter is what is formed when matter is subtracted from a vacuum. Regardless of the direction of the electric field and its induced magnetic field, the momentum will feel the same when it impacts person “S”. ES(of antimatter)/c = +ES(of matter)/c pxS(of antimatter) = +pxS(of matter) Two particles move down the road and pass person “S”. The matter particle has “S(of matter) R”, “S(of matter) = M”, and “mBmatter > 0”. ES(of matter)/c = mBmatter*c*cosh(S(of matter)) = mBmatter*c*cosh(M) pxS(of matter) = mBmatter*c*sinh(S(of matter)) = mBmatter*c*sinh(M) Antimatter particle has “S(of antimatter) = M + i*”, and “mBantimatter < 0”. ES(of antimatter)/c = mBantimatter*c*cosh(S(of antimatter)) = mBantimatter*c*cosh(M + i*) = mBantimatter*c*cosh(M) = mBmatter*c*cosh(M) = ES(of matter)/c pxS(of antimatter) = mBantimatter*c*sinh(S(of antimatter)) = mBantimatter*c*sinh(M + i*) = mBantimatter*c*sinh(M) = mBmatter*c*sinh(M) = pxS(of matter) Newtonian Mechanics with AntiMatter. “mBantimatter = mBmatter” should not be put into the context of mass times acceleration equals force, but rather into the context of force equals the time derivative of momentum, as Newton originally presented his second law. Per “pxS(of antimatter) = +pxS(of matter)” the negative mass of antimatter is inconsequential. Negative rest mass also affects Newton’s Law of Gravity, but General Relativity, which uses energy and not mass, supersedes it, and per “ES(of antimatter)/c = +ES(of matter)/c” the negative mass of antimatter is
69 CHAPTER 2 – PARTICLES inconsequential. Note that antimatter has not been produced in a quantity large enough to measure force due to gravity. Reverse Parity of AntiMatter. A bus in reverse parity is perhaps best visualized using a righthand glove. Fingers point to positive “x”. Pull fingers through the open end to turn it inside out. Fingers now point to negative “x” and the glove looks like a lefthand glove. That same operation is not possible with a bus or particle. Rather, imagine the bus is an illustration on a sheet of paper as one page in a book sitting flat on a table. The page is turned by lifting it up and placing it upside down on the other side of the book to create a mirror image. If ever antimatter is made from its matter counterpart, then it will have been rotated through another dimension, just as the page had to be lifted out of the plane of the table. This pageturning visualization is a classic interpretation of the reverse parity of antimatter. Antimatter Electric Current Density. Electric charge “QB” is in a space derivative (with respect to “xB” as a second reference to reference frame “B”) to form electric charge density “Bmatter = dQB/dxB” as the hyperbolicradius of the current density invariant “2J”. Bmatter = dQBmatter/dxBmatter
;
2J
= 1M*Bmatter*exp(qx*M)
The ratio of antimatter electric charge to location, formed as a derivative, has a double negative, and, therefore, no negative. QBantimatter = QBmatter
;
dQBantimatter = dQBmatter
xBantimatter = xBmatter
;
dxBantimatter = dxBmatter
dQBantimatter/dxBantimatter = dQBmatter/dxBmatter = dQBmatter/dxBmatter Bantimatter = +Bmatter The Lorentz Transformation from matter observed in bus “M” to antimatter observed from the roadside “S” uses “S/M = i*”.
70 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2J
= 1M*JtM + qxM*JxM = 1M*Bmatter*exp(qx*M) = 1M*Bmatter*exp(qx*M)*exp(qx*i*)/exp(qx*i*)) = 1M*Bmatter*exp(qx*M)*(1)/(1) = (1M*/(1))*(Bmatter)*exp(qx*αM) = 1S*(Bmatter)*coshαM + qxS*(Bmatter)*sinhαM = 1S*(Bmatter)*coshαM + qxS*(Bmatter)*sinhαM = 1S*(JtM) + qxS*(JxM) = 1S*JtS + qxS*JxS
“JtS = JtM” and “JxS = JxM” state the electric charge density and the electric current density become negative. Because “Bantimatter = Bmatter”: JtS(of antimatter) = JtS(of matter)
JxS(of antimatter) = JxS(of matter)
(The analogous equations for energymomentum did not have the negative.) Two particles pass person “S”. Matter particle: “S(of matter) R”, “S(of matter) = M”, and “Bmatter > 0”. JtS(of matter) = Bmatter*c*cosh(S(of matter)) = Bmatter*c*cosh(M) JxS(of matter) = Bmatter*c*sinh(S(of matter)) = Bmatter*c*sinh(M) For the antimatter particle “S(of antimatter) = M + i*”, “Bantimatter > 0”. JtS(of antimatter) = Bantimatter*c*cosh(S(of antimatter)) = Bantimatter*c*cosh(M + i*) = Bantimatter*c*cosh(M) = Bmatter*c*cosh(M) = JtS(of matter) JxS(of antimatter) = Bantimatter*c*sinh(S(of antimatter)) = Bantimatter*c*sinh(M + i*) = Bantimatter*c*sinh(M) = Bmatter*c*sinh(M) = JxS(of matter) “JtS(of antimatter) = JtS(of matter)” says antimatter electric charge density is the negative of matter. Specifically, for the electron, the antielectron (called the positron) is observed in “S” having a positive electric charge (compared to negative charge of the electron).
71 CHAPTER 2 – PARTICLES “JxS(of antimatter) = JxS(of matter)” states antimatter electric current density is the negative of matter. A flow of positively charged particles is the negative of a flow of negatively charged particles.
Figure 24. Antimatter moving from future to past contrasted with matter moving from past to future, for “S(matter) = M” and “S(antimatter) = M i*”.
Matterwaves of Antimatter. A factory makes antimatter one particle at a time and assembles an antimatter bus, an android, and a rubber bounce ball. The (antimatter) rubber ball bounces off walls of the (antimatter) bus with smaller and smaller bounces losing energy to friction and eventually stops. Time appeared to go forward, just like for us in the factory made of matter. To explain this, we can propose that the direction of time is determined by the particle that makes the observation (which is us made of matter), and not by the particle being observed (which is the bouncing ball). This would make sense in terms of quantum mechanics because the “collapse of the wave function” in which particle properties get specified, is an observation. What would the android made of antimatter observe of us? How would it view the two slit experiment? Can time move in both directions in our cosmological model of the universe? What experiments are needed?
72 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
2.11 Distributed Material Theory A distributed material is a field of point particles, each point with an infinitesimal quantity. Examples are a fluid, a solid, an electromagnetic field, and a distributed electric charge. Each is spread through space as a continuum and varies in time. Differential calculus applies. SpaceNegative. Timespace differential gradient operator invariant “4sn” (del) is the mathematical tool for distributed material. 4
= tM*itM + xM*ixM + yM*iyM + zM*izM
translates to all number expression 4
sn
= 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM
Spacenegative operator “sn” has two aspects: Space compoundlabelnumbers “qxMsn”, “qyMsn” and “qzMsn” are negative, and the Lorentz Transformation is inverted. The inverted Lorentz Transformation compensates for the lack of a negative on “1M”. A spacenegative is necessary because time and space are in the denominator for “4”. A long rod has temperature “1T” along its length. Invariant “1T” can loosely be called a “scalar” field. The compoundlabelnumber associated with a scalar field is integer “1”. It is devoid of a reference to the unspecifiedspeedparameter “”, but is “compound”, regardless. 1T(tM,
xM) = C + a*tM + b*xM
The rod is mounted inside bus “M” and moves with the bus. “1T” has slope “b” (measured in degrees centigrade per meter) and increases at rate “a” (measured in degrees centigrade per second) when measured by a person seated on bus “M”. T/ctM = a/c
;
T/xM = b
The two gradients are placed into a timespace gradient invariant.
73 CHAPTER 2 – PARTICLES 2
sn
*1T = 1Msn*T/ctM + qxMsn*T/xM = 1Msn*(a/c) + qxMsn*b
The timespace gradient operator is 2
sn
= 1Msn*/ctM + qxMsn*/xM
Multiplication symbol “*” after operator “2sn” tells us to think of “2sn” as an invariant. Bus “M” moves at speed “vS/M” relative to roadside “S”. Gradients “T/ctS” and “T/xS” are measured. 2
sn
*1T = 1Ssn*T/ctS + qxSsn*T/xS ;
2
sn
= 1Ssn*/ctS + qxSsn*/xS
Figure 25. Temperature “T” at point “xS” decreases when the bus moves forward at speed “vS/M > 0”. Therefore, “TS/ctS < 0” for “TM/xM > 0”.
Bus “M” moves forward (“vS/M > 0”) to present colder and colder temperature to location “xS” if “a = 0” and “b > 0”, so that “T/ctS < 0”. Colder temperature requires a negative. T/ctS = (T/ctM)*(coshS/M) + (T/xM)*(sinhS/M) = (a/c)*(coshS/M) + (b)*(sinhS/M) T/xS = (T/ctM)*(sinhS/M) + (T/xM)*(coshS/M) = (a/c)*(sinhS/M) + (b)*(coshS/M)
74 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY To better see the negative, consider a nonrelativistic speed by replacing “sinhS/M” with “vS/M/c” and “coshS/M” with “1”. T/ctS = b*vS/M/c
T/xS = b
Using the above “T/ctS” and “T/xS” information, onecomponent invariant “1T” can be expressed in terms of “c*tS” and “xS”. 1T(tS,
xS) = C + (T/ctS)*(c*tS) + (T/xS)*(xS)
= C + ((T/ctM)*(coshS/M) + (T/xM)*(sinhS/M))*(c*tS) + ((T/ctM)*(sinhS/M) + (T/xM)*(coshS/M))*(xS) To prove this is correct, derive “1T(tM, xM)” from “1T(tS, xS)”. 1T(tS,
xS) = C + ((T/ctM)*(coshS/M) + (T/xM)*(sinhS/M)) *((c*tM)*(coshS/M) + (xM)*(sinhS/M)) + ((T/ctM)*(sinhS/M) + (T/xM)*(coshS/M)) *((c*tM)*(sinhS/M) + (xM)*(coshS/M)) = C + (T/ctM)*c*tM + (T/xM)*xM = 1T(tM, xM)
Gradient operators based on those equations are below. /ctS = (/ctM)*(coshS/M) + (/xM)*(sinhS/M) /xS = (/ctM)*(sinhS/M) + (/xM)*(coshS/M) tS = tM*coshαS/M  xM*sinhαS/M xS = tM*sinhαS/M + xM*coshαS/M The critical concept here is the “” sign in front of “sinhS/M”. This “” sign makes the Lorentz Transformation for “2sn” the opposite (or inverse) of the Lorentz Transformation for “2r”, given below. c*tS = (c*tM)*(coshS/M) + (xM)*(sinhS/M) xS = (c*tM)*(sinhS/M) + (xM)*(coshS/M)
75 CHAPTER 2 – PARTICLES There is no negative sign in the above “2r” invariant equations. Negatives make the two Lorentz Transformations inverses of each other. A matrix multiplied by its inverse equals one. coshS/M
sinhS/M
coshS/M
sinhS/M
* sinhS/M
coshS/M
1 0 =
sinhS/M
coshS/M
0 1
Including “” with “sinhαS/M” makes the Lorentz Transformation for “2sn” special. To show it is special, it is given symbol “sn”. Space Negative on Other Invariants. “sn” applies to other invariants. “2k = (M/c)*itM + kxM*ixM” translates to “2k = 1M*(M/c) + qxM*kxM”. S/c = (M/c)*(coshS/M) + (kxM)*(sinhS/M) kxS = (M/c)*(sinhS/M) + (kxM)*(coshS/M) “2k = (M/c)*itM  kxM*ixM” translates to“2ksn = 1Msn*(M/c) + qxMsn*kxM”. S/c = (M/c)*(coshS/M)  (kxM)*(sinhS/M) kxS = (M/c)*(sinhS/M) + (kxM)*(coshS/M) “2ksn” is abnormal (because of the negative before “ixM”). When we use “2ksn”, we are modelling a wavenumber in a special way, for example, for antimatter, if we choose that antimatter moves to the left for positive “M” when matter moves to the right for positive “M”, as will be done in the chapter on waves. “2ksn” being abnormal is unlike the timespace gradient operator “2sn”, because the spacenegative on the gradient operator “2sn” is normal (because of the positive before “ixM”). Multiplication Operation with the Space Negative. Because of the inverted matrix, space labelnumbers are negative. Also, multiplication needs a hypercomplexconjugate, for another negative. 2
sn
*1T = 1Msn*T/ctM + qxMsn*T/xM (2sn*1T)*j = T/ctM*1M*jsn + T/xM*qxM*jsn
76 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2r
= 1M*c*tM + qxM*xM xM)  C = ((2*1T)*jsn)•(2r) = (T/ctM*((1M)*jsn) + (T/xM*(qxM)*jsn))•(1M*c*tM + qxM*xM) = (T/ctM*((1M)*jsn)*(1M*c*tM) + (T/xM*(qxM)*jsn)*(qxM*xM) = (T/ctM)*(c*tM)*((1M)*jsn)*(1M) + (T/xM)*(xM)*((qxM)*jsn)*(qxM) = (T/ctM)*(c*tM) + (T/xM)*(xM)
1T(tM,
“(T/ctM)*(c*tM) + (T/xM)*(xM)” has a “+” sign between the two terms. It is different from “((2k)*j)•(2r)”, as given below, because “((2k)*j)•(2r)” has a negative sign “” between the two terms. ((2k)*j)•(2r) = (M/c)*((1M)*j) + (kxM*(qxM)*j))•(1M*c*tM + qxM*xM) = (M/c)*((1M)*j)*(1M*c*tM) + (kxM*(qxM)*j)*(qxM*xM) = (M/c)*(c*tM)*((1M)*j)*(1M) + (kxM)*(xM)*((qxM)*j)*(qxM) = M*tM  kxM*xM To create the “+” sign between the two terms in “((2 *1T)*j)•(2r)”, a negative was on the space compoundlabelnumbers introduced by the hypercomplexconjugate operation, and another negative was introduced by the spacenegative operator, for a net result of a positive. (“pxM” and “kxM” will be discussed later.) sn
(1M)sn = 1M ; (qxM)sn = qxM ; (qyM)sn = qyM ; (qzM)sn = qzM (1M)*jsn = 1M ; (qxM)*jsn = qxM*j; (qyM)*jsn = qyM*j; (qzM)*jsn = qzM*j ((1)*jsn)*1M = 1 ; ((qxM)*jsn)*qxM = +1 ((qyM)*jsn)*qyM = +1 ; ((qzM)*jsn)*qzM = +1 ((1M)*jsn)*qxM = ((qxM)*jsn)*1M = pxM ((qyM)*jsn)*qzM = ((qzM)*jsn)*qyM = i*pxM = kxM Examples of Gradient Operations with SpaceNegative. A conservation law typically uses the divergence operator, “((4)*jsn)•”. ((4)*jsn)•(4) = (tM/ctM) + (xM/xM) + (yM/yM) + (zM/zM) = 0 ((4)*jsn)•(4J) = (JtM/ctM) + (JxM/xM) + (JyM/yM) + (JzM/zM) = 0 ((4)*jsn)•(4V) = (VtM/ctM) + (VxM/xM) + (VyM/yM) + (VzM/zM) = 0
77 CHAPTER 2 – PARTICLES “4” is a density of particles (for example, a count of gravel particles in the back of a truck). “4J” is electric current density and is a special case of “4”. “4V” is voltage from the next chapter. Curl operator, “((4)*jsn)x”: ((4)*jsn)x(4J) = 6G
((4)*jsn)x(4V) = 6E
“6E” is the electromagnetic field invariant from the next chapter. What is special about the fourdimensional timespace curl operator “((4)*jsn)x” is the lack of a negative between time term and space term. This special feature is shown in the twodimensional timespace simplification given below. ((2)*jsn)x(2V) = = (/ctM*((1M)*jsn) + (/xM*(qxM)*jsn))x(1M*VtM + qxM*VxM) = (/ctM)*((1M)*jsn)*(qxM*VxM) + (/xM)*((qxM)*jsn)*(1M*VtM) = (VxM/ctM)*((1M)*jsn)*(qxM) + (VtM/xM)*((qxM)*jsn)*(1M) = (VxM/ctM + VtM/xM)*(pxM) = ExM*pxM ExM = VxM/ctM + VtM/xM ; pxM = ((1M)*jsn)*(qxM) = ((qxM)*jsn)*(1M) d’Alembert operator, what is called the harmonic operator: ((4)sn)*((4)*jsn) = ((4)sn)•((4)*jsn) = tM2  (xM2 + yM2 + zM2) = (4)2 4
2
*4V = 4J
;
4
2
*6E = 6G
TheoryDevelopmentAlgebra Mathematics. “((4)*jsn)•(4) = 0” is the same as fluid mass conservation from engineering class, where it is written “/t + 3•(*3v) = 0” for which “3v = vx*ix + vy*iy + vz*iz” is velocity. In engineering class, it was derived very pragmatically by equating what goes into and out of a cube to what is inside.
78 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY “/t + 3•(*3v) = 0” appeared strange because “+” is between time and space terms, and not “” as in “((2k)*j)•(2r) = *t  kx*x”. The “+” is now explained by use of the much less efficient nomenclature of the spacenegative in the allnumber theorydevelopmentalgebra. Theorydevelopmentalgebra is much more symbol intensive and is being used because our purpose for math has changed from engineering calculations to theory development, with the benefit being we see how mass conservation fits into the larger scheme of patterns in our world. Tensor notation calculus of General Relativity is a form of differential geometry and also is an engineeringcalculationalgebra. Covariant variables time and space are in the denominator of a derivative with the result being a contravariant vector. A contravariant vector is the tensor notation calculus analogy to the spacenegative and special rules state where to insert negatives. The challenge is to create a theorydevelopmentalgebra for General Relativity that can replace tensor notation calculus. The challenge looks difficult. Example of Particle Count Conservation. A pile of gravel moves in the positive “x” direction. Particle count starts on the left of “xB” in “dCountB/dxB = A*exp( (xB*kxB)2 )”.
Figure 26. A moving pile of gravel of height “h”.
79 CHAPTER 2 – PARTICLES Subscript “B” identifies the rest frame of the pile of gravel and is the same as “M”. “dCountB/dxB” is placed into a timespace invariant “2lump” by multiplying it by the compoundlabelnumber “1B”. 2lump
= 1B*( dCountB/dxB ) = 1B*( A*exp((xB*kxB)2) ) = 1B*lumptB
Use a Lorentz Transformation to derive “lumptS” and “lumpxS”. 2lump
= 1B*( dCountB/dxB ) = 1S*( dCountB/dxB )*exp(qx*αS/B) = 1S*( A*exp((xB*kxB)2) )*exp(qx*αS/B) = 1S*( A*exp( ((xS  vS*tS)*coshαM*kxB)2 ) )*exp(qx*αS/B) = 1S*( A*exp( ((xS  vS*tS)*coshαM*kxB)2 ) )*exp(qx*αS) = 1S*( A*exp( ((xS  vS*tS)*coshαS*kxB)2 ) )*coshαS + qxS*( A*exp( ((xS  vS*tS)*coshαS*kxB)2 ) )*sinhαS = 1S*lumptS + qxS*lumpxS lumptS = ( A*exp( ((xS  vS*tS)*coshαS*kxB)2 ) )*coshαS lumpxS = ( A*exp( ((xS  vS*tS)*coshαS*kxB)2 ) )*sinhαS 1S = 1B/exp(qx*αS/B)
Substitution “xB = (xS  vS*tS)*coshαS” was derived from the Lorentz Transformation from “B” to “S”. c*tS = c*tB*coshαS + xB*sinhαS ; c*tB = c*tS*coshαS  xS*sinhαS xS = c*tB*sinhαS + xB*coshαS ; xB = c*tS*sinhαS + xS*coshαS xB = c*tS*sinhαS + xS*coshαS ; c*tB = c*tS*coshαS  xS*sinhαS = xS*coshαS  c*tS*sinhαS = c*tS*coshαS  xS*sinhαS = xS*coshαS  c*tS*tanhαS*coshαS = c*tS*coshαS  xS*tanhαS*coshαS = (xS  tanhαS*c*tS)*coshαS = (c*tS  tanhαS*xS)*coshαS = (xS  vS*tS)*coshαS = c*(tS  vS*xS/c2)*coshαS The conservation law is applied.
80 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2
sn
= 1Ssn*tS + qxSsn*xS ; sn sn = 1S */ctS + qxS */xS
2lump
*jsn
= tS*1S*jsn + xS*qxS*jsn
= 1S*(dCountB/dxB)*coshαS + qxS*(dCountB/dxB)*sinhαS
1S*jsn*1S = 1 ; 2
2
qxS*jsn*qxS = 1
•2lump = (tS*1S*jsn)*(1S*(dCountB/dxB)*coshαS) + (xS*qxS*jsn)*(qxS*(dCountB/dxB)*sinhαS)
*jsn
= tS*((dCountB/dxB)*coshαS) + xS*((dCountB/dxB)*sinhαS) For the example above tS*((dCountB/dxB)*coshαS) = (/ctS)*(( A*exp( ((xS  vS*tS)*(coshαS)*kxB)2 ) )*coshαS) = (((xS  vS*tS)2)/ctS)*(kxB*coshαS)2 *(( A*exp( ((xS  vS*tS)*(kxB*coshαS))2 ) )*coshαS) = 2*((xS  vS*tS)/ctS)*(xS  vS*tS)*(kxB*coshαS)2 *(( A*exp( ((xS  vS*tS)*(kxB*coshαS))2 ) )*coshαS) = 2*(vS/c)*(xS  vS*tS)*(kxB2*cosh2αS) *A*exp(((xS  vS*tS)*(kxB*coshαS))2)*coshαS = 2*(sinhαS/coshαS)*(xS  vS*tS)*(kxB2*cosh2αS) *A*exp(((xS  vS*tS)*(kxB*coshαS))2)*coshαS = 2*(xS  vS*tS)*(kxB2*cosh2αS) *A*exp(((xS  vS*tS)*(kxB*coshαS))2)*sinhαS xS*((dCountB/dxB)*sinhαS) = (/xS)*(( A*exp( ((xS  vS*tS)*(kxB*coshαS))2 ) )*sinhαS) = (((xS  vS*tS)2)/xS)*(kxB*coshαS)2 *(( A*exp( ((xS  vS*tS)*(kxB*coshαS))2 ) )*sinhαS)
81 CHAPTER 2 – PARTICLES = 2*((xS  vS*tS)/xS)*(xS  vS*tS)*(kxB*coshαS)2 *(( A*exp( ((xS  vS*tS)*(kxB*coshαS))2 ) )*sinhαS) = 2*(xS  vS*tS)*(kxB2*cosh2αS) *A*exp(((xS  vS*tS)*(kxB*coshαS))2)*sinhαS The above analysis showed 2
•2lump = 0
*jsn
Conservation Law with a SpaceNegative Invariant. Abnormal spacenegative invariant “2antilumpsn” has subscript “anti” to indicate it applies to antimatter because if matter moves to the right, then the equivalent antimatter is chosen to move to the left (if observed by the same observer), and that is because the spacenegative operator means the negative of space, or, with respect to movement, the opposite direction. “2antilumpsn” and “2ksn” invariants are abnormal because the geometric translation has “+it” and “ix”, same as the abnormal “2”. This is in contrast to “+ix” of the geometric translation for normal invariants “2lump”, “2k” and “2sn”. For matter invariant “2lump”, movement in the positive “xM” direction is identified by “xB = xM  vM*tM”. In contrast, spacenegative antimatter invariant “2antilumpsn” includes “xB = xM + vM*tM”. The “+” rather than “” means movement is in the opposite direction. “xB = xM  vM*tM” was derived from the Lorentz Transformation. In contrast, “xB = xM + vM*tM” for the spacenegative is derived from the inverse matrix Lorentz Transformation for “2rsn”, rather than for “2r”. (The inverse matrix has the negatives on the “sinhαS/M”.) Alternatively, “xB = xM + vM*tM” can be derived from “2ksn*j•2r” rather than from “2k*j•2r” if speed “vM” is frequency divided by wavenumber. The overall movement of electric charge particles is the same for matter to the right or for antimatter to the left: For matter, if “q = 1” is the quantity of electric charge per particle, then the count that passes point “xS” moving right is proportional to the total electric charge and is positive. For antimatter the count is negative because “q = 1”. Using the spacenegative, those negative particles move left past point “xS” and so get subtracted, for a double negative.
82 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY “2*jsn•(2antilump)sn = 0” is shown below to be valid: dCountB/dxB = A*exp( (xB*kxB)2 ) sn 2antilump
= 1Ssn*(dCountB/dxB)*coshαS + qxSsn*(dCountB/dxB)*sinhαS = 1Ssn*(dCountB/dxB)*exp(qx*αS) = 1Ssn*( A*exp((xB*kxB)2)*exp(qx*αS) = 1Ssn*( A*exp(((xS + vS*tS)*kxB*coshαS)2) )*exp(qx*αS) = 1Ssn*( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*coshαS + qxSsn*( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS
c*tS = c*tB*coshαS  xB*sinhαS ; c*tB = c*tS*coshαS + xS*sinhαS xS = c*tB*sinhαS + xB*coshαS ; xB = c*tS*sinhαS + xS*coshαS xB = c*tS*sinhαS + xS*coshαS = xS*coshαS + c*tS*sinhαS = xS*coshαS + c*tS*tanhαS*coshαS = (xS + tanhαS*c*tS)*coshαS = (xS + vS*tS)*coshαS 2
sn
= 1Ssn*tS + qxSsn*xS = 1Ssn*/ctS + qxSsn*/xS
;
2
*jsn
= tS*1S*jsn + xS*qxS*jsn
(1S*jsn)*(1Ssn) = (1S*j)*(1S) = (1)*(1) = 1 (qxS*jsn)*(qxSsn) = (qxS*j)*(qxS) = (qx)*(qx) = 1 2
•2antilumpsn = (tS*1S*jsn)*(1Ssn*(dCount/dxB)*coshαS) + (xS*qxS*jsn)*(qxSsn*(dCountB/dxB)*sinhαS)
*jsn
= tS*((dCountB/dxB)*coshαS)  xS*((dCountB/dxB)*sinhαS) tS*((dCountB/dxB)*coshαS) = (/ctS)*(( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*coshαS) = (((xS + vS*tS)2)/ctS)*(kxB*coshαS)2 *(( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*coshαS) = 2*((xS + vS*tS)/ctS)*(xS + vS*tS)*(kxB*coshαS)2 *(( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*coshαS)
83 CHAPTER 2 – PARTICLES = 2*(vS/c)*(xS + vS*tS)*(kxB2*cosh2αS) *A*exp(((xS + vS*tS)*(kxB*coshαS))2)*coshαS = 2*(sinhαS/coshαS)*(xS + vS*tS)*(kxB2*cosh2αS) *A*exp(((xS + vS*tS)*(kxB*coshαS))2)*coshαS = 2*(xS + vS*tS)*(kxB2*cosh2αS) *A*exp(((xS + vS*tS)*(kxB*coshαS))2)*sinhαS xS*((dCountB/dxB)*sinhαS) = (/xS)*(( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS) = (((xS + vS*tS)2)/xS)*(kxB*coshαS)2 *(( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS) = 2*((xS + vS*tS)/xS)*(xS + vS*tS)*(kxB*coshαS)2 *(( A*exp(((xS + vS*tS)*(kxB*coshαS))2) )*sinhαS) = 2*(xS + vS*tS)*(kxB2*cosh2αS) *A*exp(((xS + vS*tS)*(kxB*coshαS))2)*sinhαS
2.12 Exercises Check on Text Comprehension. 1) For “c*t = 13” and “x = 12” calculate “rhyperbolic = ((c*t)2  x2)”, “coshα = c*t/rhyperbolic”, “sinhα = x/rhyperbolic”, and “tanhα = x/(c*t)”. Find “α = atanh(x/(c*t))” and confirm “1 = (cosh2α  sinh2α)”. Write “2r = 1*c*t + qx*x” and “2r = rhyperbolic*exp(qx*α)” using numbers. Plot “(c*t, x) = (13, 12)” on the hypercomplexplane and draw a straight line at hyperbolicangle “α = atanh(12/13)” and a hyperbola with hyperbolicradius “rhyperbolic = (132  122)”. 2) For “c*tM = 11”, “αM = 5” and “αS/M = 3” find “c*tS” and “xS”. For “EM/c = 7” find “ES/c” and “pxS”. What is “c*tB” and what is “mB*c”?
84 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3) How fast “vS/M/c” must bus “M” move so that its seats have half the spacing of a stationary bus? At that speed, what is the ratio of time between ticks of the bus’s clock to the roadside’s clock? What is the energy ratio of the moving bus to a stationary bus? 4) A hyperlightspeed signal of speed “vM(first) = c*coth(M(first))” is sent from Person M in the positive xdirection to Person S who walks away from Person M at a speed “vM/S” equal to positive seven feet per second. The instant the signal is received, Person S sends a signal back to the source at the same speed but in the opposite direction, “vS(second) = vM(first)”. At what speed “vM(first)” must the signal travel so that the second signal arrives at the same instant the first signal is emitted? Draw the two signals on the hypercomplexplane for Person M stationary and then, again, on the hypercomplexplane for Person S stationary. 5) An antimatter electron moves on an antimatter bus “M” at a speed “vM = c*tanhαM = 11 meters per second” with “αM R”. What is its electric charge “QBantimatter”? What is its rest mass “mBantimatter”? The bus moves relative to the road “S” with hyperbolic angle “αS/M = i*”. What is the electron’s energy “ES” and momentum “pxS”? 6) A wavy pattern for static electric charge is across the roof of a bus: “dQB/dxB = Qwavy = A*sin(xB/a)”. “dQB/dxB” is hyperbolicradius “Qwavy” of the current density invariant “2J”. Show electric charge is conserved for any speed of the bus by showing “2*jsn•2J = 0”. Write “2Jsn” (to represent an antimatter car), by analogy using the example in the text above, and show “2*jsn•2Jsn = 0”. 7) A row of spinning disks has a density “HxB+ = dHBx+/dxB” with “HBx+ > 0”. In parallel, a moving row of spinning disks has a density “HxB = dHBx/dxB” with “HBx < 0”. This second row moves at speed “vM > 0” (to the right), so that “αM > 0”. The total angular momentum of both rows together is constrained to be zero. For both rows together, what is the hyperbolic radius of the timelike angular momentum invariant for the rate of angular momentum that passes point “xM”.
85 CHAPTER 2 – PARTICLES 8) Angular momentum density “2H = 1M*qx*dHBx/dxB*exp(qx*αM)” is used in conservation law “((2)*jsn)x(2H) = 0” for which no torque is applied to change angular momentum. For “2H” below, verify the conservation law. 2H
= 1M*qx*(dHBx/dxB)*exp(qx*αM) = 1M*qx*( A*exp( ((xB)/a)2 ) )*exp(qx*αM)
Answers to Select Exercises. 1) rhyperbolic = (132  122) = 5 ; α = atanh(12/13)) = 1.60943… coshα = 13/5 = 2.6 ; (2.6)2  (2.4)2 = 6.76  5.76 = 1 sinhα = 12/5 = 2.4 ; 2r = 1*13 + qx*12 tanhα = 12/13 = 0.092307… ; 2r = 5*exp(qx*1.60943…) 2) c*tB = c*tM/cosh(αM) = 11/cosh(5) = 0.14822… xM = c*tB*sinh(αM) = 0.14822…*sinh(5) = 10.99900… c*tS = c*tB*cosh(αM + αS/M) = 0.14822…*cosh(5 + 3) = 220.9309… xS = c*tB*sinh(αM + αS/M) = 0.14822…*sinh(5 + 3) = 220.9308… c*tS = c*tM*cosh(αS/M) + xM*sinh(αS/M) = 11*cosh(3) + 10.99900…*sinh(3) = 220.9309… xS = c*tM*sinh(αS/M) + xM*cosh(αS/M) = 11*sinh(3) + 10.99900…*cosh(3) = 220.9308… mB*c = (EM/c)/cosh(αM) = 7/cosh(5) = 0.09432… pxM = mB*c*sinh(αM) = 0.09432…*sinh(5) = 6.99930… ES/c = mB*c*cosh(αM + αS/M) = 0.09432…*cosh(8) = 140.59239… pxS = mB*c*sinh(αM + αS/M) = 0.09432…*sinh(8) = 140.59235… ES/c = (EM/c)*cosh(αS/M) + pxM*sinh(αS/M) = 7*cosh(3) + 6.99930…*sinh(3) = 140.59239… pxS = (EM/c)*sinh(αS/M) + pxM*cosh(αS/M) = 7*sinh(3) + 6.99930…*cosh(3) = 140.59235…
86 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3) kxS/kxM = 2 ; αS/M = acosh(kxS/kxM) = acosh(2) = 1.31695… vS/M/c = tanh(αS/M) = 3/2 ; tS/tM = ES/EM = coshαS/M = 2 4) M(first) = αM/S/2 = atanh(vM/S/c)/2 5) and 6) solution not given 7)
2H
= 2H+ + 2H= 1M*qx*HxB+*exp(qx*αM+) + 1M*qx*HxB*exp(qx*αM) = 1M*qx*HxB+ + 1M*qx*HxB*exp(qx*αM) = 1M*qx*HxB+ + 1M*qx*HxB*(coshαM + qx*sinhαM) = 1M*qx*(HxB+ + HxB*coshαM) + 1M*HxB*sinhαM= 1M*HxB*sinhαM = 1M*(HxB+/coshαM)*sinhαM= 1M*HxB+*tanhαM = 1M*HxB+*(vM/c)
For both rows together, the hyperbolic radius of the timelike angular momentum invariant for the rate of angular momentum that passes point “xM” is “HxB+*(vM/c)”. 8)
2H
= 1M*qx*(dHBx/dxB)*coshαM + qxM*qx*(dHBx/dxB)*sinhαM = 1M*qx*(dHBx/dxB)*exp(qx*αM) = 1M*qx*( A*exp( ((xB)/a)2 ) )*exp(qx*αM) = 1M*qx*( A*exp( (((xM  vM*tM)*coshαM)/a)2 ) )*exp(qx*αM) = 1M*qx*( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*coshαM + qxM*qx*( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*sinhαM
xB = c*tM*sinhαM + xM*coshαM = xM*coshαM  c*tM*sinhαM = xM*coshαM  c*tM*tanhαM*coshαM = (xM  tanhαM*c*tM)*coshαM = (xM  vM*tM)*coshαM 2
sn
2
*jsn
= 1Msn*tM + qxMsn*xM ; = 1Msn*/ctM + qxMsn*/xM
2
*jsn
= tM*1M*jsn + xM*qxM*jsn
x2H = (tM*1M*jsn)*(1M*qx*(dHxB/dxB)*coshαM) + (xM*qxM*jsn)*(qxM*qx*(dHxB/dxB)*sinhαM)
= (tM*((dHxB/dxB)*coshαM) + xM*((dHxB/dxB)*sinhαM))*qx
87 CHAPTER 2 – PARTICLES tM*((dHxB/dxB)*coshαM) = (/ctM)*(( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*coshαM) = (((xM  vM*tM)2)/ctM)*(coshαM/a)2 *(( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*coshαM) = 2*((xM  vM*tM)/ctM)*(xM  vM*tM)*(coshαM/a)2 *(( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*coshαM) = 2*(vM/c)*(xM  vM*tM)*(cosh2αM/a2) *A*exp(((xM  vM*tM)*(coshαM/a))2)*coshαM = 2*(sinhαM/coshαM)*(xM  vM*tM)*(cosh2αM/a2) *A*exp(((xM  vM*tM)*(coshαM/a))2)*coshαM = 2*(xM  vM*tM)*(cosh2αM/a2) *A*exp(((xM  vM*tM)*(coshαM/a))2)*sinhαM xM*((dHxB/dxB)*sinhαM) = (/xM)*(( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*sinhαM) = (((xM  vM*tM)2)/xM)*(coshαM/a)2 *(( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*sinhαM) = 2*((xM  vM*tM)/xM)*(xM  vM*tM)*(coshαM/a)2 *(( A*exp( ((xM  vM*tM)*(coshαM/a))2 ) )*sinhαM) = 2*(xM  vM*tM)*(cosh2αM/a2) *A*exp(((xM  vM*tM)*(coshαM/a))2)*sinhαM 2
x2H = 0
*jsn
Questions for Further Thought.
88 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1) If unspecifiedspeedparameter “” is a rational number, then it cannot be infinite, has equal probability of being positive or negative, and has equal probability of magnitude less than one or greater than one. This probability function “P(s)” is given in Chapter 5. What alternative assumption might we make for the structure of “”? Numerator only, for equal probability of being any number on the numberline? 2) Assume “αS/M = i*” and find “c*tS”, “xS”, “vS/M/c” and “vS/c”. If “ ≠ z*/2” for “z Z”, what problem is there? Should we make a rule (as a theory of physics), so that we mimic nature? 3) Is an invariant with hyperbolicradius “dmB/dxB” or “QB” useful? 4) What does “N” in “αS/M = i**(2*N + 1)” represent physically? 5) What might an antimatter person observe of us with respect to entropy increase, cause and effect, and successive collapses of matterwave functions? Does the antimatter person see us moving backward relative to their sense of time? Propose a cosmological model of the universe. 6) Rewrite the section on antimatter electric current but substitute angular momentum for electric current. 7) Explain why “/tM + 3•(*3v) = 0” has the two terms added, in contrast to the subtraction in “(2r*j)•(2r) = (c*tM)2  (xM2 + yM2 + zM2)”. 8) Assume information transfer between detectors in the EPR experiment (see an Appendix) occurs instantly. Try to design an experiment in which detectors move away from each other so that the detectors prove time can step backward. 9) Is the angular momentum density invariant the missing spacelike energymomentum invariant as measurement units imply it is? If timelike energymomentum is formed from induced electromagnetic fields (as explained in the next chapter), what fields could produce the spacelike energymomentum invariant?
89 CHAPTER 3 – FIELDS
Chapter 3 – Fields Using his equations, Maxwell derived that the speed for light was independent of the observer’s speed. To explain it, Einstein proposed Special Relativity. Imagine light traveling between two people sidebyside on a bus. The path is longer when observed by a person on the roadside because the path has a component in the direction of bus travel. To keep the speed for light constant, a clocktower shows faster passage of time compared to a clock mounted inside the moving bus.
3.1 GeometricVector Notation The four materials in Maxwell’s Equations are physical entities in our geometric world. Bold indicates a geometricvector. Nonbold a scalar. Electric field 3E Magnetic field 3B
; ;
Current density 3j Charge density
Each exists as a distribution in time and space, as does voltage (see Panofsky, Wolfgang and Philips, Melba: Classical Electricity and Magnetism, AddisonWesley Publishing Company, Inc.; 1955). Vector voltage 3A
;
Scalar voltage
A ground for voltage is analogous to an origin for location and to an inertial reference frame for speed or momentum. Maxwell evolved his equations and settled on component equations, and a little before year 1900, Heaviside and others applied geometricunitvectors “ix”, “iy”, and “iz”, dotproduct “•”, crossproduct “x”, and gradient operator “3”. Geometricunitvectors explicitly placed components of electromagnetism into physical space. Three constants for measurement unit conversion: = 4π*107 tesla*meter/amp 1.256637*106 tesla*meter/amp ǝ 8.854188*1012 coulomb2/(newton*meter2) c 2.99792458*108 meters/second
90 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The three constants were selected so that “c2*ǝ* = 1”. A coulomb is electric charge from a specific finite quantity of protons, or that same quantity of electrons times negative one. 1 coulomb = 6.24150965(16)*1018 electron charges QBelectron = 1.6021176462*1019 coulomb Alternate units of measure are found from: 1 tesla*meter/amp = 1 newton/amp2 = 1 volt*second/(amp*meter) 1 coulomb2/(newton*meter2) = 1 farad/meter = 1 coulomb/(volt*meter) Maxwell’s Equations in geometricvector notation: 3E
= Ex*ix + Ey*iy + Ez*iz ; 3j = jx*ix + jy*iy + jz*iz = Bx*ix + By*iy + Bz*iz 3 = x*ix + y*iy + z*iz = /x*ix + /y*iy + /z*iz 3B
3•3E
= /ǝ 2 x B 3 3 = *3j + ((3E)/t)/c
3•3B
; ;
Ex/x + Ey/y + Ez/z = /ǝ Bz/y  By/z = (Ex/t)/c2 + *jx Bx/z  Bz/x = (Ey/t)/c2 + *jy By/x  Bx/y = (Ez/t)/c2 + *jz
=0 x E 3 3 = (3B)/t
; ; ; ;
Bx/x + By/y + Bz/z = 0 Ez/y  Ey/z = Bx/t Ex/z  Ez/x = By/t Ey/x  Ex/y = Bz/t
Electric Charge Conservation in geometricvector notation: /t + 3•3j = 0
/t + jx/x + jy/y + jz/z = 0
;
Voltage equations in geometricvector notation: 3A
= Ax*ix + Ay*iy + Az*iz
3B
= 3x3A
;
3E
= 3  (3A)/t
91 CHAPTER 3 – FIELDS Az/y  Ay/z = Bx Ax/z  Az/x = By Ay/x  Ax/y = Bz 3•3A
; ; ;
/x  Ax/t = Ex /y  Ay/t = Ey /z  Az/t = Ez
+ (/t)/c2 = 0 ; Ax/x + Ay/y + Az/z + (/t)/c2 = 0
(3•3)  2()/t2/c2 = /ǝ ; (3•3)3A  2(3A)/t2/c2 = *3j 2/x2 + 2/y2 + 2/z2  (2/t2)/c2 = /ǝ 2Ax/x2 + 2Ax/y2 + 2Ax/z2  (2Ax/t2)/c2 = *jx 2Ay/x2 + 2Ay/y2 + 2Ay/z2  (2Ay/t2)/c2 = *jy 2Az/x2 + 2Az/y2 + 2Az/z2  (2Az/t2)/c2 = *jz Gauss’s Law for Electricity. “3•3E = /ǝ” states divergence of an electric field is proportional to electric charge density. In other words, electric charge is the source of an electric field.
Figure 27. Gauss’s Law for Electricity. Electric field times area of a closed surface (a sphere) equals the electric charge inside. The electric field at the radius of the sphere is calculated from “Eradial*(r2**4/3) = q/ǝ”.
Gauss’s Law for Magnetism. “3•3B = 0” states there are no magnetic field charges (in contrast to the existence of electric charges). Faraday’s Law of Electric Field Induction. “3x3E = (3B)/t” states a time varying magnetic field is a source of vorticity for an induced electric field.
92 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 28. Gauss’s Law of Magnetism. Magnetic field times area of a closed surface equals zero. Ends and the curved surface of the cylinder cut in half have “B•dA = 0”. The “B•dA” of the upper rectangle equals the negative of the “B•dA” of the lower rectangle, so that all the magnetic flux out of the identified volume equals all the flux into the volume, for a net total of zero.
Figure 29. Faraday’s Law of Induction. Electric field times length along a closed curve (a circle) equals the rate of change of the magnetic field times area enclosed by the closed curve. Switch “s” closes to initiate a flow of current “i”, to create an iron magnet inside wire coils. The gap of crosssection area “A” in the iron has increasing magnitude magnetic field “dBx/dt”, which is negative. Changing magnetic field induces an electric field per “Ecircumferential*(2**r) = (dBx/dt)*A”, for “r > (A/)” (for “r” outside the gap).
Ampere’s Law of Magnetic Field Induction. “3x3B = *3j + ((3E)/t)/c2” states vorticity of an induced magnetic field is created by a time varying electric field and/or by electric current density.
93 CHAPTER 3 – FIELDS
Figure 30. Ampere’s Law of Induction. Magnetic field times length along a closed curve (a circle) equals the rate of change of the electric field times area enclosed by the closed curve, plus the electric current through the area. Switch “s” closes to initiate a current that induces a magnetic field around the wire of strength “Bcircumferential = 0*i/(2**r)”. The magnetic field continues along the wire so that it also exists around the capacitor of area “A”, per “Bcircumferential*(2**r) = (dEx/dt)*A/c2” for the radius outside the area of the capacitor.
Figure 31. Electric Charge Conservation. The flow of electric charge out a closed surface (a cube), calculated as electric current density times area of the surface, equals the negative change in electric charge inside. In other words, electric charge cannot be created or destroyed. The box on the right contains one of the two capacitor plates. The change in electric charge on the capacitor plate inside the box is “dq/dt < 0” as noticed by the increase in the electric field “Ex”. The box is penetrated by the wire out the right for which the current “i” equals the current density “Jx” times the crosssection area of the wire. The conservation equation reduces to “i + dq/dt = 0”.
94 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Electric Charge Conservation. “/t + 3•3j = 0”, rewritten “/t = 3•3j”, states the divergence of an electric current density equals the negative of the change in electric charge density.
3.2 AllNumber Notation New component symbols apply to the second step’s translation of geometry into allnumber algebra. Vt = Vx = Ax*c Vy = Ay*c Vz = Az*c
; ; ;
; ; ; ;
Kx = Bx*c Ky = By*c Kz = Bz*c
Jt = /ǝ = *(*c2) Jx = jx*(*c) = jx/(ǝ*c) Jy = jy*(*c) = jy/(ǝ*c) Jz = jz*(*c) = jz/(ǝ*c)
“*c” and “ǝ*c” are near a numerical value of one. SimpleLabelNumbers. Using components specified above, Maxwell’s Equations, the electric charge conservation equation and the voltage equation may be written using labelnumbers. jx*jy = jy*jx = jz jy*jz = jz*jy = jx jz*jx = jx*jz = jy
; ; ;
i*jx = jx*i i*jy = jy*i i*jz = jz*i
jx2 = jy2 = jz2 = i2 = 1
;
qx2 = qy2 = qz2 = 12 = +1
qx = jx/i ; qy = jy/i ; qz = jz/i qx*jy = jy*qx = qy*jx = jx*qy = qz ; qy*jz = jz*qy = qz*jy = jy*qz = qx ; qz*jx = jx*qz = qx*jz = jz*qx = qy ;
qx*qy = qy*qx = i*qz = jz qy*qz = qz*qy = i*qx = jx qz*qx = qx*qz = i*qy = jy
UnspecifiedLabelNumber. There is an unspecifiedspeedparameter “”, and there is an unspecifiedlabelnumber “” (kappa) restricted to one of “qx”, “qy” or “qz”. “” is unknown and unknowable.
95 CHAPTER 3 – FIELDS “” and “” are placed into two exponential functions: One for each side, left and right, of a simplelabelnumber “1”, “qx”, “qy” or “qz”. Each is a square root by dividing the argument by two. 1M = exp(*/2)*1*exp(*/2) = exp(*) qxM = exp(*/2)*qx*exp(*/2) qyM = exp(*/2)*qy*exp(*/2) qzM = exp(*/2)*qz*exp(*/2) Hypothetically: “ = qx” for “1M = exp(qx*)”, “qxM = qx*exp(qx*)”, “qyM = qy” and “qzM = qz” from the previous chapter. The concept of using a square root factor on both sides, to take advantage of the anticommute law “qx*qy = qy*qx”, is a classic concept in Special Relativity. 1M = exp(*/2)*1*exp(*/2) = exp(qx*/2)*1*exp(qx*/2) “ = qx” = exp(qx*/2)*exp(qx*/2) = exp(qx*) qxM = exp(*/2)*qx*exp(*/2) = exp(qx*/2)*(qx*exp(qx*/2)) “ = qx” = exp(qx*/2)*(qx*(cosh(/2)  qx*sinh(/2))) = exp(qx*/2)*((cosh(/2)  qx*sinh(/2))*qx) = exp(qx*/2)*(exp(qx*/2)*qx) = exp(qx*)*qx qyM = exp(*/2)*qy*exp(*/2) = exp(qx*/2)*(qy*exp(qx*/2)) “ = qx” = exp(qx*/2)*(qy*(cosh(/2)  qx*sinh(/2))) = exp(qx*/2)*((cosh(/2) + qx*sinh(/2))*qy) = exp(qx*/2)*(exp(qx*/2)*qy) = qy qzM = exp(*/2)*qz*exp(*/2) = exp(qx*/2)*(qz*exp(qx*/2)) “ = qx” = exp(qx*/2)*(qz*(cosh(/2)  qx*sinh(/2))) = exp(qx*/2)*((cosh(/2) + qx*sinh(/2))*qz) = exp(qx*/2)*(exp(qx*/2)*qz) = qz
96 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Invariants. Compoundlabelnumbers combine with components to form invariants. “4sn” is a spacenegative. 4V
= VtM*it + c*AxM*ix + c*AyM*iy + c*AzM*iz 4V = 1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM = *(*c2)*it + jx*(*c)*ix + jy*(*c)*iy + jz*(*c)*iz 4J = 1M*JtM + qxM*JxM + qyM*JyM + qzM*JzM 4J
3E
= ExM*ix + EyM*iy + EzM*iz c*3B = c*BxM*ix + c*ByM*iy + c*BzM*iz 6E
= pxM*ExM + pyM*EyM + pzM*EzM + kxM*KxM + kyM*KyM + kzM*KzM
4
= tM*it + xM*ix + yM*iy + zM*iz sn = 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM 4 = 1Msn*(/ctM) + qxMsn*/xM + qyMsn*/yM + qzMsn*/zM EMCompoundLabelNumbers. Two new sets of compoundlabelnumbers (“pxM”, “pyM” and “pzM” and “kxM”, “kyM” and “kzM”) have been introduced for the electromagnetic field “6E”. pxM = exp(*/2)*qx*exp(*/2) = 1M*jsn*qxM = qxM*jsn*1M = qyM*jsn*qzM/i = qzM*jsn*qyM/i = kxM/i pyM = exp(*/2)*qy*exp(*/2) = 1M*jsn*qyM = qyM*jsn*1M = qzM*jsn*qxM/i = qxM*jsn*qzM/i = kyM/i pzM = exp(*/2)*qz*exp(*/2) = 1M*jsn*qzM = qzM*jsn*1M = qxM*jsn*qyM/i = qyM*jsn*qxM/i = kzM/i kxM = exp(*/2)*jx*exp(*/2) = i*pxM = 1M*jsn*jxM = jxM*jsn*1M = qyM*jsn*qzM = qzM*jsn*qyM = jyM*jsn*jzM = jzM*jsn*jyM kyM = exp(*/2)*jy*exp(*/2) = i*pyM = 1M*jsn*jyM = jyM*jsn*1M = qzM*jsn*qxM = qxM*jsn*qzM = jzM*jsn*jxM = jxM*jsn*jzM
97 CHAPTER 3 – FIELDS kzM = exp(*/2)*jz*exp(*/2) = i*pzM = jyM*jsn*jxM = 1M*jsn*jzM = jzM*jsn*1M = qxM*jsn*qyM = qyM*jsn*qxM = jxM*jsn*jyM 1M = qxM*pxM = qyM*kzM*pxM = qyM*pyM = qyM*pzM*kxM = 1M*1 = 1M*pxM*pxM = 1M*1 = 1M*kxM*kxM = qxM*pxM qzM = 1M*pzM = 1M*kxM*pyM = 1M*pzM = 1M*pxM*kyM = jxM*pyM = qyM*pzM*pyM = qxM*kyM = qyM*kzM*kyM = 1M*pzM = qzM*pzM*pzM = 1M*i*kzM = qzM*kzM*kzM 1M*i = qyM*kyM = qyM*kzM*kxM = jyM*pxM = qyM*pzM*pxM = 1M*pxM*kxM = 1M*kxM*pxM = qxM*kxM jzM = jxM*kyM = 1M*kxM*kyM = 1M*kzM = 1M*pxM*pyM = qxM*pyM = qyM*kzM*pyM = jxM*kyM = qyM*pzM*kyM = 1M*kzM = qzM*kzM*pzM = 1M*kyM = qzM*pzM*kyM 1M = exp(*/2)*1*exp(*/2) = exp(*) qxM = exp(*/2)*qx*exp(*/2) qyM = exp(*/2)*qy*exp(*/2) qzM = exp(*/2)*qz*exp(*/2) pxM = exp(*/2)*qx*exp(*/2) pyM = exp(*/2)*qy*exp(*/2) pzM = exp(*/2)*qz*exp(*/2) kxM = exp(*/2)*jx*exp(*/2) = i*pxM kyM = exp(*/2)*jy*exp(*/2) = i*pyM kzM = exp(*/2)*jz*exp(*/2) = i*pzM = jxM*jsn*jyM Quaternion hypercomplexconjugate operation “*j” reverses the sign of each “j”, and, therefore, also reverses the sign of each “q”, and reverses order of factors. i*j = i
; 1*j = 1
jx*j = jx ; jy*j = jy ; jz*j = jz qx*j = qx ; qy*j = qy ; qz*j = qz
98 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY jxM*j = exp(*/2)*(jx)*exp(*/2) jyM*j = exp(*/2)*(jy)*exp(*/2) jzM*j = exp(*/2)*(jz)*exp(*/2) 1M*j = exp(*/2)*(1)*exp(*/2) qxM*j = exp(*/2)*(qx)*exp(*/2) qyM*j = exp(*/2)*(qy)*exp(*/2) qzM*j = exp(*/2)*(qz)*exp(*/2)
kxM*j = kxM kyM*j = kyM kzM*j = kzM
; ; ;
; ; ;
pxM*j = pxM pyM*j = pyM pzM*j = pzM
Conjugate Form of Invariants. *j 4V 4J
*j
*j 6E
4
= VtM*1M*j + VxM*qxM*j + VyM*qyM*j + VzM*qzM*j
= JtM*1M*j + JxM*qxM*j + JyM*qyM*j + JzM*qzM*j = ExM*pxM*j + EyM*pyM*j + EzM*pzM*j + KxM*kxM*j + KyM*kyM*j + KzM*kzM*j
*jsn
= tM*1M*jsn + xM*qxM*jsn + yM*qyM*jsn + zM*qzM*jsn
Governing Equations Voltage Equation
4
Lorenz Condition
4
Maxwell’s Equations
4
*jsn
x(4V) = 6E
•(4V) = 0
*jsn
*4*jsn*(4V) = 4sn*(6E) sn 4 *(6E) = 4J sn
Electric Charge Conservation Equation 4
•(4sn*4*jsn*(4V)) = 4*jsn•(4sn*(6E)) = 4*jsn•(4J) = 0
*jsn
Decomposition. Invariants in the above equations separate into pieces per nomenclature given below. Pieces alone are not invariants. Therefore, pieces are specific to an inertial reference frame even though the “M” or “S” subscript may be dropped to simplify what is written.
99 CHAPTER 3 – FIELDS 4V
= 1V + 3V 4J = 1J + 3J
; ;
1V
= 1M*VtM 1J = 1M*JtM
4
sn
3E
= pxM*ExM + pyM*EyM + pzM*EzM
; ;
3V
= qxM*VxM + qyM*VyM + qzM*VzM 3J = qxM*JxM + qyM*JyM + qzM*JzM
= 1sn + 3sn ; 1sn = 1Msn*tM sn sn sn sn 3 = qxM *xM + qyM *yM + qzM *zM 6E = 3E + 3K ;
3K
= kxM*KxM + kyM*KyM + kzM*KzM
Multiplication operation “*” decomposes into dotproduct “•” and crossproduct “x”. qx*qy = qx•qy + qxxqy = qxxqy
;
qx*qx = qx•qx + qxxqx = qx•qx
qx•qy = 0
;
qxxqx = 0
Lorenz Condition “4*jsn•(4V) = 0”. 4
*jsn
*(4V) = 4*jsn•(4V) + 4*jsnx(4V)
4
*jsn
4
*jsn
•(4V) = 1*jsn*(1V) + 3*jsn•(3V) = 0 •(4V) = tM*VtM*1M*jsn*1M + xM*VxM*qxM*jsn*qxM + yM*VyM*qyM*jsn*qyM + zM*VzM*qzM*jsn*qzM = tM*VtM + xM*VxM + yM*VyM + zM*VzM = 0
Electromagnetic Field Voltage Equation. Crossproduct “4*jsnx(4V)” results in negative of the electromagnetic field. 4
*jsn
x(4V) = 1*jsn*(3V) + 3*jsn*(1V) + 3*jsnx(3V) = 6E
1
*jsn
*(3V) + 3*jsn*(1V) = 3E
;
3
*jsn
x(3V) = 3K
100 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Electric Field Voltage Equation. 1
*(3V) + 3*jsn*(1V) = 3E
*jsn
tM*VxM*1M*jsn*qxM + xM*VtM*qxM*jsn*1M = tM*VxM*pxM + xM*VtM*pxM = pxM*ExM tM*VxM  xM*VtM = ExM tM*VyM  yM*VtM = EyM tM*VzM  zM*VtM = EzM Magnetic Field Voltage Equation. 3
*jsn
x(3V) = 3K
yM*VzM*qyM*jsn*qzM + zM*VyM*qzM*jsn*qyM = yM*VzM*kxM + zM*VyM*kxM = kxM*KxM qyM*jsn*qzM = qyM*j*qzM = jyM*j*jzM = kxM yM*VzM + zM*VyM = KxM zM*VxM + xM*VzM = KyM xM*VyM + yM*VxM = KzM TripleVectorProduct and RemnantProduct. Dotproduct and/or crossproduct do not apply to sixcomponent invariants. Use: Triplevectorproduct “■” and remnantproduct “♦”. 4
sn
*(4*jsn*(4V)) = 4sn*(4*jsn•(4V)) + 4sn*(4*jsnx(4V))
4
sn
*(4*jsnx(4V)) = 4sn■(4*jsnx(4V)) + 4sn♦(4*jsnx(4V)) = 4sn■(6E) + 4sn♦(6E)
TripleVectorProduct Gradient Identities. 4
■(4*jsnx(4V)) = 1sn*(3*jsnx(3V)) + 3snx(1*jsn*(3V)) + 3snx(3*jsn*(1V)) + 3sn•(3*jsnx(3V))
sn
101 CHAPTER 3 – FIELDS 4
■(6E) = 1sn*(3K) + 3snx(3E) + 3sn•(3K)
sn
“4sn■(4*jsnx(4V))” includes identities. 1
*(3*jsnx(3V)) + 3snx(1*jsn*(3V)) 0 *jsn *(1V)) 0 3 x(3 sn *jsn x(3V)) 0 ; 4sn■(4*jsnx(4V)) 0 3 •(3 sn sn
Electromagnetic components substitute into the sum of the first two identities to result in the first of four of Maxwell’s Equations. 1
sn
*(3*jsnx(3V)) + (3snx(1*jsn*(3V)) + 3snx(3*jsn*(1V))) 0
1
sn
*(3K) + 3snx(3E) = 0
tM*KxM*1Msn*kxM  yM*EzM*qyMsn*pzM  zM*EyM*qzMsn*pyM = 0 tM*KxM*jxM  yM*EzM*jxM + zM*EyM*jxM = 0 tM*KxM  yM*EzM + zM*EyM = 0 Electromagnetic components substitute into “3sn•(3*jsnx(3V)) 0” to result in the second of four of Maxwell’s Equations. 3
•(3*jsnx(3V)) 0
sn
;
3
•(3K) = 0
sn
xM*KxM*qxMsn*kxM  yM*KyM*qyMsn*kyM  zM*KzM*qzMsn*kzM = 0 xM*KxM + yM*KyM + zM*KzM = 0 First with second of the four Maxwell’s Equations are given below. 4
■(4*jsnx(4V)) = 4sn■(6E) = 0
sn
102 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY RemnantProduct Results. 4
♦(4*jsnx(4V)) = 4sn*(4*jsnx(4V))  4sn■(4*jsnx(4V))
sn
= 1sn*(1*jsn*(3V)) + 1sn*(3*jsn*(1V)) + 3snx(3*jsnx(3V)) + 3sn•(1*jsn*(3V)) + 3sn•(3*jsn*(1V)) Third of four of Maxwell’s Equations: 1
*(1*jsn*(3V)) + 1sn*(3*jsn*(1V)) + 3snx(3*jsnx(3V)) = 1sn*(3E) + 3snx(3K) = 3J
sn
tM*ExM*1Msn*pxM  yM*KzM*qyMsn*kzM  zM*KyM*qzMsn*kyM = qxM*JxM tM*ExM*qxM + yM*KzM*qxM  zM*KyM*qxM = qxM*JxM tM*ExM + yM*KzM  zM*KyM = JxM Fourth of four of Maxwell’s Equations: 3
•(1*jsn*(3V)) + 3sn•(3*jsn*(1V)) = 3sn•(3E) = 1J
sn
xM*ExM*qxMsn*pxM  yM*EyM*qyMsn*pyM  zM*EzM*qzMsn*pzM = 1M*JtM xM*ExM + yM*EyM + zM*EzM = JtM Third with fourth of Maxwell’s Equations: 4
♦(4*jsnx(4V)) = 4sn♦(6E) = 4J
sn
Maxwell’s Equations.
4
*(4*jsnx(4V)) = 4sn*(6E) = 4J + 0
sn
tM*KxM  yM*EzM + zM*EyM = 0 ; tM*ExM + yM*KzM  zM*KyM = JxM tM*KyM  zM*ExM + xM*EzM = 0 ; tM*EyM + zM*KxM  xM*KzM = JyM tM*KzM  xM*EyM + yM*ExM = 0 ; tM*EzM + xM*KyM  yM*KxM = JzM xM*KxM + yM*KyM + zM*KzM = 0 ; xM*ExM + yM*EyM + zM*EzM = JtM
103 CHAPTER 3 – FIELDS Electric Charge Conservation. 4
*4sn*4*jsn*(4V) = 4*jsn*(4sn*(4*jsn*(4V))) = 4*jsn*(4sn*(4*jsn•(4V))) + 4*jsn*(4sn*(4*jsnx(4V)))
*jsn
Lorenz Condition “4*jsn•(4V) = 0” and other identities apply. 4
4
*4sn*4*jsn*(4V) = 4*jsn•(4sn♦(4*jsnx(4V))) + 4*jsnx(4sn♦(4*jsnx(4V)))
*jsn
•(4sn♦(4*jsnx(4V))) = 3*jsn•(1sn*(1*jsn*(3V) + 3*jsn*(1V))) + 3*jsn•(3snx(3*jsnx(3V))) + 1*jsn*(3sn•(1*jsn*(3V) + 3*jsn*(1V)))
*jsn
3
*jsn
•(1sn*(1*jsn*(3V))) + 1*jsn*(3sn•(1*jsn*(3V))) 0 *jsn •(1sn*(3*jsn*(1V))) + 1*jsn*(3sn•(3*jsn*(1V))) 0 3 *jsn •(3snx(3*jsnx(3V))) 0 3 4
*jsn
4
*jsn
•(4sn♦(4*jsnx(4V))) 0 •(4J) = 0
tM*JtM*1M*jsn*1M + xM*JxM*qxM*jsn*qxM + yM*JyM*qyM*jsn*qyM + zM*JzM*qzM*jsn*qzM = 0 tM*JtM + xM*JxM + yM*JyM + zM*JzM = 0
104 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Summary of Component Equations. Lorenz Condition:
4
•(4V) = 0
*jsn
tM*VtM + xM*VxM + yM*VyM + zM*VzM = 0 Electric Field Voltage Equation: 1
*(3V) + 3*jsn*(1V) = 3E
*jsn
Magnetic Field Voltage Equation: 3
*jsn
x(3V) = 3K
;
tM*VxM  xM*VtM = ExM tM*VyM  yM*VtM = EyM tM*VzM  zM*VtM = EzM yM*VzM + zM*VyM = KxM zM*VxM + xM*VzM = KyM xM*VyM + yM*VxM = KzM
Maxwell’s Induced Electric Field Equation: 1
*(3K) + 3snx(3E) = 0
sn
tM*KxM  yM*EzM + zM*EyM = 0 tM*KyM  zM*ExM + xM*EzM = 0 tM*KzM  xM*EyM + yM*ExM = 0
Maxwell’s Zero Magnetic Charge Equation:
sn 3 •(3K)
=0
xM*KxM + yM*KyM + zM*KzM = 0 Maxwell’s Induced Magnetic Field Equation: 1
*(3E) + 3snx(3K) = 3J
sn
tM*ExM + yM*KzM  zM*KyM = JxM tM*EyM + zM*KxM  xM*KzM = JyM tM*EzM + xM*KyM  yM*KxM = JzM
Maxwell’s Electric Charge Equation:
3
•(3E) = 1J
sn
xM*ExM + yM*EyM + zM*EzM = JtM Electric Charge Conservation Equation:
4
•(4J) = 0
*jsn
tM*JtM + xM*JxM + yM*JyM + zM*JzM = 0
105 CHAPTER 3 – FIELDS
3.3 Gauges and SuperPotentials SuperPotentials. Voltage “4V” is called “potential”. By analogy, “4J” is the first “subpotential”. 4
•(4V) = 0 *jsn •(4J) = 0 4 *jsn
Lorenz Condition Electric Charge Conservation Equation
“4J” relates to “4V” by the square of the gradient operator. By analogy, voltage is related to the first superpotential “4U”, “4*jsn•(4U) = 0”. (4J) = (4sn•4*jsn)*(4V) = (tM2  xM2  yM2  zM2)*(4V) (4V) = (4sn•4*jsn)*(4U) = (tM2  xM2  yM2  zM2)*(4U) Superpotentials and subpotentials extend indefinitely. “(4sn•4*jsn) = (tM2  xM2  yM2  zM2)” is called the “harmonic operator” because, if “4J = 0”, then “4V” is a summation of sine waves. The physical example is light waves. Fields. Between each pair of adjacent potentials is a sixcomponent field. “6E” is between “4J” and “4V”. 4
*jsn
*(4V) = 6E ;
sn 4 *(6E)
= 4J
The first subfield “6G” has “4J” as its potential. 4
*jsn
*(4J) = 4*jsnx(4J) = 6G
4
*jsn
x(4sn♦(4*jsnx(4V))) = 4*jsnx(4sn♦(6E)) = 4*jsnx(4J) = 6G
106 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY TimeComponent Field Gauge. A potential is not unique to the field. An electromagnetic field has a potential “4V’” (“’” is “prime”) from which alternative “4V” are found using a mathematically complex field “2P”. 4V 3
= 4V’ + 4sn*2P x(3sn*2P) 0 *(3sn*2P) + 3*jsn*(1sn*2P) 0
*jsn
1
*jsn
4
*jsn
x(4sn*2P) 0
6E = 4*jsnx(4V) = 4*jsnx(4V’) + 4*jsnx(4sn*2P) = 4*jsnx(4V’) “6E” is not affected by “2P”. “2P” affects the Lorenz Condition because “2P” adds a harmonic oscillator term “(4*jsn•4sn)*2P”. To keep “4V” real, “2P” must be real. 0 = 4*jsn•(4V) = 4*jsn•(4V’) + 4*jsn•(4sn*2P) = 4*jsn•(4V’) + (4*jsn•4sn)*2P Complex number field “2P” is called the field gauge. The simplest version of “4sn*2P” is a constant, and a change to that constant is a change to the ground of the voltage time component. SpaceComponent Field Gauge does not affect the Lorenz Condition, but, rather, adds a term to the electromagnetic field “6E”. 3
*jsn
x(3snx6Q) (3*jsn•3sn)*6Q  3*jsn*(3sn•6Q)
4
*jsn
4V
•(4sn*(6Q)) = 4*jsn•(4sn■(6Q)) + 4*jsn•(4sn♦(6Q)) = 1*jsn*(3sn•(3Qi)) + 3*jsn•(1sn♦(3Qi) + 3snx(3Qr)) + 1*jsn*(3sn•(3Qr)) + 3*jsn•(1sn♦(3Qr) + 3snx(3Qi)) 0
= 4V’ + 4sn*(6Q)
107 CHAPTER 3 – FIELDS 4
•(4V) = 4*jsn•(4V’) + 4*jsn•(4sn*(6Q)) = 4*jsn•(4V’) = 0
*jsn
6E = 4*jsnx4V = 4*jsnx4V’ + 4*jsnx(4sn*(6Q)) = 4*jsnx4V’ + 4*jsnx(4■(6Q)) + 4*jsnx(4♦(6Q)) = 4*jsnx4V’ + 4*jsnx(3sn•(3Qi) + 1sn*(3Qi) + 3snx(3Qr)) + 4*jsnx(3sn•(3Qr) + 1sn*(3Qr) + 3snx(3Qi)) = 4*jsnx4V’ + 3*jsn*(3•(3Qi)) + 1*jsn*(1sn*(3Qi) + 3snx(3Qr)) + 3*jsnx(1sn*(3Qi) + 3snx(3Qr)) + 3*jsn*(3sn•(3Qr)) + 1*jsn*(1sn*(3Qr) + 3snx(3Qi)) + 3*jsnx(1sn*(3Qr) + 3snx(3Qi)) = 4*jsnx4V’ + 3*jsn*(3sn•(3Qi)) + 1*jsn*(1sn*(3Qi)) + 3*jsnx(3snx(3Qr)) + 3*jsn*(3sn•(3Qr)) + 1*jsn*(1sn*(3Qr)) + 3*jsnx(3snx(3Qi)) = 4*jsnx4V’ + 3*jsn*(3sn•(6Q)) + 1*jsn*(1sn*(6Q)) + 3*jsnx(3snx(6Q)) = 4*jsnx4V’ + (3*jsn•3sn)*(6Q) + (1*jsn*1sn)*(6Q) = 4*jsnx4V’ + (4*jsn•4sn)*(6Q) “(6E) = 4*jsnx4V’ + (4*jsn•4sn)*(6Q)” and “4*jsn•(4V) = 4*jsn•(4V’) + 4*jsn•(4sn*(6Q)) = 4*jsn•(4V’) = 0” define different electromagnetic fields for one Lorentz Condition. “2P” is a classic feature in electromagnetic theory. “6Q” was not mentioned in reference material available to the author. Electromagnetic field harmonic term “(4*jsn•4sn)*(6Q)” is similar in structure to the superfield equation given below. (6E) = (4*jsn•4sn)*(6H) = (tM2  xM2  yM2  zM2)*(6H) Concluding Statement. Superpotentials and gauge fields place Maxwell’s Equations into a larger (and mathematically beautiful) structure. Identities are easily identified in an allnumber algebra but not in the geometric algebra used when Maxwell’s Equations were first discovered.
108 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
3.4 Lorentz Transformation The Lorentz Transformation transforms components and compoundlabelnumbers from “M” (moving because seated on the bus) to “S” (stationary because standing on the roadside). The general form has two “1” factors, left and right. The technique is checked: An invariant formed as a product of two other invariants must have the same Lorentz Transformation result if the invariant is transformed directly or if the invariant is formed by multiplying two transformed invariants. FourComponent Vector Lorentz Transformation. General form: 4V = 1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM = exp(*/2)*(1*VtM + qx*VxM + qy*VyM + qz*VzM)*exp(*/2) = exp(*/2)*1*(1*VtM + qx*VxM + qy*VyM + qz*VzM)*1*exp(*/2)
= exp(*/2)*exp(qx*αS/M/2) *exp(qx*αS/M/2)*(1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM)*exp(qx*αS/M/2) *exp(qx*αS/M/2)*exp(*/2) Components: 1*VtS + qx*VxS + qy*VyS + qz*VzS = exp(qx*αS/M/2)*(1*VtM + qx*VxM + qy*VyM + qz*VzM)*exp(qx*αS/M/2) VtS = VtM*coshαS/M + VxM*sinhαS/M VxS = VtM*sinhαS/M + VxM*coshαS/M VyS = VyM ; VzS = VzM Same mathematics applies to other fourcomponent invariants. JtS = JtM*coshαS/M + JxM*sinhαS/M JxS = JtM*sinhαS/M + JxM*coshαS/M JyS = JyM ; JzS = JzM Matrix equation form:
109 CHAPTER 3 – FIELDS VtS
coshαS/M
sinhαS/M
= VxS
VtM *
sinhαS/M
coshαS/M
VxM
Lorentz Transformation of the compoundlabelnumbers: 1S = exp(*/2)*exp(qx*αS/M/2)*exp(qx*αS/M/2)*exp(*/2) = exp(*/2)*exp(qx*αS/M)*exp(*/2) qxS = exp(*/2)*exp(qx*αS/M/2)*qx*exp(qx*αS/M/2)*exp(*/2) = exp(*/2)*qx*exp(qx*αS/M)*exp(*/2) qyS = exp(*/2)*exp(qx*αS/M/2)*qy*exp(qx*αS/M/2)*exp(*/2) = exp(*/2)*exp(qx*αS/M/2)*exp(qx*αS/M/2)*qy*exp(*/2) = exp(*/2)*qy*exp(*/2) = qyM qzS = exp(*/2)*exp(qx*αS/M/2)*qz*exp(qx*αS/M/2)*exp(*/2) = exp(*/2)*exp(qx*αS/M/2)*exp(qx*αS/M/2)*qz*exp(*/2) = exp(*/2)*qz*exp(*/2) = qzM 1S*j = (exp(*/2)*exp(qx*αS/M)*exp(*/2))*j = (exp(*/2)*exp(qx*αS/M)*exp(*/2)) qxS*j = (exp(*/2)*qx*exp(qx*αS/M)*exp(*/2))*j = exp(*/2)*exp(qx*αS/M)*qx*exp(*/2) qyS*j = qyM*j ; qzS*j = qzM*j “4V” in “S” equals “4V” in “M”, proven using mathematics analogous for the same activity in twodimensional timespace. 4V
= 1S*VtS + qxS*VxS + qyS*VyS + qzS*VzS = 1M*VtM + qxM*VxM + qyM*VyM + qzM*VzM = 4V
Dotproduct of two fourcomponent invariants is an invariant. *j 4k •4r
= ((S/c)*1S*j + kxS*qxS*j + kyS*qyS*j + kzS*qzS*j) •(1S*c*tS + qxS*xS + qyS*yS + qzS*zS)
110 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = ((S/c)*1M*j*1S*c*tS + kxS*qxS*j*qxS*xS + kyS*qyS*j*qyS*yS + kzS*qzS*j*qzS*zS) = (S/c)*c*tS  kxS*xS  kyS*yS  kzS*zS = ((M/c)*coshαS/M + kxM*sinhαS/M)*(c*tM*coshαS/M + xM*sinhαS/M)  ((M/c)*sinhαS/M + kxM*coshαS/M)*(c*tM*sinhαS/M + xM*coshαS/M)  kyM*yM  kzM*zM = ((M/c)*c*tM*(cosh2αS/M  sinh2αS/M) + (kxM*xM*(sinh2αS/M  cosh2αS/M) + (M/c)*xM*(sinhαS/M*coshαS/M + coshαS/M*sinhαS/M) + (kxM*c*tM*(coshαS/M*sinhαS/M + sinhαS/M*coshαS/M)  kyM*yM  kzM*zM = (M/c)*c*tM  kxM*xM  kyM*yM  kzM*zM = 4k*j•4r SpaceNegative Lorentz Transformation. The inverted matrix for the spacenegative Lorentz Transformation is complemented by an inverted matrix for the spacenegative compoundlabelnumbers. tS = tM*coshαS/M  xM*sinhαS/M xS = tM*sinhαS/M + xM*coshαS/M yS = yM ; zS = zM 1Ssn = (1M*coshαS/M  qxM*sinhαS/M)sn = 1Msn*coshαS/M + qxMsn*sinhαS/M qxSsn = (1M*sinhαS/M + qxM*coshαS/M)sn = 1Msn*sinhαS/M + qxMsn*coshαS/M 4
sn
= 1Ssn*tS + qxSsn*xS + qySsn*yS + qzSsn*zS
= (1M*coshαS/M + qxM*sinhαS/M)sn*(tM*coshαS/M  xM*sinhαS/M) + (1M*sinhαS/M + qxM*coshαS/M)sn*(tM*sinhαS/M + xM*coshαS/M) + qyMsn*yM + qzMsn*zM = (1Msn*coshαS/M + qxMsn*sinhαS/M)*(tM*coshαS/M  xM*sinhαS/M) + (+1Msn*sinhαS/M + qxMsn*coshαS/M)*(tM*sinhαS/M + xM*coshαS/M) + qyMsn*yM + qzMsn*zM
111 CHAPTER 3 – FIELDS = (1Msn*tM + qxMsn*xM)*(cosh2αS/M  sinh2αS/M) + (qxMsn*tM + 1Msn*xM)*(coshαS/M*sinhαS/M  coshαS/M*sinhαS/M) + qyMsn*yM + qzMsn*zM = 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM = 4sn “4*jsn•(4V) = 0” and “4*jsn•(4J) = 0” are valid after the Lorentz Transformation. 4
•(4J) = (tS*1S*jsn + xS*qxS*jsn + yS*qyS*jsn + zS*qzS*jsn) •(1S*JtS + qxS*JxS + qyS*JyS + qzS*JzS)
*jsn
= tS*1M*jsn*1S*JtS + xS*qxS*jsn*(qxS)*JxS + yS*qyS*jsn*(qyS)*JyS + zS*qzS*jsn*(qzS)*JzS = tS*JtS + xS*JxS + yS*JyS + zS*JzS = (tM*coshαS/M  xM*sinhαS/M)*(JtM*coshαS/M + JxM*sinhαS/M) + (tM/c*sinhαS/M + xM*coshαS/M)*(JtM*sinhαS/M + JxM*coshαS/M) + yM*JyM + zM*JzM = tM*JtM*(cosh2αS/M  sinh2αS/M) + xM*JxM*(sinh2αS/M + cosh2αS/M) + tM*JxM*(sinhαS/M*coshαS/M  coshαS/M*sinhαS/M) + xM*JtM*(coshαS/M*sinhαS/M + sinhαS/M*coshαS/M) + yM*JyM + zM*JzM = tM*JtM + xM*JxM + yM*JyM + zM*JzM =0
Electromagnetic Field Lorentz Transformation. On the right is “exp(qx*αS/M/2)” from the voltage invariant. On the left is the reciprocal “exp(qx*αS/M/2)” from the gradient invariant, but with the argument changed to positive by the conjugate operation, and then back to negative by the spacenegative. Begin with the general form of the Lorentz Transformation.
112 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 6E
= pxM*ExM + pyM*EyM + pzM*EzM + kxM*KxM + kyM*KyM + kzM*KzM = exp(*/2)*1*(qx*ExM + qy*EyM + qz*EzM + jx*KxM + jy*KyM + jz*KzM)*1*exp(*/2)
= exp(*/2)*exp(qx*αS/M/2) *exp(qx*αS/M/2)*(qx*ExM + qy*EyM + qz*EzM + jx*KxM + jy*KyM + jz*KzM) *exp(qx*αS/M/2) *exp(qx*αS/M/2)*exp(*/2) The equation to find “S” component values for “6E” is below. qx*(ExS + i*KxS) + qy*(EyS + i*KyS) + qz*(EzS + i*KzS) = exp(qx*αS/M/2)*( qx*(ExM + i*KxM) + qy*(EyM + i*KyM) + qz*(EzM + i*KzM) )*exp(qx*αS/M/2) Derivations: qy*EyS = cosh(αS/M/2)*( qy*EyM )*cosh(αS/M/2) + qx*sinh(αS/M/2)*( qy*EyM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qz*i*KzM )*qx*sinh(αS/M/2) + qx*sinh(αS/M/2)*( qz*i*KzM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qy*EyM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qy*EyM ) + 2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qz*i*KzM ) = coshαS/M*( qy*EyM )  qx*sinhαS/M*( qz*i*KzM ) = qy*coshαS/M*EyM  qx*qz*i*sinhαS/M*KzM = qy*(coshαS/M*EyM + sinhαS/M*KzM) qx*qz*i = ( (jx/i)*(jz/i)*i ) = jx*jz*i = jy*i = qy qy*i*KyS = cosh(αS/M/2)*( qy*i*KyM )*cosh(αS/M/2) + qx*sinh(αS/M/2)*( qy*i*KyM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qz*EzM )*qx*sinh(αS/M/2) + qx*sinh(αS/M/2)*( qz*EzM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qy*i*KyM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qy*i*KyM )  2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qz*EzM ) = coshαS/M*( qy*i*KyM )  qx*sinhαS/M*( qz*EzM ) = qy*i*coshαS/M*KyM  qx*qz*sinhαS/M*EzM = qy*i*(coshαS/M*KyM  sinhαS/M*EzM)
113 CHAPTER 3 – FIELDS qx*qz = (jx/i)*(jz/i) = jy = qy*i qz*EzS = cosh(αS/M/2)*( qz*EzM )*cosh(αS/M/2) + qx*sinh(αS/M/2)*( qz*EzM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qy*i*KyM )*qx*sinh(αS/M/2) + qx*sinh(αS/M/2)*( qy*i*KyM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qz*EzM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qz*EzM ) + 2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qy*i*KyM ) = coshαS/M*( qz*EzM )  qx*sinhαS/M*( qy*i*KyM ) = qz*coshαS/M*EzM  qx*qy*i*sinhαS/M*KyM = qz*(coshαS/M*EzM  sinhαS/M*KyM) qx*qy*i = ( (jx/i)*(jy/i)*i ) = jx*jy*i = jz*i = qz qz*i*KzS = cosh(αS/M/2)*( qz*i*KzM )*cosh(αS/M/2) + qx*sinh(αS/M/2)*( qz*i*KzM )*qx*sinh(αS/M/2) + cosh(αS/M/2)*( qy*EyM )*qx*sinh(αS/M/2) + qx*sinh(αS/M/2)*( qy*EyM )*cosh(αS/M/2) = cosh(αS/M/2)*cosh(αS/M/2)*( qz*i*KzM ) + qx2*sinh(αS/M/2)*sinh(αS/M/2)*( qz*i*KzM ) + 2*qx*sinh(αS/M/2)*cosh(αS/M/2)*( qy*EyM ) = coshαS/M*( qz*i*KzM )  qx*sinhαS/M*( qy*EyM ) = qz*i*coshαS/M*KzM  qx*qy*sinhαS/M*EyM = qz*i*(coshαS/M*KzM + sinhαS/M*EyM) qx*qy = (jx/i)*(jy/i) = jz = qz*i ExS = ExM KxS = KxM EyS = EyM*coshαS/M + KzM*sinhαS/M KyS = EzM*sinhαS/M + KyM*coshαS/M EzS = EzM*coshαS/M  KyM*sinhαS/M KzS = EyM*sinhαS/M + KzM*coshαS/M EyS + i*KyS
coshαS/M
i*sinhαS/M
= EzS + i*KzS
EyM + i*KyM *
i*sinhαS/M
coshαS/M
EzM + i*KzM
114 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 4
*jsn
The equation to find “S” compoundlabelnumbers for “6E = *(4V)” uses the left and right denominators. pxS = pxM pyS = exp(*/2)*exp(qx*αS/M/2)*qy*exp(qx*αS/M/2)*exp(*/2) = exp(*/2)*qy*exp(qx*αS/M)*exp(*/2) pzS = exp(*/2)*exp(qx*αS/M/2)*qz*exp(qx*αS/M/2)*exp(*/2) = exp(*/2)*qz*exp(qx*αS/M)*exp(*/2) kxS = kxM kyS = exp(*/2)*jy*exp(qx*αS/M)*exp(*/2) kzS = exp(*/2)*jz*exp(qx*αS/M)*exp(*/2)
A check ensured “6E = 6E”: To check that the theory is correct, “M” Lorentz Transformation components and labelnumbers for “4*jsn” and “4V” substitute into the “S” version of “6E = 4*jsnx(4V)”, so that the subscripts are all “M” and not “S”. Components and compoundlabelnumbers are grouped to “6E” expressed in “M”. Also, the component transformations are confirmed by reference to Page 71 of Introduction to Modern Physics by Richtmyer, Kennard and Lauritsen, McGrawHill Book Company, Inc., 1955 and by reference to Page 210 of Methods of Theoretical Physics Part I by Morse and Feshbach, McGrawHill Book Company, Inc., 1953. The third check ensures “4sn♦(6E) = 4J”. (as calculated in “S” from components in “M”) matches current density (as calculated from “4sn” and “6E” in “S”). 1S*JtS + qxS*JxS + qyS*JyS + qzS*JzS = (1Ssn*tS + qxSsn*xS + qySsn*yS + qzSsn*zS) ♦(pxS*ExS + pyS*EyS + pzS*EzS + kxS*KxS + kyS*KyS + kzS*KzS)*(1) 1S*JtS = xS*qxSsn*pxS*ExS  yS*qySsn*pyS*EyS  zS*qzSsn*pzS*EzS qxS*JxS = tS*1Ssn*pxS*ExS  yS*qySsn*kzS*KzS  zS*qzSsn*kyS*KyS qyS*JyS = tS*1Ssn*pyS*EyS  zS*qzSsn*kxS*KxS  xS*qxSsn*kzS*KzS qzS*JzS = tS*1Ssn*pzS*EzS  xS*qxSsn*kyS*KyS  yS*qySsn*kxS*KxS
115 CHAPTER 3 – FIELDS Compoundlabelnumbers multiply as 1S = qxSsn*pxS = qySsn*pyS = qzSsn*pzS qxS = 1Ssn*pxS*(1) = qySsn*kzS = qzSsn*kyS*(1) qyS = 1Ssn*pyS*(1) = qzSsn*kxS = qxSsn*kzS*(1) qzS = 1Ssn*pzS*(1) = qxSsn*kyS = qySsn*kxS*(1) JtS = xS*ExS + yS*EyS + zS*EzS JxS = tS*ExS + yS*KzS  zS*KyS JyS = tS*EyS + zS*KxS  xS*KzS JzS = tS*EzS + xS*KyS  yS*KxS To prove each of the four component equations is correct, “M” component expressions substitute for “S” component expressions. After manipulation of the equations, previously stated component Lorentz Transformation equations for current density “4J” are found to apply. Time “t” component:
JtS = xS*ExS + yS*EyS + zS*EzS
(JtM*coshαS/M + JxM*sinhαS/M) = (tM*sinhαS/M + xM*coshαS/M)*(ExS) + (yM)*(EyM*coshαS/M + KzM*sinhαS/M) + (zM)*(EzM*coshαS/M  KyM*sinhαS/M) JtM = xM*ExS + yM*EyM + zM*EzM JxM = tM*ExS + yM*KzM  zM*KyM “x” component:
JxS = tS*ExS + yS*KzS  zS*KyS
(JtM*sinhαS/M + JxM*coshαS/M) = (tM*coshαS/M  xM*sinhαS/M)*(ExS) + (yM)*(EyM*sinhαS/M + KzM*coshαS/M)  (zM)*(EzM*sinhαS/M + KyM*coshαS/M) JtM = xM*ExS + yM*EyM + zM*EzM JxM = tM*ExS + yM*KzM  zM*KyM “y” component:
JyS = tS*EyS + zS*KxS  xS*KzS
JyM = (tM*coshαS/M  xM*sinhαS/M)*(EyM*coshαS/M + KzM*sinhαS/M) + zM*KxM  (tM*sinhαS/M + xM*coshαS/M)*(EyM*sinhαS/M + KzM*coshαS/M)
116 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY JyM = (tM*EyM  xM*KzM)*(cosh2αS/M  sinh2αS/M) + zM*KxM + (tM*KzM  xM*EyM)*(2*sinhαS/M*coshαS/M  2*sinhαS/M*coshαS/M) JyM = tM*EyM  xM*KzM + zM*KxM “z” component:
JzS = tS*EzS + xS*KyS  yS*KxS
JzM = (tM*coshαS/M  xM*sinhαS/M)*(EzM*coshαS/M  KyM*sinhαS/M) yM*KxM +(tM*sinhαS/M + xM*coshαS/M)*(EzM*sinhαS/M +KyM*coshαS/M) JzM = (tM*EzM + xM*KyM)*(cosh2αS/M  sinh2αS/M)  yM*KxM + (tM*KyM + xM*EzM)*(2*sinhαS/M*coshαS/M  2*sinhαS/M*coshαS/M) JzM = tM*EzM + xM*KyM  yM*KxM
3.5 BiotSavart Law The BiotSavart Law for the electromagnetic field of a moving particle is derived from the electric field of a stationary particle. 6E
= (q/(4**ǝ*rM2))*((xM/rM)*pxM + (yM/rM)*pyM + (zM/rM)*pzM)
ExM = (q/(4**ǝ*rM2))*(xM/rM) EyM = (q/(4**ǝ*rM2))*(yM/rM) EzM = (q/(4**ǝ*rM2))*(zM/rM)
; ; ;
KxM = 0 KyM = 0 KzM = 0
ExS = ExM = (q/(4**ǝ*rM3))*xM EyS + i*KyS
coshαS/M =
EzS + i*KzS
(q/(4**ǝ*rM2))*(yM/rM)
i*sinhαS/M *
i*sinhαS/M
coshαS/M
(q/(4**ǝ*rM2))*(zM/rM)
EyS = (q/(4**ǝ*rM2))*(yM/rM)*(coshαS/M) KyS = (q/(4**ǝ*rM2))*(zM/rM)*(sinhαS/M) EzS = (q/(4**ǝ*rM2))*(zM/rM)*(coshαS/M) KzS = (q/(4**ǝ*rM2))*(yM/rM)*(sinhαS/M)
117 CHAPTER 3 – FIELDS “rM2” must be independently Lorentz Transformed into “S” from “M”. In “M” there is no time “tM” difference between the electrically charged particle and point (xM, yM, zM) where the electromagnetic field is being measured. locationM2 = xM2 + yM2 + zM2  c2*tM2 = rM2  c2*tM2 Equations in “S” have that same assumption of no time discrepancy, but in “S” not “M”. Those surrounding points in “S” are at different times with respect to “M”. xS2 + yS2 + zS2  c2*tS2 = xM2 + yM2 + zM2  c2*tM2 rS2  c2*tS2 = rM2  c2*tM2 rS2 = rM2  c2*tM2 ; rM2 = rS2 + c2*tM2 c*tS
coshαS/M
sinhαS/M
sinhαS/M
coshαS/M
= xS c*tM
* coshαS/M
xM
sinhαS/M
= xM
c*tM
c*tS *
sinhαS/M
coshαS/M
xS
c*tM = c*tS*coshαS/M  xS*sinhαS/M c2*tM2 = c2*tS2*cosh2αS/M + xS2*sinh2αS/M  c2*tS*xS*sinh(2*αS/M) = xS2*sinh2αS/M rM2 = rS2 + c2*tM2 = rS2 + xS2*sinh2αS/M xM = xS*coshαS/M  c*tS*sinhαS/M = xS*coshαS/M yS = yM
;
zS = zM
ExS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*xS*coshαS/M ; KxS = 0 EyS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*yS*coshαS/M KyS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*zS*sinhαS/M EzS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*zS*coshαS/M KzS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*yS*sinhαS/M
118 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 6E
= pxS*(ExS + i*KxS) + pyS*(EyS + i*KyS) + pzS*(EzS + i*KzS)
(The author hasn’t seen the above component equations in a book or paper. That statement is common in this book, but with the expectation there is a book or paper not yet seen.) From Wikipedia (“BiotSavart Law”) attributed to Oliver Heaviside in year 1888 (before the discovery of Special Relativity in 1905): ExSW = (q/(4**ǝ))*((1  vS/M2/c2)/(1  sin2*(vS/M2/c2))3/2)*(xS/rS3) KxSW = 0 EySW = (q/(4**ǝ))*((1  vS/M2/c2)/(1  sin2*(vS/M2/c2))3/2)*(yS/rS3) KySW = (vS/M/c)*EzS EzSW = (q/(4**ǝ))*((1  vS/M2/c2)/(1  sin2*(vS/M2/c2))3/2)*(zS/rS3) KzSW = (vS/M/c)*EyS cos = xS/rS ; sin2 = (yS2 + zS2)/rS2 ; rS2 = xS2 + yS2 + zS2 The two sets of electromagnetic component expressions in “S” appear very different. Component values deviate with slightly higher values for Heaviside’s expressions (with the “W” subscript) for relativistic speeds and for “xS” locations far from the origin. For example, at “S/M = 6” and “xS = yS = 1” with “zS = 0” there is “ExS = EyS = 0.0000246” and “ExSW = EySW = 0.0000548”, for “q/(4**ǝ) = 1”. The deviation between “EyS” and “EySW” may possibly be attributed to a nonrelativistic approximation in Heaviside’s derivation (for example “sinh2αS/M vS/M2/c2”). That guess has not been confirmed. Translation from AllNumber Algebra to Geometry. We could use measurements to resolve the discrepancy. Translate allnumber algebra into geometry per the third step. “ixS”, “iyS” and “izS” substitute for “pxS”, “pyS” and “pzS” and for “kxS”, “kyS” and “kzS”.
119 CHAPTER 3 – FIELDS
3.6 Electric EnergyMomentum of an Electron Energymomentum invariant “4p” is calculated from electromagnetic field invariant “6E” using empirically derived relationships. Bus “M” moves at speed “vS/M” relative to roadside “S”. “6E” components are used to find energymomentum invariant “4p” components. As a check, energymomentum invariant (“4p”) components in “S” found from “6E” must also be found from energymomentum invariant (“4p”) components in “M”.
Figure 32. Left: Two parallel capacitor plates. Right: Two concentric capacitor plates.
Select the simplest geometry possible. •
For a field parallel with motion, the two capacitor plates are perpendicular to motion.
•
For a field perpendicular to motion, the two capacitor plates are cylindrical concentric circular surfaces with the axis parallel to motion.
Electric charge may be placed onto capacitor plates using a battery. Alternatively, the plates are close together and then electrically charged, and pulled apart: Energy equals force times distance. Bus mass converts to electric field energy and back again. Energy and momentum (and therefore speed) remain constant during the mass to electric field energy conversion.
120 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Electromagnetic Field Perpendicular to Motion. EyM = A*y/(y2 + z2) = A*y/r2
;
EzM = A*z/(y2 + z2) = A*z/r2
The outer cylinder has an excess of negatively charged electrons because the electric field points outward from positive to negative. (Electric field is negative of gradient of voltage, and, therefore, the inner cylinder has higher voltage.) Energy density “tM” (sigma) was discovered empirically through mathematical modeling of measured energy in experiments. tM = ǝ*(ExM*ExM + EyM*EyM + EzM*EzM + KxM*KxM + KyM*KyM + KzM*KzM)/2 Symbol “⊥” applies to fields perpendicular to motion. ⊥tM = ǝ*(EyM*EyM + EzM*EzM)/2 = ǝ*((A*y/r2)*(A*y/r2) + (A*z/r2)*(A*z/r2))/2 = ǝ*(A2*(y2 + z2))/((r2)2)/2 = ǝ*(A2/r2)/2 EyS + i*KyS
coshαS/M
i*sinhαS/M
i*sinhαS/M
coshαS/M
= EzS + i*KzS
A*y/r2 *
EyS = cosh(αS/M)*A*y/r2 EzS = cosh(αS/M)*A*z/r2
; ;
A*z/r2
KyS = sinh(αS/M)*A*z/r2 KzS = sinh(αS/M)*A*y/r2
⊥tS = ǝ*(EyS*EyS + EzS*EzS + KyS*KyS + KzS*KzS)/2 = ǝ*((y2 + z2)*(cosh2(αS/M) + sinh2(αS/M))*A2/((r2)2)/2 = ǝ*(cosh(2*αS/M)*A2/r2)/2 To complement energy density “⊥tS”, there is energy per area per time Poynting Vector “⊥xS*ixS + ⊥yS*iyS + ⊥zS*izS”, empirically found equal to the cross product of electric and magnetic fields, “ǝ*(3Ex3B)”. ⊥xS = ǝ*(EyS*KzS  EzS*KyS) ⊥yS = ǝ*(EzS*KxS  ExS*KzS) ⊥zS = ǝ*(ExS*KyS  EyS*KxS)
121 CHAPTER 3 – FIELDS ⊥xS = ǝ*(EyS*KzS  EzS*KyS) = ǝ*(y2 + z2)*cosh(αS/M)*sinh(αS/M)*A2/(r2)2 = ǝ*(sinh(2*αS/M)*A2/(y2 + z2))/2 = ǝ*(sinh(2*αS/M)*A2/r2)/2 Energy density “⊥tS” is positive because of squares. And, energy per area per time “⊥xS” is positive for positive “αS/M”, so that energy moves in the direction of motion. Because they were empirically derived, “⊥tS” and “⊥xS” are combined without a compoundlabelnumber. ⊥tS + qx*⊥xS = ǝ*((A2/r2)/2)*(cosh(2*αS/M) + qx*sinh(2*αS/M)) = ǝ*((A2/r2)/2)*exp(qx*2*αS/M) = ǝ*((A2/r2)/2)*exp2(qx*αS/M) exp(qx*2*αS/M) = cosh(2*αS/M) + qx*sinh(2*αS/M) = cosh2(αS/M) + sinh2(αS/M) + qx*2*sinh(αS/M)*cosh(αS/M) = cosh2(αS/M) + qx*qx*sinh2(αS/M) + qx*2*sinh(αS/M)*cosh(αS/M) = (cosh(αS/M) + qx*sinh(αS/M))2 = exp2(qx*αS/M) To remove the square operation in “exp2(qx*αS/M)”, “⊥tS + qx*⊥xS” is multiplied by volume between plates. Multiplication by volume must be an integral because of radius “r” dependency in “⊥tS” and “⊥xS”. We continue the empirical analysis using spacenegative of volume, and by making the volume invariant mathematically real rather than imaginary. Differential volume element “d(4Volsn)” consists of an “LxM” factor in the xdirection, a factor in the radial direction, and a factor in the tangential direction, “d(4Volsn) = 1S*(LxM)*(2**r)*dr*exp(qx*αS/M)”. The integration was from “rinnerplate” to “routerplate”. d4p = (⊥tS + qx*⊥xS)*d(4Volsn)/c = ǝ*((A2/r2)/2)*exp2(qx*αS/M)*(1S*dVolB*exp(qx*αS/M))/c = ǝ*((A2/r2)/2)*exp2(qx*αS/M)*(1S*(LxM)*(2**r)*dr*exp(qx*αS/M))/c = 1S*(ǝ/c)*(A2/2)*exp(qx*αS/M)*LxM*(2*)*(dr/r) 4p
= 1S*(ǝ/c)*(A2/2)*LxM*(2*)*ln(routerplate/rinnerplate)*exp(qx*αS/M) = 1S*mB*c*exp(qx*αS/M)
122 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The above expression is the energymomentum invariant for an electromagnetic field perpendicular to motion and formed from two cylindrical concentric circular capacitor plates. Rest mass “mB*c”: mB*c = (ǝ/c)*(A2/2)*LxM*(2*)*ln(routerplate/rinnerplate) We conclude, in general, any electromagnetic field perpendicular to the direction of motion may be replaced with an equivalent rest mass. An Electromagnetic Field Parallel to Motion. ExS = ExM
;
KxS = KxM
Positive “ExM” points right, in the direction of motion, because of a deficit of electrons on the left plate, and excess electrons on the right. (Left plate has the higher voltage.) Per “ExS = ExM”, a positive test charge “qTest” stationary with roadside “S” experiences the same force “qTest*ExS” regardless of speed “vS/M” of the bus. But note that energy must be applied to the bus to keep its speed constant because the test charge is repulsed by the left capacitor plate. If the test charge is stationary with respect to the bus, there is force “FxM = qTest*ExM” observed with both time and space components from the roadside because “xM = xS*cosh(αS/M)  c*tS*sinh(αS/M)”. To satisfy energy and momentum conservation, use the same basic form of energy density equations given for the case of the perpendicular electric field. (The below equations weren’t found in a book or paper.) tS = ǝ*(ExS*ExS + KxS*KxS)*cosh(2*αS/M)/2 = ǝ*(ExM*ExM + KxM*KxM)*cosh(2*αS/M)/2 xS = ǝ*(ExS*ExS + KxS*KxS)*sinh(2*αS/M)/2 = ǝ*(ExM*ExM + KxM*KxM)*sinh(2*αS/M)/2 “tS” is positive because of squares and the “cosh” function. Energy per area per time “xS” (that passes a location “xS”) is positive for positive “αS/M”, so that energy moves in the direction of motion.
123 CHAPTER 3 – FIELDS tS + qx*xS = ǝ*(ExM2/2)*(cosh(2*αS/M) + qx*sinh(2*αS/M)) = ǝ*(ExM2/2)*exp(qx*2*αS/M) = ǝ*(ExM2/2)*exp2(qx*αS/M) VolM = LxM*LyM*LzM 4Vol
sn
= 1Ssn*VolM*exp(qx*αS/M) = 1S*VolM*exp(qx*αS/M)
4p
= (tS + qx*xS)*4Volsn/c = ǝ*(ExM2/2)*exp2(qx*αS/M)*(1S*VolM*exp(qx*αS/M))/c = 1S*ǝ*(ExM2/2)*(VolM)*exp(qx*αS/M)/c
4p
= 1S*mB*c*exp(qx*αS/M)
;
mB*c = (ǝ/c)*(ExM2/2)*(VolM)
Electric Field of a Stationary Electron. Radius of an electron is calculated from rest mass and electric charge using a macroscopic model. Assume negative electric charge is evenly distributed on the surface of a spherical electron. Electric field lines of force extend radially inward to the surface. Electric field magnitude decreases inversely with the square of the distance from the center. ExM = A*(xM/rM3) A = q/(4**ǝ)
;
EyM = A*(yM/rM3) ;
;
EzM = A*(zM/rM3)
rM2 = xM2 + yM2 + zM2
tM = ǝ*(ExM*ExM + EyM*EyM + EzM*EzM + KxM*KxM + KyM*KyM + KzM*KzM)/2 = ǝ*(ExM*ExM + EyM*EyM + EzM*EzM)/2 = ǝ*(A2*(xM2 + yM2 + zM2)/rM6)/2 = ǝ*(A2/rM4)/2 Energy density “tM = ǝ*(A2/rM4)/2” is integrated from infinite radius inward (following electric field lines) to the assumed classical radius “re” of the electron. “dVol = 4**rM2*drM”. d(EB/c) = (ǝ*(A2/rM4)/2)*dVol/c = ((ǝ/c)*(A2/rM4)/2)*4**rM2*drM = (1/2)*((ǝ/c)*(q/(4**ǝ))2/rM4)*4**rM2*drM = (1/2)*(q2/(4**ǝ*c))*(drM/rM2)
124 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Integrand “(1/2)*(q2/(4**ǝ*c))*(drM/rM2)” is integrated to the below rest energy (divided by speedoflight) of an electron. mB*c = (1/2)*(q2/(4**ǝ*c))*(1/re) Substitutein measured rest mass “mB = 9.11*10^31 kg” and measured electric charge “q = 1.60*10^19 C” to calculate the classical radius “re”. re = (1/2)*(q2/(4**ǝ*c))*(1/(mB*c)) = 1.409… *1015 meters If an electron had a classical radius less than “re”, then more electric field would cause more rest mass than what is measured. •
Inside the classical radius is the electric charge. If we count electrons, then we are counting a quantity of electric charge.
•
Outside the classical radius is the electric field. Inertia of the energymomentum invariant is, per this model, due to electromagnetic field induction.
The classical radius macroscopic model of the electron fails when extended to the magnetic field caused by rotation, to suggest the classical radius is not a physically real surface. Mu and tau particles differ from the electron by having more rest mass and by being unstable. Rather than calculate a smaller classical radius for these particles, the extra mass can be attributed to whatever field causes the instability. The whatever field behaves like mass inertia, as does the electromagnetic field, and so probably creates an induced field when in motion. A similar statement applies to protons, etc. EnergyMomentum Density for the BiotSavart Law. A = q/(4**ǝ) ExS = (A/(rS2 + xS2*sinh2αS/M)3/2)*xS*coshαS/M KxS = 0 EyS = (A/(rS2 + xS2*sinh2αS/M)3/2)*yS*coshαS/M KyS = (A/(rS2 + xS2*sinh2αS/M)3/2)*zS*sinhαS/M EzS = (A/(rS2 + xS2*sinh2αS/M)3/2)*zS*coshαS/M KzS = (A/(rS2 + xS2*sinh2αS/M)3/2)*yS*sinhαS/M
125 CHAPTER 3 – FIELDS ⊥tS = ǝ*(EyS*EyS + EzS*EzS + KyS*KyS + KzS*KzS)/2 = ǝ*A2*(yS2 + zS2)*(cosh2αS/M + sinh2αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*(yS2 + zS2)*cosh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 ⊥xS = ǝ*(EyS*KzS  EzS*KyS) = ǝ*A2*(yS2 + zS2)*(coshαS/M*sinhαS/M)*(1/(rS2 + xS2*sinh2αS/M)3) = ǝ*A2*(yS2 + zS2)*sinh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 tS = ǝ*(ExS*ExS + KxS*KxS)*cosh(2*αS/M)/2 = ǝ*A2*(xS2)*cosh2αS/M*cosh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 xS = ǝ*(ExS*ExS + KxS*KxS)*sinh(2*αS/M)/2 = ǝ*A2*(xS2)*cosh2αS/M*sinh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 tS = ⊥tS + tS = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*cosh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 xS = ⊥xS + xS = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*sinh(2*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 tS + qx*xS = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*exp(2*qx*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*(xS2*cosh2αS/M + yS2 + zS2)*exp2(qx*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*(xM2 + yM2 + zM2)*exp2(qx*αS/M)*(1/(rS2 + xS2*sinh2αS/M)3)/2 = ǝ*A2*rM2*exp2(qx*αS/M)*(1/(rS2 + (c*tM)2)3)/2 = ǝ*A2*rM2*exp2(qx*αS/M)*(1/(rM2)3)/2 = ǝ*((A2/rM4)/2)*exp2(qx*αS/M) Hyperbolicradius “ǝ*(A2/rM4)/2” is the same as energy density of the stationary electron, “tM = ǝ*(A2/rM4)/2”.
126 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
3.7 Maxwell’s Wave Equation Maxwell’s Wave Equation is given in three forms below. 4
2
*(6E) = 0
;
(4*jsn•4sn)*(6E) = 0 (tM2  (xM2 + yM2 + zM2))*(6E) = 0
“42” is called the “harmonic” operator because it applies to functions that are sine waves. (tM2  xM2)*sin(kx*(x  c*t)) = 0 (sin(kx*(x  c*t)))/x = kx*cos(kx*(x  c*t)) (kx*cos(kx*(x  c*t)))/x = kx2*sin(kx*(x  c*t)) (sin(kx*(x  c*t)))/ct = kx*cos(kx*(x  c*t)) (kx*cos(kx*(x  c*t)))/ct = kx2*sin(kx*(x  c*t)) An electromagnetic field “6E” that satisfies Maxwell’s Wave Equation “42*(6E) = 0” must be a summation of sine waves. These include cosine waves (because ninety degrees can be added into the argument) and any motion at the speed of light (because its wave form can be a summation using different wavenumbers per a Fourier transform). Maxwell’s Wave Equation is derived by taking the gradient of both sides of Maxwell’s Equations “4sn*(6E) = 4J”. 4
*4sn*(6E) = 4*jsn*(4J)
*jsn
Apply identity “(4*jsnx4sn)*(6E) 0” to the left side and apply the electric charge conservation equation “4*jsn•(4J) = 0” to the right side. 4
*4sn*(6E) = (4*jsn•4sn)*(6E)
*jsn
;
4
*(4J) = 4*jsnx(4J)
*jsn
Set both sides equal to zero. Typically, “4J = 0”. (4*jsn•4sn)*(6E) = 0
;
4
*jsn
x(4J) = 0
127 CHAPTER 3 – FIELDS Spiral Waves. Each spiral wave is the sum of two perpendicular waves. First Spiral Wave (6Efirst) = Eamp*(kyM + pzM)*exp(i*(kxM*(xM + c*tM))) (EyM + i*KyM)first = i*Eamp*exp(i*(kxM*(xM + c*tM))) (EzM + i*KzM)first = Eamp*exp(i*(kxM*(xM + c*tM))) EyMfirst = Eamp*sin(kxM*(xM + c*tM)) KyMfirst = Eamp*cos(kxM*(xM + c*tM)) EzMfirst = Eamp*cos(kxM*(xM + c*tM)) KzMfirst = Eamp*sin(kxM*(xM + c*tM)) Second Spiral Wave (6Esecond) = Eamp*(kyM  pzM)*exp(i*(kxM*(xM  c*tM))) (EyM + i*KyM)second = i*Eamp*exp(i*(kxM*(xM  c*tM))) (EzM + i*KzM)second = Eamp*exp(i*(kxM*(xM  c*tM))) EyMsecond = Eamp*sin(kxM*(xM  c*tM)) KyMsecond = Eamp*cos(kxM*(xM  c*tM)) EzMsecond = Eamp*cos(kxM*(xM  c*tM)) KzMsecond = Eamp*sin(kxM*(xM  c*tM)) Third Spiral Wave (6Ethird) = Eamp*(kyM + pzM)*exp(i*(kxM*(xM + c*tM))) (EyM + i*KyM)third = i*Eamp*exp(i*(kxM*(xM + c*tM))) (EzM + i*KzM)third = Eamp*exp(i*(kxM*(xM + c*tM))) EyMthird = Eamp*sin(kxM*(xM + c*tM)) KyMthird = Eamp*cos(kxM*(xM + c*tM)) EzMthird = Eamp*cos(kxM*(xM + c*tM)) KzMthird = Eamp*sin(kxM*(xM + c*tM)) Fourth Spiral Wave (6Efourth) = Eamp*(kyM  pzM)*exp(i*(kxM*(xM  c*tM))) (EyM + i*KyM)fourth = i*Eamp*exp(i*(kxM*(xM  c*tM))) (EzM + i*KzM)fourth = Eamp*exp(i*(kxM*(xM  c*tM))) EyMfourth = Eamp*sin(kxM*(xM  c*tM)) KyMfourth = Eamp*cos(kxM*(xM  c*tM)) EzMfourth = Eamp*cos(kxM*(xM  c*tM)) KzMfourth = Eamp*sin(kxM*(xM  c*tM)) To check that the Fourth Spiral Wave is valid, first check it against Maxwell’s Wave Equation by taking derivatives:
128 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2(6Efourth)/(ctM)2 = kxM2*(6Efourth) 2(6Efourth)/(xM)2 = kxM2*(6Efourth) 2(6Efourth)/(ctM)2  2(6Efourth)/(xM)2 = 0 The above check only confirmed the Fourth Spiral Wave equation was constructed of sine waves that move at the speedoflight. A second check confirms electric and magnetic fields induce each other, to ensure there is not a misplaced negative: “1sn*(6E) + 3snx(6E) = 0” 1
sn
3
sn
*(6Efourth) = (1Msn)*(6Efourth)/(ctM) = (1M)*(kyM  pzM)*Eamp*(exp(i*(kxM*(xM  c*tM))))/(ctM) = (i*kxM)*Eamp*(jyM  qzM)*exp(i*(kxM*(xM  c*tM))) x(6Efourth) = (qxMsn)*(6Efourth)/(xM) = Eamp*(qxMsn*kyM  qxMsn*pzM)*(exp(i*(kxM*(xM  c*tM))))/(xM) = (i*kxM)*Eamp*(qzM + jyM)*exp(i*(kxM*(xM  c*tM))) qxMsn*kyM = qxM*kyM = qzM qxMsn*pzM = qxM*kzM/i = qzM*kxM/i = qyM*(i) = jyM Alternatively, as a quick check on the math, the Poynting Vector gives the direction of energy travel and must point in the direction of motion. The Poynting Vector is the cross product of the electric field by the magnetic field. For example, for the First Spiral Wave there are “EyMfirst = Eamp*sin(kxM*(xM + c*tM))” and “KzMfirst = Eamp*sin(kxM*(xM + c*tM))” as two components of a plane wave that induce each other. For a slightly positive “xM” at “tM = 0” there is “EyMfirst > 0” and “KzMfirst < 0”. Righthand fingers pass through positive “yM” and then through negative “zM” and the thumb points in negative “xM”, which is correct for the First Spiral Wave. Therefore, induction is correctly modeled. RightHanded and LeftHanded Spiral Waves. Point the thumb in the direction of wave travel (“+x” for “xM  c*tM” and “x” for “xM + c*tM”). The maximum of the helix of the wave spiral is similar to threads on a screw. Curl fingers to ride the maximum as the thumb moves in the direction of wave travel.
129 CHAPTER 3 – FIELDS The coordinate system is a righthand coordinate system with “x” the front table edge going right, with “y” the left table edge going away, and with “z” up. The example is the Fourth Spiral Wave. 3Efourth
= Eamp*(pzM*cos(kxM*(xM  c*tM))  pyM*sin(kxM*(xM  c*tM)))
Set “tM = 0”. Fingers pass positive “Ez”, then at “kxM*xM = /2”, negative “Ey” using a right hand. The Fourth Spiral Wave is righthanded. The Third Spiral Wave is left hand because fingers pass negative “Ez” at “xM = 0”, and then positive “Ey” at “kxM*xM = /2”. 3Ethird
= Eamp*(pzM*cos(kxM*(xM + c*tM))  pyM*sin(kxM*(xM + c*tM))) Second Spiral Wave is lefthanded.
3Esecond
= Eamp*(pzM*cos(kxM*(xM  c*tM)) + pyM*sin(kxM*(xM  c*tM)))
First Spiral Wave is righthanded. 3Efirst
= Eamp*(pzM*cos(kxM*(xM + c*tM)) + pyM*sin(kxM*(xM + c*tM)))
A (nonspinning) source disintegrates into two photons with the same handedness, like a nut that separates from a bolt that depart in opposite directions with the same handedness of spin. RightHanded Photon Pair (first and fourth): 6E+fourthfirst
= Eamp*(kyM  pzM)*exp(i*(kxM*(xM  c*tM)))
LeftHanded Photon Pair (third and second): 6E+secondthird
= Eamp*(kyM  pzM)*exp(i*(kxM*(xM  c*tM)))
Quantum Effect on Spin. The four spiral waves each represent an idealized hypothetical photon’s wave. They are idealized because direction is known, as is frequency, and spin axis is known to be parallel or else antiparallel to direction of motion. Actual photon waves, per quantum effects, have spin in the direction of motion only if the
130 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY measurement of the axis of spin is in the direction of motion, and when spin axis is not being measured, it includes a component perpendicular to direction of motion, and same with an electron. Matter and AntiMatter for Electromagnetic Waves. Because a photon has zero length due to the extreme of length contraction, its back side (taillights) is confounded with its front side (headlights), so there is no distinguishing between matter and antimatter. Regardless, they are paired with direction of motion: First with third and second with fourth. Lorentz Transformation of a Spiral Wave. The example below has wave fronts that move in the positive “x” direction because of “xM  c*tM”. (6Efourth) = Eamp*(kyM  pzM)*exp(i*(kxM*(xM  c*tM))) (EyM + i*KyM) = i*Eamp*exp(i*(kxM*(xM  c*tM))) (EzM + i*KzM) = Eamp*exp(i*(kxM*(xM  c*tM))) (EyS + i*KyS)
coshαS/M
i*sinhαS/M
= (EzS + i*KzS)
(EyM + i*KyM) *
i*sinhαS/M
coshαS/M
(EzM + i*KzM)
(EyS + i*KyS) = Eamp*exp(i*(kxM*(xM  c*tM)))*(i*coshαS/M  i*sinhαS/M) = i*Eamp*exp(i*(kxM*(xM  c*tM)))*exp(αS/M) = i*Eamp*exp(i*(kxS*(xS  c*tS)))*exp(αS/M) (EzS + i*KzS) = Eamp*exp(i*(kxM*(xM  c*tM)))*(i*i*sinhαS/M  coshαS/M) = Eamp*exp(i*(kxM*(xM  c*tM)))*exp(αS/M) = Eamp*exp(i*(kxS*(xS  c*tS)))*exp(αS/M) Observer “S” sees amplitude increased by factor “exp(αS/M)” compared to “M”, accompanied by an increase in frequency “kxS > kxM”. “kxS*(xS  c*tS)” equals “kxM*(xM  c*tM)” because it is an invariant.
131 CHAPTER 3 – FIELDS GeometricVector Notation. Induction equations “3x3E = (3B)/t” and “3x3B = ((3E)/t)/c2” are satisfied for the fourth wave. 3E
= ExM*ixM + EyM*iyM + EzM*izM ; 3B = BxM*ixM + ByM*iyM + BzM*izM = Eamp*sin(kxM*(xM  c*tM))*iyM = (Eamp/c)*cos(kxM*(xM  c*tM))*iyM + Eamp*cos(kxM*(xM  c*tM))*izM + (Eamp/c)*sin(kxM*(xM  c*tM))*izM 3x3E
= (/xM*ixM)x(Eamp*cos(kxM*(xM  c*tM))*izM) + (/xM*ixM)x(Eamp*sin(kxM*(xM  c*tM))*iyM) = kxM*Eamp*sin(kxM*(xM  c*tM))*(iyM) + kxM*Eamp*cos(kxM*(xM  c*tM))*(+izM) = kxM*Eamp*sin(kxM*(xM  c*tM))*(iyM) + kxM*Eamp*cos(kxM*(xM  c*tM))*(izM)
(3B)/t = (/tM)((Eamp/c)*cos(kxM*(xM  c*tM))*iyM) + (/tM)((Eamp/c)*sin(kxM*(xM  c*tM))*izM) = kxM*Eamp*sin(kxM*(xM  c*tM))*(iyM) + kxM*Eamp*cos(kxM*(xM  c*tM))*(izM) = kxM*Eamp*sin(kxM*(xM  c*tM))*(iyM) + kxM*Eamp*cos(kxM*(xM  c*tM))*(izM) 3x3B
= (/xM*ixM)x((Eamp/c)*sin(kxM*(xM  c*tM))*izM) + (/xM*ixM)x((Eamp/c)*cos(kxM*(xM  c*tM))*iyM) = kxM*(Eamp/c)*cos(kxM*(xM  c*tM))*(iyM) + kxM*(Eamp/c)*sin(kxM*(xM  c*tM))*(+izM) = kxM*(Eamp/c)*cos(kxM*(xM  c*tM))*(iyM) + kxM*(Eamp/c)*sin(kxM*(xM  c*tM))*(izM)
((3E)/t)/c2 = (/tM)((Eamp)*sin(kxM*(xM  c*tM))*iyM)/c2 + (/tM)((Eamp)*cos(kxM*(xM  c*tM))*izM)/c2 = kxM*c*Eamp*cos(kxM*(xM  c*tM))*(iyM)/c2 + kxM*c*Eamp*sin(kxM*(xM  c*tM))*(izM)/c2 = kxM*(Eamp/c)*cos(kxM*(xM  c*tM))*(iyM) + kxM*(Eamp/c)*sin(kxM*(xM  c*tM))*(izM)
132 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
3.8 Forces Using GeometricVector Notation Evidence the electromagnetic field “6E” and electric charge density “4J” are physically real comes from measurements of force. Force causes a reaction force or else accelerates test electric charge “q”.
Figure 33a. “Ez” is negative because “Ez = Vt/z < 0”. “q > 0” imposes a force “Fz = q*Ez < 0” on whatever is holding it stationary. Figure 33b. Magnetic field “Bx < 0” surrounds electric charge “q > 0” that moves at speed “vy > 0” to create force “Fz = q*vy*Bx > 0”, which accelerates the particle so that it travels in a circle. More mass “m” means less curvature.
Imaginary Force. There is no measurable force due to a stationary electric charge in a magnetic field, or a moving charge in an electric field. But, “q*3B” and “(q*3v)x3E” do not simply equal zero. 3Fr
= q*3E + (q*3v)x3B
Frx = q*Ex + q*vy*Bz  q*vz*By Fry = q*Ey + q*vz*Bx  q*vx*Bz Frz = q*Ez + q*vx*By  q*vy*Bx 3F
= 3Fr i*3Fi
; 3Fi = q*c*3B  (q*3v/c)x3E ; Fix = q*c*Bx  ((q*vy/c)*Ez  (q*vz/c)*Ey) ; Fiy = q*c*By  ((q*vz/c)*Ex  (q*vx/c)*Ez) ; Fiz = q*c*Bz  ((q*vx/c)*Ey  (q*vy/c)*Ex)
133 CHAPTER 3 – FIELDS Energy Rate. A current of positive charge flows from a high voltage “+” battery terminal to the “” at a lower value of “x”, “q*vx < 0”. “Ex < 0” because “Ex = Vt/x”. Energy transfers out from the electrical system into heat when both “vx” and “Ex” are the same sign (both are negative) because “Ftr = q*(3v/c)•3E > 0”. An imaginary rate of energy change “Fti” that cannot be measured occurs when current flows parallel to a magnetic field. Ft = Ftr i*Fti = q*(3v/c)•3E i*q*3v•3B
Figure 34. Energy is lost in a resistor so that “Ftr = q*(3v/c)•3E > 0”.
3.9 Force Density Invariant “3F” and “Ft” are to be expressed using allnumber algebra. First guess: 4g
= (4J)*(6E) = (4J)■(6E)  (4J)♦(6E) = ((3J)•(3K) + (3J)x(3E) + (1J)*(3K))  ((3J)•(3E) + (3J)x(3K) + (1J)*(3E)) Two terms of “4g” are expanded out. 1M*gtM = (JxM*KxM*qxM*kxM + JyM*KyM*qyM*kyM + JzM*KzM*qzM*kzM) + (JxM*ExM*qxM*pxM + JyM*EyM*qyM*pyM + JzM*EzM*qzM*pzM) = 1M*((JxM*KxM + JyM*KyM + JzM*KzM)*i + (JxM*ExM + JyM*EyM + JzM*EzM)) qxM*kxM = jxM*kxM*(i) = (1M)*(i) = 1M*i qxM*pxM = jxM*kxM*(1) = (1M)*(1) = 1M
134 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY qxM*gxM = ((JyM*EzM*qyM*pzM + JzM*EyM*qzM*pyM) + JtM*KxM*1M*kxM) + ((JyM*KzM*qyM*kzM + JzM*KyM*qzM*kyM) + JtM*ExM*1M*pxM) = qxM*((JyM*EzM  JzM*EyM) + JtM*KxM))*i + ((JyM*KzM  JzM*KyM) + JtM*ExM)) qyM*pzM = jyM*kzM = jxM = qxM*(i) qyM*kzM = jyM*kzM*i = jxM*i = qxM “4g” has the correct arrangement of plus and minus signs. gtr = Jx*Ex + Jy*Ey + Jz*Ez ; gti = Jx*Kx + Jy*Ky + Jz*Kz ;
Frt = q*(vx*Ex + vy*By + vz*Ez)/c Fit = q*(vx*Bx + vy*Ey + vz*Bz)
gxr = (Jy*Kz  Jz*Ky) + Jt*Ex ; Frx = (q*vy*Bz  q*vz*By) + q*Ex gxi = (Jy*Ez  Jz*Ey) + Jt*Kx ; Fix = (q*(vy/c)*Ez  q*(vz/c)*Ey) + q*c*Bx Applying Maxwell’s Equations. If the field from test charge “4J” were included in the externally applied electromagnetic field “6E”, then the force would be there, regardless. Substitute “4sn*(6E) = (4J)” into “4g”. 4g
= (4sn*(6E))*(6E) = (4sn*6E)*6E = ((4sn*6E)♦6E + (4sn*6E)■6E) = ((4sn♦6E)♦6E + (4sn■6E)♦6E + (4sn♦6E)■6E + (4sn■6E)■6E)
= ((3sn•3E)*3E + (1sn*3E + 3snx3K)x3K + (1sn*3E + 3snx3K)•3E + (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K + (1sn*3K + 3snx3E)•3E + (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E + (1sn*3E + 3snx3K)•3K + (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E + (1sn*3K + 3snx3E)•3K) First and fourth rows have mathematically real components. Second and Third have imaginary components. The first row sums to the real portion of “4g”. The third row sums to the imaginary portion of “4g”. The second and fourth rows each sum to zero but individual terms in the second and fourth rows are not zero, for example, “(1sn*3K)•3K 0”.
135 CHAPTER 3 – FIELDS Time component terms of the first row are given below. Gradient operator “” only applies to the component factor immediately behind it, and not to both field component factors, to simplify what is written. Portion of (1M*gtrM) = (1sn*3E + 3snx3K)•3E = (tM*ExM*ExM*1Msn*pxM*pxM + tM*EyM*EyM*1Msn*pyM*pyM + tM*EzM*EzM*1Msn*pzM*pzM) + (yM*KzM*ExM*qyMsn*kzM*pxM + zM*KyM*ExM*qzMsn*kyM*pxM + zM*KxM*EyM*qzMsn*kxM*pyM + xM*KzM*EyM*qxMsn*kzM*pyM + xM*KyM*EzM*qxMsn*kyM*pzM + yM*KxM*EzM*qyMsn*kxM*pzM) = 1M*((tM*ExM*ExM + tM*EyM*EyM + tM*EzM*EzM) + (yM*KzM*ExM + zM*KyM*ExM) + (zM*KxM*EyM + xM*KzM*EyM) + (xM*KyM*EzM + yM*KxM*EzM)) qyMsn*kzM*pxM = jyM*kzM*kxM = jxM*kxM = 1M A Negative is Needed. “gtrM”, like “Frt”, models the loss of energy from the system, per the example of the battery and resistor, and this is properly modelled by the “(1sn*3E)•3E” term in the first row with the example “(tM*ExM)*ExM = tM*(ExM*ExM)/2”. A positive value is a loss of energy, and the term is positive because energy “ExM*ExM” is positive. The gradient is negative because it is a loss of energy, and the “1” makes “gtM” positive. The math is as expected. Good. “(1sn*3K)•3K” of the fourth row has “(tM*KxM)*KxM*i2 = tM*(KxM*KxM)/2”. “i2” was introduced by labelnumbers because “3K” is mathematically imaginary if “3E” is mathematically real. The extra negative is bad because “3K•3K = K2” is subtracted from “3E•3E = +E2” in “3E•3E + 3K•3K = E2  K2”, in contradiction to experiments for which total energy density is “E2 + K2”. To fix this, a negative is needed for “3K”. To help specify how to insert the negative, second row term “3E•3K” plus third row term “3K•3E” should subtract to cancel. And, first row “3Ex3K” needs to subtract from its negative “3Kx3E” of the fourth row for conformance to the experimentally derived Poynting Vector. The conclusion is that the fourth row needs a negative relative to the first row, and that negative is acceptable because the fourth row sums to zero. The second row also sums to zero and it needs a negative
136 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY relative to the third row. Also, because the imaginary force of the third row is not measurable, the second and third rows can be either plus or minus relative to the first row. Place “1” with the first “■” and “1” with the second “■”. 4f
= ((4sn♦6E)♦6E (4sn■6E)♦6E (4sn♦6E)■6E  (4sn■6E)■6E)
The components of “4f” separate into models of physics with two examples being electromagnetic field energy density and the Poynting vector. Note that “4fr = 4gr”. Also, “4f = 4fr i*4fi”. Mathematically Ugly. Because of the inserted negatives, “4f” cannot be expressed using “*” operator gradients, and that means “4f” is ugly. By analogy, the dot product in the exponential function, “4k*j•4r”, is ugly because crossproduct “4k*jx4r” is not added to it. This ugliness is explained by equating “4k*j•4r” to the product of hyperbolic radii. The ugliness of “4f”, too, goes away. “4f” becomes beautiful using the proposed algebra in this book’s last chapter. Terms of the Force Density Invariant. 4f = ((3sn•3E)*3E + (1sn*3E + 3snx3K)x3K + (1sn*3E + 3snx3K)•3E) ( (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K + (1sn*3K + 3snx3E)•3E ) ( (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E + (1sn*3E + 3snx3K)•3K )  ( (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E + (1sn*3K + 3snx3E)•3K ) = ( (1J)*3E + (3J)x3K + (3J)•3E ) ( (10)*3E + (30)x3K + (30)•3E ) ( (1J)*3K + (3J)x3E + (3J)•3K )  ( (10)*3K + (30)x3E + (30)•3K ) 1f = (3J)•3E (30)•3E (3J)•3K  (30)•3K = ( (1sn*3E + 3snx3K)•3E ) ( (1sn*3K + 3snx3E)•3E ) ( (1sn*3E + 3snx3K)•3K )  ( (1sn*3K + 3snx3E)•3K )
137 CHAPTER 3 – FIELDS ftr = (Jx*Ex + Jy*Ey + Jz*Ez) = ( (Ex/ct)*Ex + (Ey/ct)*Ey + (Ez/ct)*Ez )  ( (Kx/ct)*Kx  (Ky/ct)*Ky  (Kz/ct)*Kz ) + ( (Kz/y + Ky/z)*Ex + (Kx/z + Kz/x)*Ey + (Ky/x + Kx/y)*Ez )
 ( (Ey/z  Ez/y)*Kx + (Ez/x  Ex/z)*Ky + (Ex/y  Ey/x)*Kz )
y*Kz*Ex*(qyM)*kzM*pxM = y*Kz*Ex*(qyM)*pyM = y*Kz*Ex*1M z*Ey*Kx*(qzM)*pyM*kxM = z*Ey*Kx*(qzM)*(pzM) = z*Ey*Kx*1M t*Ex*Ex*(1M)*pxM*pxM = t*Ex*Ex*(1M)*(1) = t*Ex*Ex*1M t*Kx*Kx*(1M)*kxM*kxM = t*Kx*Kx*(1M)*(1) = t*Kx*Kx*1M Jx*Ex*qxM*pxM = Jx*Ex*1M fti = (Jx*Kx + Jy*Ky + Jz*Kz) = ( (Kx/ct)*Ex + (Ky/ct)*Ey + (Kz/ct)*Ez ) ( (Ex/ct)*Kx  (Ey/ct)*Ky  (Ez/ct)*Kz ) ( (Kz/y  Ky/z)*Kx + (Kx/z  Kz/x)*Ky + (Ky/x  Kx/y)*Kz ) ( (Ey/z  Ez/y)*Ex + (Ez/x  Ex/z)*Ey + (Ex/y  Ey/x)*Ez ) y*Kz*Kx*(qyM)*kzM*kxM = y*Kz*Kx*(qyM)*(kyM) = i*y*Kz*Kx*1M z*Ey*Ex*(qzM)*pyM*pxM = z*Ey*Ex*(qzM)*(kzM) = i*z*Ey*Ex*1M t*Kx*Ex*(1M)*kxM*pxM = t*Kx*Ex*(1M)*(i) = i*t*Kx*Ex*1M t*Ex*Kx*(1M)*pxM*kxM = t*Kx*Kx*(1M)*(i) = i*t*Ex*Kx*1M Jx*Kx*qxM*kxM = i*Jx*Kx*1M 3f = ( (1J)*3E + (3J)x3K ) ( (0)*3E + (30)x3K ) ( (1J)*3K + (3J)x3E )  ( (0)*3K + (30)x3E ) = ( (3sn•3E)*3E + (1sn*3E + 3snx3K)x3K ) ( (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K ) ( (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E )  ( (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E ) fzr = Jt*Ez + (Jy*Kx  Jx*Ky) = ( (Ky/ct)*Ex  (Kx/ct)*Ey ) + ( (Ex/ct)*Ky  (Ey/ct)*Kx ) + ( (Kz/y + Ky/z)*Ky )  ( (Kz/x  Kx/z)*Kx ) + ( (Ez/y + Ey/z)*Ey )  ( (Ez/x  Ex/z)*Ex ) + ( (Ex/x + Ey/y + Ez/z)*Ez )  ( (Kx/x + Ky/y + Kz/z)*Kz )
138 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY t*Ky*Ex*(1M)*kyM*pxM = t*Ky*Ex*(1M)*(pzM) = t*Ky*Ex*qzM t*Ex*Ky*(1M)*pxM*kyM = t*Ex*Ky*(1M)*(pzM) = t*Ex*Ky*qzM y*Ez*Ey*(qyM)*pzM*pyM = y*Ez*Ey*(qyM)*kxM = y*Ez*Ey*qzM y*Kz*Ky*(qyM)*kzM*kyM = y*Kz*Ky*(qyM)*(kxM) = y*Kz*Ky*qzM x*Ex*Ez*(qxM)*pxM*pzM = x*Ex*Ez*(qxM)*kyM = x*Ex*Ez*qzM x*Kx*Kz*(qxM)*kxM*kzM = x*Kx*Kz*(qxM)*(kyM) = x*Kx*Kz*qzM Jx*Ky*qxM*kyM = Jx*Ky*qzM Jt*Ez*1M*pzM = Jt*Ez*qzM fzi = Jt*Kz (Jy*Ex  Jx*Ey) = ( (Ky/ct)*Kx + (Kx/ct)*Ky ) ( (Ex/ct)*Ey  (Ey/ct)*Ex ) (( (Kz/y + Ky/z)*Ey )  ( (Kz/x  Kx/z)*Ex )) (( (Ez/y  Ey/z)*Ky ) + ( (Ez/x  Ex/z)*Kx )) ( (Ex/x + Ey/y + Ez/z)*Kz ) ( (Kx/x + Ky/y + Kz/z)*Ez ) t*Ky*Kx*(1M)*kyM*kxM = t*Ky*Kx*(1M)*(kzM) = t*Ky*Kx*jzM t*Ex*Ey*(1M)*pxM*pyM = t*Ex*Ey*(1M)*(kzM) = t*Ex*Ey*jzM y*Ez*Ky*(qyM)*pzM*kyM = y*Ez*Ky*qyM*pxM = y*Ez*Ky*jzM y*Kz*Ey*(qyM)*kzM*pyM = y*Kz*Ey*qyM*pxM = y*Kz*Ey*jzM x*Ex*Kz*(qxM)*pxM*kzM = x*Ex*Kz*(1M)*kzM = x*Ex*Kz*jzM x*Kx*Ez*(qxM)*kxM*pzM = x*Kx*Ez*(i*1M)*(pzM) = x*Kx*Ez*jzM Jx*Ey*qxM*pyM = Jx*Ey*jzM Jt*Kz*1M*kzM = Jt*Kz*jzM As a check on compoundlabelnumber products, component force equations have as factors Maxwell’s Equations. Component Maxwell’s Equations: Kx/ct  Ez/y + Ey/z = 0 Ky/ct  Ex/z + Ez/x = 0 Kz/ct  Ey/x + Ex/y = 0 Kx/x + Ky/y + Kz/z = 0
; ; ; ;
Ex/ct + Kz/y  Ky/z = Jx Ey/ct + Kx/z  Kz/x = Jy Ez/ct + Ky/x  Kx/y = Jz Ex/x + Ey/y + Ez/z = Jt
In the component equation for “fzr” are six of Maxwell’s Equations. • • •
“Jt*Ez” equates to “(Ex/x + Ey/y + Ez/z)*Ez” “Jy*Kx” equates to “(Ey/ct)*Kx  (Kz/x  Kx/z)*Kx” “Jx*Ky” equates to “(Ex/ct)*Ky + (Kz/y + Ky/z)*Ky”
139 CHAPTER 3 – FIELDS • • •
“0” equates to “(Kx/x + Ky/y + Kz/z)*Kz” “0” equates to “(Ky/ct)*Ex  (Ez/x  Ex/z)*Ex” “0” equates to “(Kx/ct)*Ey + (Ez/y + Ey/z)*Ey”
In the component equation for “fzi” are these other six Equations that also conform to Maxwell’s Equations. • • • • • •
“Jt*Kz” equates to “(Ex/x + Ey/y + Ez/z)*Kz” “Jy*Ex” equates to “(Ey/ct)*Ex (Kz/x  Kx/z)*Ex” “ Jx*Ey” equates to “(Ex/ct)*Ey (Kz/y + Ky/z)*Ey” “0” equates to “(Kx/x + Ky/y + Kz/z)*Ez” “0” equates to “(Ky/ct)*Kx (Ez/x  Ex/z)*Kx” “0” equates to “(Kx/ct)*Ky (Ez/y  Ey/z)*Ky”
Example Use of First Case Force Density. Gradient components of the negative of the Third Spiral Wave Solution are given below. Third Spiral Wave (6Ethird) = Eamp*(kyM + pzM)*exp(i*(kxM*(xM + c*tM))) (EyM + i*KyM)third = i*Eamp*exp(i*(kxM*(xM + c*tM))) (EzM + i*KzM)third = Eamp*exp(i*(kxM*(xM + c*tM))) EyMthird = Eamp*sin(kxM*(xM + c*tM)) KyMthird = Eamp*cos(kxM*(xM + c*tM)) EzMthird = Eamp*cos(kxM*(xM + c*tM)) KzMthird = Eamp*sin(kxM*(xM + c*tM)) 3E
= Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) K = Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM))) 3 E = 6 3E + 3K 1
sn
3
sn
*3E = 1M*/ct*3E = Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM))  qzM*sin(kxM*(xM + c*tM))) Using: 1M*pyM = qyM 1M*pzM = qzM x3E = (qxM*/x  qyM*/y  qzM*/z)x3E = Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM))) Using: qxM*pyM = jxM*kyM = jzM qxM*pzM = jxM*kzM = jyM
140 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3
sn
•3E = (qxM*/x  qyM*/y  qzM*/z)•3E = 0 1
sn
3
sn
3
sn
*3K = 1M*/ct*3K = Eamplitude*kxM*(jyM*sin(kxM*(xM + c*tM))  jzM*cos(kxM*(xM + c*tM))) Using: 1M*kyM = jyM 1M*kzM = jzM x3K = (qxM*/x  qyM*/y  qzM*/z)x3K = Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM))  qyM*cos(kxM*(xM + c*tM))) Using: qxM*kyM = jxM*kyM/i = qzM qxM*kzM = jxM*kzM/i = qyM •3K = (qxM*/x  qyM*/y  qzM*/z)•3K = 0 (1sn*3E)•3E = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM))  qzM*sin(kxM*(xM + c*tM)))) • (Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(1M  1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qyM*pyM = 1M qzM*pzM = 1M (1sn*3E)x3E = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM))  qzM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(jxM)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(jxM) Using: qyM*pzM = jyM*kzM = jxM qzM*pyM = jyM*kzM = jxM (3snx3E)•3E = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) •(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*i*1M*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*i*1M Using: jzM*pzM = i*1M jyM*pyM = i*1M
141 CHAPTER 3 – FIELDS (3snx3E)x3E = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(qxM  qxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: jzM*pyM = jyM*kzM/i = qxM jyM*pzM = jyM*kzM/i = qxM (1sn*3E)•3K = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM))  qzM*sin(kxM*(xM + c*tM)))) •(Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(i*1M)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(i*1M) Using: qyM*kyM = i*jyM*kyM = i*1M qzM*kzM = i*1M (1sn*3E)x3K = (Eamplitude*kxM*(qyM*cos(kxM*(xM + c*tM))  qzM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(pxM  pxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qyM*kzM = pxM qzM*kyM = pxM (3snx3E)•3K = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) •(Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(1M  1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: jzM*kzM = 1M jyM*kyM = 1M (3snx3E)x3K = (Eamplitude*kxM*(jzM*cos(kxM*(xM + c*tM)) + jyM*sin(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*jxM*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*jxM Using: jzM*kyM = jyM*kzM = jxM jyM*kzM = jxM (1sn*3K)•3E = (Eamplitude*kxM*(jyM*sin(kxM*(xM + c*tM))  jzM*cos(kxM*(xM + c*tM)))) •(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(i*1M)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(i*1M) Using: jyM*pyM = i*1M jzM*pzM = i*1M
142 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY (1sn*3K)x3E = (Eamplitude*kxM*(jyM*sin(kxM*(xM + c*tM))  jzM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(qxM  qxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: jyM*pzM = qxM jzM*pyM = qxM (3snx3K)•3E = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM))  qyM*cos(kxM*(xM + c*tM)))) •(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*(1M  1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qzM*pzM = 1M qyM*pyM = 1M (3snx3K)x3E = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM))  qyM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(pyM*sin(kxM*(xM + c*tM)) + pzM*cos(kxM*(xM + c*tM))) = Eamplitude2*kxM*jxM*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*jxM Using: qzM*pyM = jyM*kzM = jxM qyM*pzM = jyM*kzM = jxM (1sn*3K)•3K = (Eamplitude*kxM*(jyM*sin(kxM*(xM + c*tM)) + jzM*cos(kxM*(xM + c*tM)))) •(Eamplitude*(kyM*cos(kxM*(xM + c*tM)) + kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(1M  1M)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: jyM*kyM = 1M jzM*kzM = 1M (1sn*3K)x3K = (Eamplitude*kxM*(jyM*sin(kxM*(xM + c*tM))  jzM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(jxM)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(jxM) Using: jyM*kzM = jxM jzM*kyM = jxM (3snx3K)•3K = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM))  qyM*cos(kxM*(xM + c*tM)))) •((Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM))) = Eamplitude2*kxM*(i*1M)*(cos2(kxM*(xM + c*tM)) + sin2(kxM*(xM + c*tM))) = Eamplitude2*kxM*(i*1M) Using: qzM*kzM = i*1M qyM*kyM = i*1M
143 CHAPTER 3 – FIELDS (3snx3K)x3K = (Eamplitude*kxM*(qzM*sin(kxM*(xM + c*tM))  qyM*cos(kxM*(xM + c*tM)))) x(Eamplitude*(kyM*cos(kxM*(xM + c*tM))  kzM*sin(kxM*(xM + c*tM)))) = Eamplitude2*kxM*(qxM  qxM)*sin(kxM*(xM + c*tM))*cos(kxM*(xM + c*tM)) = 0 Using: qzM*kyM = qyM*kzM = qxM qyM*kzM = qxM 4f = 1f + 3f = ( (1sn*3E + 3snx3K)•3E ) W ( (1sn*3K + 3snx3E)•3E ) W ( (1sn*3E + 3snx3K)•3K )  ( (1sn*3K + 3snx3E)•3K ) + (( (3sn•3E)*3E + (1sn*3E + 3snx3K)x3K ) W ( (3sn•3K)*3E + (1sn*3K + 3snx3E)x3K ) W ( (3sn•3E)*3K + (1sn*3E + 3snx3K)x3E )  ( (3sn•3K)*3K + (1sn*3K + 3snx3E)x3E )) = (1sn*3E)•3E + (3snx3K)•3E W (1sn*3K)•3E W (3snx3E)•3E W (1sn*3E)•3K W (3snx3K)•3K  (1sn*3K)•3K  (3snx3E)•3K + (3sn•3E)*3E + (1sn*3E)x3K + (3snx3K)x3K W (3sn•3K)*3E W (1sn*3K)x3K W (3snx3E)x3K W (3sn•3E)*3K W (1sn*3E)x3E W (3snx3K)x3E  (3sn•3K)*3K  (1sn*3K)x3E  (3snx3E)x3E = 0 + 0 W Eamplitude2*kxM*i*1M W Eamplitude2*kxM*i*1M W Eamplitude2*kxM*i*1M W Eamplitude2*kxM*i*1M  0  0 + 0 + 0 + 0 W 0 W Eamplitude2*kxM*jxM W Eamplitude2*kxM*jxM W 0 W Eamplitude2*kxM*jxM W Eamplitude2*kxM*jxM  0  0  0 = 0 “4f = 0” because “4J = 0”.
3.10 Area and Volume Differential Operators • • • • •
Per nothing Gradient (per length) differential operator “4sn” (del) Per area differential operator “6sn” (theta) Per volume differential operator “4sn” (xi) Per world volume differential operator “1” (omega)
144 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The count of terms matches a row of Pascal’s Triangle: 1, 4, 6, 4, 1. The first “1” is for a OneComponent Invariant. It is Lorentz Transformed by Lorentz Transforming components it depends on. Electric current density “4J” uses the per volume differential operator. Finding Wavenumber. “4sn” can be made equivalent to a wavenumber/frequency invariant “4k” with consideration as to what function “1T” is. The first example is “1T = A*sin(kxM*xM  M*tM)”.
=
sn sn 4 *(A*sin(kxM*xM  M*tM)) = A*(4 *sin(kxM*xM  M*tM)) sn sn sn sn A*(1M */ct + qxM */x + qyM */y+qzM */z)*sin(kxM*xM  M*tM) = A*(1Msn*/ctM + qxMsn*/xM)*sin(kxM*xM  M*tM) = A*(1Msn*(M/c) + qxMsn*(kxM))*cos(kxM*xM  M*tM)
= A*(1M*(M/c) + qxM*(kxM))*cos(kxM*xM  M*tM) = A*(4k)*cos(kxM*xM  M*tM) “1T = A*exp(i*(kxM*xM  M*tM))” avoids the change from sine to cosine. 4
sn
*(A*exp(i*(kxM*xM  M*tM))) = A*(4sn*exp(i*(kxM*xM  M*tM))) = A*(1Msn*/ctM + qxMsn*/xM)*exp(i*(kxM*xM  M*tM)) = A*(1Msn*(i*M/c) + qxMsn*(i*kxM))*exp(i*(kxM*xM  M*tM)) = A*i*(1M*(M/c) + qxM*(kxM))*exp(i*(kxM*xM  M*tM)) = i*(4k)*A*exp(i*(kxM*xM  M*tM))
“4sn = i*4k” (which has terms “1Msn*/ctM = 1M*(i*M/c)” and “qxM */xM = qxM*(i*kxM)”) is an abbreviation of “4sn*(1T) = i*4k*(1T) and is exclusive to “1T = A*exp(i*(kxM*xM  M*tM))”. “i*4sn = 4k” is used in the next chapter on waves. sn
* Note: The remaining portion of this chapter can be bypassed A Linear Scalar Field’s Gradient. A count (zero, one, two, three, four, and higher) is equally spaced along the “x” axis. Count is an invariant scalar field represented by the below expression. The compoundlabelnumber in “1Count” is “1”. 1Count
= b*xM
145 CHAPTER 3 – FIELDS Visualize “1Count” as a long rod on which numbers are written: Zero at the back of the bus and increasing forward (“b > 0”).
=
sn sn 4 *(1Count) = 4 *(b*xM) sn sn (1M */ctM + qxM */xM + qyMsn*/yM = (qxMsn*/xM)*(b*xM) = qxMsn*b
+ qzMsn*/zM)*(b*xM)
“4sn*(1Count) = qxMsn*b” states “1Count” varies by gradient “b” in the “qxMsn” direction (that is, positive “ixM” direction). “1Count” is Lorentz Transformed from “1Count = b*xM” to “1Count = b*(xS*cosh(αS/M)  c*tS*sinh(αS/M))”, after which a gradient “4sn” is taken. 4
*(b*xM) = 4sn*(b*(xS*cosh(αS/M)  c*tS*sinh(αS/M))) = b*(1Msn*/ctM*c*tS*sinh(αS/M) + qxMsn*/xM*xS*cosh(αS/M)) = b*(1Msn*sinh(αS/M) + qxMsn*cosh(αS/M))
sn
Alternatively, “4sn*(1Count) = qxMsn*b” is Lorentz Transformed to the result “4sn*(1Count) = 1Ssn*b*sinh(αS/M) + qxSsn*b*cosh(αS/M)”, per the matrix equation below. “/ctM*(1Count) = 0”. /ctS*(1Count)
/ctM*(1Count)
cosh(αS/M) sinh(αS/M) =
/xS*(1Count)
* sinh(αS/M)
/xM*(1Count)
cosh(αS/M)
cosh(αS/M) sinh(αS/M) =
0 *
sinh(αS/M)
cosh(αS/M)
b*sinh(αS/M) =
b
b*cosh(αS/M)
Either method has the same result. 4
sn
*(1Count) = 1Ssn*b*(sinh(αS/M)) + qxSsn*b*cosh(αS/M)
Time component “b*sinh(αS/M)” of “4sn*(1Count)” is negative because “1Count” decreases at rate “b*sinh(αS/M)” when observed from one location “xS” (for “αS/M > 0”). If we are standing on roadside “S” looking at numbers, then the numbers decrease.
146 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Rather than care about numbers, we can care about spacing of the numbers and the rate numbers pass us. Spacing of numbers, called wavenumber, is a positive value (if we select it to be positive). The rate at which numbers pass us, called frequency, is positive (per that selection). We continue to use the gradient operator, but we must transition away from the gradient operator invariant with its spacenegative to the wavenumberfrequency invariant that has no spacenegative (because now “qxM” translates to “ixM”). Remove the spacenegative in “4sn*(1Count) = qxMsn*b” by replacing “qxMsn” with “qxM” so that “4sn*(1Count) = qxM*b = 4k”. 4
sn
*(1Count) = 4k
;
4k
= qxM*kxM = qxM*b = 1S*b*sinh(αS/M) + qxS*b*cosh(αS/M) = 1S*S/c + qxS*kxS
PerArea Differential Operator “6sn” has six terms. sn 6
= pxMsn*(2/(xM*ctM) + i*(2/(yM*zM))) + pyMsn*(2/(yM*ctM) + i*(2/(zM*xM))) + pzMsn*(2/(zM*ctM) + i*(2/(xM*yM))) = pxMsn*(xrM + i*xiM) + pyMsn*(yrM + i*yiM) + pzMsn*(zrM + i*ziM)
On the floor inside bus “M” is a rectangular array of particles. On each particle is a sequential number in the “x” direction and a sequential number in the “y” direction. The first particle counted is in the rear of the bus on the right side, “x = 0” and “y = 0”. The count of particles is “1Count = (bx*xM)*(by*yM)”. (“bx > 0”, “by > 0”) “6sn” operates on “1Count = (bx*xM)*(by*yM)” to quantify the change in count relative to area. sn 6 *(1Count)
= pxMsn*(2/(xM*ctM) + i*(2/(yM*zM)))*1Count + pyMsn*(2/(yM*ctM) + i*(2/(zM*xM)))*1Count + pzMsn*(2/(zM*ctM) + i*(2/(xM*yM)))*1Count
147 CHAPTER 3 – FIELDS = pxMsn*(2/(xM*ctM) + i*(2/(yM*zM)))*(bx*xM)*(by*yM) + pyMsn*(2/(yM*ctM) + i*(2/(zM*xM)))*(bx*xM)*(by*yM) + pzMsn*(2/(zM*ctM) + i*(2/(xM*yM)))*(bx*xM)*(by*yM) = pzMsn*(i*(2/(xM*yM)))*(bx*xM)*(by*yM) = pzMsn*i*bx*by “6sn*(1Count) = pzMsn*i*bx*by” states the change in count with respect to area equals “bx*by”, as observed by a person seated on bus “M”. The “i” factor in “pzMsn*i” means the “pzMsn*i = kzMsn” label number translates to the “x / y” plane and not to the “z” direction. The result “6sn*(1Count)” is Lorentz Transformed. (yrS + i*yiS)*(1Count) cosh(αS/M) i*sinh(αS/M) (yrM + i*yiM)*(1Count) = * (zrS + i*ziS)*(1Count) i*sinh(αS/M) cosh(αS/M) (zrM + i*ziM)*(1Count) Rightside column vector has “(yrM + i*yiM)*(1Count) = 0” and “(zrM + i*ziM)*(1Count) = i*bx*by”. The matrix operator is a space negative. Terms of the left side column vector, “(yrS + i*yiS)*(1Count) = (i*bx*by)*(i*sinh(αS/M))” and “(zrS + i*ziS)*(1Count) = (i*bx*by)*(cosh(αS/M))”, are in the invariant expression below. sn 6 *(1Count)
= pySsn*(yrS+i*yiS)*(1Count) + pzSsn*(zrS+i*ziS)*(1Count) = pySsn*(bx*by*sinh(αS/M)) + pzSsn*(i*bx*by*cosh(αS/M))
“pzSsn*(i*bx*by*cosh(αS/M))” of “6sn*(1Count)” states there are more particles for the same amount of x*y area if we use “xS” rather than “xM” for that x*y area, by a factor of “cosh(αS/M)”. Term “pySsn*(bx*by*sinh(αS/M))” of “6sn*(1Count)” is a measure of perlength and pertime, because it is real and is not imaginary. Perlength is in the “y” direction, perpendicular to the direction of motion. Pertime is associated with the negative. The count as written on each particle decreases relative to a person standing on roadside “S”.
148 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The complementary invariant to the perarea differential operator “6 ” is wavenumberarea invariant “6kk”. There is no spacenegative on “6kk”. sn
6kk
= pzM*i*kkziM = pzM*i*bx*by = pyS*bx*by*sinh(αS/M) + pzS*i*bx*by*cosh(αS/M) = pyS*kkyrS + pzS*i*kkziS
“kkziS” component is spacing of particle rows in the “x” direction, and “kkziS” includes a factor “by” for density along particle rows. “kkyrS” component is frequency of particle rows that pass a person standing on roadside “S” (and “kkyrS” includes a factor “by” for the density of particles along rows). The Lorentz Transformation used the below matrix equation. The matrix operator below is not a spacenegative, as identified by the different location of the negative on offdiagonal terms. (kkyrS + i*kkyiS)
cosh(αS/M) i*sinh(αS/M) =
(kkzrS + i*kkziS)
(kkyrM + i*kkyiM) *
i*sinh(αS/M)
cosh(αS/M)
(kkzrM + i*kkziM)
The above matrix operator was also used for the Lorentz Transformation for electromagnetic field and area invariants. EyS + i*KyS
coshαS/M
i*sinhαS/M
i*sinhαS/M
coshαS/M
coshαS/M
i*sinhαS/M
= EzS + i*KzS ByS + i*AyS
*
= BzS + i*AzS
EyM + i*KyM EzM + i*KzM ByM + i*AyM *
i*sinhαS/M
coshαS/M
BzM + i*AzM
PerVolume Differential Operator “4sn” (xi) has four terms. sn 4
= 1Msn*(3/(xM*yM*zM)) + qxMsn*(3/(ctM*yM*zM)) + qyMsn*(3/(ctM*xM*zM)) + qzMsn*(3/(ctM*xM*yM)) = 1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM
149 CHAPTER 3 – FIELDS A long rectangular prism array of particles sits on the floor of bus “M”. On each are three numbers: “x” (to the front of the bus), “y” (from right to left), and “z” (up). At the rear, right side and on the floor is the particle labelled “x = 0”, “y = 0” and “z = 0”. The count of particles is given by “1Count = (bx*xM)*(by*yM)*(bz*zM)”. (“bx > 0”, “by > 0”, and “bz > 0”) Pervolume differential operator “4sn” operates on particle count scalar “1Count = (bx*xM)*(by*yM)*(bz*zM)” to quantify the change in count relative to volume. sn 4 *(1Count)
= 1Msn*(3/(xM*yM*zM))*(1Count) + qxMsn*(3/(ctM*yM*zM))*(1Count) + qyMsn*(3/(ctM*xM*zM))*(1Count) + qzMsn*(3/(ctM*xM*yM))*(1Count)
= 1Msn*(3/(xM*yM*zM))*(bx*xM)*(by*yM)*(bz*zM) + qxMsn*(3/(ctM*yM*zM))*(bx*xM)*(by*yM)*(bz*zM) + qyMsn*(3/(ctM*xM*zM))*(bx*xM)*(by*yM)*(bz*zM) + qzMsn*(3/(ctM*xM*yM))*(bx*xM)*(by*yM)*(bz*zM) = 1Msn*(3/(xM*yM*zM))*(bx*xM)*(by*yM)*(bz*zM) = 1Msn*bx*by*bz “4sn*(1Count) = 1Msn*bx*by*bz” states the change in count with respect to volume equals “bx*by*bz”, as observed by a person seated on bus “M”. “bx*by*bz” is density of counts: Counts per Volume or counted particles per volume. “4sn*(1Count)” is Lorentz Transformed. tS*(1Count) xS*(1Count)
tM*(1Count)
cosh(αS/M) sinh(αS/M) =
* sinh(αS/M)
cosh(αS/M)
xM*(1Count)
Rightside column vector has “tM*(1Count) = bx*by*bz” and “xM*(1Count) = 0”. The matrix operator is space negative. Terms of the left side column vector, “tS*(1Count) = bx*by*bz*cosh(αS/M)” and “xS*(1Count) = bx*by*bz*sinh(αS/M)”, are in the invariant expression below.
150 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY sn 4 *(1Count)
= 1Ssn*(tS*(1Count)) + qxSsn*(xS*(1Count)) = 1Ssn*bx*by*bz*cosh(αS/M) + qxSsn*(bx*by*bz*sinh(αS/M))
“1Ssn*bx*by*bz*cosh(αS/M)” of “4sn*(1Count)” states there are more particles for the same amount of x*y*z volume if we use “xS” rather than “xM” for that x*y*z volume, by a factor “cosh(αS/M)”. “qxSsn*(bx*by*bz*sinh(αS/M))” of “4sn*(1Count)” is a measure of perarea and pertime. Perarea is in the “y*z” direction, perpendicular to the direction of motion. Pertime is associated with the negative because the xdirection count as written on each particle decreases relative to a person standing on roadside “S”. The complementary invariant is currentdensity invariant “4” (rho). There is no spacenegative on “4”. 4
= 1M*tM = 1M*bx*by*bz = 1S*bx*by*bz*cosh(αS/M) + qxS*bx*by*bz*sinh(αS/M) = 1S*tS + qxS*xS
“tS” is density. “xS” is flow per area in the “x” direction due to movement of bus “M” relative to roadside “S”. There was no spacenegative in the Lorentz Transformation matrix operator. 4
= 1S*bx*by*bz*cosh(αS/M) + qxS*bx*by*bz*sinh(αS/M)
compared to sn 4 *(1Count)
= 1Ssn*bx*by*bz*cosh(αS/M) + qxSsn*(bx*by*bz*sinh(αS/M))
shows “4 = 4sn*(1Count)”. It applies to a homogeneous material. If “bx = by = bz”, then it is a solid homogeneous block. Electric Current Density Invariant. “Q” is electric charge per electron particle, “1Charge = Q*1Count”. “4J = Q*4” is found using pervolume operator “4sn” (for a homogeneous material): 4J
= 4sn*(1Charge)
151 CHAPTER 3 – FIELDS PerWorldVolume Differential Operator “1” (omega) has one term. 1
= 1*(4/(ctM*xM*yM*zM))
Inside a long rectangular prism on bus “M” is an array of lights that flash. Four numbers are printed onto each light. The first number is a count of flashes the light has had. The second number is a sequential number in the “x” direction, third in the “y” direction, fourth “z”. The initial light flash in the rear of the bus, on the right side and on the floor corresponds to “c*t = x = y = z = 0”. (“bt, bx, by, bz > 0”) “1Count = (bt*c*tM)*(bx*xM)*(by*yM)*(bz*zM)”. “1” operates on “1Count = (bt*c*tM)*(bx*xM)*(by*yM)*(bz*zM)” to quantify the change in count relative to worldvolume. 1*(1Count) 4
= 1*(4/(ctM*xM*yM*zM))*(1Count) = 1*( /(c*tM*xM*yM*zM))*((bt*c*tM)*(bx*xM)*(by*yM)*(bz*zM)) = bt*bx*by*bz
“1*(1Count) = bt*bx*by*bz” is the density of counts per worldvolume. The same result “1*(1Count)” is observed by a person standing on roadside “S” because length contraction balances time dilation as “cosh(αS/M)/cosh(αS/M) = 1”. With respect to the roadside, lights do not all flash at the same time, but, rather, appear to move forward as a pulse faster than the speedoflight. Perworldvolume density “1” is not complementary to, but rather, is equal to “1*(1Count) = bt*bx*by*bz”.
3.11 Exercises Text Comprehension Exercises. 1) Prove “6E = 4*jsnx(4V)”. Use “4*j” and “4V” expressed in “S”. Substitute in “M” expressions for compoundlabelnumbers and components of “4*j” and “4V”. Reduce the expression to the result of “6E” expressed in “M”. 2) Prove the following identities.
152 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY a. b. c. d. e. f. g. h. i. j. k. l. m.
1
sn
*(3*jsnx(3V)) + 3snx(1*jsn*(3V)) 0 sn *jsn *(1V)) 0 3 x(3 sn *jsn •( x( 3 3 3V)) 0 sn *jsn x(4V)) 0 4 ■(4 *jsn •(1sn*(1*jsn*(3V))) + 1*jsn*(3sn•(1*jsn*(3V))) 0 3 *jsn •(1sn*(3*jsn*(1V))) + 1*jsn*(3sn•(3*jsn*(1V))) 0 3 *jsn •(3snx(3*jsnx(3V))) 0 3 *jsn •(4sn♦(4*jsnx(4V))) 0 4 j (4* snx4sn)*(6E) 0 *jsn x(3sn*2P) 0 3 *jsn *(3sn*2P) + 3*jsn*(1sn*2P) 0 1 *jsn x(4sn*2P) 0 4 *jsn x(3snx6Q) (3*jsn•3sn)*6Q  3*jsn*(3sn•6Q) 3
n.
4
*jsn
•(4sn*(6Q)) = 4*jsn•(4sn■(6Q)) + 4*jsn•(4sn♦(6Q)) = 1*jsn*(3sn•(3Qi)) + 3*jsn•(1sn♦(3Qi) + 3snx(3Qr)) + 1*jsn*(3sn•(3Qr)) + 3*jsn•(1sn♦(3Qr) + 3snx(3Qi)) 0
3) Confirm “1sn*(6Ethird) + 3snx(6Ethird) = 0”. 4) For “2P = a*(c*t)2 + b*x2”, find “4sn*2P”. Find “4*jsn•(4sn*2P)”, find “(4*jsn•4sn)*2P”, and identify the relationship between “a” and “b” for “(4*jsn•4sn)*2P = 0”. For any values of “a” and “b” show that “4*jsnx(4sn*2P) = 0”. 5) Confirm “6E = Eamp*(jyM + qzM)*cos(kxM*(xM  c*tM))” complies with Maxwell’s Equations. Find the simplest representation of “4V” for “6E”. What is the effect on “6E” and on the Lorenz Condition of adding a “4Vconstants” (that has components that are each a constant relative to time and to space) to “4V”? Confirm that specific scalar gauge function “2P = Pmax*exp(i*n*kM*(xM  c*tM))” satisfies the criteria “(4*jsn•4sn)*2P = 0”. Find “4sn*2P”. 6) Show “ftr” and “fti” component equations conform to Maxwell’s Equations
153 CHAPTER 3 – FIELDS Select Exercises Solutions. 1) Not Given 2) Not Given 3)
1
sn
3
sn
*(6Ethird) = (1Msn)*(6Ethird)/(ctM) = (1M)*(kyM + pzM)*Eamp*(exp(i*(kxM*(xM + c*tM))))/(ctM) = (i*kxM)*Eamp*(jyM + qzM)*exp(i*(kxM*(xM + c*tM)))
x(6Ethird) = (qxMsn)*(6Ethird)/(xM) = Eamp*(qxMsn*kyM + qxMsn*pzM)*(exp(i*(kxM*(xM + c*tM))))/(xM) = (i*kxM)*Eamp*(qzM  jyM)*exp(i*(kxM*(xM + c*tM)))
qxMsn*kyM = qxM*kyM = qzM qxMsn*pzM = qxM*kzM/i = qzM*kxM/i = qyM*(i) = jyM
4)
1
sn
*(6Ethird) + 3snx(6Ethird) = 0 OK
2P
= a*(c*t)2 + b*x2
4
sn
4
*jsn
*2P = (1M*t + qxM*x + qxM*x + qxM*x)*(a*(c*t)2 + b*x2) = 1M*2*a*c*t + qxM*2*a*x •(4sn*2P) = (t*1M*jsn + x*qxM*jsn + y*qyM*jsn + z*qzM*jsn) •(1M*2*a*c*t + qxM*2*b*x) = (t*(2*a*c*t))*(1M*jsn*1M) + (x*2*b*x)*(qxM*jsn*qxM) = 2*a + 2*b
(4*jsn•4sn)*2P = (t2  x2  y2  z2)*(a*(c*t)2 + b*x2) = 2*a + 2*b a = b 4
x(4sn*2P) = (t*1M*jsn + x*qxM*jsn + y*qyM*jsn + z*qzM*jsn) x(1M*2*a*c*t + qxM*2*b*x)
*jsn
= (t*(2*b*x))*(1M*jsn*qxM) + (x*2*a*c*t)*(qxM*jsn*1M) = 0*pxM + 0*pxM = 0
154 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 5)
6E
= Eamp*(jyM + qzM)*cos(kxM*(xM  c*tM))
KyM = Eamp*cos(kxM*(xM  c*tM)) EzM = Eamp*cos(kxM*(xM  c*tM)) tM*KyM + xM*EzM = ((1)*(kxM) + kxM)*Eamp*sin(kxM*(xM  c*tM)) = 0 OK First attempt at “4V”: VxM = z*KyM = z*Eamp*cos(kxM*(xM  c*tM)) VtM = y*EzM = y*Eamp*cos(kxM*(xM  c*tM)) 4V = 1M*VtM + qxM*VxM = 1M*VtM + qxM*VxM EzM = zM*VtM = Eamp*cos(kxM*(xM  c*tM)) OK KyM = zM*VxM = Eamp*cos(kxM*(xM  c*tM)) OK *jsn •4V = tM*VtM + xM*VxM 4 = ((y)*(kxM) + z*kxM)*Eamp*sin(kxM*(xM  c*tM)) ≠ 0 not OK Second attempt at “4V”: VzM = KyM/kxM = (1/kxM)*Eamp*cos(kxM*(xM  c*tM)) VyM = EzM/kxM = (1/kxM)*Eamp*cos(kxM*(xM  c*tM)) 4V = qyM*VyM + qzM*VzM EzM = tM*VzM = (kxM)*(1/kxM)*Eamp*cos(kxM*(xM  c*tM)) OK KyM = xM*VzM = (kxM)*(1/kxM)*Eamp*cos(kxM*(xM  c*tM)) OK *jsn •4Vsn = yM*VyM + zM*VzM = 0 + 0 = 0 OK 4 4Vconstants
= 1M*Konstt + qxM*Konstx + qyM*Konsty + qzM*Konstz
“4*jsnx4Vconstants = 0” and “4*jsnx4V = 0”, therefore there is no effect on “6E” or on Lorenz Condition if “4V + 4Vconstants” is substituted for “4V”. 2P
= Pmax*exp(i*n*kM*(xM  c*tM))
(4*jsn•4sn)*2P = (tM2  xM2  yM2  zM2)*2P = (i*n*kM)2*2P  (i*n*kM)2*2P = 0 OK
155 CHAPTER 3 – FIELDS 4
sn
4
*jsn
4
*jsn
*2P = (1Msn*tM + qxMsn*xM + qyMsn*yM + qzMsn*zM)*2P = (1M*(i*n*kM)  qxM*(i*n*kM))*2P = (1M + qxM)*(i*n*kM)*2P
•(4sn*2P) = (tM*1M*jsn + xM*qxM*jsn + yM*qyM*jsn + zM*qzM*jsn)•(4sn*2P) *jsn = ((1M *1M)*(i*n*kM)2 + (qxM*jsn*qxM)*(i*n*kM)*(i*n*kM))*2P = ((1)*(i*n*kM)2 + (1)*(i*n*kM)*(i*n*kM))*2P = 0 OK x(4sn*2P) = (tM*1M*jsn + xM*qxM*jsn + yM*qyM*jsn + zM*qzM*jsn)x(4sn*2P) = (tM*1M*jsn + xM*qxM*jsn)x((1M + qxM)*(i*n*kM)*2P) + (yM*qyM*jsn + zM*qzM*jsn)x((1M + qxM)*(i*n*kM)*2P) *jsn = ((1M *qxM)*(i*n*kM)2 + (qxM*jsn*1M)*(i*n*kM)*(i*n*kM))*2P = ((pxM)*(i*n*kM)2 + (pxM)*(i*n*kM)*(i*n*kM))*2P = 0 OK 6) Solution is almost identical to the text given for “fzr” and “fzi”. Further Thought. 1) “KyS = (q/(4**ǝ*(rS2 + xS2*sinh2αS/M)3/2))*zS*sinhαS/M” of the BiotSavart Law reaches a maximum when the two particles are closest. Should the moving particle be a little past the other? 2) Can we violate the Lorenz Condition and still have a consistent mathematics for electromagnetism? 3) The author could not find an allnumber identity that combined the below identities, and the reason is speculated to be that the force density math of the last chapter is needed. 3A 3
= Ax*ix + Ay*iy + Az*iz ;
3B
= Bx*ix + By*iy + Bz*iz
= x*ix + y*iy + z*iz = /x*ix + /y*iy + /z*iz ; 1 = t = /(c*t)
156 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 3•(3Ax3B)
(3x3A)•3B + (3x3B)•3A 1*(3A•3B) (1*3A)•3B + (1*3B)•3A 1*(3Ax3B) (1*3A)x3B  (1*3B)x3A x( Ax 3 3 3B) ((3B •3)*3A)  ((3A•3)*3B) + (3•3B)*3A  (3•3A)*3B 3*(3A•3B) ((3B •3)*3A) + ((3A•3)*3B) + (3x3B)x3A + (3x3A)x3B 4) Proposed invariant “4S” has time component “U = ǝ*(3E•3E + c2*3B•3B)/2” and space components “3S = ǝ*c*(3Ex3B  3Bx3E)/2”. Try to apply compound label numbers and find “(pxM)*i ≠ pxM” for “*i” applied to quaternions. Try to perform a Lorentz Transformation. Explain why “4S” is not an actual invariant. “1f = 4sn•4S” with respect to components, but why “4f 4sn*4S”? Use “6Efourth = Eamp*(kyM  pzM)*exp(i*(kxM*(xM  c*tM)))” and “4S = (3E  3K)*(3E + 3K)/2”. Notice there’s no mathematically beautiful way to create “4S” from “6E”. 5) Try to find a linear combination of two of “qx”, “qy” or “qz” for “”. 6) How can antimatter be worked into a macroscopic approximation theory that inertial mass we measure is electromagnetic field energy of an electron? 7) Triplevectorproduct identity “4sn■(4*jsnx(4V)) 0” includes two spacecomponent identities that add together to create Faraday’s Law of Induction. Is there significance to “1sn*(3K) + 3snx(3E) = 0” being formed from two identities? 1
sn
*(3*jsnx(3V)) + 3snx(1*jsn*(3V)) 0 ; 3snx(3*jsn*(1V)) 0
1
sn
1
sn
1
sn
*(3*jsnx(3V)) + (3snx(1*jsn*(3V)) + 3snx(3*jsn*(1V))) 0 *(3*jsnx3V) + 3snx(1*jsn*3V + 3*jsn*1V) 0 *(3K) + 3snx(3E) = 0
157 CHAPTER 3 – FIELDS 8) To better understand spacenegative, review how spacenegative behaves in the nonrelativistic approximation example given below. A temperature gradient is given in “B” and is then Lorentz Transformed to “M” and to “S”. The very simple nonrelativistic approximation example is given so that spacenegative is the only complexity. Is there a more optimal algebraic technique that can replace messy spacenegative and that performs the same function? In “B”: 1T = b*xB ; 2sn*1T = qxBsn*b = qxBsn*1T/xB (1T/ctB = 0) sn sn In “M”: 1T = b*(xM  vM*tM) ; 2 *1T = 1M *(b*vM/c) + qxMsn*b In “S”: 1T = b*(xS  vS*tS) ; 2sn*1T = 1Ssn*(b*vS/c) + qxSsn*b Because of the nonrelativistic approximation: • • •
tB = tM = tS vS = vM + vS/M vM*vS/M/c2 = 0
1T/ctM 1T/xM
1
* vM/c
1T/ctS
1
1
1T/xS
* vS/M/c
1T/ctS
1T/xB 1T/ctM
vS/M/c
= 1T/xS
1T/ctB
vM/c
=
1
1T/xM
1
vS/M/c
=
1
vM/c
* vS/M/c
1
vM/c
* 1
1T/ctB 1T/xB
1
1T/ctB
vS/c
=
* vS/c
1
1T/xB
158 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
159 WAVES
Chapter 4 – Waves The Dirac Equation is our relativistic model for dynamics of an electron. An electron is so small Newton’s Second Law (force equals mass times acceleration) does not apply. Newton’s Second Law was presented in Chapter 1 by writing its geometricvector equation and then explaining force, mass, and acceleration. In contrast, the Dirac Equation is a set of four first order differential equations that needs to be developed slowly emphasizing a logical thought process.
4.1 Differential Operator Mechanical EnergyMomentum. An electron’s mechanical energy and momentum combine in the timespace momentum invariant “4p”. 4p
= exp(*/2)*(EM/c + qx*pxM + qy*pyM + qz*pzM)*exp(*/2)
“EM” is mechanical energy. “pxM”, “pyM”, and “pzM” are mechanical momentum components. “4p” is timelike because energy is modeled using “cosh” and momentum “sinh”. 4p
= exp(*/2)*(mB*c*coshαM + q*mB*c*sinhαM)*exp(*/2) = exp(*/2)*mB*c*exp(q*αM)*exp(*/2)
“αM” relates to electron speed by “vM = c*tanhαM”. Subscript “M” identifies the inertial reference frame of the observer. “B” is rest frame of the particle so that “mB” is rest mass. “c” is speedoflight. “q” is made general through use of knowable circularangles “”. q = qx*cos(x/yz) + (qy*cos(y/z) + qz*sin(y/z))*sin(x/yz)
q*q = 1
“1M” and “qM” are compoundlabelnumbers. “” is the unknown and unknowable unspecified simplelabelnumber. “” contrasts with “q” because “q” is knowable.
160 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1M = exp(*)
; qM = exp(*/2)*q*exp(*/2)
qM = qxM*cos(x/yz) + (qyM*cos(y/z) + qzM*sin(y/z))*sin(x/yz) 4p
qM*j*qM = 1
= 1M*mB*c*coshαM + qM*mB*c*sinhαM
Electrical EnergyMomentum. The electrical energymomentum invariant is “4q = QB*4V”. “QB” is electron electric charge. “4V” is external voltage. The analogy for energy component “QB*1V” is potential energy of a car at the top of a hill. To visualize electrical momentum component “QB*3V” think about what happens when a wire with direct current is cut. The magnetic field around the wire provides inertia to maintain the electric current, typically by ionizing air to make air conductive. Total EnergyMomentum. Per de Broglie relations, total energy is proportional to frequency “” (or “1k”) and total momentum is proportional to wavenumber “k” (or “3k”) with the constant of proportionality Planck’s constant “ħ” (hbar) (“ħ = h/(2*)”). ħ = 1.054571800(13)*1034 Joule*seconds
(angular momentum)
Total “ħ*4k” equals mechanical plus electrical. ħ*4k = 4p + QB*4V The next task is to associate an actual wave to the particle. Gradient Substitution for Frequency/WaveNumber. Total energymomentum invariant “ħ*4k” is replaced with a differential gradient operator. ħ*4k = i**ħ*4sn
and
ħ*4ksn = i**ħ*(4sn)sn = i**ħ*4
161 WAVES Replacement of frequency “4k” by gradient operator “i**4sn” is justified by use of an invariant wave function “1T” of the form below. Assume “” equals “+1”. 1T
= exp(i**(kxM*xM  M*tM))
4k*j•4r = kxM*xM  M*tM (written for “kyM = kzM = 0”) 4k*jsn•4r = kxM*xM  M*tM (written for “kyM = kzM = 0”) “4k = i**4sn” (and “4ksn = i**4”) 1M*M/c is replaced by i**1sn sn 3k is replaced by i**3
; 1Msn*M/c is replaced by i**1 sn ; 3k is replaced by i**3
Per the above equations, the spacenegative alternative is redundant and so will be dropped along with “”. “i**1sn” is substituted for the total energymomentum invariant. i**ħ*(4sn) = 4p + QB*4V = 1M*mB*c*coshαM + qM*mB*c*sinhαM + QB*(1V + 3V) i**ħ*(4sn) = i**ħ*(1)  i**ħ*(3) 1M*mB*c*coshαM = i**ħ*(1)  QB*1V qM*mB*c*sinhαM = i**ħ*(3)  QB*3V To make the above two equations component equations, divide them left and right by “exp(*/2)”. 1*mB*c*coshαM = i*ħ**tM  QB*VtM q*mB*c*sinhαM = qx*(i*ħ**xM  QB*VxM) + qy*(i*ħ**yM  QB*VyM) + qz*(i*ħ**zM  QB*VzM) “4sn” requires wave function “1T” so that the above component equations apply to something that is both a particle and a wave.
162 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
AntiMatter Visualized as the SpaceNegative. Antimatter was modeled with “αS/M = i*”. Here, antimatter is suggested to be the spacenegative. The two models for antimatter are independent, such that matter could be modeled with the spacenegative.
4.2 Development of the Dirac Equation Algebraic Matrix Equation for mechanical components. 12 = exp(q*αM)*exp(q*αM) 12 = (coshαM  q*sinhαM)*(coshαM + q*sinhαM) 12 = cosh2αM  q2*sinh2αM 12  cosh2αM = q2*sinh2αM (12  cosh2αM) = q2*sinh2αM “(12  cosh2αM) = q2*sinh2αM” becomes four equations. +: +: : :
(1 + coshαM)*(1 + coshαM) = (q*sinhαM)*(q*sinhαM) (1  coshαM)*(1  coshαM) = (q*sinhαM)*(q*sinhαM) (1  coshαM)*(1  coshαM) = (q*sinhαM)*(q*sinhαM) (1 + coshαM)*(1 + coshαM) = (q*sinhαM)*(q*sinhαM)
“+” is redundant to “+”, and “” to “”. Address both “+” and “” with a “” sign. The “” equation is split by introducing enabler functions “PPM” and “QQM”. ((1 + coshαM)*PPM)*((1 + coshαM)*QQM) = ((q*sinhαM)*QQM)*((q*sinhαM)*PPM) ((1 + coshαM)*QQM) = ((q*sinhαM)*PPM) ((1 + coshαM)*PPM) = ((q*sinhαM)*QQM) (1 + coshαM)*PPM + (q*sinhαM)*QQM = 0 (q*sinhαM)*PPM + (1 + coshαM)*QQM = 0
163 WAVES (1 + coshαM)
(q*sinhαM)
PPM *
(q*sinhαM) (1 + coshαM)
0 =
QQM
0
“” sign in front of “q*sinhαM” (“q*sinhαM = q*sinh(αM)” and “coshαM = cosh(αM)”) relates to motion being right or left per “vM/c = tanhαM = tanh(αM)” because the “” of “” refers to spacenegative. “0”’s on the right make it a “singular” algebraic matrix equation. Algebraic Solutions to the Matrix Equation. In general, an algebraic 2x2 singular matrix equation has two independent solution pairs: pair “1” and pair “2”. Because of angle identities, there are three equivalent ways of writing the two pairs. cosh(αM/2) = cosh(αM  αM/2) = coshαM*cosh(αM/2)  sinhαM*sinh(αM/2) sinh(αM/2) = sinh(αM  αM/2) = sinhαM*cosh(αM/2)  coshαM*sinh(αM/2) (1 + coshαM)/sinhαM = sinhαM/(1 + coshαM) = cosh(αM/2)/sinh(αM/2) ((PPM, QQM)1)nothalfanglesA = ((q*sinhαM), 1 + coshαM) ((PPM, QQM)2)nothalfanglesA = (sinhαM, (q)*(1 + coshαM)) ((PPM, QQM)1)nothalfanglesB = ((q)*(1 + coshαM), sinhαM) ((PPM, QQM)2)nothalfanglesB = (1 + coshαM, (q*sinhαM)) (PPM, QQM)1 = ((q*sinh(αM/2)), cosh(αM/2)) (PPM, QQM)2 = (sinh(αM/2), (q*cosh(αM/2))) “(q)” is preferred to be a factor on “sinhαM”. But, as is obvious in the three forms above, this preference cannot be satisfied. “sinh(αM/2)” and “cosh(αM/2)” together form mechanical momentum per “sinhαM = 2*sinh(αM/2)*cosh(αM/2)” and mechanical energy per “coshαM = cosh2(αM/2) + sinh2(αM/2)”. A general solution for a singular algebraic matrix equation is a linear combination formed by multiplying an arbitrary constant, “f” or “g”, by each of the two solutions.
164 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY PPM1 = (q)*PPM2
and
QQM1 = (q)*QQM2
(PPM, QQM) = f*(PPM, QQM)1 + g*(PPM, QQM)2 = f*(PPM, QQM)1 g*(q)*(PPM, QQM)1 = (f (q)*g)*(PPM, QQM)1 The half angle is useful because a half in an exponent represents a square root operation. It implies a need for a square operation later in the analysis. The square operation, in the form of a complex number multiplied by its conjugate, is required for calculating a measurable particle property from a quantum mechanics wave function solution. Substitute Differential Operators into the Matrix Equation. First, multiply by rest mass “mB*c”. (mB*c + mB*c*coshαM) (q*mB*c*sinhαM)
(q*mB*c*sinhαM) PPM 0 * = (mB*c + mB*c*coshαM) QQM 0
(mB*c + i**ħ*tM  QB*VtM) (i**ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) PPM 0 * = (i**ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) (mB*c + i**ħ*tM  Q*VtM) QQM 0
“1T = exp(i**(kxM*xM  M*tM))” had a handedness specified by “ = +1”. Opposite handedness is specified by “ = 1”. For completeness, “” has remained in the analysis until now, when it will be identified as irrelevant. A change in handedness (in this development of the Dirac Equation) is a change from matter to antimatter (or from antimatter to matter). To make that change: Change “” to “”. Multiply all four components of the above 2x2 matrix by “1”. Swap the sign of the electric charge so that “QB” is replaced by its negative “QB” and swap the sign of the rest mass so that “mB” is replaced by “mB”. Look at what remains. See it is the same as what was started with. The choice of “” was irrelevant and therefore “ = +1” is used.
165 WAVES (mB*c + i*ħ*tM  QB*VtM) (i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) PPM 0 * = (i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) (mB*c + i*ħ*tM  QB*VtM) QQM 0
The transformation of the mathematical model from particle to wave occurred when mechanical energy and mechanical momentum were replaced by total energy (minus electrical) and total momentum (minus electrical), respectively, by use of differential operators. Now that the equation pertains to waves and not to particles, the eitheror plus or minus separately sign “” is replaced by the both plus and minus but also neither plus nor minus separately sign “N”. (mB*c + i*ħ*tM  QB*VtM) N(i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) PM 0 * = N(i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) (mB*c + i*ħ*tM  QB*VtM) QM 0
Substitutein Matrix Isomorphs. “qx”, “qy” and “qz” have 2x2 matrix isomorph equivalents which are very similar to Pauli Spin Matrices (with the difference being “R2” is negative). “R1 = G1/i”, “R2 = G2/i”, “R3 = G3/i”, and “1 = i/i”. 0 i G1 => i 0
0 1 i G2 => G3 => 1 0 0
0 1 R1 =>
0 k R2 =>
1 0
0 i
1
0
0
1
R3 => k 0
1 0 1 => 0 1
(“k” is the same as “i” but is different symbolically for tracking.) The traditional substitution is “qx => R1”, “qy => R2”, and “qz => R3”. Enabler functions are replaced by column vectors. Dirac Equation:
166 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 1000 1000 0001 0100 0100 0010 mB*c* + (i*ħ*/ctM  QB*VtM)* N (i*ħ*/xM  QB*VxM)* 0 0 1 0 0010 0100 0 0 0 1 0001 1000 001 0 1M 0 0 0 1 2M N (i*ħ*/yM  QB*VyM)* N (i*ħ*/zM  QB*VzM)* * = 0 k00 1 000 3M k 0 0 0 0 1 0 0 4M 000 k 0 0 k 0
0 0 0 0
The “Dirac Spinor” is the columnvector with four components “1M”, “2M”, “3M”, “4M”. These four component symbols are shorthand for more expanded symbols “z1M”, “z2M”, “z3M”, “z4M”. “z” identifies “qz” has nonzero major diagonal elements. “” identifies “N” is included. An algebraic 4x4 singular matrix equation has four independent solutions. The Dirac Equation is a differential 4x4 singular matrix equation with eight solutions: Four for matter and four for antimatter.
4.3 Solutions to the Dirac Equation In general, finding solutions to the Dirac Equation is difficult. But finding solutions is easy for the specific case of motion in the positive “x” direction with no external voltage applied. For that case, the Dirac Equation is reduced to two simple pairs of equations of identical form. 1M + (iħ/mBc)*1M/ctM  N(iħ/mBc)*4M/xM = 0 4M + (iħ/mBc)*4M/ctM  N(iħ/mBc)*1M/xM = 0 2M + (iħ/mBc)*2M/ctM  N(iħ/mBc)*3M/xM = 0 3M + (iħ/mBc)*3M/ctM  N(iħ/mBc)*2M/xM = 0 The first pair has four Dirac Spinor solutions analogous, one for one, with the four electromagnetic spiral waves.
167 WAVES First Dirac Spinor Solution (for “z+1Mfirst” and “z+4Mfirst”) 1Mfirst = amp*cosh(αM/2)*exp(i*(NkxMam*xM + Mam*tM)) 4Mfirst = amp*sinh(αM/2)*exp(i*(NkxMam*xM + Mam*tM)) 4Mfirst = tanh(αM/2)*1Mfirst Second Dirac Spinor Solution (for “z+1Msecond” and “z+4Msecond”) 1Msecond = amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 4Msecond = amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 4Msecond = coth(αM/2)*1Msecond Third Dirac Spinor Solution (for “z+1Mthird” and “z+4Mthird”) 1Mthird = amp*sinh(αM/2)*exp(i*(NkxMam*xM + Mam*tM)) 4Mthird = amp*cosh(αM/2)*exp(i*(NkxMam*xM + Mam*tM)) 4Mthird = coth(αM/2)*1Mthird Fourth Dirac Spinor Solution (for “z+1Mfourth” and “z+4Mfourth”) 1Mfourth = amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 4Mfourth = amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 4Mfourth = tanh(αM/2)*1Mfourth (For engineering, “N” will be “+”. For theory development retain “N”, and do not place it as in the below alternative.) Alternative First Dirac Spinor Solution (do not use) 1Mfirst = amp*cosh(αM/2)*exp(i*(kxMam*xM + Mam*tM)) 4Mfirst = Namp*sinh(αM/2)*exp(i*(kxMam*xM + Mam*tM)) 4Mfirst = Ntanh(αM/2)*1Mfirst
Proof the First Solution is Correct. coshαM = ħ*M/(mB*c2) ;
sinhαM = ħ*kxM/(mB*c)
(for 4V = 0)
Substitute the first solution into the pair of differential equations. “amp” and “exp(i*(NkxM*xM + M*tM))” divide out.
168 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 0 = 1Mfirst + (iħ/mc)*1Mfirst/ctM  N(iħ/mc)*4Mfirst/xM = cosh(αM/2) + (iħ/mc2)*(i*M)*cosh(αM/2)  N(iħ/mc)*(i*NkxM)*sinh(αM/2) = cosh(αM/2)  coshαM*cosh(αM/2) + sinhαM*sinh(αM/2) = 0 0 = 4Mfirst + (iħ/mc)*4Mfirst/ctM  N(iħ/mc)*1Mfirst/xM = sinh(αM/2) + (iħ/mc2)*(i*M)*sinh(αM/2)  N(iħ/mc)*(i*NkxM)*cosh(αM/2) = sinh(αM/2)  coshαM*sinh(αM/2) + sinhαM*cosh(αM/2) = 0 Single Speed in this Simple Example. Regardless of there being only one speed represented by “αM”, the concept of an interference group applies. Matter and Antimatter. The plus “+” or the minus “” sign (for “N” “+”) in front of “M” determines if the solution is antimatter or is matter, respectively. “kxMam” and “Mam” are for antimatter. “kxMm” and “Mm” are for matter. Antimatter requires a spacenegative in a Lorentz Transformation. The actual numerical value of “kxM” and “M” is not dependent on it being matter or antimatter. kxMam = kxMm = kxM
;
Mam = Mm = M
Matter. Second and fourth solutions pertain to matter (for “N” “+”) with “mB > 0” and “QB < 0”. “kxMm*xM  Mm*tM = 4k*j•4r” describes wave crests and nodes that move in the positive “xM” direction at phase speed “vpM”. Phase speed “vpM/c”, for the simple case of no voltage, is the reciprocal of the electron particle’s group speed “vM/c = tanhαM”. vpMm = M/kxM = Mm/kxMm = c*coshαM/sinhαM vpMm/c = c/vMm = cothαM (matter) AntiMatter. First and third solutions (for “N” “+”) have “mB < 0” and “QB > 0” for an antimatter electron (positron). “kxMam*xM + Mam*tM = *jsn •4r” describes wave crests and nodes that move in the negative “xM” 4k direction at speed “vpM”. vpMam/c = (M/c)/(kxM) = (Mam/c)/(kxMam) = coshαM/sinhαM = c/vMam = cothαM (antimatter)
169 WAVES In the headlight/taillight visualization both electrons and positrons have headlights pointing toward more positive “x” (right). And, electric current for both electrons and positrons is the same. Spin. The “+”/“” sign in front of “i” determines if angular momentum spin is righthand or lefthand. In the car visualization a rightside steering wheel represents righthand spin. A positron with a left steering wheel is the antimatter counterpart to an electron with a right steering wheel, and that pairs first with fourth and second with third. Glove Visualization for Both Antimatter and Spin. Second/fourth solutions (matter) are visualized as a left/right glove pair with fingers pointing to positive “x”. Pull the gloves inside out, and now the gloves represent the first/third solutions (antimatter) pointing to negative “x”. Second became first and fourth became third. Rotate these to point to positive “x” and place them inside the original gloves, first inside fourth and third inside second. Excess energy separates them with second/fourth moving to more positive “x” and third/first moving to more negative “x”, for equal and opposite linear momentum and angular momentum.
4.4 Particle Properties Because Dirac Spinor waves cannot be measured directly, the solution is postprocessed to replace wave nature with particle nature, using the method proposed by Max Born for Schrödinger’s Equation solutions. Multiply the complexconjugate of the Dirac Spinor solution wave function, “4*i”, by the Dirac Equation, and perform the opposite operation. Add the two and, alternatively, subtract the two. Subtraction drops the mathematically real terms. Addition drops imaginary terms. Subtraction Equation Resulting in Electric Current Density. The sum equals zero because of the zeros on the right side of the Dirac Equation. (“M” subscript is dropped to not clutter the equation.)
170 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 0 = 1*i*(mc*1 + iħ*1/ct  QVt*1  Niħ*4/x  NQVx*4  Nk*iħ*4/y  Nk*QVy*4  Niħ*3/z  NQVz*3) + 2*i*(mc*2 + iħ*2/ct  QVt*2  Niħ*3/x  NQVx*3  N(k)*iħ*3/y  N(k)*QVy*3  N(1)*iħ*4/z  N(1)* QVz*4) + 3*i*(mc*3 + iħ*3/ct  QVt*3  Niħ*2/x  NQVx*2  Nk*iħ*2/y  Nk*QVy*2  Niħ*1/z  NQVz*1) + 4*i*(mc*4 + iħ*4/ct  QVt*4  Niħ*1/x  NQVx*1  N(k)*iħ*1/y  N(k)*QVy*1  N(1)*iħ*2/z  N(1)*QVz*2)  (mc*1*i  iħ*1*i/ct  QVt*1*i + Niħ*4*i/x  NQVx*4*i + N(k)*iħ*4*i/y  N(k)*QVy*4*i + Niħ*3*i/z  NQVz*3*i)*1  (mc*2*i  iħ*2*i/ct  QVt*2*i + Niħ*3*i/x  NQVx*3*i + N(+k)*iħ*3*i/y  N(k)*QVy*3*i + N(1)*iħ*4*i/z  N(1)*QVz*4*i)*2  (mc*3*i  iħ*3*i/ct  QVt*3*i + Niħ*2*i/x  NQVx*2*i + N(k)*iħ*2*i/y  N(k)*QVy*2*i + Niħ*1*i/z  NQVz*1*i)*3  (mc*4*i  iħ*4*i/ct  QVt*4*i + Niħ*1*i/x  NQVx*1*i + N(k)*iħ*1*i/y  N(k)*QVy*1*i + N(1)* iħ*2*i/z  N(1)*QVz*2*i)*4
“mc” and “QV” terms subtract away. Only “iħ” terms remain. 0 = 1*i*(iħ*1/ct  Niħ*4/x  Nk*iħ*4/y  Niħ*3/z) + 2*i*(iħ*2/ct  Niħ*3/x  N(k)*iħ*3/y  N(1)*iħ*4/z) + 3*i*(iħ*3/ct  Niħ*2/x  Nk*iħ*2/y  Niħ*1/z) + 4*i*(iħ*4/ct  Niħ*1/x  N(k)*iħ*1/y  N(1)*iħ*2/z)  (iħ*1*i/ct + Niħ*4*i/x + N(k)*iħ*4*i/y + Niħ*3*i/z)*1  (iħ*2*i/ct + Niħ*3*i/x + Nk*iħ*3*i/y N iħ*4*i/z)*2  (iħ*3*i/ct + Niħ*2*i/x + N(k)*iħ*2*i/y + Niħ*1*i/z)*3  (iħ*4*i/ct + N(iħ*1*i/x + Nk*iħ*1*i/y N iħ*2*i/z)*4 “iħ” terms are combined using the chain rule. For example: 4*i*(2/z) + (4*i/z)*2 = (4*i*2)/z 4*i*((1)*iħ*2/z) + ((1)*iħ*4*i/z)*2 = (1)*iħ*(4*i*2)/z 0 = iħ*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M)/ctM  Niħ*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M)/xM
171 WAVES  Nk*iħ*(1M*i*4M  2M*i*3M + 3M*i*2M  4M*i*1M)/yM  Niħ*(1M*i*3M  2M*i*4M + 3M*i*1M  4M*i*2M)/zM Substitute “+” for “N” because the subtraction equation is a particle equation and is not a wave equation. 0 = tM*JtM + xM*JxM + yM*JyM + zM*JzM = JtM/ctM + JxM/xM + JyM/yM + JzM/zM JtM = QB*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M) JxM = QB*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M) JyM = QB*k*(1M*i*4M  2M*i*3M + 3M*i*2M  4M*i*1M) JzM = QB*(1M*i*3M  2M*i*4M + 3M*i*1M  4M*i*2M) “” components are “(particle count per volume)” and “QB” is “electric charge per particle” to make measurement units on “J” “electric charge per volume”. First Dirac Spinor Solution example: JtM = QB*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M) = QB*amp2*(cosh2(αM/2) + sinh2(αM/2)) = QB*amp2*coshαM JxM = QB*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M) = QB*amp2*(2*cosh(αM/2)*sinh(αM/2)) = QB*amp2*sinhαM JyM = 0 4J
JzM = 0
= QB*amp2*(1M*coshαM  qxM*sinhαM) = QB*amp2*1M*exp(qx*αM)
The first solution above was for a positron that moves to negative “x” for “αM > 0”, per the “qx*αM”. “QB > 0” so net current is to negative “x”.
172 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Addition Equation has the last four lines of the subtraction equation started by a “+” rather than a “”. It pertains to the field outside the classical radius of the electron. 0 = 1*i*(mc*1  QVt*1  NQVx*4  Nk*QVy*4  NQVz*3) + 2*i*(mc*2  QVt*2  NQVx*3  N(k)*QVy*3  N(1)*QVz*4) + 3*i*(mc*3  QVt*3  NQVx*2  Nk*QVy*2  NQVz*1) + 4*i*(mc*4  QVt*4  NQVx*1  N(k)*QVy*1  N(1)*QVz*2) + (mc*1*i  QVt*1*i  NQVx*4*i  N(k)*QVy*4*i  NQVz*3*i)*1 + (mc*2*i  QVt*2*i  NQVx*3*i  Nk*QVy*3*i  N(1)*QVz*4*i)*2 + (mc*3*i  QVt*3*i  NQVx*2*i  N(k)*QVy*2*i  NQVz*1*i)*3 + (mc*4*i  QVt*4*i  NQVx*1*i  Nk*QVy*1*i  N(1)*QVz*2*i)*4 + 1*i*(iħ*1/ct  Niħ*4/x  Nk*iħ*4/y  Niħ*3/z) + 2*i*(iħ*2/ct  Niħ*3/x  N(k)*iħ*3/y  N(1)*iħ*4/z) + 3*i*(iħ*3/ct  Niħ*2/x  Nk*iħ*2/y  Niħ*1/z) + 4*i*(iħ*4/ct  Niħ*1/x  N(k)*iħ*1/y  N(1)*iħ*2/z) + (iħ*1*i/ct + Niħ*4*i/x + N(k)*iħ*4*i/y + Niħ*3*i/z)*1 + (iħ*2*i/ct + Niħ*3*i/x + Nk*iħ*3*i/y  Niħ*4*i/z)*2 + (iħ*3*i/ct + Niħ*2*i/x + N(k)*iħ*2*i/y + Niħ*1*i/z)*3 + (iħ*4*i/ct + Niħ*1*i/x + Nk*iħ*1*i/y  Niħ*2*i/z)*4 = 2*(m*c)*(1*i*1 + 2*i*2  3*i*3  4*i*4) + 2*(Q*Vt)*(1*i*1 + 2*i*2 + 3*i*3 + 4*i*4) + 2*(QVx)*(1*i*4 + 2*i*3 + 3*i*2 + 4*i*1) + 2*k*(QVy)*(+1*i*4  2*i*3 + 3*i*2  4*i*1) + 2*(QVz)*(1*i*3  2*i*4 + 3*i*1  4*i*2) + 1*i*(iħ*1/ct  Niħ*4/x  Nk*iħ*4/y  Niħ*3/z) + 2*i*(iħ*2/ct  Niħ*3/x  N(k)*iħ*3/y  N(1)*iħ*4/z) + 3*i*(iħ*3/ct  Niħ*2/x  Nk*iħ*2/y  Niħ*1/z) + 4*i*(iħ*4/ct  Niħ*1/x  N(k)*iħ*1/y  N(1)*iħ*2/z) + (iħ*1*i/ct + Niħ*4*i/x + N(k)*iħ*4*i/y + Niħ*3*i/z)*1 + (iħ*2*i/ct + Niħ*3*i/x + Nk*iħ*3*i/y  Niħ*4*i/z)*2 + (iħ*3*i/ct + Niħ*2*i/x + N(k)*iħ*2*i/y + Niħ*1*i/z)*3 + (iħ*4*i/ct + Niħ*1*i/x + Nk*iħ*1*i/y  Niħ*2*i/z)*4
173 WAVES Specific to the Fourth Dirac Spinor Solution, the addition equation results in the following equation. 0 = 2*(m*c)*(1Mfourth*i*1Mfourth  4 Mfourth*i*4 Mfourth) + iħ*1Mfourth*i*1Mfourth/ct  Niħ*1Mfourth*i*4Mfourth/x + iħ*4Mfourth*i*4Mfourth/ct  Niħ*4Mfourth*i*1Mfourth/x + iħ*1Mfourth*i/ct*1Mfourth + Niħ*4Mfourth*i/x*1Mfourth + iħ*4Mfourth*i/ct*4Mfourth + Niħ*1Mfourth*i/x*4Mfourth = 2*(m*c)*amplitude2*(cosh2(αM/2)  sinh2(αM/2))  ħ*(M/c)*amplitude2*(cosh2(αM/2) + cosh2(αM/2)) + Nħ*(kxM)*amplitude2*(Ncosh(αM/2)*sinh(αM/2) + Nsinh(αM/2)*cosh(αM/2)) + Nħ*(kxM)*amplitude2*(Ncosh(αM/2)*sinh(αM/2) + Nsinh(αM/2)*cosh(αM/2))
 ħ*(M/c)*amplitude2*(sinh2(αM/2) + sinh2(αM/2)) = 2*(m*c)*amplitude2 + ħ*(M/c)*2*amplitude2*cosh(αM)  ħ*(kxM)*2*amplitude2*sinh(αM) = 2*(m*c)*amplitude2 + 2*(m*c)*amplitude2*(cosh2(αM)  sinh2(αM)) = 2*(m*c)*amplitude2*(1  (cosh2(αM)  sinh2(αM)) =0
4.5 Two Alternative Arrangements The “x” Arrangement has “qy => R1”, “qz => R2”, and “qx => R3”. 1000 0100
1000 0100
0001 0010 m*c* + (i*ħ*/ctM  q*VtM)* N (i*ħ*/yM  q*VyM)* 0 0 1 0 0010 0100 0 0 0 1 0001 1000 000 k 001 0 x1M 0 0 0 k 0 0 0 0 1 x2M 0 N (i*ħ*/zM  q*VzM)* N (i*ħ*/xM  q*VxM)* * = 0 k00 1 000 x3M 0 k 0 0 0 0 1 0 0 x4M 0
For the case of “x” direction motion and no external voltage:
174 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY x1M + (iħ/mc)*x1M/ctM  N(iħ/mc)*x3M/xM = 0 x3M + (iħ/mc)*x3M/ctM  N(iħ/mc)*x1M/xM = 0 x2M + (iħ/mc)*x2M/ctM + N(iħ/mc)*x4M/xM = 0 x4M + (iħ/mc)*x4M/ctM + N(iħ/mc)*x2M/xM = 0 First Dirac Spinor Solution x1Mfirst = amp*cosh(αM/2)*exp(i*(NkxM*xM + M*tM)) x3Mfirst = amp*sinh(αM/2)*exp(i*(NkxM*xM + M*tM)) x3Mfirst = x1Mfirst*tanh(αM/2) Second Dirac Spinor Solution x1Msecond = amp*sinh(αM/2)*exp(i*(NkxM*xM  M*tM)) x3Msecond = amp*cosh(αM/2)*exp(i*(NkxM*xM  M*tM)) x3Msecond = x1Msecond*coth(αM/2) Third Dirac Spinor Solution x1Mthird = amp*sinh(αM/2)*exp(i*(NkxM*xM + M*tM)) x3Mthird = amp*cosh(αM/2)*exp(i*(NkxM*xM + M*tM)) x3Mthird = x1Mthird*coth(αM/2) Fourth Dirac Spinor Solution x1Mfourth = amp*cosh(αM/2)*exp(i*(NkxM*xM  M*tM)) x3Mfourth = amp*sinh(αM/2)*exp(i*(NkxM*xM  M*tM)) x3Mfourth = x1Mfourth*tanh(αM/2) The “y” Arrangement has “qz => R1”, “qx => R2”, and “qy => R3”. 1000 0100
1000 0100
0001 0010 m*c* + (i*ħ*/ctM  q*VtM)* N (i*ħ*/zM  q*VzM)* 0 0 1 0 0010 0100 0 0 0 1 0001 1000 000 k 001 0 0 0 k 0 0 0 0 1 N (i*ħ*/xM  q*VxM)* N (i*ħ*/yM  q*VyM)* 0 k00 1 000 k 0 0 0 0 1 0 0
y1M 0 y2M 0 * = y3M 0 y4M 0
175 WAVES For the case of “x” direction motion and no external voltage: y1M + (iħ/mc)*y1M/ctM  Nk*(iħ/mc)*y4M/xM = 0 y4M + (iħ/mc)*y4M/ctM N k*(iħ/mc)*y1M/xM = 0 y2M + (iħ/mc)*y2M/ctM N k*(iħ/mc)*y3M/xM = 0 y3M + (iħ/mc)*y3M/ctM  Nk*(iħ/mc)*y2M/xM = 0 First Dirac Spinor Solution y1Mfirst = amp*cosh(αM/2)*exp(i*(NkxM*xM + M*tM)) y4Mfirst = k*amp*sinh(αM/2)*exp(i*(NkxM*xM + M*tM)) y4Mfirst = k*y1Mfirst*tanh(αM/2) Second Dirac Spinor Solution y1Msecond = amp*sinh(αM/2)*exp(i*(NkxM*xM  M*tM)) y4Msecond = (k)*amp*cosh(αM/2)*exp(i*(NkxM*xM  M*tM)) y4Msecond = k*y1Msecond*coth(αM/2) Third Dirac Spinor Solution y1Mthird = amp*sinh(αM/2)*exp(i*(NkxM*xM + M*tM)) y4Mthird = k*amp*cosh(αM/2)*exp(i*(NkxM*xM + M*tM)) y4Mthird = k*y1Mthird*coth(αM/2) Fourth Dirac Spinor Solution y1Mfourth = amp*cosh(αM/2)*exp(i*(NkxM*xM  M*tM)) y4Mfourth = (k)*amp*sinh(αM/2)*exp(i*(NkxM*xM  M*tM)) y4Mfourth = k*y1Mfourth*tanh(αM/2) “z”, “x”, and “y” arrangements appear redundant because they result in the same measurable charge density spacetime invariant “2J”.
4.6 Lorentz Transformation of a Dirac Spinor The Lorentz Transformation for a Dirac Spinor uses half angle “αM/2”. Proof of validity of the Lorentz Transformation is that the Dirac Spinor in “S” can be obtained by either the Lorentz Transformation of the Dirac
176 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Spinor directly, or by solving the Dirac Equation with its component variables specified in “S” rather than in “M”. “4J”, too, must be properly affected by the Lorentz Transformation. The fourth solution is the example because it is for matter. “1Mfourth” and “4Mfourth” are associated with compoundlabelnumbers “e1M” and “e4M”, respectively, with “e1M = qx*e4M”. 1 0 e1 = qx*e4 =
0 0 0 1 0 0 1 0 =
0 0
0 0 *
0 1 0 0 1 0 0 0
0 1
In the next chapter is an explanation for anticommutative operations “e1M = e4M*qx” and “e4M = e1M*qx”. Condensing components into a single expression for “4fourth”. 4fourth
= e1M*1M + e4M*4M = e1M*1Mfourth + e4M*4Mfourth
= (e1M*amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) + (e4M*amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) = (e1M*amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) + (e1M*qx*amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) = e1M*amp*exp(qx*αM/2)*exp(i*(NkxMm*xM  Mm*tM)) “e1M*amp*exp(qx*αM/2)*exp(i*(NkxM*xM  M*tM))” is convenient because “e1M” changes to “e1S”, “αM” changes to “αS = αM + αS/M”, and “NkxMm*xM  Mm*tM” to “NkxSm*xS  Sm*tS”. We begin with the general form of the Lorentz Transformation. 4fourth
= e1M*amp*exp(qx*αM/2)*exp(i*(NkxMm*xM  Mm*tM)) = e1M*amp*exp(qx*αM/2)*exp(i*(NkxMm*xM  Mm*tM)) *exp(qx*αS/M/2)/exp(qx*αS/M/2)
177 WAVES = (e1M/exp(qx*αS/M/2)) *amp*exp(qx*αM/2)*exp(qx*αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) = e1S*amp*exp(qx*(αM + αS/M)/2)*exp(i*(NkxMm*xM  Mm*tM)) = e1S*amp*exp(qx*αS/2)*exp(i*(NkxMm*xM  Mm*tM)) = e1S*amp*exp(qx*αS/2)*exp(i*(NkxSm*xS  Sm*tS)) = e1S*1Sfourth + e4S*4Sfourth cosh(αM/2)*cosh(αS/M/2) + sinh(αM/2)*sinh(αS/M/2) = cosh(αM/2  αS/M/2) cosh(αM/2)*sinh(αS/M/2) + sinh(αM/2)*cosh(αS/M/2) = sinh(αM/2  αS/M/2) e1*1Sfourth + e4*4Sfourth = (e1*1Mfourth + e4*4Mfourth)*exp(qx*αS/M/2) = (e1*1Mfourth + e4*4Mfourth)*(cosh(αS/M/2) + qx*sinh(αS/M/2)) = e1*1Mfourth*cosh(αS/M/2) + e4*qx*4Mfourth*sinh(αS/M/2) + e1*qx*1Mfourth*sinh(αS/M/2) + e4*4Mfourth*cosh(αS/M/2) = e1*(1Mfourth*cosh(αS/M/2)  4Mfourth*sinh(αS/M/2)) + e4*(1Mfourth*sinh(αS/M/2) + 4Mfourth*cosh(αS/M/2)) 1Sfourth = 1Mfourth*cosh(αS/M/2)  4Mfourth*sinh(αS/M/2) = amp*cosh(αM/2)*cosh(αS/M/2)*exp(i*(NkxMm*xM  Mm*tM))  amp*sinh(αM/2)*sinh(αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) = amp*cosh(αM/2)*cosh(αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) + amp*sinh(αM/2)*sinh(αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) = amp*cosh(αM/2  αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) = amp*cosh(αS/2)*exp(i*(NkxMm*xM  Mm*tM)) = amp*cosh(αS/2)*exp(i*(NkxSm*xS  Sm*tS)) = amp*cosh(αS/2)*exp(i*(NkxSm*xS  Sm*tS))
178 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 4Sfourth = 1Mfourth*sinh(αS/M/2) + 4Mfourth*cosh(αS/M/2) = amp*cosh(αM/2)*sinh(αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) + amp*sinh(αM/2)*cosh(αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) = amp*sinh(αM/2  αS/M/2)*exp(i*(NkxMm*xM  Mm*tM)) = amp*sinh(αS/2)*exp(i*(NkxMm*xM  Mm*tM)) = amp*sinh(αS/2)*exp(i*(NkxSm*xS  Sm*tS)) = amp*sinh(αS/2)*exp(i*(NkxSm*xS  Sm*tS)) e1M = e1*exp(qx*/2) = e1*cosh(/2)  e1*qx*sinh(/2) = e1*cosh(/2) + e4*sinh(/2) e4M = e4*exp(qx*/2) = e4*qx*sinh(/2) + e4*cosh(/2) = e1*sinh(/2) + e4*cosh(/2) e1S = e1M/exp(qx*αS/M/2) = e1M*cosh(αS/M/2) + e4M*sinh(αS/M/2) e4S = e4M/exp(qx*αS/M/2) = e1M*sinh(αS/M/2) + e4M*cosh(αS/M/2) e1S = e4S*qx e4S = e1S*qx Complexconjugate is needed for electric current density. 4fourth
*i
= (exp(i*(NkxMm*xM  Mm*tM)))*i*(exp(qx*αM/2))*i*amp*i*e1M*i
1Sfourth*i = 1Mfourth*i*cosh(αS/M/2) + 4Mfourth*i*sinh(αS/M/2) 4Sfourth*i = 1Mfourth*i*sinh(αS/M/2) + 4Mfourth*i*cosh(αS/M/2) JtS = QB*(1S*i*1S + 2S*i*2S + 3S*i*3S + 4S*i*4S) = QB*(1Sfourth *i*1Sfourth + 4Sfourth*i*4Sfourth) = QB*((1Mfourth*i*cosh(αS/M/2) + 4Mfourth*i*sinh(αS/M/2)) *(1Mfourth*cosh(αS/M/2) + 4Mfourth*sinh(αS/M/2)) + (1Mfourth*i*sinh(αS/M/2) + 4Mfourth*i*cosh(αS/M/2)) *(1Mfourth*sinh(αS/M/2) + 4Mfourth*cosh(αS/M/2))) = QB*(1Mfourth*i*1Mfourth*cosh2(αS/M/2) + 1Mfourth*i*4Mfourth*cosh(αS/M/2)*sinh(αS/M/2)
179 WAVES + 4Mfourth*i*4Mfourth*sinh2(αS/M/2) + 4Mfourth*i*1Mfourth*cosh(αS/M/2)*sinh(αS/M/2) + 1Mfourth*i*1Mfourth*sinh2(αS/M/2) + 1Mfourth*i*4Mfourth*cosh(αS/M/2)*sinh(αS/M/2) + 4Mfourth*i*4Mfourth*cosh2(αS/M/2) + 4Mfourth*i*1Mfourth*cosh(αS/M/2)*sinh(αS/M/2) = QB*(1Mfourth*i*1Mfourth*(cosh2(αS/M/2) + sinh2(αS/M/2)) + 1Mfourth*i*4Mfourth*2*cosh(αS/M/2)*sinh(αS/M/2) + 4Mfourth*i*4Mfourth*(sinh2(αS/M/2) + cosh2(αS/M/2)) + 4Mfourth*i*1Mfourth*2*cosh(αS/M/2)*sinh(αS/M/2) = QB*(1Mfourth*i*1Mfourth + 4Mfourth*i*4Mfourth)*coshαS/M + QB*(1Mfourth*i*4Mfourth + 4Mfourth*i*1Mfourth)*sinh(αS/M) = QB*amp2*(coshαM/2*coshαM/2 + sinh(αM/2)*sinh(αM/2))*coshαS/M +QB*amp2(coshαM/2*sinh(αM/2) + sinh(αM/2)*coshαM/2)*sinh(αS/M) = QB*amp2*coshαM*coshαS/M + QB*amp2*sinh(αM)*sinh(αS/M) = QB*amp2*cosh((αM + αS/M)) = QB*amp2*cosh(αS) JxS = QB*(1S*i*4S + 2S*i*3S + 3S*i*2S + 4S*i*1S) = QB*(1Sfourth *i*4Sfourth + 4Sfourth*i*1Sfourth) = QB*((1Mfourth*i*cosh(αS/M/2) + 4Mfourth*i*sinh(αS/M/2)) *(1Mfourth*sinh(αS/M/2) + 4Mfourth*cosh(αS/M/2)) + (1Mfourth*i*sinh(αS/M/2) + 4Mfourth*i*cosh(αS/M/2)) *(1Mfourth*cosh(αS/M/2) + 4Mfourth*sinh(αS/M/2))) = QB*(1Mfourth*i*4Mfourth*cosh2(αS/M/2) + 1Mfourth*i*1Mfourth*cosh(αS/M/2)*sinh(αS/M/2) + 4Mfourth*i*1Mfourth*sinh2(αS/M/2) + 4Mfourth*i*4Mfourth*cosh(αS/M/2)*sinh(αS/M/2) + 1Mfourth*i*4Mfourth*sinh2(αS/M/2) + 1Mfourth*i*1Mfourth*cosh(αS/M/2)*sinh(αS/M/2) + 4Mfourth*i*1Mfourth*cosh2(αS/M/2) + 4Mfourth*i*4Mfourth*cosh(αS/M/2)*sinh(αS/M/2)
180 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = QB*(1Mfourth*i*4Mfourth*(cosh2(αS/M/2) + sinh2(αS/M/2)) + 1Mfourth*i*1Mfourth*2*cosh(αS/M/2)*sinh(αS/M/2) + 4Mfourth*i*1Mfourth*(sinh2(αS/M/2) + cosh2(αS/M/2)) + 4Mfourth*i*4Mfourth*2*cosh(αS/M/2)*sinh(αS/M/2)) = QB*(1Mfourth*i*4Mfourth + 4Mfourth*i*1Mfourth)*coshαS/M  QB*(1Mfourth*i*1Mfourth + 4Mfourth*i*4Mfourth)*sinh(αS/M) = QB*(amp2(coshαM/2*sinh(αM/2) + sinh(αM/2)*coshαM/2)*coshαS/M + QB*(amp2(coshαM/2*coshαM/2 + sinh(αM/2)*sinh(αM/2))*sinh(αS/M) = QB*amp2*sinh(αM)*coshαS/M + QB*amp2*coshαM*sinh(αS/M) = QB*amp2*sinh((αM + αS/M)) = QB*amp2*sinh(αS) 4J
= QB*amp2*(1S*coshαS  qxS*sinh(αS)) = QB*amp2*1S*exp(qx*αS) (N becomes + for a particle)
In “4J = QB*amp2*1S*exp(qx*αS)”, “amp2 > 0” and “QB < 0”. “exp(qx*αS)” represents motion to positive “x” (right) for positive “αS” and “QB” makes currentdensity of that motion negative. Similarly, “QB < 0” makes the charge density time term negative. The conclusion is that the Lorentz Transformation was correct. AntiMatter. For antimatter there is an “am” subscript. Sam/c
coshαS/M
sinhαS/M
sinhαS/M
coshαS/M
= kxSam Sm/c
*
coshαS/M
kxMam Mm/c
sinhαS/M
= kxSm
Mam/c
* sinhαS/M
coshαS/M
kxMm
The above process can be repeated for the First Dirac Spinor Solution (that is, for antimatter), to the result “αS = αM  αS/M”.
181 WAVES
4.7 Exercises Text Comprehension Exercises. 1) Start with “4p” and “QB*4V”, and end with 1*mB*c*coshαM = i*ħ**tM  QB*VtM q*mB*c*sinhαM = qx*(i*ħ**xM  QB*VxM) + qy*(i*ħ**yM  QB*VyM) + qz*(i*ħ**zM  QB*VzM) 2) Start with “12 = exp(q*αM)*exp(q*αM)” and end with (mB*c + i*ħ*tM  QB*VtM) N(i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) PM 0 * = N(i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) (mB*c + i*ħ*tM  QB*VtM) QM 0
3) Write the Dirac Equation as four first order differential equations. 4) Prove the Fourth Dirac Spinor Solution to the Dirac Equation. 5) Make the analogy that a matter electron with lefthand spin is a car with its steering wheel on the left, as in France. Two very high energy photons collide over the channel between Dover and Dunkerque to create a matter car and an antimatter car. Explain the analogy between cars and electrons. 6) Find electric charge current density spacetime invariant “4J” for the Second Dirac Spinor Solution. 1Msecond = amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 4Msecond = amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 7) Find the four solutions for the second pair of equations for “x” direction motion for “x”arrangement and “y”arrangement Dirac Equations. Prove correct and calculate current density.
182 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 8) Write the general form of the Lorentz Transformation for the Second Dirac Spinor Solution. “e1M = e4M*qx” and “e4M = e1M*qx”. 1Msecond = amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 4Msecond = amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) Select Exercise Solutions. 1)
4p
= exp(*/2)*(EM/c + qx*pxM + qy*pyM + qz*pzM)*exp(*/2) = exp(*/2)*(mB*c*coshαM + q*mB*c*sinhαM)*exp(*/2) = 1M*mB*c*coshαM + qM*mB*c*sinhαM
ħ*4k = 4p + QB*4V = i**ħ*4sn = i**ħ*(1)  i**ħ*(3) ħ*4ksn = 4psn + QB*4Vsn = i**ħ*(4sn)sn = i**ħ*4 = i**ħ*(1) + i**ħ*(3) 1M*m*c*coshαM = i**ħ*(1)  QB*1V 1*mB*c*coshαM = i*ħ**tM  QB*VtM qM*m*c*sinhαM = i**ħ*(3)  QB*3V q*mB*c*sinhαM = qx*(i*ħ**xM  QB*VxM) + qy*(i*ħ**yM  QB*VyM) + qz*(i*ħ**zM  QB*VzM) 2) 12 = exp(q*αM)*exp(q*αM) 12 = (coshαM  q*sinhαM)*(coshαM + q*sinhαM) 12 = cosh2αM  q2*sinh2αM 12  cosh2αM = q2*sinh2αM (12  cosh2αM) = q2*sinh2αM (1 + coshαM)*(1 + coshαM) = (q*sinhαM)*(q*sinhαM) ((1 + coshαM)*PM)*((1 + coshαM)*QM) = ((q*sinhαM)*QM)*((q*sinhαM)*PM) (1 + coshαM)*PM = (q*sinhαM)*(QM) + 0 (1 + coshαM)*(QM) = (q*sinhαM)*PM  0
183 WAVES
(1 + coshαM)
(q*sinhαM) PM * = (q*sinhαM) (1 + coshαM) QM
0 0
(mB*c + mB*c*coshαM) (q*mB*c*sinhαM)
(q*mB*c*sinhαM) PM 0 * = (mB*c + mB*c*coshαM) QM 0
(mB*c + i*ħ*tM  QB*VtM) N(i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) PM 0 * = N(i*ħ*(qx*xM+qy*yM+qz*zM)QB*(qx*VxM+qy*VyM+qz*VzM)) (mB*c + i*ħ*tM  QB*VtM) QM 0
3) Solution: mB*c*1M + i*ħ*1M/ctM  QB*VtM*1M N i*ħ*4M/xM N QB*VxM*4M N i*ħ*k*4M/yM N QB*VyM*k*4M N i*ħ*3M/zM N QB*VzM*3M = 0 mB*c*2M + i*ħ*2M/ctM  QB*VtM*2M N i*ħ*3M/xM N QB*VxM*3M N i*ħ*k*3M/yM N QB*VyM*k*3M N i*ħ*4M/zM N QB*VzM*4M = 0 mB*c*3M + i*ħ*3M/ctM  QB*VtM*3M N i*ħ*2M/xM N QB*VxM*2M N i*ħ*k*2M/yM N QB*VyM*k*2M N i*ħ*1M/zM N QB*VzM*1M = 0 mB*c*4M + i*ħ*4M/ctM  QB*VtM*4M N i*ħ*1M/xM N QB*VxM*1M N i*ħ*k*1M/yM N QB*VyM*k*1M N i*ħ*2M/zM N QB*VzM*2M = 0
4) 1Mfourth + (iħ/mBc)*1Mfourth/ctM  N(iħ/mBc)*4Mfourth/xM = 0 cosh(αM/2) + (iħ/mBc2)*(i*M)*cosh(αM/2)  N(iħ/mBc)*(i*NkxM)*sinh(αM/2) = 0 cosh(αM/2)  coshαM*cosh(αM/2) + sinhαM*sinh(αM/2) = 0 0=0
184 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 4Mfourth + (iħ/mBc)*4Mfourth/ctM  N(iħ/mBc)*1Mfourth/xM = 0 sinh(αM/2) + (iħ/mBc2)*(i*M)*sinh(αM/2)  N(iħ/mBc)*(i*NkxM)*cosh(αM/2) = 0 sinh(αM/2) + coshαM*sinh(αM/2)  sinhαM*cosh(αM/2) = 0 0=0 5) Car Visualization for Both Antimatter and Spin: In the pair production event, a lefthand spin matter electron is modeled as a car made of matter with steering wheel on the left. The car faces Dunkerque, France. In the same physical space at the instant of production a reverseparity second car is also produced. This second car made of antimatter also has steering wheel on the left and faces Dunkerque, France. Any energy in excess of the rest energy of the two cars is equally applied as kinetic energy that pushes the matter car toward Dunkerque and the antimatter car toward Dover. The steering wheel on the left of both cars means both cars have a lefthand spin, for zero total angular momentum (per the analogy of a nut and bolt that unscrew and separate, retaining their spin), just as there needs to be zero total linear momentum. The antimatter car has reverseparity. To unreverse the reverseparity of the antimatter car, flatten it front to back and go further to stretch it full length, so that the front points toward Dover, England, and see the steering wheel is on the right, as cars are in England. The right steering wheel means righthanded spin. Reverse parity has reverse spin. 6) 1Msecond*i = amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) 4Msecond*i = amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM)) JtM = QB*(1M*i*1M + 2M*i*2M + 3M*i*3M + 4M*i*4M) = QB*amp2*(sinh2(αM/2) + cosh2(αM/2)) = QB*amp2*coshαM JxM = QB*(1M*i*4M + 2M*i*3M + 3M*i*2M + 4M*i*1M) = QB*amp2*(2*cosh(αM/2)*sinh(αM/2)) = QB*amp2*sinhαM
185 WAVES JyM = 0 4J
JzM = 0
= QB*amp2*(1M*coshαM + qxM*sinhαM) = QB*amp2*1M*exp(qx*αM)
7) Solution not given. See text for similar solutions. 8)
4second
= e1M*1M + e4M*4M = e1M*1Msecond + e4M*4Msecond
= (e1M*amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) + (e4M*amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) = (e4M*qx*amp*sinh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) + (e4M*amp*cosh(αM/2)*exp(i*(NkxMm*xM  Mm*tM))) = e4M*amp*exp(qx*αM/2)*exp(i*(NkxMm*xM  Mm*tM)) = e4M*amp*exp(qx*αM/2)*exp(i*(NkxMm*xM  Mm*tM)) *exp(qx*αS/M/2)/exp(qx*αS/M/2) = e4S*amp*exp(qx*(αM + αS/M)/2)*exp(i*(NkxMm*xM  Mm*tM)) = e4S*amp*exp(qx*αS/2)*exp(i*(NkxMm*xM  Mm*tM)) = e4S*amp*exp(qx*αS/2)*exp(i*(NkxSm*xS  Sm*tS)) = e1S*1Ssecond + e4S*4Ssecond Further Thought 1) Perform the Lorentz Transformation of the Fourth Dirac Spinor Solution with “αS/M = (i  jx)*/2” (for the transformation of timelike to spacelike), and again with “αS/M = i*/2” (for sublightspeed motion of a Dirac Spinor wave crests and nodes) with a factor “i” applied to the hyperbolicradius, and then again with “αS/M = i*” (for antimatter) with a factor “1” applied to the hyperbolicradius. Explain what the results represent physically and speculate if anything in nature fits that description.
186 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2) The equation that relates the currentdensity invariant to the Dirac Spinor invariant has components of the Dirac Spinor invariant multiplied by each other. The implication is that Dirac Spinor space is a square root of geometric space. If this implication is valid, then other fourcomponent invariants of our geometricvector space should also be able to be expressed as a form of a square of Dirac Spinor space. Try to quantify this thought by creating a more general theory that relates Dirac Spinor space to geometric space. 3) Geometry of Dirac Spinor Space. The mathematical existence of the four “” components of the Dirac Equation suggest there is a translation from numbers to geometry that creates a physically real “” space. We expect some sort of physical reality to “” space because the intention of theorydevelopmentalgebra is to mathematically model physics in all its intricate detail, and physics is real, a reality that we traditionally suppose is geometrically real. In this book we ignore the possible physical reality of “” space per the excuse of the Process from Descartes, in which we do not revert to the geometry of step 3 until prepared to take a measurement. This excuse supposes there is no geometry other than what we perceive because numbers are fundamental, not geometry. Forget the Process from Descartes and make a guess at what the geometry of “” Dirac Spinor Space is. How might the guess be verified by experiment? If there is a geometry applicable to Dirac Spinor “” space, then a translation to geometry is needed inside Dirac Spinor “” space. Think about this further after reading the next chapter in which electromagnetic field theory is combined into the Dirac Equation through the requirement that precision resolution is restricted to a finite value in geometry, any geometry, including Dirac spinor space.
187 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS
Chapter 5 – Proposed Theory The proposed Theory of NonFinite Numbers replaces real numbers with “localreal numbers”. A set of localreal numbers is local to a finite maximum count of known or knowable placevalue digits before and after the decimal point. •
Localreal numbers are derived from the new proposed
•
Real numbers are dropped after proving Cantor’s Continuum Hypothesis is incompatible with the proposed new axiom
•
Localreal numbers are applied using the Lorentz Transformation to create the proposed Theory of Special Relativity with NonFinite Numbers
•
Verified because electromagnetic force density (with energy and momentum density) derives from the Dirac Equation
reciprocalofzero axiom
5.1 LocalReal Numbers Finite. A finite number results from addition, subtraction, multiplication and division (not by zero) operations that can be counted. An irrational number is finite because it is bounded by rational numbers. Three Dots and Six Dots. Three dots “…” represents a finite count using natural numbers. Three dots does not apply to the quantity of zeros after the decimal point for an integer because there are more zeros than can be counted, and that’s because wherever a count ends one more can be added. Six dots “……” is a new symbol to represent the quantity of zeros after the decimal point for an integer. To create an integer, the number is written in placevalue digit form, and then the number is cut at the
188 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY decimal point and all contributions after the decimal point are discarded. This process, called truncation, is a bulk process in which placevalue digits are not addressed individually, and therefore there is no count. “……” represents a quantity larger than “…”. Three dots represents a finite count that is indefinite in its magnitude, what has traditionally been called the finite potential infinity, “L”. In contrast, six dots is given symbol “” (omega). L=1+1+1+…
;
= 1 + 1 + 1 + ……
PlaceValue Digit Notation. A placevalue digit notation number, for example “56.45” may be rewritten as a power series, “5*101 + 6*100 + 4*101 + 5*102”. In general …… + an*sn + … + a1*s1 + a0*s0 + a1*s1 + … + an*sn + …… The outer two “……” represent a quantity that extends beyond what can be counted. It applies to the zeros after the decimal for an integer, and it applies to the nonpattern of an irrational number, for example, squarerootoftwo, “2 = 1.414213562373……”. Starting at “m < n”, “am = 0” all the way up to “an”, and for “> n” the “a”’s are not specified or specifically addressed, same as “< n”. Irrational Numbers. An irrational number is proven to not be rational. The example is “log23 = ln3/ln2 = 1.09861……/0.69314…… = 1.58496……”, as used in “21.5849625…… = 3”. Assume “log23 = p/q” with “p, q N”. log23 = p/q ; q*log23 = p ; log2(3^q) = p ; 2^log2(3^q) = 2^p ; 3^q = 2^p Odd number “3^q” cannot equal even number “2^p”. Therefore, “p, q N” is impossible, and “log23 = p/q” is not rational. “q*log23 = p” and “3^q = 2^p” are valid if “p = q = 0”: “0*log23 = 0” and “3^0 = 2^0 = 1” to suggest a relationship between irrational numbers “p/q = 0/0” and “log23”. The sequence of equations “log23 = p/q ; (1/p)*log23 = 1/q ; 3^(1/p) = 2^(1/q)” with “p = q = 0” gives “(1/0)*log23 = 1/0” and “3^(1/0) = 2^(1/0)”, to include irrational number “1/0” in that relationship.
189 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS Proposed ReciprocalofZero Axiom. Division by zero is prohibited per the definition of rational numbers. In contrast, division by zero has a relationship to irrational numbers, and is needed. Because there is no division by zero in ZFC axiomatic set theory on which algebra is based, we introduce an axiom called the new proposed reciprocalofzero axiom. •
No operation that includes division by zero may possibly result in a nonzero finite number. Per the new axiom, we define properties for “1/0”:
•
.1. Selection of “0” in “1/0” being positive or else negative is unknown and unknowable
•
.2. “1/0” is so large it represents the neverending feature of large number magnitude
•
.3. Permitted Operation: Addition of finite number “q” to “1/0”. 1/0 + q = 1/0 + (0*q)/0 = (1 + 0*q)/0 = 1/0
•
qQ
.4. Prohibited Operation: Addition of “1/0” to another “1/0”. Each “1/0” has an independent selection of being positive or negative. 1/0 + 1/0 = 1/0  1/0 ≠ a result
•
.5. Permitted Operation: Distributive property of multiplication over addition using “1/0” as the common factor but with the condition two “1/0”s on the right have dependent (same) positive or negative property. (1/0)*2 = (1/0 + 1/0) Both “0”’s have the same positive or negative (1/0)*(a + b) = (1/0)*a + (1/0)*b a, b Q a, b ≠ 0
•
.6. Permitted Operation: Distributive property for addition of zero. It requires “0/0” not equal “1/0”. (1/0)*(1 + 0) = (1/0) + (1/0)*0 = 1/0 + 0/0 = 1/0
qQ
190 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •
.7. Permitted Operation: Multiplication of finite “q” to “1/0”. (1/0)*q = q/0 = 1/(0/q) = 1/0
•
q Q and q ≠ 0
.8. Prohibited Operation: Multiplication of zero “0” by “1/0”. (1/0)*0 = 0/0 ≠ a result
•
.9. Permitted Operation: Multiplication of “1/0” to another “1/0”. (1/0)*(1/0) = 1/0
•
.10. Permitted Operation: “1/0” base to “n” exponent. (1/0)^n = 1/0
•
nN
(n = 0 is Prohibited)
.11. Permitted Operation: Reciprocal of “1/0”. (1/0)^(1) = 1/(1/0) = 0
•
.12. Permitted Operation: Base “n” to “1/0” exponent. The visualization is “n*n*n*n*……” with the count represented by “……” being “1/0”, in analogy to “1 + 1 + 1 + ……” being “1/0”. n^(1/0) = 1/0 or else 0 with the selection unknown and unknowable n^(1/0) = 0 or else 1/0 with selection unknown and unknowable n N or n = e (“n = e” means “exp(x)” applies, not “e^x”.)
•
.13. Permitted Operation: Logarithm of zero. 1/0 = logn(0) or else +logn(0) with the selection being unknown and unknowable n N or n = e
•
.14. Permitted Operation: Logarithm of an exponent. logn(n^1/0) = (1/0)*(logn(n)) = (1/0)*1 = 1/0 , n N or n = e
191 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS •
.15. Permitted Operation: Finite “n” cannot equal “1/0”. nN
0 < n < 1/0 and n ≠ 1/0 and 1/0 < n < 0 •
.16. Prohibited Operation: Inequality of “1/0” to itself. “1/0 + q = 1/0 + (0*q)/0 = (1 + 0*q)/0 = 1/0”. Therefore,“1/0” cannot be greater than or less than “1/0”.
An algebra specifies a logical system that always gives the same calculated result. If a violation is found, then insert another prohibition to avoid it. +
0
Q
1/0
*
0
Q
1/0
0
0
Q
1/0
0
0
0
No
Q
Q
2*Q
1/0
Q
0
Q^2
1/0
1/0
1/0
1/0
No
1/0
No
1/0
1/0

0
Q
1/0
/
0
Q
1/0
0
0
Q
1/0
0
No
0
0
Q
Q
0
1/0
Q
1/0
1
0
1/0
1/0
1/0
No
1/0
1/0
1/0
No
^
0
n
1/0
log
0
n
1/0
0
1
0
No
0
No
0?
No
n
1
n^n
1/0 or 0
n
1/0 or 1/0
1
1/0 or 1/0
1/0
No
1/0
No
1/0
No
No
No
Table 4. Crude algebra operations involving “1/0” (“Q Q”, “n N”).
Applying the Crude Algebra of “1/0” to “log23”. Use the power series of “log23” to evaluate “(1/0)*log23 = 1/0”. Per the six dots, the tailend terms of the “……” are multiplied by “1/0” in one bulk operation to a sum result of “1/0”. Therefore, “(1/0)*log23 = 1/0”. There is no end to placevalue digits of “log23” after the decimal point, and there is no end to the magnitude of “1/0”. Because “1/0” cannot be less than “1/0”, the two concepts counter each other so that “(1/0)*log23 = 1/0” is valid, such that
192 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY “1/0” on the right does not have a contribution of placevalue digits after the decimal point. Also, per the new axiom, “3^(1/p) = 2^(1/q)” is valid for “1/p = 1/q = 1/0” but not for smaller values of “1/p” and “1/q”. Because six dots “……” has been used to represent the quantity of nonpattern placevalue digits for an irrational number, the six dots represent “l/0”, “ = 1 + 1 + 1 + …… = 1/0”. Before we had the new proposed reciprocalofzero axiom, we had no formal means of stating in a constructive sense that an irrational number had no end to its nonrepeating pattern. Equations for proven irrational numbers, “q*log23 = p”, “q*2 = p”, “q* = p”, and “q*e = p”, are each satisfied only if “q” and “p” equal “l/0” and nothing less, except “0”. The association of “1/0” with irrational numbers per the six dots is theoretical because, in our geometric, physical world, we are only able to put measurement to finite quantities, per the three dots. Only a rational approximation can be realized. Base Two Representation of Numbers rather than base ten is preferred. •
Base two has only ones and zeros for placevalue digits
•
The quantity of sets “P(a)” created from a quantity of possible members “a” in a set is found using a base of two, “P(a) = 2^a”
•
Base two has “1 + 20 + 21 + 22 = 23 = 8”
Truncated Numbers. A power series representation is divided into two portions. •
The first portion is a known or knowable rational number, called a truncated number, identified with three dots to a count of “Lmax N” placevalue digits before and after the decimal point
•
The second portion, imprecision term “” (xi), is the unknown and unknowable portion split between what is larger and smaller than the rational number, as identified with six dots on both ends of the power series
193 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS Truncated Numbers “TN”: •
Written in base two. Positive or negative or zero. Specific to “Lmax” count of possibly nonzero placevalue digits before and after the decimal point. “TN Q”
•
“1/(2^Lmax)” is the smallest. “2^Lmax” largest. “((2^Lmax)*(2^Lmax) + 1)*2  1”
Quantity is
For “Lmax = 0” truncated numbers are “1”, “0”, and “1”. The eight (“2^Lmax = 8”) truncated numbers between zero (inclusive) and one (exclusive) for “Lmax = 3” are on the left. The nine “((2^Lmax)*(2^Lmax) + 1)*2  1 = 9” truncated numbers for “Lmax = 1” are on the right. 10.000000…… 0.1110000…… 1.100000…… 0.1100000…… 1.000000…… 0.1010000…… 0.100000…… 0.1000000…… 0.000000…… 0.0110000…… 0.100000…… 0.0100000…… 1.000000…… 0.0010000…… 1.100000…… 0.0000000…… 10.000000…… A rational number with a denominator not a power of two has a nonzero repeating pattern and is not in a set of truncated numbers. LocalZero, “AminA0”, is the unknown and unknowable placevalue digits smaller than a truncated number. •
Zeros before the decimal point. Zeros after the decimal point to a finite count of “Lmax” (in base two). “AminA0 < 2^Lmax”
•
“d = b  b” for each placevalue digit after a count of “Lmax” after the decimal point (in base two) AminA0 = 0.000…(zeros to a count of Lmax)…000ddddd…… AminA0 = 0.000dddd…… for Lmax = 3 minA A 0 = 0.dddd…… for Lmax = 0
194 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY “b” represents “box” (or “both”). We do not know if the box contains “0” or “1” until opened. A box is both “0” and “1” and is not “0” or “1” separately. “b” is the mathematical analogy for the box containing Schrödinger’s Cat. Until we open the box, we do not know if Schrödinger’s Cat is dead (1) or alive (0). Inside the box is a radioactive nucleus. If the nucleus decays, then the emitted particle breaks a container of poison and the cat dies. The decay both happened (state = “1”) and did not happen (state = “0”) until observed by the box being opened. That is the condition of each “b”. “b” is both one and zero, and “b” is neitherseparately one nor zero until the placevalue digit is observed, after which it becomes a one or else a zero. Each placevalue digit symbol “b” is a separate box that is opened independently of any other box. (Notice the time dependency.) “AminA0 = 0.dddd……” is the result of “0.bbbbb……” subtracted from “0.bbbbb……”. Both are positive, therefore, difference “AminA0 = 0.bbbbb……  0.bbbbb……” can be either positive or negative, depending on the first nonzero value of “d” in the sequence. The subscript of “A” indicates it represents the concept of positive and negative, both and neitherseparately. “minA0” has “min” to indicate it is smaller than truncated numbers (“AminA0 < 2^Lmax”). LocalInfinity. There is “0.25” chance a “d” will be “1 (= 0  1)”, “0 (= 0 0)”, “0 (= 1  1)”, or “1 = (1  0)”. Therefore, “0.5 = 1/2” chance the first “d” will become zero “0” and “0.52 = 1/(22) = 0.25” chance, both it and the next “d” will become zero. The quantity of “d”’s has no end, for “1/(21/0)”, which equals either “0” or “1/0” using the proposed new reciprocalofzero axiom. “0” is the relevant solution. Because a localzero cannot equal integer zero, a division reciprocal, called a localinfinity, “AmaxA = 1/AminA0”, exists. It is larger in magnitude than a truncated number (“AmaxA > 2^Lmax”). In the general power series given above, it is the six dots on the left.
195 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS LocalReal Numbers. A localreal number consists of a truncated number added to imprecision term “” (xi), which is either localzero or else localinfinity. “C” is used because “A” pertains to “” being either “CmaxC” (“A” “+”) or else “CminC0” (“A” “”). Continuum quantities in the geometrically real world are localreal numbers. = CmaxC or else CminC0 Two physically real quantities cannot be equal. A geometric continuum quantity, for example, length, subtracted from a geometrically independent version of itself, does not equal integer zero, and that means the four sides of a unit length square are not the ideal of integer one, but, rather, each have a localzero adder for finite imprecision. And, a “2” diagonal is not perfectly “2” because it has a localzero adder. The localzeros added to each of two sides of a unit length square are proven to be different: There is a finite chance the first “d” placevalue digit will be the same between the two localzeros, and that quantity squared both it and the next “d” will be the same, and zero chance all the “d”’s will be the same. A “1” followed by a “1” has the same result as a “0” followed by a “1”, and that is accounted for. Continuum. There is an even chance a localreal number with placevalue digits “d” randomly selected to be “1”, “0”, “0”, or “1” will become any (irrational) number. Because there is an even chance, the localreal numbers form a “continuum”. This is a different definition of “continuum” when compared to the continuum in Cantor’s theory of infinite sets for real numbers, but the concept is the same, and, therefore, the same word “continuum” is used. No Positive Actual Infinity. Localreal numbers have been defined without use of positive actual infinity. Positive actual infinity, as a count of placevalue digits after the decimal point, was proposed by Cantor as required for real numbers. Therefore, localreal numbers are different from real numbers.
196 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
5.2 Cantor’s Theory of Infinite Sets Cantor made the formal assumption that a positive infinity was the quantity of members in the set of natural numbers and a larger positive infinity was the quantity of members in the set of real numbers. The two infinities were forced to be actual infinities and not large finite number potential infinities because, per his Continuum Hypothesis, no set can have a quantity of members between the two infinities. Cantor proposed his theory of infinite sets late in the 1800’s, several years before 1905, when Einstein developed his theory of Special Relativity. Special Relativity, like every other mathematical model of physics, does not directly use Cantor’s two infinities. The disconnect between Cantor’s infinities and Special Relativity prompted the author’s search for a proper replacement in applied mathematics for positive actual infinity. Cantor’s infinity theory was critically important. •
It formed the foundation for evolving more theory on how numbers are constructed
•
It showed the importance of defining numbers using placevalue digits as opposed to ratios (for rational numbers) or nebulously (for irrational numbers)
•
It used base two to an exponent, and that suggested a switch to base two placevalue digit nomenclature
•
It placed positive actual infinity into a theory against which alternative theories of infinity could be contrasted
•
It generated plenty of literature so that several perspectives on the theory could be found by reading, rather than by inventing
•
It became an axiom in set theory, which suggested an alternative to Cantor’s Continuum Hypothesis would be an alternative axiom
•
It lacked an algebra by which to calculate with positive actual infinities
197 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS •
It was absent from mathematical models of physics and that meant it probably wasn’t real and highlighted that an improvement in the theory of infinity was needed and was possible
Cantor’s Theory of Infinite Sets – Countable Sets. Positive actual infinity “N0” (aleph null) is the quantity of members in “N”. N = {1, 2, 3, …, N0} Cantor counted integers to prove the set of integers “Z” has the same quantity “N0” as does the set of natural numbers “N”: “0” was first, “1” was second, “1”, was third, “2” was fourth, “2” was fifth, etc. The person reading the proof must extrapolate from a finite count to an actual infinity quantity. The word “countable” means members of the set can be identified onetoone with the set of natural numbers. Examples of countable sets are integers, prime numbers, rational numbers (per Cantor’s diagonal proof), squares of natural numbers, even numbers, and products of two integers. The set of real numbers is not “countable”. Dedekind Cut. The Dedekind Cut was the precedent to Cantor’s theory of infinite sets. The Dedekind Cut is best described by Dedekind himself in a quote from Essays on the Theory of Numbers by Richard Dedekind, 1963 (originally 1901) by Dover Publications, Inc. Page 15: “From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.” If the cut is at a rational number, then the high side number is the rational number and the low side number is infinitely close, but just less. Alternatively, if the cut is at an irrational number, then both the high side and low side numbers are infinitely close to each other, and the irrational number is between them. Per the definition of the Dedekind Cut, we ignore the removed material of a saw cut, and we pretend the cut is a scissors cut (which does not remove material) by making the two numbers “essentially” the same number.
198 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY A true scissors cut would be an idealized cut because high and low numbers differ by integer zero. It appears Dedekind was attempting to make the cut nonidealized, as if the cut were physical in our geometric world where the numberline exists as an actual line. To visualize the Dedekind cut, use “Lmax = 3” for the count of base two placevalue digits after the decimal point. A Dedekind Cut at half (“.100” in base two) has “.100” on the high side and “.100  2^Lmax = .100  .001 = .011” on the low side. To satisfy the word “essentially”, the difference “2^Lmax” must be smaller than a rational number, and that requires “Lmax” to be a positive actual infinity, “N0”, for “2^ N0”. This contrasts with the idealized extreme case of “Lmax = 1/0” for which “2^(1/0) = 0” and both numbers are “.100”. A Dedekind Cut at irrational number squarerootoftwo “2 = 1.4142135……” has high/low number pairs at the cut (1.4, 1.5), (1.41, 1.42), (1.414, 1.415), (1.4142, 1.4143), etc., for the largest count number “Lmax” increasing through “1”, “2”, “3”, “4”, etc., respectively. When “Lmax” reaches an actual infinity “N0”, then the high and low numbers and all numbers between them, including the irrational number squarerootoftwo, are essentially the same number, because the span between the low and high numbers is the positive infinitesimal “(10^N0) > 0”. If sequential rational numbers “1/Lmax” and “1/(Lmax  1)” differ by “1/(Lmax*(Lmax  1))”, then a difference “2^(Lmax)” of sequential real numbers is smaller, and more so if “2^(Lmax)” becomes “2^N0”. Uncountable Sets. “Cardinality” of a set is the quantity of members in the set. Cardinality is given operator notation of two “” lines. A textbook for Cantor’s theory of infinite sets is Mathematical Proofs, A Transition to Advanced Mathematics, Second Edition by Chartrand, Polimeni, and Zhang, Pearson Addison Wesley, 2008, beginning on Page 221. From the top of Page 236: “Indeed, if A is any denumerable set, then A = N0. The set R of real numbers is also referred to as the continuum and its cardinality is denoted by c. Hence R = c and from what we have seen, N0 < c. It was the German mathematician Georg Cantor who helped to put the theory of sets on a firm foundation.” Cantor proved “N0 < c” using quantity “c” real numbers spanning zero (inclusive) to one (exclusive) and using quantity “N0” of placevalue
199 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS digits after the decimal point. He created several different real numbers by writing random placevalue digits after the decimal point. 0.5928609813… 0.5232… 0.7290165316… 0.3991… 0.7831994831… 0.8482401809… Cantor took the first placevalue digit from the first number, second from the next, and so on, to form a new number. And he varied the process to create other new numbers. Because new numbers could be formed by this process, he concluded “N0 < c”. The person must extrapolate from finite to infinite. As a refinement to “N0 < c”, Cantor proposed “c = 2^N0”. To visualize “c = 2^N0”, substitute “Lmax = 3” for “N0”. The set of three members “{.1, .01, .001}” implies eight sets per “8 = 23”: {}, {.1}, {.01}, {.001}, {.1, .01}, {.1, .001}, {.01, .001}, {.1, .01, .001} There are “2 = 2^1” numbers “0.0” and “0.1” if we consider only the first placevalue digit after the decimal point. “4 = 2^2” numbers “0.00”, “0.01”, “0.10” and “0.11” for the first two. “8 = 2^3” as given above for the first three. “2^Lmax” numbers for the first “Lmax”. This visualization requires base two, and that’s why base two is in “c = 2^N0”. “c = 2^N0” as a quantity of real numbers applies to the span from negative infinity “N0” to positive infinity “N0”, in analogy to “N0” applying to any countable set. Properties of the two Actual Infinities. From Cantor’s proofs, we identify the following properties for “N0” and “c”. Finite numbers satisfy these properties. •
“N0” and “c” have no contribution after the decimal point, in analogy to natural numbers used in counting
•
“N0” and “c” are positive, also in analogy to natural numbers used in counting, and as required by “N0 < c” and “c = 2^N0”
•
“N0 < c” per Cantor’s proof, and “c = 2^N0”
200 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Perhaps these other properties also apply: •
“c” is an even number because “c = 2^N0”, by furthering the analogy of Cantor’s infinities with finite numbers
•
“0 < N0 < c < 1/0”, “N0 ≠ 1/0” and “c ≠ 1/0” if we accept the proposed new reciprocalofzero axiom
Attempt to place a “1” at the “N0”th position before the decimal point by writing an everincreasing string of zeros to form “c = 2^N0”. Per this visualization, “c = 2^N0” cannot be negative and therefore cannot equal “1/0”. Analogously, “1/c = 2^N0” cannot equal integer zero. Cantor’s Continuum Hypothesis. Included in Cantor’s theory of infinite sets is the conjecture that there is no set “S” for which N0 < S < c Inequality “N0 < S < c” states the quantity of members in a set may equal “N0”, as applies to countable sets, and may equal “c”, as applies to uncountable sets, but the quantity of members in a set cannot be “S” between “N0” and “c”. There cannot be a count up from “N0” toward “c” because any number in a count is the maximum number in the set of numbers from one to that number. There is no “N0 + 1”, “2*N0”, “c/2”, or “c  1”. The Continuum Hypothesis (that “N0 < S < c” is impossible) cannot apply if “N0” and/or “c” are finite. Therefore, it is the Continuum Hypothesis that creates mathematically the property of positive actual infinity for “N0” and for “c”. And it is the Continuum Hypothesis that justifies multisetapplicability of “N0” and of “c”. The Continuum Hypothesis is only a conjecture because it has not been proven. Attempts have been made. In the referenced textbook (middle of page 236) is a description of the attempts: “However, in 1931 the Austrian mathematician Kurt Gödel proved that it was impossible to disprove the Continuum Hypothesis from the axioms on which the theory of sets is based. In 1963 the American mathematician Paul Cohen took it one step further by showing that it was
201 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS also impossible to prove the Continuum Hypothesis from these axioms. Thus the Continuum Hypothesis is independent of the axioms of set theory.” An almost identical quote is given on Page 137 of THE
PHILOSOPHY OF SET THEORY, An Historical Introduction to Cantor’s Paradise, by Mary Tiles, from Dover Publications, Inc, 2004 (Originally
1989 from Basil Blackwell Ltd.). A very readable summary of the effect the Continuum Hypothesis has had on axiomatic set theory is given in the article “Dispute over Infinity Divides Mathematicians – To determine the nature of infinity, mathematicians face a choice between two new logical axioms. What they decide could help shape the future of mathematical truth” by Natalie Wolchover, Quanta Magazine, December 3, 2013. The “axioms of set theory” are listed in Mary Tiles’ book, Pages 121123.
Axiom of extensionality Null set axiom Pair set axiom Sum set axiom Axiom of infinity
Axiom of foundation Subset axiom and replacement axiom Power set axiom Power Axiom of choice
Per page 125, the axiom of infinity identifies the unboundedness of natural numbers because one can always be added. Unboundedness pertains to the finite potential infinity (defined per Aristotle) and not to the completed (defined per Aristotle) or actual infinity “N0”. Rather, “N0” is created when the Continuum Hypothesis is used as an axiom, as is suggested as okay by the two proofs (by Gödel and by Cohen). The book and article emphasize that axiomatic set theory with the actual infinity forms “holes” that pure mathematicians attempt to close. Applying Binary Operations to an Actual Infinity. If calculation “c = 2^N0” is accepted, then, it follows “N0 = ln(c)/ln(2)” should be, too. But equations such as “N0*ln(2) + = 1 + 1/2 + 1/3 + … + 1/(c  1)” require a violation of the Continuum Hypothesis and suggest any attempt at an algebra will fail. In the tables below, binary operations consistent with the Continuum Hypothesis are on the left, and binary operations consistent
202 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY with finite numbers are on the right. If, on the rightside tables, finite numbers “N0” and “c” (“C”) are both replaced by “1/0”, then the crude nonnumber algebra applies. + 0 Q N0 C
0 0 Q N0 C
Q Q 2*Q N0 C
N0 N0 N0 N0 c
C C C C C
+ 0 Q N0 c
0 0 Q N0 c
Q Q 2*Q N0+Q c+Q
N0 N0 N0+Q 2*N0 c+N0
c c c+Q c+N0 2*c
* 0 Q N0 C
0 0 0 0 0
Q 0 Q*Q N0 C
N0 0 N0 N0 c
C 0 C C C
* 0 Q N0 c
0 0 0 0 0
Q 0 Q*Q N0*Q c*Q
N0 0 N0*Q N0^2 c*N0
c 0 c*Q c*N0 c^2
^ 0 2 N0 c
0 1 1 1 1
2 0 2^2 N0 C
N0 0 c ? ?
C 0 2^c ? ?
^ 0 2 N0 c
0 1 1 1 1
2 0 2^2 N0^2 c^2
N0 0 2^N0 N0^N0 c^N0
c 0 2^c N0^c c^c
Table 5. Comparison of binary operations. Cantor’s Continuum Hypothesis is on the left. In contrast, actual infinities being finite are on the right (“Q Q”).
The table does not include inverse operations because “log2(N0)” (in analogy to “log2(c) = N0”) does not seem to have a result. Tables on the left for actual infinities are not sufficient for calculations. An algebra for actual infinities might exist, but we presently do not know rules of that algebra. Higher Order Actual Infinities. Cantor replaced “c” with “N1” (aleph one) because “N1 = 2^N0” suggests “N2 = 2^N1”, “N3 = 2^N2”, …. Is there an actual infinity subscript? Perhaps that question illustrates the holes.
203 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS Incompatibility of “N0” with Irrational Numbers. If “2^c” is even and “3^c” is odd, then “log23” is not in the set of real numbers. That result was expected because, in the discussion on the Dedekind cut, an irrational number resided inside a “2^N0” interval, and the Dedekind cut appears to be the basis of Cantor’s “c = 2^N0” real numbers over an interval. Because an irrational number resides inside a “1/c” interval, “c*log23 ≠ c”, by which is stated a positive number “c” (without contribution after the decimal point) cannot be multiplied by “log23” to equal any other “c” (also without contribution after the decimal point), and that’s because the nonpattern for “log23” cannot end at a positive quantity “N0” of placevalue digits (per the proposed new reciprocalofzero axiom by which “log23” has more placevalue digits after the decimal point than “c” has before the decimal point). Because irrational numbers are removed from real numbers, infinities “N0” and “c” lose relevance and the Continuum Hypothesis does not apply. From another perspective: Because “N0” and “c” are positive, we can speculate “0*N0 = 0” and “0*c = 0”, and (as if they are finite) 1/0 + N0 = 1/0 + 0*N0/0 ; 1/0 + c = 1/0 + 0*c/0 = (1 + 0*N0)/0 = (1 + 0)/0 = 1/0 = (1 + 0*c)/0 = (1 + 0)/0 = 1/0 Replacing Real Numbers with LocalReal Numbers. In our geometric world of the numberline, there is no “1/0”, and so no irrational numbers, and, per the argument above, there is no positive actual infinity “N0”, and so no real numbers, either. What remains is the finite potential infinity “Lmax” and the proposed localreal numbers. But, to dispose of positive actual infinity “N0”, its mysteriousness had to be tamed by defining it using finite number properties, after which it was proven incompatible with a proposed new axiom. The properties and even the axiom could be oversimplified or otherwise in error, and so, it’s inappropriate to celebrate. Instead, we honor Cantor because his innovations provided necessary historical precedent. And, most humbling, we expect another new theory to come along soon because the proposed new axiom intentionally ignored identified complexity at infinity (see a “Further Thought” question) and did so per the excuse we only needed enough selfconsistency for some fairly mundane applied mathematics on nearly hundred year old physics.
204 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Infinity in Mathematical Models of Physics. Finite imprecision of localreal numbers applies where a localinfinity replaces a division by zero singularity in a mathematical model of physics. Motion at the speedoflight is a divisionbyzero singularity in Special Relativity. But first we need to place the localinfinity into an algebra field, or, at least, have defined enough properties to use it.
5.3 Algebra Field for LocalReal Numbers FinishedCalculation and FinalResult. Localreal numbers with finite precision pertain to the geometric world of continuum quantities in which the problem to be solved is set up. The problem is then translated into all number algebra after which operations are not inhibited by finite precision. In preparation for a measurement, the result is translated back into geometry and restricted to finite precision. •
FinishedCalculation is at the end of the allnumber second step in the Process from Descartes
•
FinalResult. Impose finite precision to find the geometric “finalresult” in the third step in the Process from Descartes
For example: “AminA0” times “AminA0” has “Lmax” zeros after the decimal point in each of the two factors and has “2*Lmax + 1” zeros in the positive product as the finishedcalculation. Translation into a finalresult limits the finishedcalculation to “Lmax” zeros and is written as another localzero “CminC0”. Probability Functions. Swap “maximum” for “minimum” for “Lmax = 0”. AmaxA0 = 0.dddd……
AminA = 1/AmaxA0
Probability Distribution for “AmaxA0”. For “s = AmaxA0”, the calculated probability distribution “dPmax0/ds” is given below.
205 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS dPmax0/ds = 0 dPmax0/ds = s + 1 dPmax0/ds = 1  s dPmax0/ds = 0
Pmax0 = 0 Pmax0 = (s + 1)2/2 Pmax0 = (1 + 2*s  s2)/2 Pmax0 = 1
for s < 1 for 1 < s < 0 for 0 < s < 1 for 1 < s
Probability Curve for the LocalInfinity. “s = 1/AmaxA0” dPmin/ds(s) = s2 + s3 dPmin/ds(s) = 0 dPmin/ds(s) = s2  s3
Pmin(s) = s1  s2/2 Pmin(s) = 1/2 Pmin(s) = 1  s1 + s2/2
Probability Curve for the Quotient. “s = AminA0/BminB0” Pmin0/min0 = 1/(3*s)  1/(12*s2) Pmin0/min0 = 1/2 + s/3 + s2/12 Pmin0/min0 = 1/2 + s/3  s2/12 Pmin0/min0 = 1  1/(3*s) + 1/(12*s2)
for s < 1 for 1 < s < 0 for 0 < s < 1 for 1 < s
dPmin0/min0/ds = 1/(3*s2) + 1/(6*s3) dPmin0/min0/ds = 1/3 + s/6 dPmin0/min0/ds = 1/3  s/6 dPmin0/min0/ds = 1/(3*s2)  1/(6*s3)
for s < 1 for 1 < s < 0 for 0 < s < 1 for 1 < s
Figure 35. Probability Distributions.
for  < s < 1 for 1 < s < 1 for 1 < s <
206 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 36. Right side is Probability Distribution for a ratio of two localzeros, “s = AminA0/BminB0”.
Base Two Exponential Function. Assuming only a real number result, “2^(AmaxA)” is calculated by counting zeros as placevalue digits left and right of the decimal point, in base two: “ > 2^Lmax” (lambda) for “A” “+”, and “1/ < 2^Lmax” for “A” “”, both outside truncated numbers. 2^(AmaxA) = *(1/2 + A1/2) + (1/)*(1/2  A1/2) The finalresult of “2^(AmaxA)” is truncated number zero added to imprecision term “” (xi). “” equals either localinfinity “CmaxC” or else localzero “CminC0 = 1/CmaxC” depending on “A” being “+” or “A” “”, respectively. A localreal number is a truncated number with “” added to it. Also: exp(AmaxA) = ()*(1/2 + A1/2) + (1/)*(1/2  A1/2)
207 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS Imprecision Term “” for Special Relativity. “αM” of “αM = atanh(vM/c)” is a truncated number to which “” is added using a Lorentz Transformation. A hyperbolicradius is also a truncated number to which “” is added. Both are included in “αS/M” using the “(1  qx)” factor. αS/M = (1  qx)* = (1  qx)*CmaxC or else = (1  qx)*CminC0
First Case Second Case
First Case pertains to “CmaxC”. For “C” “+”, “CmaxC” becomes “calcC”. exp(qx*αS/M) = exp(qx*(1  qx)*) = exp(qx*(1  qx)*CmaxC) First Case = exp(qx*(1  qx)*calcC) “C” positive, “+” = exp((1 + qx)*calcC) = exp(calcC)*exp(qx*calcC) = exp(calcC)*(cosh(calcC) + qx*sinh(calcC)) = exp(calcC)*(exp(calcC)  sinh(calcC) + qx*sinh(calcC)) = exp(calcC)*(exp(calcC)  (1  qx)*sinh(calcC)) = 1  (1  qx)*exp(calcC)*sinh(calcC) = 1  (1  qx)*exp(calcC)*(exp(calcC)  exp(calcC))/2 = 1  (1  qx  (1  qx)*exp2(calcC))/2 = (1 + qx + (1  qx)*exp2(calcC))/2 = (1 + qx)/2 + (1  qx)*exp2(calcC)/2 (1 + qx)/2 “exp(qx*(1  qx)*calcC) (1 + qx)/2” conforms to an exponent with a singular matrix having unit magnitude result. Division Reciprocal (which is, effectively, “C” negative, “”): 1/exp(qx*αS/M) = exp(qx*αS/M) = exp(qx*(1  qx)*) = exp(qx*(1  qx)*calcC) = exp((1  qx)*calcC) = exp(calcC)*exp(qx*calcC) = exp(calcC)*(cosh(calcC) + qx*sinh(calcC)) = exp(calcC)*(exp(calcC)  sinh(calcC) + qx*sinh(calcC)) = exp(calcC)*(exp(calcC)  (1  qx)*sinh(calcC))
208 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY = 1  (1  qx)*exp(calcC)*sinh(calcC) = 1  (1  qx)*exp(calcC)*(exp(calcC)  exp(calcC))/2 = 1  (1  qx  (1  qx)*exp2(calcC))/2 = (1 + qx)/2 + (1  qx)*exp2(calcC)/2 (1  qx)*exp2(calcC)/2 “exp(qx*(1  qx)*calcC) (1  qx)*exp2(calcC)/2” is reciprocal of the portion of “exp(qx*(1  qx)*calcC)” that deviated away from “(1 + qx)/2”. “((1 + qx)/2)*exp(qx*α) = ((1 + qx)/2)*exp(α)” applied. Second Case pertains to “CminC0”. For “C” “+”, “CminC0” becomes “0calcC”. exp(qx*αS/M) = exp(qx*(1  qx)*) = exp(qx*(1  qx)*CminC0) Second Case = exp(qx*(1  qx)*0calcC) “C” positive, “+” = exp((1 + qx)*0calcC) = exp(0calcC)*exp(qx*0calcC) = exp(0calcC)*(cosh(0calcC) + qx*sinh(0calcC)) = exp(0calcC)*(exp(0calcC)  sinh(0calcC) + qx*sinh(0calcC)) = exp(0calcC)*(exp(0calcC)  (1  qx)*sinh(0calcC)) = 1  (1  qx)*exp(0calcC)*sinh(0calcC) 1  (1  qx)*(1  0calcC)*0calcC 1 Division Reciprocal (which is, effectively, “C” negative, “”): 1/exp(qx*αS/M) = exp(qx*αS/M) = exp(qx*(1  qx)*) = exp(qx*(1  qx)*0calcC) = exp((1  qx)*0calcC) = exp(0calcC)*exp(qx*0calcC) = exp(0calcC)*(cosh(0calcC) + qx*sinh(0calcC)) = exp(0calcC)*(exp(0calcC)  sinh(0calcC) + qx*sinh(0calcC)) = exp(0calcC)*(exp(0calcC)  (1  qx)*sinh(0calcC)) = 1  (1  qx)*exp(0calcC)*sinh(0calcC) 1  (1  qx)*(1 + 0calcC)*(0calcC) 1 “exp(qx*αS/M) 1” and “1/exp(qx*αS/M) 1”.
209 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS
Fields and Waves use a Lorentz Transformation with division by two in the argument. First Case FinishedCalculation: exp(AmaxA) = calcC “C” “+” exp(qx*αS/M/2) = exp(qx*(1  qx)*calcB/2) First Case, “C” “+” = (1 + qx)/2 + (1  qx)*exp2(calcC/2)/2 (1 + qx)/2 1/exp(qx*αS/M/2) = exp(qx*(1  qx)*calcC/2) = (1 + qx)/2 + (1  qx)*exp2(calcC/2)/2 (1  qx)*exp2(calcC/2)/2 exp(qx*αS/M/2) = exp(qx*(1  qx)*0calcC/2) Second Case, “C” “+” = 1  (1  qx)*exp(0calcC/2)*sinh(0calcC/2) 1  (1  qx)*(1  0calcC/2)*0calcC/2 1 1/exp(qx*αS/M/2) = exp(qx*(1  qx)*0calcC/2) Second Case, “C” “+” = 1  (1  qx)*exp(0calcC/2)*sinh(0calcC/2) 1 + (1  qx)*(1 + 0calcC/2)*(0calcC/2) 1 Criteria of a Group. FinishedCalculations use group theory as it pertains to rational numbers. Inverse operations do not apply to the finalresult because of deletion of the portion of the finishedcalculation that was smaller and larger than truncated numbers.
5.4 Lorentz Transformation with NonFinite Numbers Lorentz Transformation for the proposed Theory of Special Relativity with NonFinite Numbers. Imprecision term “” is added to truncated number “αM” using “αS/M = (1  qx)*” in “αS = αM + αS/M”. The “(1  qx)” factor makes First Case components finite. αS/M = (1  qx)*
;
vS/M/c = tanhαS/M = tanh((1  qx)*)
210 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY vS/MFirstCase/c = tanh((1  qx)*calcC) First Case, “C” “+” = (tanh(calcC)  qx*tanh(calcC))/(1  qx*tanh(calcC)*tanh(calcC)) = ((1  qx)*tanh(calcC))/(1  qx*tanh(calcC)*tanh(calcC)) (1  qx)*(1  exp2(calcC))/(1  qx*(1  exp2(calcC))2) (1  qx)*(1  exp2(calcC))/(1  qx*(1  2*exp2(calcC))) 1 “vS/MFirstCase/c” equals one minus a number smaller in magnitude than the smallest positive truncated number “2^Lmax” so that the finalresult equals one plus a localzero. It means the First Case bus “M” moves at the speedoflight. Our visualization is the photon. vS/MSecondCase/c = tanh((1  qx)*0calcC) Second Case, “C” “+” = (tanh(0calcC)  qx*tanh(0calcC))/(1  qx*tanh(0calcC)*tanh(0calcC)) = (1  qx)*tanh(0calcC)/(1  qx*tanh(0calcC)*tanh(0calcC)) (1  qx)*0calcC/(1  qx*0calcC*0calcC) 0 “vS/MSecondCase/c” finalresult equals a localzero for no motion. The bus is not moving. Our visualization is the electron. General Form of the Lorentz Transformation. 2r
= 1M*(c*tB)*exp(qx*αM) = 1M*(c*tB)*exp(qx*αM)*1 = 1M*(c*tB)*exp(qx*αM)*exp(qx*αS/M)/exp(qx*αS/M) = 1M*(c*tB)*exp(qx*αM) *exp(qx*(1  qx)*)/exp(qx*(1  qx)*) = (1M/exp(qx*(1  qx)*)) *(c*tB)*exp(qx*αM)*exp(qx*(1  qx)*) = 1S*(c*tB)*exp(qx*αS) = 1S*(c*tS + qx*xS)
1S = 1M/exp(qx*(1  qx)*) (c*tB)*exp(qx*αS) = (c*tB)*exp(qx*αM)*exp(qx*(1  qx)*)
211 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS Component Transformations. standard process.
“c*tS” and “xS” are found using the
c*tS + qx*xS = (c*tB)*exp(qx*αM)*exp(qx*(1  qx)*) First Case FinishedCalculation: = calcC (with “C” “+”) c*tS + qx*xS = (c*tB)*exp(qx*αM)*exp(qx*αS/M) = (c*tB)*exp(qx*αM)*exp(qx*(1  qx)*) = (c*tB)*exp(qx*αM)*exp(qx*(1  qx)*calcC) First Case and “C” “+” 2 = (c*tB)*exp(qx*αM)*((1 + qx)/2 + (1  qx)*exp (calcC)/2) = (c*tB)*(exp(qx*αM)*(1 + qx)/2 + exp(qx*αM)*(1  qx)*exp2(calcC)/2) = (c*tB)*(exp(αM)*(1 + qx)/2 + exp(αM)*(1  qx)*exp2(calcC)/2) c*tS = (c*tB)*(exp(αM)/2 + exp(αM)*exp2(calcB)/2) First Case (c*tB)*exp(αM)/2 xS = (c*tB)*(exp(αM)/2  exp(αM)*exp2(calcB)/2) (c*tB)*exp(αM)/2
First Case
“c*tS xS” because “c*tS” and “xS” have the same finite truncated number portion “(c*tB)*exp(αM)/2” in the finished calculation, and that is because hyperbolicradius “(c*tB)*exp(calcC)” is small, effectively zero, and the other factor “exp(qx*calcC)” is large (infinite), to compensate. Roadside observer “S” measures time “c*tS” when an object of speed “vM” inside a bus passes location “xS”. For the First Case, the bus travels at speedoflight, “vS/MFirstCase = c” and per “c*tS (c*tB)*exp(αM)/2”, measured time “tS” increases proportionally to the advancement of time “tB” on the object. For example, if “αM = 0”, then “c*tS c*tB/2”. Observing an Object that Moves at the SpeedofLight. c*tS (c*tB)*exp(αM)/2 ; = (c*tB)*(coshαM + sinhαM)/2 = (c*tM + xM)/2
xS (c*tB)*exp(αM)/2 = (c*tB)*(coshαM + sinhαM)/2 = (c*tM + xM)/2
212 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY If “xM” identifies the back of a baseball and “xM + xM” identifies the front of a baseball, then, because “(c*tM + xM)back” equals “(c*tM + xM)front”, front is earlier than back by “xM/c”. c*tS = xS = (c*tM + xM)back/2 = (c*tM + xM)front/2 c*tMfront = c*tMback  (xMfront  xMback) = c*tMback  xM All locations in “M” back to front along the baseball are at one location “xS” at one time “c*tS” because each location is at a different “tM” time, per length contraction. Time “tB” displayed on the clock mounted on the moving object (baseball) is not stopped when observed from “S”. This special feature of the mathematical model contrasts with our traditional expectation that there is no passage of time on a photon. Second Case FinishedCalculation: = 0calcC (with “C” “+”) c*tS + qx*xS = (c*tB)*exp(qx*αM)*exp(qx*(1  qx)*) = (c*tB)*exp(qx*αM)*exp(qx*(1  qx)*0calcC) Second Case and “C” “+” = (c*tB)*exp(qx*αM)*(1  (1  qx)*exp(0calcC)*sinh(0calcC)) = (c*tB)*exp(qx*αM)  (c*tB)*exp(qx*αM)*(1  qx)*exp(0calcC)*sinh(0calcC) = (c*tB)*exp(qx*αM)  (c*tB)*exp(αM)*(1  qx)*exp(0calcC)*sinh(0calcC) (c*tB)*exp(qx*αM)  (c*tB)*exp(αM)*(1  qx)*(1  0calcC)*0calcC (c*tB)*exp(qx*αM) Second Case c*tS = c*tM  (c*tB)*exp((αM + 0calcC)*sinh(0calcC) c*tM  (c*tB)*exp((αM + 0calcC)*0calcC c*tM Second Case xS = xM + (c*tB)*exp((αM + 0calcC)*sinh(0calcC) xM + (c*tB)*exp((αM + 0calcC)*0calcC xM Second Case For the Second Case, components measured in “S” are the same as components measured in “M”. There is only one roadside “S” where the observer stands. Also, there is only one bus “M” where a different observer sits. There are two
213 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS “S/M” speeds for the bus: “S/MFirstCase” and “S/MSecondCase”. Because of the two “S/M” speeds, there are two separate observations from the one roadside “S” of the one bus “M”. CompoundLabelNumbers. 1S = 1M/exp(qx*αS/M) = 1M*exp(qx*αS/M) = 1M*exp(qx*(1  qx)*) First Case FinishedCalculation: = calcC (with “C” “+”) 1S = 1M*exp(qx*(1  qx)*) = 1M*exp(qx*(1  qx)*calcC) First Case with “C” “+” 2 = 1M*((1 + qx)/2 + (1  qx)*exp (calcC)/2) = 1M*(1 + qx)/2 + 1M*(1  qx)*exp2(calcC)/2 1M*(1  qx)*exp2(calcC)/2 qxS = qx*1S = qx*1M*(1 + qx)/2 + qx*1M*(1  qx)*exp2(calcC)/2 = 1M*(1 + qx)/2  1M*(1  qx)*exp2(calcC)/2 1M*(1  qx)*exp2(calcC)/2 = 1S Translation from First Case finishedcalculation “1S” to finalresult “itS” has “exp2(calcC)/2” replaced by a localinfinity. And, for the First Case, “ixS = itS”. Second Case FinishedCalculation: = 0calcC (with “C” “+”) 1S = 1M*exp(qx*(1  qx)*) = 1M*exp(qx*(1  qx)*0calcC) Second Case with “C” “+” = 1M*(1  (1  qx)*exp(0calcC)*sinh(0calcC)) 1M*(1  (1  qx)*(1 + 0calcC)*(0calcC)) 1M qxS = qx*1S qxM
214 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Relative to a measurement (that is, when translated into geometry), the Second Case compoundlabelnumbers create the same geometricunitvectors for “S” as for “M”. Two Locations at One Time. Two people stand on roadside “S”. A person at the speed limit sign observed the Second Case at the same time a person down the road observed the First Case. Per the proposed theory, the two observations are of the same particle because the photon is a projection of the electron. If we think in terms of geometric reality, our traditional notion is that one particle cannot be in two places. But, per this math, it’s true. The suggestion is that numbers alone are fundamental, and not geometric reality. Step three is not a reversion to fundamental physics, but is only preparation for a measurement. If “C” is negative “”, then First Case (“ = CmaxC”) component magnitudes are infinite and compoundlabelnumbers are finite. There might not be theoretical significance to “C” “” if “” is the exclusively positive finishedcalculation “*(1/2 + A1/2) + (1/)*(1/2  A1/2)” rather than “(CmaxC)*(1/2 + A1/2) + (CminC0)*(1/2  A1/2)”. Other General Form. 2r
= 1M*(c*tB)*exp(qx*αM)*exp(qx*αS/M)/exp(qx*αS/M) = 1M*T*exp(ln(c*tB/T)*exp(qx*αM)*exp(qx*αS/M)/exp(qx*αS/M) = 1S*T*exp(ln(c*tB/T)*exp(qx*αM)*exp(qx*(1  qx)*) = 1S*T*exp(ln(c*tB/T)  )*exp(qx*(αM + )) = 1S*T*exp(ln(c*tB/T))*exp(qx*αM)*exp()*exp(qx*) = 1S*T*exp(ln(c*tB/T))*exp(qx*αM)*(cosh()/exp() + qx*sinh()/exp()) = 1S*(c*tB)*exp(qx*αM)*(cosh()/exp() + qx*sinh()/exp()) = 1S*(c*tB)*exp(qx*αM)*((1 + exp2()) + qx*(1  exp2())/2 = 1S*(c*tB)*exp(qx*αM)*((1 + qx)/2 + (1  qx)*exp2()/2)
Other Invariants. Other invariants also have a First Case.
215 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS Location (timelike) Location (spacelike)
2r
= 1S*c*tB*exp(qx*αS) 2s = 1S*qx*sxB*exp(qx*αS)
Frequency (timelike) Wavenumber (spacelike)
2/c
EnergyMomentum Charge Density
2p
= 1S*(B/c)*exp(qx*αS) 2k = 1S*qx*kxB*exp(qx*αS) = 1S*mB*c*exp(qx*αS) 2J = 1S*B*exp(qx*αS)
Wavenumber and Frequency Observed at the SpeedofLight. On the “2r” hypercomplexplane “2/c = 1B*(B/c)” is plotted as stationary horizontal evenly spaced parallel “wave crest” lines that extend left and right for all of “xB” space. Lines are closer for higher frequency “B”. As a visualization, two long rods along the floor of the bus, front to back, bounce side to side to make a bang sound everyone seated on the bus hears. The bang sound is the horizontal lines. And people standing along the roadside raise their hands in unison for each bang. Then, the bus moves forward with “αS/M > 0”. 2/c
= 1S*(B/c)*exp(qx*αS/M) = 1S*(B/c)*cosh(αS/M) + qxS*(B/c)*sinh(αS/M) = 1S*(S/c) + qxS*kxS
Because the bus is moving, wave crest lines slope up and to the right and are spaced closer together by factor “cosh(αS/M)” (because of time dilation and “S/c = (B/c)*cosh(αS/M)”). People on the roadside each raise their hand when they hear the bang sound of the rods. The speed of the hands is faster than the speedoflight. The bus speeds up to (nearly) the speedoflight so that time dilation is nearly infinite. Hands raise only once and the motion of raising the hand moves at the speedoflight.
216 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 37a. “2/c = 1S*(B/c)” is illustrated on the left. On the right, invariant “2/c = 1S*(B/c)*exp(qx*αS)” is illustrated, for “αS > 0”.
Figure 37b. “2/c = 1S*(B/c)*exp(qx*αS)” for “αS = +” is illustrated on the left. Right has “2/c = 1S*(B/c)*exp(qx*αS)” for the First Case “αS = (1  qx)*”, “C” “+”.
For the First Case, frequency and wavenumber components are finite and equal, “S/c kS/c (B/c)/2”. Hands raise at the speedoflight and do so repeatedly. S/c = (B/c)*exp(calcC)*cosh(calcC) First Case “C” “+” = (B/c)*exp(calcC)*(exp(calcC) + exp(calcC))/2 = (B/c)*(1 + exp(2*calcC))/2 (B/c)/2 People on the roadside also hear the Second Case, for which the bus is stationary. For the Second Case they all raise their hands simultaneously. Both the First Case and Second Case together are visible in the raising of hands.
217 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS Momentum Observed at the SpeedofLight. First Case “ES/c” and “pxS” are finite. For “αM = 0”: ES/c = (mB*c)*exp(calcC)*cosh(calcC) First Case “C” “+” = (mB*c)*exp(calcC)*(exp(calcC) + exp(calcC))/2 = (mB*c)*(1 + exp(2*calcC))/2 (mB*c)/2 pxS = (mB*c)*exp(calcC)*sinh(calcC) First Case “C” “+” = (mB*c)*exp(calcC)*(exp(calcC)  exp(calcC))/2 = (mB*c)*(1  exp(2*calcC))/2 (mB*c)/2 For “αM 0” energy and momentum as observed from roadside “S” equal the mean average of energy and momentum as observed inside bus “M”. ES/c (mB*c)*exp(αM)/2 First Case “C” “+” = (EB/c)*exp(αM)/2 = ((EB/c)*cosh(αM) + (EB/c)*sinh(αM))/2 = (EM/c + pxM)/2 pxS (EM/c + pxM)/2
First Case
“C” “+”
Photons have the property of energy equal to momentum. For the case of very low energy photons, “αM 2^Lmax”) from the hyperbolicangle, for a product that is a finite finishedcalculation result. Therefore, finalresult components “S/c” and “kxS” in “4 = 1S*S/c + qxS*kxS” are finite, so that the photon’s precession frequency “S” observed through measurements is finite and is not zero (regardless of the hyperbolicradius “B*(QB*/(mB*c)))” being so small it may be considered to be a zero). Precession frequency “S” is the frequency of the electromagnetic wave. Imagine a bar magnet with its magnetic field spiraling through space pointing radially outward to create the magnetic field component of a spiral electromagnetic wave. Per this proposed model, spin of a photon due to precessing motion has a factor of two compared to frequency of an electron. That appears to be why a photon has a spin (angular momentum) of one unit (of Planck’s constant) and an electron has a spin half of one unit. More mathematical rigor is needed, but the point is made that photon spin is not required to be zero (or a half) by the proposed Theory of Special Relativity with NonFinite Numbers.
248 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
5.8 Exercises Exercises for Text Comprehension 1) Prove squarerootoftwo “2” is irrational. algebra for “1/0” to prove “1/0*2 = 1/0”.
Apply the crude
2) Find the rational approximation for “log23” in base two for thirtytwo placevalue digits after the decimal point. 3) What are truncated numbers for “Lmax = 0” and “Lmax = 1”? How many truncated numbers are for “Lmax = 9”, and what are the largest and smallest? 4) Write localzero “AminA0” for “Lmax = 9”. 5) Prove a localzero cannot equal integer zero. 6) Write Cantor’s proof that both the set of natural numbers and the set of integers have the same quantity of members. 7) Write Cantor’s proof that the quantity of real numbers from zero (inclusive) to one (exclusive) is a larger quantity than the quantity of natural numbers. Use placevalue digit notation in base ten with at least five representative strings of placevalue digits. 8) Write Cantor’s Continuum Hypothesis and explain how it requires its two infinities to be actual infinities and not finite, and for real numbers to have actual infinitesimal precision with an infinite quantity over a finite interval. 9) Review equations for “dPmax0(s)/ds” and “Pmax0(s)”. Write equations for “dPmin0(s)/ds” and “Pmin0”. What is the area under curve “dPmin0(s)/ds” from negative one to positive one?
249 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS 10) Prove localreal numbers form a continuum per the criteria that there be an equal chance of a randomly selected number being in any interval of the same length along the numberline. 11) In the Dirac Equation form for Maxwell’s Wave Equation there is a “N”. What does “N” represent? Answers to Select Exercises. 1) Per “p2 = 2*q2”, “p2” has factor “2” and “p” has factor “2”, and therefore “q2” has factor “2” and “q” has factor “2”. Our first observation is that both “p” and “q” must be even numbers. Our second observation is that at least one of “p” or “q” must be able to be odd, and that is because “p” and “q” are in a ratio such that they both can be divided by two until one of them is odd. The two observations are incompatible. Therefore, the original assumption “2 = p/q”, with “p” and “q” formed by starting with one and adding one repeatedly, is incorrect. Set “p = 1/0” and “q = 1/0” and apply the crude nonnumber algebra so that both observations are satisfied by “p” and “q” both being even and, because “1/0 = 1/0 + 1” one of them can be odd, too. 2) log1011 1.10010101110000000001101000111001…… 3) For “Lmax = 0” the count of truncated numbers is “((1)*(1) + 1)*2  1 = 3”, and truncated numbers are “1”, “0” and “1”. For “Lmax = 1” the count of truncated numbers is “((2)*(2) + 1)*2  1 = 9”, and truncated numbers (written in base two) are “10.0”, “1.1”, “1.0”, “0.1”, “0.0”, “0.1”, “1.0”, “1.1”, “10.0”. For “Lmax = 9” the count of truncated numbers is “((2^9)*(2^9) + 1)*2  1 = 534389”, and the largest truncated number is (written in base two) “1000000000.0” (“512”), and the positive smallest truncated number is “0.000000001” (“1/512”). 4) AminA0 = 0.000000000dddddd…… 5) See book text 6) See book text
250 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 7) Use the table below to form new numbers, by selecting a placevalue digit from each number given in the table. .224744871 .581138830 .870828693 .121320344 .345207880
.549409757 .738612788 .915475947 .082207001 .240370349
New: .280309741 .571202909
8) The Continuum Hypothesis requires no set have a quantity of members between the quantity of members in the set of natural numbers and the quantity of members in the set of real numbers. Any number is the quantity of members in a set for which it is the largest, and so to forbid those intermediate numbers from existing, the two identified quantities must not be finite, but, rather, actual infinity. And, the two quantities cannot be reciprocalofzero, per the crude algebra, because they are not equal and must be positive. For the set of real numbers to have a quantity base two to actual infinity, real numbers have an infinitesimal difference one to the next. Because real numbers are defined with a dependency on actual infinity, and because the property of actual infinity is derived from the Continuum Hypothesis, the Continuum Hypothesis forms the basis of the set of real numbers. 9) Write equations for “dPmin0(s)/ds” and “Pmin0”. dPmin0/ds = 0 dPmin0/ds = (s/(2^Lmax) + 1)*2^Lmax dPmin0/ds = (1  s/(2^Lmax))*2^Lmax dPmin0/ds = 0 Pmin0 = 0 Pmin0 = (s/(2^Lmax) + 1)2/2 Pmin0 = (2  (1  s/(2^Lmax))2)/2 Pmin0 = 1
for s/(2^Lmax) < 1 for 1 < s/(2^Lmax) < 0 for 0 < s/(2^Lmax) < 1 for 1 < s(2^Lmax)
for s/(2^Lmax) < 1 for 1 < s/(2^Lmax) < 0 for 0 < s/(2^Lmax) < 1 for 1 < s/(2^Lmax)
Area under “dPmin0/ds” curve equals one, “Pmin0(s = ) = 1”. 10) Answer not provided.
251 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS 11) “N” “+” models movement in positive “x”direction. “N” “” models movement in the negative “x”direction. Because of “N”, “N” says both possibilities apply together, not one or the other. Further Thought. 1)
Attempt to prove “2” is included in Cantor’s real numbers by “c*2” equaling “c”. Is this a correct check? Must “c” be even and so cannot be odd? Mathematics is unambiguous deductive logic yet asking to prove “2” is in Cantor’s set of real numbers introduces subjectivity and wishywashy ambiguity that fails independent verification. Do you agree?
2)
Prove two localzeros cannot be the same by using a proof similar to the proof a localzero cannot equal integer zero.
3)
Set “T” equal to the infinite “……” sum of reciprocal natural numbers and subtract “T/2” from it, twice to derive “ln(2)  ln(2) = 0”. The second “ln(2)” is a quantity “1/0” of “0”’s. How can the crude algebra be expanded to include theory for this second “ln(2)”? If we don’t do this, then we are ignoring what could be a necessary expansion of our mathematical tools.
4)
Einstein’s two theories of Relativity each include a division by zero, a singularity. What other mathematical models of physics include a division by zero? Can the proposed Theory of NonFinite Numbers be applied to those mathematical models of physics?
5) Apply the proposed Theory of NonFinite Numbers to the singularity at the center of a black hole. The General Theory of Relativity is written as “Rab  (R*gab)/2 + *gab = 8**G*Tab”. 6)
Unknown and unknowable placevalue digits become known or knowable as time progresses, along with Entropy increase, cause and effect, and collapse of the wave function. Is “now” (and “here”) specific to a value of “Lmax”? How might the transition from unknowable to knowable with regard to numbers be related to the wave function collapse of quantum mechanics, and to Hugh
252 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Everett’s 1957 proposed “many worlds interpretation” of quantum mechanics in which all possibilities occur in a divergence of reality. How is “now” unique, or isn’t it, in a theory, yet? 7)
With regard to the above question, if “Lmax” increases with time, then how do we reconcile “Lmax” with antimatter if time is in reverse for antimatter ?
8)
How does the new proposed reciprocalofzero axiom pertain to the exotic Lorentz Transformation for motion faster than the speedoflight with respect to division by zero for the speed?
9)
Using this chapter’s new theory for force density, can we find a theory by which energy and momentum “” for the electric field around an electron are derived through mathematics, rather than empirically derived through observations of experiments?
10) One value of “Lmax” applies, per the above models of quantum mechanics, when one particle observes another particle. Does each particle have a unique value of “Lmax” with respect to each other particle? Or, perhaps, does “Lmax” apply to a collection of particles, or to an inertial reference frame? 11) If a photon is one in the same particle as the electron that emitted it, then timespace is distorted by there being two locations for the one particle. And, the simultaneous validity of the First Case with the Second Case appears to create an equivalence between a set of labelnumbers in “S” with singularlabelnumbers in “M”. How is space structured? 12) Try to define properties so that “2^c” and “3^c” are equal, so that irrational number “log23” is a ratio of two versions of “c”. Retain “c < 2^c < 1/0” and no contribution after the decimal point. Try to define a positive actual infinity or two for which “log23” or “2” equals the ratio of that infinity or of those infinities. The important point is to make the mathematics unambiguous, and to not depend on mysterious, vague, and remote properties in your positive actual infinity.
253 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS 13) If the new proposed reciprocalofzero axiom is included in axiomatic set theory, then the Continuum Hypothesis must be rejected. Do you agree? 14) What is the definition of a real number per Descartes? What is the definition of a real number per Cantor? What is next? 15) Speedoflight in Special Relativity is mathematically analogous to the event horizon of a nonrotating black hole in General Relativity. The analogy is more complete if the radius of the black hole is infinite, to create a flat space event horizon. Per the theory by Hawking / Bekenstein, temperature of a black hole’s surface decreases with radius of the black hole, so that a more complete analogy of an infinite radius has zero temperature at the event horizon. Per the evolving theory of loop gravity in which General Relativity is reconciled with quantum mechanics, as developed by Eugenio Bianchi for black holes, quantum fluctuations on the surface of the black hole decrease as radius is increased. Quantum fluctuations generate temperature. Might “Lmax” (or its rate of increase) be very large for cold nearly flat space and very small for the curved event horizon of a hot small radius black hole? To keep “Lmax” finite must there be curvature in the universe? What about the other extreme, for which the smallest black hole has a diameter of the event horizon on the scale of Matvei Bronstein’s Planck length, “LP = (ħ*G/c3)”? Was “Lmax = 0” held constant before the big bang? 16) Electrons are fermions of half spin that repel each other, per the Pauli Exclusion Principle, with the example of shells of an atom. Photons are as opposite as possible. Photons are bosons of full spin that have a tendency to coincide, as in a laser. Electrons are the material and photons are the force field. Because they are different to an extreme, combining electrons with photons into one mathematical model feels strange, feels incorrect, feels like it should be impossible. And, photons are generated by protons, and photons are created by pair annihilation / pair production. How can the proposed Theory of Special Relativity with NonFinite Numbers be generalized to include protons and pair production? What are your thoughts?
254 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 17) The theory of the Dirac Spinor matterwave was developed from rest mass inertia which, per the classical radius model of the electron, results from the electromagnetic field. It’s not surprising the electromagnetic field is also a Dirac Spinor, per the new theory presented in this book, because the two phenomena are made of the same stuff, it seems. Where next to take this theory? 18) A proposed experiment is to simultaneously measure the photon particle and the electron particle that emitted it, to check for correlated properties, in analogy with the EPR experiment. How could such an experiment be set up? What would it measure? 19) In the book Reality is Not What It Seems – The Journey to Quantum Gravity by Carlo Rovelli, Riverhead Books, 2017 (2014 in Italian), Page 245/246, is the quote below. In the two points, positive actual infinity is removed from mathematical relevance, and the quantity of information from one particle observing another particle increases with time. How can the proposed Theory of NonFinite Numbers fit into quantum mechanics per this quote? “In fact, the entire structure of quantum mechanics can be read and understood in terms of information, as follows. A physical system manifests itself only in interacting with another. The description of a physical system, then, is always given in relation to another physical system, the one with which it interacts. Any description of a system is therefore always a description of the information a system has about another system, that is to say, the correlation between the two systems. The mysteries of quantum mechanics become less dense if interpreted this way, as the description of the information that physical systems have about one another. The description of a system, in the end, is nothing other than a way of summarizing all the past interactions with it, and using them to predict the effect of future interactions. The entire formal structure of quantum mechanics can be in large measure expressed in two simple postulates: 1. The relevant information in any physical system is finite 2. You can always obtain new information on a physical system”
255 CHAPTER 5  PROPOSED THEORY OF NONFINITE NUMBERS 20) AntiMatter Photon. Per the First Case Lorentz Transformation, observer “S” measures passage of time “tS” proportional to passage of time “tB” for a photon. Regardless of the passage of time, a photon has complete length contraction. Also, per experiments in which two photons annihilate each other, it appears a photon is its own antimatter particle. How can a photon be its own antimatter particle if the direction of its passage of time is dependent on the electron that emitted it being either matter or antimatter? 21) The two zeros on the right side of the First Case Dirac Equation were replaced by nonzero value “a” to model electric charge. Try to find justification to create a theory for what electric charge really is. 22) As time progresses, a photon gets absorbed and disappears, so that the First Case disappears as the hyperbolicangle becomes more precise (with more zeros before the decimal point) with respect to an observing particle. The photon disappears because the hyperbolicangle is small relative to “2^Lmax” and stays small. A succession of electron and photon emission and absorption events occurs through an increase in “Lmax”. Try to make this theory quantitative to make it useful. 23) Step three in the Process from Descartes is a change from numbers to geometry in preparation to take a measurement. It is not a change to a more fundamental concept of reality. The suggestion is that numbers are fundamental to reality, and not objects like electrons and photons. What we perceive as our physical reality is the adjustment of numbers which become more accurate/precise relative to each other. That adjustment appears to follow the discovered Dirac Equation, and there’s likely much more complexity beyond that equation. The author (me) is biased to math (rather than physics). What do you think is fundamental, math or physics?
256 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
257 APPENDIX A  OCTONIONS AND SEDONIONS
Appendix A – Octonions and Sedonions Octonions were proposed by Caylay immediately after Hamilton proposed quaternions, in 1843. Quaternions anticommute “jx*jy = jy*jx = jz”. Likewise, octonions anticommute. When Hamilton selected “jx*jy = jz”, he selected against equating “jx*jy” to “jz”. For octonions, there is more than one selection required when setting up the multiplication scheme. One possible multiplication scheme for octonions is selected below. Each row is a “triple”. j1*j2 = j3 ; j2*j4 = j6 ; j3*j6 = j5 ; j4*j3 = j7 ; j5*j1 = j4 ; j6*j7 = j1 ; j7*j5 = j2
Figure 40. Traditional triangle model for octonion multiplication. *
1
j1
j2
j3
1
1*1 = 1
1*j1 = j1
1*j2 = j2
1*j3 = j3
j1
j1*1 = j1
j1*j1= 1
j1*j2 = j3
j1*j3 = j2
j2
j2*1 = j2
j2*j1 = j3
j2*j2 = 1
j2*j3 = j1
j3
j3*1 = j3
j3*j1 = j2
j3*j2 = j1
j3*j3 = 1
j4
j3*1 = j3
j4*j1 = j5
j4*j2 = j6
j4*j3 = j7
j5
j3*1 = j3
j5*j1 = j4
j5*j2 = j7
j5*j3 = j6
j6
j3*1 = j3
j6*j1 = j7
j6*j2 = j4
j6*j3 = j5
j7
j3*1 = j3
j7*j1 = j6
j7*j2 = j5
j7*j3 = j4
258 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY *
j4
j5
j6
j7
1
1*j4 = j4
1*j5 = j5
1*j6 = j6
1*j7 = j7
j1
j1*j4 = j5
j1*j5 = j4
j1*j6 = j7
j1*j7 = j6
j2
j2*j4 = j6
j2*j5 = j7
j2*j6 = j4
j2*j7 = j5
j3
j3*j4 = j7
j3*j5 = j6
j3*j6 = j5
j3*j7 = j4
j4
j4*j4 = 1
j4*j5 = j1
j4*j6 = j2
j4*j7 = j3
j5
j5*j4 = j1
j5*j5 = 1
j5*j6 = j3
j5*j7 = j2
j6
j6*j4 = j2
j6*j5 = j3
j6*j6 = 1
j6*j7 = j1
j7
j7*j4= j3
j7*j5 = j2
j7*j6 = j1
j7*j7 = 1
Table 6. Multiplication table for octonions.
0 1 2 3 4 5 6 7
1 0 3 2 5 4 7 6
2 3 0 1 6 7 4 5
3 2 1 0 7 6 5 4
4 5 6 7 0 1 2 3
5 4 7 6 1 0 3 2
6 7 4 5 2 3 0 1
7 6 5 4 3 2 1 0
Table 7. Short form multiplication table for octonions.
Criteria of an algebraic group for octonions “O = {1, j1, j2, j3, j4, j5, j6, j7}” with respect to multiplication, “{O, *(anticommute)}”: •
Closure: No holes in the 16x16 multiplication table
•
Identity: Identity element is integer one
•
Commutative Property: Applies, but with two different of the seven octonions anticommuting
•
Associative Property: Applies, but with three different octonions (that are not all three in the same triple) antiassociating
259 APPENDIX A  OCTONIONS AND SEDONIONS •
Inverse: Ratio of any two numbers is in the set of numbers
An inverse property example is “j1” divided by “j4”. First, substitute two factors for the numerator, then, second, apply the (anti) commutative law (if needed) followed by the (anti)associative law (if needed) to form a ratio of like numbers. The ratio of like numbers is then replaced with the number one. Each combination must equal “j1*j4”. j1/j4 = (j2*j3)/j4 = (j2*(j7*j4))/j4 = ((j2*j7)*j4)/j4 = (j2*j7)*(j4/j4) = (j2*j7) = j5 j1/j4 = (j4*j5)/j4 = (j5*j4)/j4 = j5*(j4/j4) = j5 j1/j4 = (j6*j7)/j4 = (j6*(j4*j3))/j4 = (j6*(j3*j4))/j4 = (j6*j3)*(j4/j4) = (j6*j3) = (j3*j6) = j5 Degradations. Each higher order of hypercomplexity includes a new breakdown in symmetry through introduction of another negative. A breakdown in symmetry is called (in this book) a “degradation”. The pattern of degradations only goes to the fourth order, if only because the degradation of the fourth order is too severe to go on. •
Real numbers (hypercomplex order “N = 0”) have no degradation (other than their unboundedness which implies a lack of closure)
•
AntiIdentity. Complex numbers (“N = 1”) have the complex number labelnumber square to negative one i2 = 1
•
AntiCommutative. Quaternions (“N = 2”). jy*jz = jx = (jx) = jz*jy
•
AntiAssociative. Octonions (“N = 3”). (j1*j2)*j4 = j3*j4 = j7 = j1*j6 = j1*(j2*j4)
260 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •
AntiInverse. Sedonions (“N = 4”) j14s/j6s = (j8s*j6s)/j6s = j8s*(j6s/j6s) = j8s = (j8s) = (j13s*j5s) = (j13s*j5s)*(j6s/j6s) = ((j13s*j5s)*j6s)/j6s = (j13s*(j5s*j6s))/j6s = (j13s*(j6s*j5s))/j6s = (j13s*j3s)/j6s = j14s/j6s
Because “j14s/j6s = j14s/j6s”, it is said sedonion algebra is not a division algebra. (Alternatively, we may say sedonion algebra is not yet properly specified.) Matrix Isomorphs for Octonions. Pauli Spin Matrices, multiplied or divided by “i”, result in the below traditional set of 2x2 matrix isomorphs for Hamilton’s quaternions. 1 0
0 i
1=
jx = 0 1
0 1 jy =
i 0
i
0
jz = 1 0
0 i
The above four matrices are (traditionally) placed into a general matrix multiplication scheme using complexconjugate “*i” (“i*i = i”) on the rightside columnvector. a
b*i
b
a*i
c
d*i
d
c *i
*
=
a*c  b*i*d
a*d*i  b*i*c*i
b*c + a*i*d
b*d*i + a*i*c*i
e =
f*i
f e*i
Letters “a” and “b” (also “c” and “d”, and “e” and “f”) correspond to terms of the 2x2 matrix isomorphs of the quaternions by these substitutions: 1: a = 1, b = 0 ;
jx: a = 0, b = i ;
jy: a = 0, b = 1 ;
jz: a = i, b = 0
Factors are rearranged because complex numbers commute. a
b*i
b
a*i
c
d*i
d
c *i
*
=
a*c  d*b*i
d*i*a  b*i*c*i
c*b + a*i*d
b*d*i + c*i*a*i
e =
f*i
f e*i
261 APPENDIX A  OCTONIONS AND SEDONIONS The arrangement above is more correct because “e*i = b*d*i + c *a ” has factors in each term reversed compared to “e = a*c  d*b*i”. The new arrangement, with the reversed order of factors and terms in the product 2x2 matrix, specifies a new multiplication scheme between two 2x2 matrices. The new multiplication scheme is necessary for factors that do not commute, that is, if factors are quaternions. Hypercomplexconjugate operation “*j” replaces complexconjugate operation “*i”. “jx*j = jx”, “jy*j = jy”, and “jz*j = jz” with factors reversed in order. The below matrix multiplication operation applies to matrices with quaternion hypercomplex numbers as elements/terms of the matrix. *i
*i
a
b*j
b
a*j
c
d*j
d
c*j
*
1: a = 1, b = 0 ; j1: a = 0, b = 1 ;
=
a*c  d*b*j
d*j*a  b*j*c*j
c*b + a*j*d
b*d*j + c*j*a*j
j2: a = jx, b = 0 ; j3: a = 0, b = jx ;
j4: a = jy, b = 0 ; j5: a = 0, b = jy ;
e =
f*j
f e*j
j6: a = jz, b = 0 j7: a = 0, b = jz
(The above 2x2 matrix equation for octonion matrix isomorphs was discovered by the author because a search did not find a previously known set of matrix isomorphs for octonions.) Examples for “j2*j4 = j6” and “j4*j3 = j7”, respectively. jx
0
jy
0
* 0
jx
jy
0
0
 jy
0
jx
* 0
jy
jx*jy
0
=
0
0
0
 jz
0 (jy*jx) 0
jx*jy
= jx
jz =
0
jz
jz
0
= jy*jx
0
“jx” can be replaced with its 2x2 matrix isomorph and “i” and “1” by their 2x2 matrix isomorphs to create 8x8 matrices (but with a messy multiplication scheme). 0 1 1 0 i = 1= 1 0 0 1
262 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY General Rule for Matrix Multiplication with nonCommuting Elements. A C
E
G
* B D
A*E + F*C
G*A + C*H
F*B + D*E
B*G + H*D
= F
H
VectorAft Multiplication: A C
E *
B D
A*E + F*C =
F
E*B + D*F
VectorFront Multiplication: E
A C *
F
E*A + C*F =
B
D
B*E + F*D
Vectorfront multiplication anticommutes when compared to vectoraft multiplication. For example: In “j5*j3 = j6” use the left column of “j3” to result in the left column of “j6”. Now, reverse the order of the factors to “j3*j5”. Again, only use the left column of “j3”. Per the vectorfront multiplication operation, the result of “j3*j5” is negative of the left column of “j6”. As a side note, we will not use the below alternative. A C
E
G
* B
D
E*A + C*F
A*G + H*C Do not use
= F
H
B*F + E*D
G*B + D*H
Rotation of a SevenDimensional Object. Sevendimensional ultraspace modeled with octonions is organized as seven threedimensional spaces so that there are three planes of rotation around each axis of rotation. Rotation around the “j1” axis has rotation in “j2”/ “j3”, “j4”/ “j5”, and “j6”/ “j7” planes. Rotation angles in each plane are “23”, “45”, and “67”, respectively. “23” begins at positive “j2” axis and is measured towards positive “j3” axis. (Rotations in 7d space might be new with this book.)
263 APPENDIX A  OCTONIONS AND SEDONIONS 0
1
1
0
j1o1 = jx*cos23
jx*sin23
j2o1 =
; jx*sin23
jx*cos23
jy*cos45
jy*sin45
j4o1 =
jx*sin23
jx*cos23
jx*cos23
jx*sin23
j3o1 =
jy*sin45
jy*cos45
jy*cos45
jy*sin45
jz*sin67
jz*cos67
jz*cos67
jz*sin67
; j5o1 = jy*sin45
jy*cos45
jz*cos67
jz*sin67
j6o1 =
; j7o1 = jz*sin67
jz*cos67
Multiplications “j1o1*j2o1 = j3o1”, “j5o1*j1o1 = j4o1”, and “j6o1*j7o1 = j1o1” do not involve angle addition, and, therefore, are relatively trivial. The other four triples “j2o1*j4o1 = j6o1”, “j3o1*j6o1 = j5o1”, “j4o1*j3o1 = j7o1”, and “j7o1*j5o1 = j2o1” involve addition of angles. “j3o1*j6o1”: jx*sin23
jx*cos23
jx*cos23
jx*sin23
j3o1*j6o1 =
jz*cos67
jz*sin67
* jz*sin67
jz*cos67
(jx*jz)*sin23*cos67 + (jz*jx)*cos23*sin67 (jz*jx)*sin23*sin67 + (jx*jz)*cos23*cos67 j3o1*j6o1 = (jz*jx)*cos23*cos67 + (jx*jz)*sin23*sin67 (jx*jz)*cos23*sin67 + (jz*jx)*sin23*cos67
jy*(sin23*cos67 + cos23*sin67) jy*(sin23*sin67  cos23*cos67) j3o1*j6o1 = jy*(cos23*cos67  sin23*sin67) jy*(cos23*sin67 + sin23*cos67)
264 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY jy*sin(23 + 67) j3o1*j6o1 = jy*cos(23 + 67)
jy*cos(23 + 67) jy*sin(23 + 67)
jy*sin45
jy*cos45
jy*cos45
jy*sin45
j3o1*j6o1 =
= j5o1
The above matrix multiplication that resulted in “j3o1*j6o1 = j5o1” required “23 + 45 + 67 = 0”. “23 + 45 + 67 = 0” is also valid for “j2o1*j4o1 = j6o1”, “j4o1*j3o1 = j7o1”, and “j5o1*j2o1 = j7o1”. jz*(cos23*cos45  sin23*sin45) jz*(cos23*sin45  sin23*cos45) j2o1*j4o1 = jz*(sin23*cos45  cos23*sin45) jz*(sin23*sin45  cos23*cos45) jz*cos(23 + 45) j2o1*j4o1 = jz*sin(23 + 45) jz*cos67 j2o1*j4o1 = jz*sin67
jz*sin(23 + 45) jz*cos(23 + 45)
jz*sin67 = j6o1 jz*cos67
…… …… …… …… …… …… …… …… …… …… …… …… …… jy*cos45
jy*sin45
jy*sin45
jy*cos45
j4o1*j3o1 =
jx*sin23
jx*cos23
jx*cos23
jx*sin23
*
(jy*jx*cos45*sin23 + jx*jy*sin45*cos23) (jx*jy*cos45*cos23 + jy*jx*sin45*sin23) j4o1*j3o1 = (jx*jy*sin45*sin23 + jy*jx*cos45*cos23) (jy*jx*sin45*cos23 + jx*jy*cos45*sin23)
265 APPENDIX A  OCTONIONS AND SEDONIONS
jz*(cos45*sin23 + sin45*cos23) jz*(cos45*cos23  sin45*sin23) j4o1*j3o1 = jz*(sin45*sin23 + cos45*cos23) jz*(sin45*cos23  cos45*sin23) jz*sin(45 + 23) j4o1*j3o1 = jz*cos(45 + 23) jz*sin67 j4o1*j3o1 = jz*cos67
jz*cos(45 + 23) jz*sin(45 + 23) jz*cos67 = j7o1 jz*sin67
…… …… …… …… …… …… …… …… …… …… …… …… …… jz*(sin45*cos23 + cos45*sin23) jz*(sin45*sin23 + cos45*cos23) j5o1*j2o1 = jz*(cos45*cos23  sin45*sin623) jz*(cos45*sin23  sin45*cos23) jz*sin(45 + 23) j5o1*j2o1 = jz*cos(45 + 23) jz*sin67 j5o1*j2o1 = jz*cos67
jz*cos(45 + 23) jz*sin(45 + 23)
jz*cos67 = j7o1 jz*sin67
A sevendimensional object is modeled as seven sticks.
266 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 41. Sevendimensional space as seven threedimensional spaces.
Some sticks are rotated ninety degrees and comply with “23 + 45 + 67 = 0”, per the illustration below.
Figure 42. Rotation in 23 plane causes a counter rotation in 45 plane.
A sevendimensional object can also rotate around an axis in threedimensional space, for example, the yaxis. The below substitutions may be visualized using 4x4 matrix isomorphs for octonions. 0 1 jyqy =
i*coszx
i*sinzx
; jzqy = 1 0
i*sinzx
i*coszx
i*coszx
i*sinzx
; jxqy = i*sinzx i*coszx
267 APPENDIX A  OCTONIONS AND SEDONIONS Unlike octonions and quaternions, there is no rotation possible for complex numbers, because there is only one label number for complex numbers. General Model for LabelNumbers. •
Real numbers “N = 0” have the trivial 1x1 matrix isomorph.
•
Complex numbers “N = 1” have “2N = 21 = 2” labelnumbers, “1” and “i”, and a 2x2 matrix isomorph.
•
Quaternions “N = 2” have “2N = 22 = 4” labelnumbers and a 4x4 matrix isomorph. (1 + jx)*(1 + jy) = 1 + jx + jy + jx*jy = 1 + jx + jy + jz
•
= (1 + jy)*(1 + jz) = (1 + jz)*(1 + jx)
Octonions “N = 3” have “2N = 23 = 8” labelnumbers and an 8x8 matrix isomorph. (1 + j1 + j5 + j6)*(1 + j2) = 1 + j1 + j2 + j3 + j4 + j5 + j6 + j7 (1 + j2 + j4 + j5)*(1 + j3) = 1 + j1 + j2 + j3 + j4 + j5 + j6 + j7 There are twentyone restructures of the two octonion equations above, one restructure for each number in the 7x3 twentyone number table.
•
Sedonions “N = 4” have sixteen labelnumbers. The multiplication table for sedonions is not yet finalized.
VectorSpace. Vector space is so far useful in applied mathematics to “N = 2”. A more complex notion of vector space may be needed for “N = 3” and higher orders. •
“N = 1” introduced complex labelnumber “i”. Subtract away the real number so the quantity of labelnumbers in an “N = 1” vectorspace is “2N  1 = 1”.
268 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY •
Fourdimensional timespace of Special Relativity was created by combining “N = 1” with “N = 2” to form the “N = 2” vectorspace: “(21  1) + (22  1) = 1 + 3 = 4” terms. An example number is “658*i + 89*jx + 57*jy + 456*jz”.
•
The natural extension to octonions is elevendimensional vectorspace: “11 = 1 + 3 + 7 = (21  1) + (22  1) + (23  1)”. The seven dimensions of ultraspace are as different from space as space is from time. The example number is “658*i + 89*jx + 57*jy + 456*jz + 26*j1 + 44*j2 + 785*j3 + 963*j4 + 76*j5 + 659*j6 + 154*j7”.
•
“N = 4” creates a twentysixdimensional vectorspace “26 = 1 + 3 + 7 + 15”.
•
After that is “N = 5” for “57 = 1 + 3 + 7 + 15 + 31”, for a fiftysevendimensional vectorspace, and so on.
Sedonions. A simple attempt at sedonion algebra applies the same matrix multiplication operation applied to octonion algebra. The octonionconjugate operation “*jo” requires the negative of the octonions, “j1*jo = j1”, …, “j7*jo = j7”, and requires reverse order of factors. a
b*jo
b
a*jo
c
d*jo
d
c*jo
*
=
a*c  d*b*jo
d*jo*a  b*jo*c*jo
c*b + a*jo*d
b*d*jo + c*jo*a*jo
e
f*jo
f
e*jo
=
1: a = 1, b = 0 ; j2s: a = j1, b = 0 ; j4s: a = j2, b = 0 ; j6s: a = j3, b = 0 j1s: a = 0, b = 1 ; j3s: a = 0, b = j1 ; j5s: a = 0, b = j2 ; j7s: a = 0, b = j3 j8s: a = j4, b = 0 ; j10s: a = j5, b = 0 ; j12s: a = j6, b = 0 ; j14s: a = j7, b = 0 j9s: a = 0, b = j4 ; j11s: a = 0, b = j5 ; j13s: a = 0, b = j6 ; j15s: a = 0, b = j7 j1s*j2s = j3s j1s*j4s = j5s j1s*j6s = j7s j1s*j8s = j9s j1s*j10s = j11s j1s*j12s = j13s j1s*j14s = j15s
; ; ; ; ; ; ;
j2s*j4s = j6s ; j2s*j8s = j10s ; j2s*j12s = j14s ; j4s*j8s = j12s ; j12s*j10s = j6s ; j14s*j8s = j6s ; j14s*j10s = j4s ;
j3s*j6s = j5s ; j3s*j10s = j9s ; j3s*j14s = j13s ; j5s*j12s = j9s ; j13s*j6s = j11s ; j15s*j6s = j9s ; j15s*j4s = j11s ;
j4s*j3s = j7s ; j8s*j3s = j11s ; j12s*j3s = j15s ; j8s*j5s = j13s ; j10s*j13s = j7s ; j8s*j15s = j7s ; j10s*j15s = j5s ;
j7s*j5s = j2s j11s*j9s = j2s j15s*j13s = j2s j13s*j9s = j4s j7s*j11s = j12s j7s*j9s = j14s j5s*j11s = j14s
269 APPENDIX A  OCTONIONS AND SEDONIONS 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
0
3
2
5
4
7
6
9
8
11
10
13
12
15
14
2
3
0
1
6
7
4
5
10
11
8
9
14
15
12
13
3
2
1
0
7
6
5
4
11
10
9
8
15
14
13
12
4
5
6
7
0
1
2
3
12
13
14
15
8
9
10
11
5
4
7
6
1
0
3
2
13
12
15
14
9
8
11
10
6
7
4
5
2
3
0
1
14
15
12
13
10
11
8
9
7
6
5
4
3
2
1
0
15
14
13
12
11
10
9
8
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
9
8
11
10
13
12
15
14
1
0
3
2
5
4
7
6
10
11
8
9
14
15
12
13
2
3
0
1
6
7
4
5
11
10
9
8
15
14
13
12
3
2
1
0
7
6
5
4
12
13
14
15
8
9
10
11
4
5
6
7
0
1
2
3
13
12
15
14
9
8
11
10
5
4
7
6
1
0
3
2
14
15
12
13
10
11
8
9
6
7
4
5
2
3
0
1
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Table 8. Shorthand multiplication table for sedonions (first attempt).
For rotation around the “j1s” axis, define seven rotation angles “23s”, “45s”, “67s”, “89s”, “1011s”, “1213s” and “1415s”. These angles are analogous to the previously defined angles “23”, “45”, and “67” in the discussion on octonions. 0
1
1
0
j1s1 = j1*cos23s
j1*sin23s
j2s1 =
j1*sin23s
j1*cos23s
j1*cos23s
j1*sin23s
j2*sin45s
j2*cos45s
j2*cos45s
j2*sin45s
; j3s1 = j1*sin23s
j1*cos23s
j2*cos45s
j2*sin45s
j4s1 =
; j5s1 = j2*sin45s
j2*cos45s
270 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY j3*cos67s
j3*sin67s
j6s1 =
j3*sin67s
j3*cos67s
; j7s1 = j3*sin67s
j3*cos67s
j3*cos67s
j3*sin67s
j4*cos89s
j4*sin89s
j4*sin89s
j4*cos89s
j4*sin89s
j4*cos89s
j4*cos89s
j4*sin89s
j5*cos1011s
j5*sin1011s
j5*sin1011s
j5*cos1011s
j5*cos1011s
j5*sin1011s
j6*sin1213s
j6*cos1213s
j6*cos1213s
j6*sin1213s
j7*sin1415s
j7*cos1415s
j7*cos1415s
j7*sin1415s
j8s1 =
; j9s1 =
j10s1 =
; j11s1 = j5*sin1011s
j5*cos1011s
j6*cos1213s
j6*sin1213s
j12s1 =
; j13s1 = j6*sin1213s
j6*cos1213s
j7*cos1415s
j7*sin1415s
j14s1 =
; j15s1 = j7*sin1415s
Octonions:
Sedonions:
j7*cos1415s
xyz
jx jy jz 23 45 67 123 j2 j4 j6 246 j3 j5 j7 365 437 j1 j2 j3 j4 j5 j6 j7 514 23s 45s 67s 89s 1011s 1213s 1415s 671 j2s j4s j6s j8s j10s j12s j14s 752 j3 j5s j7s j9s j11s j13s j15s
The octonion/sedonion table above shows multiplication of “j1*j2 = j3” internal to sedonion 2x2 matrices is analogous to “jx*jy = jz” internal to octonion 2x2 matrices. Also “j2*j4 = j6”, “j5*j1 = j4”, and “j6*j7 = j1” are analogous to “jx*jy = jz” because numbers increase in value: 123, 246, 145, and 167. The other three triples are not: 743, 654, and 752, because numbers decrease in value. Each of seven triples pertains to a plane perpendicular to the “1” axis. Each plane is associated with four triples.
271 APPENDIX A  OCTONIONS AND SEDONIONS 23s + 45s + 67s = 0, j2s1*j4s1 = j6s1 ;
(123) j3s1*j6s1 = j5s1 ; j4s1*j3s1 = j7s1 ;
23s + 89s + 1011s = 0, (145) j2s1*j8s1 = j10s1 ; j3s1*j10s1 = j9s1 ; j8s1*j3s1 = j11s1 ;
j7s1*j5s1 = j2s1 j11s1*j9s1 = j2s1
23s + 1213s + 1415s = 0, (167) j2s1*j12s1 = j14s1 ; j3s1*j14s1 = j13s1 ; j12s1*j3s1 = j15s1 ; j15s1*j13s1 = j2s1 45s + 89s + 1213s = 0, (246) j4s1*j8s1 = j12s1 ; j5s1*j12s1 = j9s1 ; j8s1*j5s1 = j13s1 ; j13s1*j9s1 = j4s1 1213s + 1011s + 67s = 0, (653) j12s1*j10s1 = j6s1 ; j13s1*j6s1 = j11s1 ; j10s1*j13s1 = j7s1 ; j7s1*j11s1 = j12s1 1415s + 89s + 67s = 0, (743) j14s1*j8s1 = j6s1 ; j15s1*j6s1 = j9s1 ; j8s1*j15s1 = j7s1 ; j7s1*j9s1 = j14s1 1415s + 1011s + 45s = 0, (752) j14s1*j10s1 = j4s1 ; j15s1*j4s1 = j11s1 ; j10s1*j15s1 = j5s1 ; j5s1*j11s1 = j14s1 Angles “s23”, “s45”, “s67”, “s89”, “s1011”, “s1213” and “s1415” must all be zero for the seven angle equations to each equal zero, and that means the math is incorrect as a model for rotation in fifteendimensional ultraultra space. The four normal planes can be thought of as righthand rotation. The three antinormal planes can be thought of as lefthand rotation. The switch of hands made rotation impossible. To investigate lefthand planes: “j6s*j9s = j15s” uses “j4*j3 = j7” because “j6s*j9s” has the product “j3*j4 = j7” and “j4*j3” has the product “jy*jx = jz”. “j6s*j9s”:
j3*cos67s
j3*sin67s
j6s1*j9s1 =
j4*sin89s
j4*cos89s
j4*cos89s
j4*sin89s
* j3*sin67s
j3*cos67s
272 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY (j3*j4*cos67s*sin89s + j4*j3*sin67s*cos89s) (j4*j3*cos67s*cos89s + j3*j4*sin67s*sin89s) j6s1*j9s1 = (j4*j3*sin67s*sin89s + j3*j4*cos67s*cos89s) (j3*j4*sin67s*cos89s + j4*j3*cos67s*sin89s)
j7*(cos67s*sin89s + sin67s*cos89s) j7*(cos67s*cos89s  sin67s*sin89s) j6s1*j9s1 = j7*(sin67s*sin89s + cos67s*cos89s) j7*(sin67s*cos89s + cos67s*sin89s) j7*sin(67s + 89s)
j7*cos(67s + 89s)
j7*cos(67s + 89s)
j7*sin(67s + 89s)
j6s1*j9s1 =
j7*sin1415s
j7*cos1415s
j7*cos1415s
j7*sin1415s
j6s1*j9s1 =
= j15s1
Based on that example, left hand angle equations apply: “1213s + 1011s + 67s = 0”, “1415s + 89s + 67s = 0”, and “1415s + 1011s + 45s = 0”, such that there should not be negative.
Figure 43. A failed attempt at a tetrahedron triangle diagram for “N = 4” multiplicationdivision. Fill in circles and arrows and find the task impossible.
273 APPENDIX A  OCTONIONS AND SEDONIONS Antiinverse degradation of the above sedonion algebra may be experienced algebraically by attempting to create a 15x7 table for sedonion 7spaces analogous to the 7x3 table for octonion 3spaces. A 15x7 table for sedonions would divide fifteendimensional space into fifteen different sevendimensional spaces. The lack of a division algebra for this simple attempt at sedonion algebra is proven in the Hurwitz Theorem. See the last two “Thought Exercises” questions for the beginning of other attempts for 2x2 matrix formulation of sedonion algebra. Thought Exercises 1)
Write a 2x2 matrix isomorph for octonion “a + b*j1 + c*j2 + d*j3 + e*j4 + f*j5 + g*j6 + h*j7”.
2)
Write a matrix multiplication table for octonions “j1”, “j2”, “j3”, “j4”, “j5”, “j6”, “j7” and complex number factor “i”. Use “q1”, “q2”, “q3”, “q4”, “q5”, “q6”, “q7” with “q1 = j1/i”, etc. Address negatives with a note.
3)
Write the 2x2 matrix isomorph of the octonioncomplex number “a + b*q1 + c*q2 + d*q3 + e*q4 + f*q5 + g*q6 + h*q7” in which components “a”, “b”, “c”, “d”, “e”, “f”, “g”, and “h” are mathematically complex.
4)
In Chapter 1 is a section titled “QuaternionComplexHypercomplex Numbers”. Follow the format of that section for octonions and octonionscomplex. Why can’t octonionsquaternions or sedonions be placed into that format?
5)
“j1”, “j2”, “j3”, “j4”, “j5”, “j6”, “j7” were each assigned a 2x2 matrix isomorph, per the text above. Find all alternative 2x2 matrix isomorphs to “j1”, “j2”, “j3”, “j4”, “j5”, “j6”, “j7” holding to the criteria that the same 7x3 table of triples applies? Are there other tables?
6)
There is no algebra by which to find real number magnitude of “658*i + 89*jx + 57*jy + 456*jz + 26*j1 + 44*j2 + 785*j3 + 963*j4 + 76*j5 + 659*j6 + 154*j7” because quaternionoctonion products
274 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY have no specified algebra. This issue will likely get resolved through an application in applied mathematics. Find that application. 7)
Find octonion 2x2x2 matrix isomorphs that use complex labelnumbers “1” and “i” for terms/elements. The major diagonal of a 2x2x2 matrix would likely be elements “(ar + i*ai)” and “(ar + i*ai)*i”, in analogy to the 2x2 matrix isomorphs for quaternions. The three minor diagonals would likely each be analogous to the one minor diagonal of the 2x2 matrix isomorphs for quaternions. The challenge is to develop a 2x2x2 matrix multiplication scheme for which the 7x3 table of triples for octonion multiplication is satisfied. (ar + i*ai) (br + i*bi) (cr + i*ci)
(dr + i*di) (cr + i*ci)*i (dr + i*di)*i (br + i*bi)*i (ar + i*ai)*i
1: ar = 1 ; j2: cr = 1 ; j4: ai = 1 ; j6: ci = 1 j1: br = 1 ; j3: dr = 1 ; j5: bi = 1 ; j7: di = 1 8)
Try to find a 2x2x2x2 matrix structure for sedonions. Four corners are adjacent to each of the two corners of the major diagonal. And six corners are not adjacent to the two corners.
9)
Propose an expansion of the Dirac Equation that uses octonion matrix isomorphs in addition to, or as a substitute for, quaternion 2x2 matrix isomorphs. A clue in the text was justification for “e4*qx = qx*e4”. Another clue is that quaternions are inside matrix isomorphs of octonions.
10) The 7x3 table for octonions implies the first column is an “x” dimension, the second “y”, and the third “z”. But that implication is not used in the algebra. It suggests the algebra is incomplete. 11) Using triples “j13s = j3s*j14s”, “j5s = j3s*j6s”, “j13s = j8s*j5s” and “j6s = j14s*j8s”, the violation of the antiassociative property that led to the antiinverse property is:
275 APPENDIX A  OCTONIONS AND SEDONIONS j14s = (j3s)*j13s = (j6s*j5s)*j13s j14s = j6s*(j8s) = j6s*(j5s*j13s) = +(j6s*j5s)*j13s Can matrix multiplication be modified to remove the antiassociative violation? See the matrix multiplication operation below, in which new groupings apply (for example, “F(CI)” rather than “(FC)I”). The new groupings would affect octonions and not quaternions, and the operations become trinary and not binary. A C ((
E G )*(
B D
I K
))*( F H
(AE+FC)I+J(GA+CH) K(AE+FC)+(GA+CH)L )= J L I(EB+DF)+(BH+HD)J (EB+DF)K+L(BG+HD)
(AE)I+(FC)I+J(GA)+J(CH) K(AE)+K(FC)+(GA)L+(CH)L = I(EB)+I(DF)+(BH)J+(HD)J (EB)K+(DF)K+L(BG)+L(HD)
12) Can sedonion algebra be redesigned to require an octonion rotation in 7dimensional space when there is a sedonion rotation in 15dimensional space, such that the octonions inside the matrix isomorphs for sedonions change to make the sedonions change, too? Like the above attempt to make sedonion algebra work, it changes matrix isomorphs, and so is more than simple sedonion labelnumber manipulations, and that means the restrictive Hurwitz theorem does not apply.
276 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
277 APPENDIX B  SPOOKY ACTION AT A DISTANCE
Appendix B – Spooky Action at a Distance EPR Experiment Set Up. Two photons created as a pair have coordinated properties and are called “entangled”. The two entangled photons travel a macroscopic distance, perhaps a meter or further in opposite directions, and are each detected. Each detector is comprised of a polarizing film followed by a photographic film. The two polarizing films are parallel and both photographic films detect the photons, to show us polarity of the photons was the same, as expected, because they are entangled. To explain this expected experiment result, we might venture to guess polarity of the two photons was determined at the time of creation at the emission source that sits between the two detectors. Other phases of the EPR experiment have the polarizing films at a different angles. Results are interpreted using Bell’s Inequality (from year 1964) and lead to the conclusion polarization of the two entangled photons is determined at the moment of detection, not emission. For that to happen, the direction of polarization is coordinated over the macroscopic distance that separates the two detectors and coordinated instantly over that macroscopic distance: Not at the speedoflight but faster, instantly. Einstein called the instant communication “spooky action at a distance”.
Figure 44. EPR Experiment for photons.
278 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY The EPR experiment may also be performed with electrons. Electrons do not have polarization, but, rather, electrons have spin. Axis of spin (using the righthand rule) is measured as a direction perpendicular to the plane of rotation. Two entangled electrons have the axis of spin of one electron antiparallel with respect to the other electron, as required for a total of zero angular momentum for the pair.
Figure 45. EPR Experiment for electrons.
For a visualization, substitute baseballs for electrons. Two baseballs were resting beside each other. A little expanding spring sent the baseballs in opposite directions. The little expanding spring was not perfectly aligned between the two centers, and, therefore, created baseball spin. The axis of spin must be equal and opposite, so that the angular momentum of both baseballs together equals zero. We watched the release of the expanding spring. We saw the spin of the two baseballs as they moved in opposite directions. We watched the two baseballs each pass their spin direction detector. The spin detector display confirmed what we had been seeing. A baseball is a particle, always a particle. A baseball is a particle before its spin is detected, and the baseball is a particle when its spin is detected. Electrons are different from baseballs because electrons have particle/wave duality. An electron is a wave until it is detected as a particle. Unlike particles, waves do not have spin. Therefore, waves do not have an axis for spin. Per particle/wave duality of quantum mechanics, axis for spin of an electron can only become specified when the electron transitions from being an unobserved wave into being an observed particle, a transition that occurs at the detector. Photons, too, have particle/wave duality. Polarization of a photon is a property of the particle. For visualization, imagine the wave of a
279 APPENDIX B  SPOOKY ACTION AT A DISTANCE single photon approaching a polarizing film. Polarizing film is comprised of long stretched molecules that have thin gaps between molecules. The photon’s electric field can only pass those long stretched molecules if the electric field is exactly parallel to the gaps. In contrast, if the electric field is exactly perpendicular to the gaps, then stretched molecules absorb the photon by vibrating back and forth (like a guitar string) due to energy gained by absorbing the photon’s electric field energy. This vibration energy is the energy of the photon, now fully absorbed. And then, the vibration energy is lost to heat. On the back side of the polarizing film is a photographic film detector with an electron that absorbs the photon with a jiggle vibration, if the photon passed the polarizing film to reach it. The jiggle of the electron causes an exposure spot on photographic film. The polarizing film forces the single nondivisible photon to be completely polarized with gaps (such that it passes) or completely polarized perpendicular to gaps (such that it is absorbed). The photon, as a particle, cannot be a combination of perpendicular and parallel, because the photon is quantized as an allornothing particle. Half the photons pass the polarizing film, and half are absorbed. An electron, when used in the EPR experiment, likewise, is a single particle at the detector. Electron spin sensed by the detector forces the spin to become parallel or else antiparallel to the detector device (analogous to the photon electric field being forced to be parallel or else perpendicular to the polarizing film of the detector device) with no intermediate spin direction possible. Lack of the particle property of polarization for photons and lack of the particle property of spin direction for electrons, up until the moment of detection, was a debated issue, until the EPR experiment with Bell’s Inequality settled the debate. Bell’s Inequality is explained with an example. Consider a set of bikes in a large garage. Each bike has (A) an engine or not, has (B) five pounds of fuel or not, and has (C) blue paint or not. Notice that correlated properties may be included. For example, it doesn’t matter that a bike without an engine would naturally not have fuel. Three properties “A”, “B”, and “C” each have values of “Yes” or “No”, for eight combination possibilities (because 8 = 23). Each quantity (QTY) is found by counting.
280 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY QTY1(YYY), QTY2(NYY), QTY3(YNY), QTY4(YYN) QTY5(YNN), QTY6(NYN), QTY7(NNY), QTY8(NNN) (YYN) means A=Yes, B=Yes, and C=No, and (NYN) means A=No, B=Yes, and C=No. If a letter A, B, or C is in the parentheses, then that property can be either Yes or No. For example, QTY(AYY) has A=Yes or No, B=Yes, and C=Yes. If there are ten objects, then the eight quantities listed above sum (as QTY(ABC)) to ten. As an example, consider this set of ten objects: (YNN), (YYN), (YNN), (NYN), (NNN), (YYY), (YNN), (NYN), (YYN), (NNY) QTY1(YYY)=1 QTY3(YNY)=0
QTY5(YNN)=3 QTY7(NNY)=1
QTY2(NYY)=0 QTY4(YYN)=2
QTY6(NYN)=2 QTY8(NNN)=1
Bell’s Inequality is QTY(YNC) + QTY(AYN) QTY(YBN) In words: The quantity of objects with A=Yes and B=No plus the quantity of objects with B=Yes and C=No is greater than or equal to the quantity of objects with A=Yes and C=No. The first term QTY(YNC) has no consideration as to the state of “C”. It can be expanded, as can the other two terms: QTY(YNC) = QTY3(YNY) + QTY5(YNN) QTY(AYN) = QTY4(YYN) + QTY6(NYN) QTY(YBN) = QTY4(YYN) + QTY5(YNN) QTY(YNC) + QTY(AYN) QTY(YBN) QTY3(YNY) + QTY5(YNN) + QTY4(YYN) + QTY6(NYN) QTY4(YYN) + QTY5(YNN) QTY3(YNY) + QTY6(NYN) 0
281 APPENDIX B  SPOOKY ACTION AT A DISTANCE The last statement is true because all quantities, including QTY3(YNY) and QTY6(NYN), are greater than or equal to zero. Therefore, Bell’s Inequality is proven. Using the ten objects of our example: QTY(YNC) = (0+3) = 3 QTY(AYN) = (2+2) = 4 QTY(YBN) = (2+3) = 5 Because 3 + 4 = 7 5, Bell’s Inequality is satisfied. In words per the example: The number of bikes with an engine and less that five pounds of fuel plus the number of bikes with more than five pounds of fuel and no blue paint is greater than or equal to the number of bikes with an engine and no blue paint. In words the inequality is not obvious or intuitive. That is why we need the math. The important point about Bell’s Inequality is that it must be satisfied if the objects have properties. If Bell’s Inequality is not satisfied, then the objects do not have properties. Predicting How Bell’s Inequality Applies to a Baseball. In the plane perpendicular to direction of motion is the center point of the baseball and through that center point is a line for the projection of the axis of rotation onto the plane. Use the righthand rule so that the axis of rotation projected onto this plane has an arrowhead designating the direction of the thumb. From the center point of the baseball, draw the “x”axis and, at ninety degrees, the “y”axis on the plane. Angle “” from the “x”axis to the arrowhead is always known for the baseball because, for a baseball, the direction of the axis of rotation exists even when we are not detecting it.
Figure 46. Properties A, B, C used in Bell’s Inequality.
282 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Figure 47. Combined properties used in Bell’s Inequality: QTY(YNC), QTY(AYN), QTY(YBN), respectively. QTY(YNC) + QTY(AYN) QTY(YBN).
Figure 48. EPR experiment results with baseballs and not with electrons.
“A” is “Yes” if axis of rotation is in the hemisphere of “+y”, 0 < 180o, and “No” corresponds to 180o < 360o. “B” is “Yes” if axis of rotation is in the hemisphere with both “+x” and “+y”, 45o < 135o, and “No” corresponds to 135o < 315o. “C” is “Yes” for “+x”, 90o < 90o, and “No” for 90o < 270o. Each baseball pair is given “Yes” or “No” for each of A, B, and C. Baseballs that satisfy A=Yes and B=No have 135o < 180o (one eighth of a circle). Baseballs that satisfy B=Yes and C=No have 90o
283 APPENDIX B  SPOOKY ACTION AT A DISTANCE < 135o (one eighth of a circle). Baseballs that satisfy A=Yes and C=No have 90o < 180o (one quarter of a circle). One eighth plus one eighth equals one quarter, and therefore Bell’s Inequality is satisfied. We do the experiment (hypothetically) three times. The first set of 1,000,000 has 125,032 A=Yes with B=No. The second set of 1,000,000 has 124,992 B=Yes with C=No. The third set of 1,000,000 has 250,005 A=Yes with C=No. QTY(YNC) + QTY(AYN) QTY(YBN) 125,032 + 124,992 = 250,024 250,005 Bell’s Inequality was satisfied. Because the three numbers were from three different experiment events, there was a chance a statistically explainable variation could have made “QTY(YNC) + QTY(AYN)” slightly less than “QTY(YBN)”. That possibility alerts us to the need to have a very large number of runs in the experiment. Predicting How Bell’s Inequality Does Not Apply to an Electron. The right detector (which is the first to make a detection) reads if the axis of rotation of the electron is parallel with the “+y”axis. The left detector has the same reading as the right detector (after compensating for the expected axis of rotation to be antiparallel).
Figure 49. Both detectors set to “A”. Electrons are detected 50% for “A” equal to “Yes” in both.
Now, in the second phase of the experiment, the left detector (second detector) is rotated so that it reads if the axis of rotation is parallel (or else antiparallel) with “+x”axis. This reading is instantaneously after the right detector (“+y”axis) reading. The word “instantaneous” means a signal at the speedoflight would not have time to reach the second detector.
284 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Per quantum mechanics, the event of reading “A” (if the axis of rotation is in the “+y” direction) forces the electron’s axis of rotation to be exactly in the “+y” direction, or else in the “y” direction, and nothing else. The second detection for reading “C” (on the left, an instant later) forces the rotation axis of the entangled electron to now move parallel or else antiparallel to “+x”axis. “+x”axis is perpendicular to both “+y”axis and “y”axis. Therefore, half the detections for the second detection (on left) read “+x”axis and half the detections read “x”axis.
Figure 50. Left detector set to “C”. Electrons are detected 50% for “C” equal to “Yes”. QTY(YBY) equals one quarter of the total.
Now, as a third phase in the experiment, have the first (right) detector aligned with “+y”axis, and have the second (left) detector aligned with “y”axis. In this case, the left detector reads a “No” for each “Yes” on the right detector. This is an extreme case for which angle “” (phi) equals zero. The general rule is that the second (left) detector has probability “sin2(/2)” of a detection with “” measured from the (opposite) direction of the previous measurement of the entangled pair of electrons. If the two detectors are parallel (the first phase), then “=180o” so that “sin2(/2) = sin2(180o/2) = sin2(90o) = 12 = 1”. If the two detectors are antiparallel (the third phase), then “=0o” so that “sin2(/2) = sin2(0o/2) = 0”. And, if the two detectors are perpendicular (the second phase), then “=90o” so that “sin2(/2) = sin2(90o/2) = sin2(45o) = (1/2)2 = 1/2”. The last two phases of the experiment have “=45o” (so that “sin2(/2) = sin2(45o/2) = sin2(22.5o) = .38268…2 = 0.146447…”) and “=135o” (so that sin2(/2) = sin2(135o/2) = sin2(67.5o) = .92388…2 = 0.85355…”). All particles are accounted for between what is counted as inconformance and notinconformance, per “cos2(/2) + sin2(/2) = 1”.
285 APPENDIX B  SPOOKY ACTION AT A DISTANCE The EPR Experiment with Electrons. Do the experiment (hypothetically) with pairs of electrons. There are three experimental set ups.
Figure 51. EPR Experiment with electrons. Bell’s Inequality is not satisfied because 7% + 7% ≱ 25%.
“73,111” events in “1,000,000” have A=Yes with B=No. The number “73,111” conforms to “1,000,000*0.5*sin2(45o/2)”. The “0.5” factor applies to A=Yes and the factor “sin2(45o/2) = 0.146447…” applies to the subsequent detection for B=No. After that, “73,056” events have B=Yes with C=No. And then, “249,986” events have A=Yes with C=No. We do a statistical evaluation to explain away minor error in the numbers and conclude Bell’s inequality is not satisfied because: QTY(YNC) + QTY(AYN) ≱ QTY (YBN) for electrons 73,111 + 73,056 = 146,167 ≱ 249,986 In the laboratory, Bell’s inequality was not satisfied using photons, electrons, protons, or any other subatomic particles.
286 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Properties Determined at Instant of Detection. We expected Bell’s inequality to be unsatisfied because of the “sin2(/2)” rule, which was derived from the requirement that spin of an electron becomes parallel or else antiparallel with the axis of measurement (which makes an electron different from a baseball). Because Bell’s Inequality is unsatisfied, we conclude there are no hidden variables or other memory mechanisms by which direction of spin of an electron was determined when emitted and simply carried as information to the detectors. It means direction of spin is determined at the moment of detection. A typical emission is one particle, not two entangled particles as in the EPR experiment. When this one particle is detected, perhaps by your eye, or perhaps by a scientific device that we were monitoring, we are unsure if that particle traversed the distance from its emission time and place to the detection time and place as a particle with distinct particle properties (for example, a spin axis direction), or not. When the particle travels it is a wave, and a wave should not be a carrier of particle properties, but subatomic particles are a bit mysterious. Before we had EPR experiment results, it was thought perhaps waves did carry particle properties. And that is because, when detecting one particle, we did not have a means to know if spin was established at the time of emission or at the time of detection. Now, because of entanglement, we know. Instantaneous Information Travel. The second conclusion is that information traveled a macroscopic distance instantaneously. There is a macroscopic distance between the two detectors, and those two detectors each detect spin direction simultaneously. Direction of spin is determined only at the instant of detection, and that means readings on the two detectors are mysteriously coordinated one to the other. It means information traveled from one detector to the other at instantaneous speed, not as fast as speedoflight but faster, faster to the ultimate speed: instantaneous, “v = 1/0”. For the typical situation of one emitted particle (and not two entangled particles), spooky action at a distance is a “collapse of the wave function”, in which the wave disappears from everywhere the particle is not (relative to an observing particle), when the particle is detected. The challenge is to explain spooky action at a distance with a mathematical model.
287 APPENDIX B  SPOOKY ACTION AT A DISTANCE New Theory. An electron and the observed photon from that electron are both oneinthesame particle, per the proposed Theory of Special Relativity with NonFinite Numbers. Entangled particles, too, are one particle: A whole light cone for two entangled photons is oneinthesame as the electrons at its center, relative to an observing particle, until an observation event (or lack of observation event) of a photon is made by an observing particle. More rigor is needed, but the one particle theory has promise in explaining EPR experiment results. It is through the EPR experiment that quantum mechanics is brought to a macroscopic scale, to where we can physically see how strange quantum mechanics is. Therefore, naturally, we want to use the EPR experiment to see with our own eyes any proposed new theory.
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289 APPENDIX C – DISCOVERING AN ABSTRACTION
Appendix C – Discovering an Abstraction About the Author. The author, Paul C Daiber, is a mechanical engineer who designs and services combustion turbine engines. Outside professional employment, the engineer chose to solve this puzzle: Define infinity by finding an inertial frame of reference in which a blue photon has a faster speed than a red photon. An engineering spec/procedure was written for developing new theory: 1) Abstractions. How we understand our world. •
Definitions. A person categorizes patterns in the physical, real, natural world by creating a definition for each identified pattern. Each definition is an abstraction. An abstraction is defined in words or other symbols for people to use.
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Ours. Abstractions as definitions of categories are for our use. We define abstractions as we find useful or otherwise as we want. No external entity gives us our abstractions or defines our abstractions.
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Application and Ambiguity. Definitions of abstractions have some degree of ambiguity. Definitions are valid only to the extent definitions are consistently applied to the real world by the definition’s prescribed rules.
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Names. Abstractions are each named. Each is accompanied by a counterset of abstractions, each of which also get a name.
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Start Simple. Begin development of definitions of new abstractions by first finding patterns that are obvious. Develop more definitions by finding less obvious patterns.
290 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY 2) Discovery. A discovery is outside what can logically be derived from abstractions, used as axioms, we already have. •
Research. Become familiar with what are assumed to be inherently valid abstractions by which we know our world.
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Guess a new definition by investigating an observation, or variations, extrapolations, combinations, tangents, violations, or alternatives.
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Analysis. Build a bridge of logic from old definitions to the guess or from the guess to old definitions by evolving the guess to conform to logic.
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Finish. Write it up to communicate the discovered abstraction to other people. Definitions of abstractions can be communicated and comprehended only by physical, real, natural world example experiences other people have with that abstraction, through the use of prescribed rules of the definition.
The method for discovering an abstraction was inspired from a passage in Morris Kline’s book A History of Mathematical Thought, Volume 1 (Oxford University Press, 1972) in which he paraphrased Descartes who said logic could not be used to investigate unknown fundamental truths. It seemed Descartes thought that thoughts from outside logic were needed when searching for something rules of logic were not designed to find. Elsewhere in that chapter, Morris Kline said mathematicians at the beginning of the age of algebra developed theory without discipline or rigor. In contrast to that era four hundred years ago, algebra of today is full of discipline and rigor. If we were to rethink algebra enough to include infinity in applied mathematics, then, it seemed, we had to remove ourselves from today’s discipline and rigor. Being undisciplined, we grab what we want. We violate an axiomatic rule to create something illogical, because we really want it. (The value of grabbing what is wanted was perhaps learned from Star Trek in the 1960’s: Captain Kirk got what he wanted by leading his crew to do what he wanted, regardless of rules and regulations imposed
291 APPENDIX C – DISCOVERING AN ABSTRACTION on him. The intelligent, unemotional and exclusively logical first officer, Spock, could not lead by violating rules. Therefore, Spock could only be a support person to his captain.) To solve the puzzle of blue and red photons, existing theories of physics had to be learned. Maxwell’s Equations was learned in engineering school using Halliday and Resnick’s textbook. Schrodinger’s Equation had to be selftaught. Fortunately, Eisberg’s and Resnick’s textbook was on someone’s bookshelf at work, and Gasiorowicz’s textbook was found in a usedbook store. Real treasure came from the author’s dad’s college textbooks, from the mid1950’s, which were intercepted on their way to the trash when the author was home from college. In those books was Methods of Theoretical Physics Part I by Morse and Feshbach, McGrawHill Book Company, Inc., 1953. Unlike what was found on the internet, this old book’s development of the Dirac Equation was written for engineers and easily learned. Next, the Dirac Equation and Maxwell’s Equations had to be placed into one algebra. The most effective algebra had been assumed to be geometric algebra in Chapter 6 of Hildebrand’s textbook Advanced Calculus for Applications from college. But geometric algebra didn’t work well. Quaternions were found better because of the ease of writing identities and tracking gauge invariance, and because electric and magnetic fields could be combined into one invariant having complex components. Reading math textbooks became an enjoyable hobby. That might sound strange, but the author was recently with two old college roommates (Gator and Sonny) and they went straight to the math section when the three of them entered the Half Price Books bookstore in the (Walnut Creek, CA) square. Talking in Starbucks (coffee shop) that evening, it appeared, in general: People seek answers to the mysteriousness of the world around them by learning rational and logical techniques of math. Infinity was one such mystery. What was actual infinity? Extensive reading did not find an answer. One of the best books on infinity was Aristotle’s book Physics. He summarized what was suspected about infinity at that time, twentyfour hundred years ago. A generation or two prior to Aristotle, the Pythagoreans had proven irrationality of squarerootoftwo, and that proof forced the Pythagoreans to admit that infinity, as something
292 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY different from finite natural numbers, had mathematical relevance. The Pythagoreans disliked infinity because they attributed to infinity unwanted properties “mysterious, vague, and remote” (these three words are a remembered quote from a book that cannot be found), and, in general, the Pythagoreans disliked the contamination of something proven to be irrational in their system of rational deduction. Archimedes, too, disliked infinity (per his story The Sand Reckoner) as did most of the great men of science in ancient Greece, Aristotle included. The general dislike of positive actual infinity in ancient Greece grew on the author like good advice from a parent. The feeling was that actual infinity existed in our mathematics as a temporary substitute for a theory presently unknown or unavailable to us. Early attempts to learn Cantor’s theory of infinite sets were unsuccessful because Cantor’s real numbers seemed to not include irrational numbers. Other people had seen this issue, with evidence this quote: “Cantor took N1 to be the power of the continuum. This question, however, remains open, and for the present we see no trace of a path to its solution.” (from The World of Mathematics Volume Three by Newman, 1956, Simon and Schuster, page 1599 in the article Infinity by Hans Hahn.) That book opened the door to seeking alternatives to what Cantor had specified to be the real number continuum. One early attempt (by the author) gave an irrational number a positive infinity of placevalue digits after the decimal point, and then unwritten and unspecified placevalue digits after that. It was close to what became localreal numbers, but at the time it led nowhere. In the last two years of the project, the author’s son’s college pure mathematics textbook, written by Chartrand and others, was found in a box in the basement, and Cantor’s theory could finally be learned as a full and complete theory. Once the bigger picture of what Cantor intended was learned, then his theory could be adapted into what was becoming the solution to the blue/red photon puzzle. A second lesson from Cantor was that he grabbed a definition for infinity, and then built a bridge of logic to it. His process was good. Long before Cantor’s theory was understood, at a time when there wasn’t much hope of success, an idea for infinity came from the phrase “forever and a day”. Reciprocalofzero was forever. To add one to it, hyperbolic trig functions were used: “cosh(1/0) = 1 + 1/0” and “sinh(1/0) = 1/0”. From that guess Maxwell’s Equations were derived from the Dirac Equation. It was the first big break.
293 APPENDIX C – DISCOVERING AN ABSTRACTION “1/0” meant the mathematics was invalid, but, but, there was truth down inside it, truth somewhere deeply hidden. Suspected truth became certain when the correct electromagnetic field force density was calculated using the complex conjugate. A burden of responsibility was felt, because, if a bridge of logic did not get built, then this discovery would likely become lost and forgotten, and that was because anyone who cared would not get to hear about it because any mathematics dependent on division by zero cannot be published. Division by zero is famous as a tool of deception and is called “pseudomathematics”. If division by zero could not be removed, then, most likely, the discovery would not be stumbled upon again for many, many years, perhaps for as long as it had already lay hidden, from when the Dirac Equation was understood (1930) to then (2010). Formal prohibition against division by zero is in the definition of rational numbers, and that meant, maybe, “1/0” and “0/0” could be changed into the wanted infinity by investigating irrational numbers. “2” was ripped apart every way possible. After some years the focus changed to “” (zeta) in “1 = 1 + 5” and “1 = 2 + 3”, which were found by equating lengths thirty degrees apart on the equilateral triangle spiral. A rational approximation to “” was calculated for each of the two equations. Was the portion notyetcalculated the same with regards to the geometric construction of the triangle spiral? It took many years to develop that question, and even longer to answer it. It was guessed the two “”’s could only be precise to a positive actual infinity quantity of placevalue digits. Their difference was called real number zero. The unknown placevalue digits starting at the actual infinity count meant real number zero was different from integer zero, and that meant it could be a denominator. The division reciprocal of a real number zero became the infinity that applied to the hyperbolicangle for motion at the speedoflight. This was another moment of celebration. With division by integer zero gone, the theory seemed complete enough to actually tell someone. A downloadable early version of this book (titled: Infinity Applied to Special Relativity) was placed on the internet, to make it available to the public in case the project never finished. The book told a somewhat logical story from beginning to end but it lacked key essentials, for example, it lacked the exotic Lorentz Transformation needed to model
294 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY motion at the speedoflight. It wasn’t advertised and no copies sold, and tallies showed only a person or two reviewed the first several free pages. Real number zero became “localzero” when Aristotle’s potential infinity, a finite number, replaced positive actual infinity in the count of placevalue digits. With no actual infinity in the numbers, there was worry the proposed system of numbers had deviated too far from what was generally accepted as numbers. At the very end of the project a new proposed axiom substantiated the removal of positive actual infinity, and the bridge of logic was completed. Per the method, we guessed at an abstraction (which was the combination of Maxwell’s Equations into the Dirac Equation using division by zero), and then ideas evolved until the core idea was found (which was the proposed new reciprocalofzero axiom). To communicate the discovery in a logical progression, per the method’s last bullet, reverse activities (with the proposed new reciprocalofzero axiom explained first, and the derivation of Maxwell’s Equations from the Dirac Equation last). But reversed activities in the communication erased the process of discovery that was actually followed. To not lose that excitement, with its guesses, its stress, finding critical books, and its little victories, this “About the Author” was written. Early versions for this math book had little more than an initial guess at what infinity could be. A guess cannot be the logical foundation of a math book, regardless of how strongly a person claims the guess is correct. Discipline came from Paul’s brother Andy J Daiber, another engineer. Andy was/is a hobby mathematician, but Andy was not someone like the author who searched for playgrounds in numbers. Rather, Andy was someone who appreciated the serious core values pure mathematicians have. (Andy sent the book by Mary Tiles, the Teach Yourself book on group theory, Paul Cohen’s Set Theory and the Continuum Hypothesis, and over the years he sent maybe a dozen other pure mathematics books to the author.) The author is grateful to Andy for reading this book every several years, when the book was incomplete and illogical. Andy mercilessly shot down every idea presented without substantiation. Our human tendency is to think our claims are correct, and to defend our claims when subjected to criticism. In mathematics, only logic can be a defense. When digging deeper to find fundamental evidence to prove one’s claims, we stumble upon fantastic discoveries that are unexpectedly interesting and useful. That generalization of an
295 APPENDIX C – DISCOVERING AN ABSTRACTION interactive process summarizes the human side of the development of the proposed Theory of NonFinite Numbers. The attitude, or the sense of values, that fundamentally led to the need to write this book came from something said by a Citizenship in the Community merit badge counselor, Mr. O’Brien, paraphrased as: Don’t bother reading something unless you intend to do something with that knowledge. Now, forty years later and no longer a child, this rule for living is changed to: Success in life is not measured by how much knowledge you take to the grave with you, but rather by what good you did with that knowledge. I hope some good comes out of this effort to insert infinity into applied mathematics. What I really want is for your curiosity to have become ignited, and for you to further grow theory for nonfinite numbers. And, I hope you enjoyed reading this book. \\//,
The Storybook In its first draft, storybook Alien Invasion Math Story was a collection of essays and calculations written during the hunt for a useful infinity. The storyline, with its exaggerated emotional extremes, portrayed the confusion, frustration, misdirection and stress felt in the hunt. Inspiration came from the movie Race for the Double Helix (BBC), in which Crick and Watson mixed incomplete bits of data (from Franklin) with staggered inspiration. They persevered past criticism and little failures to win the race. Their moment of success came with a revelation. Me, sitting at a desk or standing in a store watching my wife shop while thinking through proofs is not nearly as exciting as Crick and Watson in their quaint apartments and labs, so my storyline is fiction. But, at least the math and the essays on human will are real. And, for the revelation at the climax, well, you’ll just have to read it. \\//,
296 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Glossary Actual Infinity. Aristotle’s completed infinity. An infinity that is larger than finite. Aleph Null. Cantor’s version of the completed/actual infinity to which the Continuum Hypotheses is assigned. Algebra. A set of rules by which logic can be exercised. Algebra Field. Formed by two groups and a distributive property. A set of elements and the criteria by which operations on those elements are performed. AllNumber. An expression formed from quantity numbers, labelnumbers, the unspecifiedspeedparameter, and the unspecifiedlabelnumber. AntiMatter. The negative of matter. If matter is a something, then antimatter is the hole created by the removal of that something from nothing. Applied Mathematics. The adaptation of mathematics to mathematical models of physics. Argument. The independent variable on which a function operates. Axiom. A fundamental assumption that is accepted as a basis for subsequent derivations. Axiomatic Set Theory. The mathematical theory of a set of axioms from which algebra is derived. ZFC axiomatic set theory is one form and is most popular. Axiomatic set theory was developed in the early 1900’s. BiotSavart Law. The mathematical model of the electromagnetic field of a moving point particle. Both Plus and Minus But Also Neither Plus Nor Minus Separately. (N) The plus or minus sign is unknown and unknowable. It is in contrast to the plus or minus sign () because the implication of the plus or minus sign is it is known or knowable. Cantor’s Continuum Hypothesis. The axiom of axiomatic set theory from which the real numbers are derived. It forces the existence of an actual infinity. Cantor’s Theory of Infinite Sets. A theory of pure mathematics in which an actual infinity is assumed to exist between finite numbers and the reciprocal of the integer zero. Cardinality. The quantity of members in a set. A number is cardinal if it is the count of members in a set. This definition is part of Cantor’s theory of infinite sets. Cartesian Grid. The typical xy plot. Descartes used the Cartesian grid to plot roots of polynomials. Cause and Effect. The name for related events that are sequential in time. Collapse of the Wave Function. When an observation of one particle by another particle occurs, the perception of the observed particle changes from being a wave that is spread through space and time to being a particle that has one location in space and time. This
297 GLOSSARY AND INDEX transition from being everywhere to being in one place is informally called the collapse of the wave function. ComplexConjugate. An alternate representation of a complex or hypercomplex number. The complexlabelnumber “i” is made negative (and the order of factors and terms is reversed but only after the quaternion simplelabelnumber factors in each term are reduced to only one quaternion simplelabelnumber). Component. The factor of term in an expression that is not a direction indicator. A component is often measurable. CompoundLabelNumbers. LabelNumbers that do have a factor for gauge invariance. For a singleterm summationform allnumber expression of an invariant, the compoundlabelnumber is the same as a simplelabelnumber. Conjugate. An alternative allnumber expression. multiplication of two allnumber expressions.
The conjugate form is used in
Conservation Law. A mathematical expression that specifies how a material is created or destroyed. Continuum. The property by which spacing is not finite, but, rather, is smaller than finite. Contravariant. The alternative coordinate system representation for use with nonrectilinear or nonunitmagnitude geometricunitvectors. Cosmological Model. Our logical model of the physical universe (plethora). Count. A finite natural number. Countable. A set that has a onetoone member correspondence with the natural numbers, in Cantor’s theory of infinite sets. Crude NonNumber Algebra. The name given to the proposed algebra for the reciprocal of the integer zero based on the proposed new reciprocalofzero axiom. de Broglie Relations. Total energy equals the modified Planck’s constant times the frequency. Total momentum equals the modified Planck’s constant times the wavenumber. Dedekind Cut. A cut in the numberline for which the highside and lowside numbers at the cut, and numbers between them, are all “essentially” the same number. Per Dedekind: “From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.” Degradations. The breakdown in symmetry for higher orders of hypercomplexity. Denumerable. Same as Countable. Descartes. Mathematician and philosopher who marks the beginning of the modern era of intellectual thought. Dirac Equation. The relativistic mathematical model for the dynamics of an electron.
298 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Dirac Spinor Solution. Any of the four Dirac Spinor Solutions presented in this book. Distributed Material. A material that exists as a distribution in space and time such that a gradient exists. Distributed material may also be called a “field”, but “field” is so general a term it should be used cautiously. Distributed Material Theory. The theory (in this book) for which continuum operators, such as the gradient differential operator, may be applied. EMCompoundLabelNumbers. The labelnumbers used for a sixterm invariant. Enabler Functions. Variables inserted into an equation to accomplish a purpose. EnergyMomentum. The invariant that combines energy (as the time term) with momentum (as the space terms). EngineeringCalculation Algebra. Algebra designed to be as efficient as possible in the task of making a calculation. Entropy. The amount of disorder in the universe. The amount of entropy in the universe increases with time. EPR Experiment. The EPR experiment verified, through the use of Bell’s Inequality, that the collapse of the wave function occurs at the moment of observation and not at the moment of emission of the particle that is observed such that the properties of the particle do not exist prior to the observation. As a secondary result there is the experimental verification of “Spooky Action at a Distance”. Euclid’s Textbook. A textbook for geometry (and, to some extent, numbers) written at the beginning of the Hellenistic age (after Alexander’s conquests) that survived as the mathematics textbook until mathematics was reformulated by the work of Descartes and other contemporaries in the 1500’s using the Arabic numerals previously introduced to Europe by Fibonacci. Euler’s Constant. The difference between the sum of reciprocal positive integers and the natural logarithm of the largest positive integer in the sum plus one. Euler’s Equation. Relates the exponential function to the trigonometric functions. Exotic Lorentz Transformation. A Lorentz Transformation that uses a hyperbolicangle that is not a real (or rational) number. The exotic Lorentz Transformations take advantage of the labelnumbers in the allnumber algebra. Exponential Function. The single valued equivalent of a base of “e” to an exponent. The exponential function has a polynomial expansion to infinity as its definition. Factor. A factor in a mathematical expression is separated by multiplication and division signs, such that factors are multiplied. Feynman. A theoretical physicist who contributed extensively to QED and to the theory of electrons and photons in general. Feynman’s fame is perhaps due to his ability to present concepts through visualizations and explanations that were easily comprehendible by his intended audience.
299 GLOSSARY AND INDEX FinalResult. The third step of the Process from Descartes of a localreal number algebra operation applies the condition that the result be a localreal number (with “Lmax” maximum count of known or knowable placevalue digits before or after the decimal point). The finalresult applies to what can be measured in geometric space, with examples being the components of invariants and speed. FinishedCalculation. The second step of the Process from Descartes of a localreal number algebra operation uses probability theory to find the “finishedcalculation”. Finite. A finite number is a number from the set of natural numbers (defined by starting at one and repeatedly and unboundedly adding one), or a finite number is a number that is a result of a binary operation (addition, subtraction, multiplication or division) performed using a natural number or other finite number, per the algebra field for rational numbers. A finite (irrational) number is bounded high and low by two rational numbers. Gauge. A reference against which a perspective is made. GeometricUnitVector. One of the four geometric entities that represent mathematically a direction in space or the direction of time. Governing Equation. A mathematical model for a theory of physics. Gradient Operator. The time and space differential operator. Group. A set and the operations performed on that set, as a basis for unambiguous derivations using that set and those operations. Handedness. The fingers curl with the geometric feature and the thumb points a direction for the geometric feature. If it is the right thumb, then the handedness is righthandedness. If it is the left thumb, then the handedness is lefthandedness. In geometry the handedness is the correlation between the nomenclature of x, y, z relating to the fingers passing through positive x to positive y and the thumb pointing to positive z. In physics matter can have a geometrically righthand or a lefthand spin but, more typically, a reverse of handedness refers to a reverse of parity to the extent that matter becomes antimatter with the example being a righthand glove (matter) has the same appearance as a lefthand glove that is turned inside out (antimatter). Hypatia. Woman teacher at the end of the Hellenic / Hellenistic era. She contributed to the mathematics of conics and had other contributions to science and mathematics. Her murder marks the end of the era of the thousand years of Greek intellectual thought. HypercomplexConjugate. An alternate representation of a hypercomplex number. The quaternion simplelabelnumbers are made negative and the order of factors (and of terms) is reversed. HypercomplexPlane. Similar to the ComplexPlane, but with one of the two dimensions imaginary rather than real. An illustration of timespace on the spacespace of a sheet of paper. What is a circular rotation in timespace is a hyperbolic rotation in spacespace and so hyperbolicangles are illustrated in the hypercomplexplane using circular angles and that substitution causes a distortion in the illustration of a hypercomplexplane. Hypothesis. An educated guess.
300 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Identity. A mathematical statement that is true as a proven theorem and is stated generically without reference to a mathematical model of physics. Identity Elements. The zero for addition and the one for multiplication. Imprecision term “” (xi). Either a localzero or else a localinfinity with the selection unknown and unknowable. It is added to a rational truncated number to form a localreal number. Inertial Reference Frame. A constant speed reference frame. The word “inertial” is used because a reference frame that is accelerating because it is falling in a gravitational field may be approximated as a constant speed reference frame. Infinitesimal. The division reciprocal of infinity. Infinitely small. Infinity. Not Finished. No Finality. Unbounded. No limit. Infinity Relation Equations. The first pass function equations using a real number (or positive real number) zero in the argument. Infinity Summation Equation. A sum of reciprocal natural numbers that extends in the count of terms beyond the largest natural number. Invariant. A mathematical expression for something that is physically real, such that the expression cannot change when the physically real something is observed from a different vantage. Irrational Number. A number proven to not be rational. Kinetic Energy. The energy due to particle movement. Known or Knowable. Known or else knowable in that it may have a definite value. LabelNumbers. Numbers of unit magnitude (or zero magnitude) that represent direction rather than quantity. Length Contraction. The reduction in length when the object is moving relative to an observer. LocalInfinity. The division reciprocal of a localzero. LocalReal Number. The truncated number portion has the precision of the number limited to a finite maximum count of placevalue digits after the decimal point in base two. The same limit is before the decimal point for the truncated number portion. A localzero or else a localinfinity is added to the truncated number to form a localreal number. LocalZero. “d = b  b” for the value of each placevalue digit after a count of “Lmax” after the decimal point (in base two). Lorentz Transformation. The method for changing component and compoundlabelnumbers (or geometricunitvectors) for a different inertial reference frame. Macroscopic. The scale of objects for which quantum effects are negligible.
301 GLOSSARY AND INDEX Many Worlds Hypothesis. In traditional quantum mechanics there is only one reality with that reality being the result of particle to particle observations. It seems reasonable to assume that each observation is not unique in that whenever an observation occurs there is a split in reality such that another universe is created in which the opposite observation occurred. The repeated formation of these other universes forms the many worlds of the Many Worlds Hypothesis. Mathematical Model. A logical model for something physical. Typically, a mathematical model is used to make a prediction of a measurement or other definite observation. Equation relating quantitative values with measurement units that can be used to predict the results of an experiment or other observation in the real, natural, physical world. Matrix Equation. An equation that uses a matrix and column vectors. Matrix Isomorph. A matrix that has the same mathematical properties as a labelnumber. A matrix isomorph is used to find the mathematical properties of a labelnumber. A matrix equivalent of a labelnumber such that the set of matrices corresponding to a set of labelnumbers has the same behavior in an algebra field. Matter. The stuff of the universe. MatterWave. A wave associated with a particle per the theories of Schrödinger’s Equation and the Dirac Equation. The alternative existence of a particle when the particle is not being observed. The matterwave interference pattern with itself forms the constructive interference group. The particle location is most likely at the peak of the group. Maxwell’s Equations. The mathematical model for the electric field, the magnetic field, and electric current. The set of first order differential equations that mathematically model the electric field and the magnetic field formed by a static or moving electric charge density field. Maxwell’s Wave Equation. mathematically modeled.
The equation by which electromagnetic radiation is
Natural Numbers. The numbers formed by starting at the number one and adding one repeatedly and unboundedly. The set of natural numbers “N = {1, 2, 3, …}” is the set of numbers used for counting. Newton’s Second Law. Force equals mass times acceleration, or, as Newton said it, force equals the time derivative of momentum. NonLinear. Has products of independent variables. NumberLine. A geometric model of the real numbers. Numerator and Denominator. The fraction 5/7 has 5 as the numerator and has 7 as the denominator. Observer. In Special Relativity the properties of a particles are specified relative to the observer particle that the particle interacts with. The observer particle is personified as a person, as us with our measurement equipment.
302 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Octonions. Similar to the complex number factor but pertain to the seven directions of ultraspace. Particle. An existence with location (and time), momentum (and energy), and angular momentum. A particle may extend over a region of space, but, typically, a particle is approximated as a point particle. Pascal’s Triangle. A pyramid of rows formed by adding the two numbers immediately above, after starting with one: 1, 1 1, 1 2 1, 1 3 3 1, 1 4 6 4 1, etc. Pauli Spin Matrices. The Pauli Spin Matrices are 2x2 matrices used by Dirac in the development of the Dirac Equation. PlaceValue Digits. Placevalue digits are the integer numbers used before or after the decimal point. The value of the integer number is based on its placement relative to the decimal point, per a power series notation equivalent to placevalue digit notation. Polynomial. An algebraic summation in which the independent variable has an integer exponent in each term with the integer exponent typically nonnegative. Potential. A quantification of a sum such that the gradient of the potential is a vector field. Potential Energy. The energy due to location in a potential field. Examples are a gravity field, a pressure field, an electric voltage field, and a magnetic field. Potential Infinity. A finite number that is either so large that its value is not relevant, or else a finite number that increases instantly and unboundedly. Defined per Aristotle Process from Descartes. The threestep process that begins and ends with geometry and has allnumber algebra for the analysis in the middle. It was pioneered by Descartes in what he called analytic geometry. The process of translating a geometric problem into the abstract for analysis and then translating the result back into geometry as a prediction for a measurement. Product. The result of factors. Property. A partial definition of something. PseudoMathematics. The hidden use of the reciprocalofzero to prove true something that is not true. PseudoVector. A vector in a plane such that is depicts a rotation in the plane and not a direction perpendicular to the plane. Pure Mathematics. A system of axioms and proofs. Pure mathematics is in contrast to applied mathematics because in applied mathematics the axioms must be applicable to models of our physical, real, natural world. Pythagorean Theorem. The sum of the squares of the lengths of the two perpendicular sides of a right triangle equals the square of the length of the hypotenuse side. Quantity. A quantity is different from a count because a quantity may be an actual infinity in value. A count is limited to being finite.
303 GLOSSARY AND INDEX Quaternions. Similar to the complex number factor but pertain to the three directions of space. Multivalued square root of negative one values first proposed by Hamilton in 1843. Rational Numbers. The numbers formed as ratios of finite natural numbers. Real Numbers. The numbers that are not imaginary or complex (per Descartes), or else, the numbers that have the infinite quantity aleph one over an interval (per Cantor). ReciprocalofZero Axiom. Proposed in this book. A new proposed axiom for numbers that has its basis in the proof of irrationality of a logarithm. The purpose of this axiom is to bring axiomatic set theory into applied mathematics. Reciprocal of the Integer Zero. The reciprocal of the integer zero exists as an abstract concept for the largest magnitude number for both positive numbers and negative numbers. The nonnumber reciprocal of the integer zero is the neverending end of number magnitudes. Recursive. A recursive process has repeated steps. For example, the natural numbers are created through the recursive process of repeatedly adding one after starting at the number one. RemnantProduct. The real portion of the product of three fourcomponent timespace invariants. Reverse Parity. The complete reversal of a particle, as if the particle were a glove and the glove was insideout. A particle that is reverse parity of matter is antimatter. Rest Mass. The hyperbolicradius of the energymomentum invariant. Rest mass is thought of as the stuff of our macroscopic world, but there is no stuff of rest mass. Rather, we can think of rest mass as the fields inside a subatomic particle. Root. The root of an equation is a variable value for which the equation is satisfied. The term “square root of two” is the positive root of the equation “x2 – 2 = 0”. Scalar. Having only one term. Schrödinger’s Cat. Until we open the box, we do not know if Schrödinger’s Cat is dead or alive: Schrödinger’s Cat is a cat in a box with a glass flask of poison and with a radioactive atomic nucleus. Inside the box is a radioactive nucleus. If the nucleus decays, then the emitted particle breaks a container of poison and the cat dies. An observer outside the box does not know if the nucleus has emitted a particle to break the flask and kill the cat, or hasn’t. Relative to the observer the cat is both dead and alive as a state that is both and so is neither dead nor alive separately. The decay both happened and did not happen until the decay is observed by the box being opened. The state of the cat becomes known relative to the observer when the box is opened. Schrödinger’s Equation. The nonrelativistic predecessor to the Dirac Equation as a model for the dynamics of an electron using a wave as the intermediate observed form of the electron. It is a second order ordinary differential equation and so is quite different from the first order partial differential equation set by Dirac.
304 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Sedonions. Similar to the complex number factor but pertain to the fifteen directions of ultraultra space. SimpleLabelNumbers. LabelNumbers that do not have a factor for gauge invariance. SingularLabelNumber. A labelnumber of zero magnitude. A labelnumber that results in zero when multiplied by its hypercomplexconjugate. Singularity. A division by zero. In physics, it is where a division by zero occurs in a theory for a phenomenon, with the popular example being the center of a black hole of General Relativity. Six Dots. Repeats as the zeros repeat after the decimal point of an integer. The items are of a quantity that cannot be counted, and, therefore, cannot have individual properties and are addressed as a quantity in bulk. The quantity is not finite and is assumed to equal the nonnumber reciprocal of the integer zero. SpaceLike Invariant. A fourterm (or twoterm) invariant that has only a space term for the case of zero speed. SpaceNegative. The operator by which the Lorentz Transformation is the inverse matrix of the normal Lorentz Transformation, and, to compensate, the threespace labelnumbers are negative. Normally used for the gradient operator and normally used by convention for antimatter. An explicitly written marker that identifies an invariant or other mathematical entity as requiring an inverted matrix for the Lorentz Transformation and as requiring a negative on the quaternion compoundlabelnumbers (that is, on the space labelnumbers and not on the time labelnumber). Special Theory of Relativity. Einstein’s theory for there being no preferred inertial (constant speed) reference frame with the condition that the speedoflight be the same for every inertial reference frame. Einstein’s Special Theory of Relativity is based on the premise that there is no preferred speed of an observer and the speedoflight is the same for all observers. SpeedofLight. The speed of light over long distances in a vacuum. It is a unit conversion factor from time measurement units to space measurement units. Spin. The property of a particle by which it has angular momentum. The spin of a particle is in units of Planck’s constant. Spin may be positive or negative. Spooky Action at a Distance. Einstein’s name for the coordinated polarization (or spin) of two macroscopically separated entangled photons (or electrons) of the EPR experiment. This is Einstein’s name for the particle properties of two entangled properties to become specified at two locations that are a macroscopic distance apart. Sum. The result of terms. Term. A term in a mathematical expression is separated by plus and minus signs, such that terms are added. Theory. In physics a theory is a logical or otherwise mathematical model that is generally accepted as valid for predicting measurements within set limits. In mathematics a theory is a branch or subset of mathematics
305 GLOSSARY AND INDEX TheoryDevelopmentAlgebra. Algebra designed to explicitly represent in symbols the subtle aspects of the physics being modeled and designed explicitly to be as abstract as possible. Theory of NonFinite Numbers. Proposed in this book. The theory of numbers based on the proposed new reciprocalofzero axiom in which numbers have an unknown and unknowable portion. Theory of Special Relativity with NonFinite Numbers. Proposed in this book. The application of the proposed Theory of NonFinite Numbers to the existing Special Theory of Relativity. Three Dots. The three dots represent a count using the finite natural numbers. Each and every item being counted is individually addressed in the count. Time Dilation. The reduction in the rate of a clock’s mechanism when the clock is moving relative to an observer. TimeLike Invariant. A fourterm (or twoterm) invariant that has only a time term for the case of zero speed. TimeSpace. Also called “spacetime”. It refers to the union of time with space as a fourth dimension. Trigonometric Function. exponential function.
Sine and Cosine and other functions, as derived from the
TripleVectorProduct. The imaginary portion of the product of three fourcomponent timespace invariants. Truncated Numbers. Rational numbers of limited placevalue digits. Uncountable. A set that has a quantity of members that cannot be counted because the quantity is greater than the quantity of members in the set of natural numbers, per Cantor’s theory of infinite sets. Unknown and Unknowable. The value cannot be known because the value does not exist relative to the observer. The analogy is Schrödinger’s Cat. UnspecifiedLabelNumber. A simplelabelnumber that is unknown and unknowable but is restricted to being one of the three simplelabelnumbers and not a combination of them. The unspecifiedlabelnumber “” (kappa) is unknown and unknowable and is restricted to being exclusively one of the simplelabelnumbers “qx”, “qy”, or “qz”. UnspecifiedSpeedParameter. A hyperbolicangle that is unknown and unknowable. It differs from an independent variable “x” as an unknown because “x” is unknown but is knowable. The unspecified parameter “” specific to use in compoundlabelnumbers as a hyperbolicangle to ensure Relativistic gauge invariance (that is, to ensure there is no preferred inertial reference frame). Vector. Having multiple terms. Voltage. The potential for the electric field. The voltage as a vectorspace field includes the potential for the magnetic field.
306 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY Wave Group. A wave group is a constructive interference location of a set of individual waves. The wave group moves at a speed that may be different from the speed of the crests of the individual waves, if the waves themselves have different wave crest speeds. World Line. The path of trajectory of a particle or other object on the hypercomplexplane or the fourdimensional version of the hypercomplexplane. Or, in general, the path of a particle through fourdimensional spacetime with that world line a continuum of time and space coordinates. ZFC Set Theory. The comprehensive set theory used in mathematics, evolved from Cantor’s theory of infinite sets and other beginnings. Uses specific axioms as its bases. A field of pure mathematics
307 GLOSSARY AND INDEX
Index aleph null, xxi, 197203, 296 Ampere, 923 Android, 71 angular momentum, 55, 846, 88, 160, 169, 184, 2457, 278 antiassociative, ix, 714, 2589, 275 anticommutative, ix, 714, 176, 227, 25860 antimatter, 6471, 75, 81, 84, 88, 130, 156, 1629, 180, 1815, 252, 255, 296 Archimedes, 292 area differential operator, 1467 Aristotle, 291 axial vector, 1920, 29 axiom of infinity, x, 201 axiomatic set theory, x, 2001, 296 beautiful, 107, 136, 156, 244 Bell’s Inequality, 277, 27986, 298 Bianchi, Eugenio, 253 BiotSavart Law, 1168, 124, 155, 296 black hole, 251, 253, 304 Bronstein, Matvei (Planck length), 253 Cantor, ix, x, 195203, 248, 251, 253, 292, 296 Cantor’s Continuum Hypothesis, ix, 187, 196, 2003 cardinality, 198, 296 Cohen, 201, 294 collapse of the wave function, 71, 88, 2512, 287, 296 complex numbers, 4, 6, 8, 112, 14, 234, 29, 3334, 106, 164, 169, 178, 234, 260, 291. 297 continuum, xii, 72, 195, 198, 204, 292, 297, 306 contravariant, 18, 19, 23, 278, 78, 297 crossproduct, 3, 6, 1517, 26, 29, 89, 99, 136 crude nonnumber algebra, 191, 202, 249, 250, 251, 297 de Broglie, 160, 217, 297 Dedekind, 1978, 203, 297 denumerable / countable, 197, 198, 297 Descartes, 1, 8, 253, 290, 297 determinant 20 differential geometry, 78 dotproduct, 3, 1517, 26, 29, 76, 89, 99, 109 Einstein, 37, 46, 58, 89, 196, 251, 277, 304 EMcompoundlabelnumbers, 96
308 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY enabler functions,162, 165, 230, 298 Entropy, 88, 251, 298 EPR Experiment, 88, 254, 27787, 298, 304 Evert, Hugh, 252 exponential function, 6, 21, 24, 42, 136, 191, 2069, 298 Faraday, 912, 156 Feynman, 64, 246, 298 finalresult, 1, 204, 206, 209, 247, 298 finishedcalculation, 1, 204, 206, 209, 247, 299 finite, 187, 299 force, 23, 679, 122, 132, 159, 238, 245, 252, 301 Galileo, 36 Gauss, 912 General Relativity, 68, 78, 251, 253, 304 Gödel, 200 Hamilton, 4, 257, 303 Hawking / Bekenstein, 253 Heaviside, 4, 89, 118 Hurwitz, 273 Hypatia, 36, 299 imprecision term “” (xi), ix, 193, 2067, 209, 229, 300 irrational, xxi, 8, 1878, 192, 1978, 203, 249, 2913, 300 Kline, Morris, 290 length contraction, 478, 300 localreal numbers, xii, 187, 195, 203, 244, 249, 300 localzero, xii, 1935, 2045, 294, 300 localinfinity, xii, 1935, 2045, 294, 300 Lorentz Transformation ix, 33, 36, 44, 108, 175, 300 Lorenz Condition, 98, 100, 103, 155 magnetic moment, 246 matrix isomorph, 5, 114, 20, 31, 260, 301 nonRelativistic, 39, 42, 74, 118, 157, 304 Pascal’s Triangle, 19, 23, 29, 144, 302 Pauli Spin Matrices, xiii, xv, 5, 31, 165, 260, 302 Planck’s constant, 160, 253, 298, 302 polar vector, 19, 23, 29 Poynting Vector, 120, 128, 135 Process from Descartes, 1
309 GLOSSARY AND INDEX pseudovector, 19 QED, 246, 298 quaternion hypercomplex numbers, 431, 25775, 303 rational numbers, ix, 8, 188, 303 real numbers, x, 8, 197203 reciprocalofzero axiom, x, 189, 294, 303 remnantproduct, 179, 1002, 303 Rovelli, Carlo, 254 Schrödinger, xii, 169, 194, 301, 303, 305 six dots, xii, 1878, 192, 194, 304 spacelike, 4855, 58, 88, 304 spacenegative, 7283, 110, 146, 157, 1613, 168, 228, 304 spin, 55, 846, 129, 169, 184, 2457, 253, 27787, 304 spiral waves, 127, 167 Spooky Action at a Distance, 58, 27787, 298, 304 superpotential, 1057 tau particle, 124 Theory of NonFinite Numbers, 187, 305 Theory of Special Relativity with NonFinite Numbers, 209, 244, 305 Tiles, Mary, 201 time dilation, 46, 305 triplevectorproduct, 179, 23, 28, 1001, 156, 305 truncated numbers, xii, 192195, 2489, 305 uncountable sets, 198200, 305 unspecifiedlabelnumber, 95, 305 unspecifiedspeedparameter, 389, 72, 88, 2056, 305 vectorfront multiplication, 227, 262 vector identities, 101, 103, 152 vectorspace, 2678 volume, 18, 20, 23, 121, 14850 Wolchover, Natalie, 201 Worldvolume, 18, 151 ZFC, 201
310 SPECIAL ALGEBRA FOR SPECIAL RELATIVITY
Back Cover Applied mathematics algebra without positive actual infinity Maxwell’s Equations unite with the Dirac Equation to combine electron dynamics with photon dynamics because, by use of the algebra, an electron projects itself as a photon. Electron/photon double existence derives from Schrödinger’s Cat because each placevalue digit of a real number beyond a finite maximum in count is unknown and unknowable, analogous to the cat being both alive and dead inside its unopened box. The algebra is derived from a proposed axiom that replaces Cantor’s Continuum Hypothesis. Empirically derived energy density and the Poynting Vector unite in the force density invariant as one mathematical model. That unity suggests quantities in our geometric world actually do have finite imprecision, and that the new algebra applies to more modern theories of physics. Visualizations and exercises help comprehension. The mathematics is simple enough to be understood by a high school student who has taken first year level college math and physics classes (and is familiar with trigonometry and logarithms, complex numbers, matrix multiplication, geometricunitvectors, and partial differential equations). One particle at two places violates a preconceived notion that that isn’t possible. The one particle is material (fermion electron) and, its opposite, force (boson photon). Take this radical notion further by supposing perceived reality results from numbers, alone from objects, interacting by becoming more precise with respect to each other, to form patterns we see as the Dirac Equation and other mathematical models of physics. The universe is fundamentally numbers.