Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics 3031237617, 9783031237614

Table of contents :
Prologue
Contents
1 Preliminary Introduction
References
2 Basic Definitions and Statements of the Main Results
2.1 Generalized-Lorentz's Structures with Time Direction and Global Time
2.1.1 Pseudo-Lorentzian Coordinate Systems
2.2 Kinematical Lorentz's Structure with Global Time
2.3 Kinematical and Dynamical Generalized-Lorentz Structures with Time Direction
2.4 Lagrangian of the Motion of a Classical Point Particle in a Given Pseudo-metric with Time Direction
2.5 Lagrangian of the Electromagnetic Field in a Given Pseudo-metric
2.6 Correlated Pseudo-metrics
2.7 Kinematically Correlated Models of the Genuine Gravity
2.8 Lagrangian for Dynamical Time Direction and Its Limiting Case
2.9 Lagrangian of the Genuine Gravity
References
3 Mass, Charge and Lagrangian Densities and Currents of the System of Classical Point Particles
4 The Total Simplified Lagrangian of the Gravity in a Cartesian Coordinate System
4.1 The Total Simplified Lagrangian in (2.9.23), (2.9.24), for the Limiting Case of (2.9.20) in a Cartesian Coordinate System
5 The Euler-Lagrange for the Lagrangian of the Motion of A Classical Point Particle in a Cartesian Coordinate System
6 The Euler-Lagrange for the Lagrangian of the Gravitational and Electromagnetic Fields in a Cartesian Coordinate System
6.1 The Euler-Lagrange for the Lagrangian in (4.1.71) in a Cartesian Coordinate System
7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate System Which Is Cartesian and Inertial Simultaneously
7.1 Certain Curvilinear Coordinate System in the Case of Stationary Radially Symmetric Gravitational Field and Relation to the Schwarzschild Metric
References
8 Newtonian Gravity as an Approximation of Our Model
8.1 Newtonian Gravity as an Approximation of (6.1.52)
9 Polarization and Magnetization
9.1 Polarization and Magnetization in a Cartesian Coordinate System
10 Detailed Proves of the Stated Theorems, Propositions and Lemmas
Appendix Some Technical Statements

Citation preview

Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics

Arkady Poliakovsky

Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics

Arkady Poliakovsky Department of Mathematics Ben-Gurion University of the Negev Be’er Sheva, Israel

ISBN 978-3-031-23761-4 ISBN 978-3-031-23762-1 (eBook) https://doi.org/10.1007/978-3-031-23762-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Prologue

The idea of a four-dimensional space-time, rather than separate three-dimensional space and one-dimensional time, is the cornerstone of all relativistic theories. Furthermore, the idea of covariance (in tensor formulation) of the most fundamental Physical Laws under a change of general coordinate systems, given by arbitrary (both linear and smooth nonlinear) transformations, is the central feature of the theory of general relativity. However, both these ideas do not really contradicts the non-relativistic considerations of space-time! For example, both the Galilean transformations and the Lorentz’s transformations can be considered with equal right as two different particular cases of general linear transformations of the space-time (note that one can call a particular coordinate frame system by the word inertial if it is obtained by a general linear transformation from the other, commonly considered as inertial, coordinate frame). The existence of a global time scalar function in the four-dimensional space-time also implies no contradiction, since this scalar equals to the first coordinate of the space-time only in very special coordinate systems, while we assume the general covariance of the Physical Laws. The central goal of the present research is to show that both relativistic and non-relativistic approaches to space-time can be unified. In particular, the global universal time function, or at least the four-covector field of the time direction at every point of the space-time, can be concordantly introduced to relativistic theories. This is useful, in particular, when we are talking about the absolute age of the universe. This is also useful for the Hamiltonian formulation of the Dirac equation, and moreover, it also provides the way to formulate the Dirac equation for multiple particles, similarly to what was done for the Schrödinger equation of the non-relativistic quantum mechanics. In this research we postulate that all real physical processes appear in some valid pseudo-metric, describing a generalized gravity field, correlated with some fourcovector of time direction. Furthermore, we distinguish two types of generalized gravity. The first type is the fictitious gravity which we call inertia. This type of gravity depends only on the flat Geometry of empty space-time via the choice of specific coordinate system, and it is independent on the surrounding real matter consisting of gravitational masses or other real physical fields. The second type of gravity is the genuine (real) gravity, which depends essentially on the real physical matter, v

vi

Prologue

especially on gravitational masses. We assume that this type of gravity vanishes away from essential gravitational masses and strong real physical fields. So, at every point of space-time we deal with two different pseudo-metrics simultaneously. The first is the kinematical flat Minkowski’s pseudo-metric, describing the Geometry of the virtually emptied space-time. The second is the dynamical pseudometric, which combines both effects of inertia and the genuine gravity, depending both on the kinematical pseudo-metric and on the genuine gravity field. In this sense our model is bimetrical. On the other hand, the dynamical pseudo-metric is assumed to preserve the same four-dimensional volume as the kinematical one, so that our model is unimodular. Similarly, we distinguish two types of four-covector fields of the time direction. The first type is the kinematical time direction, which equals to the four-dimensional gradient of the kinematical global time, depending only on the flat Geometry of empty space-time via the choice of specific coordinate system. The second type of time direction is the dynamical time direction, which depends on the real physical fields, similarly to the genuine gravity. Then we assume that the vacuum expectation value of the dynamical time direction equals to the gradient of the kinematical global time, while the vacuum expectation value of the dynamical pseudo-metric equals to the kinematical one, so that the vacuum expectation value of the genuine gravity vanishes. The difference between the dynamical and the kinematical pseudo-metrics is assumed to depend on the four-covector of genuine gravity and the four-covector of the dynamical time direction. Both for genuine gravity and for the dynamical time direction we consider Proca-type Lagrangian actions. In particular, in contrast to the inertia and the kinematical global time, one can further quantify the fields of genuine gravity and the dynamical time direction, similarly to what can be done for the electromagnetic field. Since the common name for the quant of the electromagnetic field is photon, one can name the quant of the genuine gravity graviton and the quant of the dynamical time direction timion. Then, in the framework of our model, the graviton can be considered as the spin-one neutral massless particle, similarly to photon, while the timion can be considered as the spin-one neutral particle having a negligible but not completely vanishing mass. In the approximating case of our model, where certain constants tend to infinity, the dynamical time direction tends to be equal to the four-dimensional gradient of the kinematical global time, and we get the simplified model where the difference between the dynamical and the kinematical pseudo-metrics is assumed to depend on the four-covector of genuine gravity only! Then, the dynamical pseudo-metric genuinely depends on one four-covector only, which by unimodularity condition has only three genuinely independent components. In this sense our model is much more simple than the theory of general relativity, where the symmetric 4 × 4-pseudometric depends on the full ensemble of all ten independent components. Even our full model depends only on two four-covectors (the genuine gravity and the dynamical time direction), which by the unimodularity condition has only seven genuinely independent components. However, we are able to prove that in the framework of our simplified model, the gravitational field, generated by some massive body, resting and spherically symmetric in certain coordinate system, is given by the dynamical pseudo-metric, such that there exists some curvilinear coordinate system in the

Prologue

vii

space-time, where this pseudo-metric coincides with the well-known Schwarzschild metric from general relativity! In particular, all the optical effects (like Sagnac, Michelson-Morley, etc.) that we find in the framework of our model coincide with the effects considered in the framework of general relativity for the Schwarzschild metric. Finally, all the mechanical effects (like Mercury motion in the Sun gravity field, etc.) will be the same in the framework of our model, like in the case of general relativity for the Schwarzschild metric, provided that the time does not appear explicitly in these effects. Furthermore, we also prove that gravitational field, governed by our model, generated by a general slowly (non-relativistically) moving massive matter in certain inertial coordinate system can be well approximated, by the classical model of the Newtonian Gravity. Gravitational waves are also possible both in the full and simplified models presented here. Finally, in this research we give a covariant formulation of the Electrodynamics of the moving dielectric and para/diamagnetic continuum mediums in arbitrary dynamical pseudo-metric. The Lorentz’s covariant theory of the moving para/diamagnetic continuum mediums in the flat Lorentz’s pseudo-metric was first introduced by H. Minkowski (1908). Here we formulate the general covariant theory in a different alternative way, that is, suitable for formulation in a general pseudo-metric including the presence of the genuine gravity. In particular, we show that our model is completely consistent with the well-known Fizeau’s experiment. In this research we do not consider the quantum consideration of the Lagrangian action. This would be the topic of a further research.

Contents

1

Preliminary Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5

2

Basic Definitions and Statements of the Main Results . . . . . . . . . . . . . 2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Pseudo-Lorentzian Coordinate Systems . . . . . . . . . . . . . . . . 2.2 Kinematical Lorentz’s Structure with Global Time . . . . . . . . . . . . . 2.3 Kinematical and Dynamical Generalized-Lorentz Structures with Time Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Lagrangian of the Motion of a Classical Point Particle in a Given Pseudo-metric with Time Direction . . . . . . . . . . . . . . . . . 2.5 Lagrangian of the Electromagnetic Field in a Given Pseudo-metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Correlated Pseudo-metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Kinematically Correlated Models of the Genuine Gravity . . . . . . . 2.8 Lagrangian for Dynamical Time Direction and Its Limiting Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Lagrangian of the Genuine Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3 4

5

Mass, Charge and Lagrangian Densities and Currents of the System of Classical Point Particles . . . . . . . . . . . . . . . . . . . . . . . .

7 17 25 26 28 32 33 38 42 46 53 55

The Total Simplified Lagrangian of the Gravity in a Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Total Simplified Lagrangian in (2.9.23), (2.9.24), for the Limiting Case of (2.9.20) in a Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

The Euler-Lagrange for the Lagrangian of the Motion of A Classical Point Particle in a Cartesian Coordinate System . . . .

77

65

ix

x

6

Contents

The Euler-Lagrange for the Lagrangian of the Gravitational and Electromagnetic Fields in a Cartesian Coordinate System . . . . . 6.1 The Euler-Lagrange for the Lagrangian in (4.1.71) in a Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81

7

Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate System Which Is Cartesian and Inertial Simultaneously . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.1 Certain Curvilinear Coordinate System in the Case of Stationary Radially Symmetric Gravitational Field and Relation to the Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . . 96 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8

Newtonian Gravity as an Approximation of Our Model . . . . . . . . . . . 105 8.1 Newtonian Gravity as an Approximation of (6.1.52) . . . . . . . . . . . . 105

9

Polarization and Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9.1 Polarization and Magnetization in a Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Appendix: Some Technical Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Chapter 1

Preliminary Introduction

In the classical theories of special and general relativity the inertia and the gravity are described by certain pseudo-metric of signature {1, −1, −1, −1} in the fourdimensional space-time. On the other hand, in the framework of the Newton-Cartan theory (see [1–6]) the Geometry of the space-time is (incompletely) described by the two-times contravariant symmetric degenerate tensor {h mn }m,n=0,1,2,3 of signature {0, 1, 1, 1} and a covector (w0 , w1 , w2 , w3 ) of time direction, everywhere nonvanishing and satisfying 3 

hm j w j = 0

∀ m = 0, 1, 2, 3.

(1.0.1)

j=0

Moreover, in the case that there exists a scalar field τ satisfying wm =

∂τ ∂xm

∀ m = 0, 1, 2, 3,

(1.0.2)

this field can serve as a global time in R4 . In this paper we unify both these approaches and build the model, completely describing the Geometry and the gravity in R4 , which includes both the pseudo-metric and the global time scalar field (or more generally the covector of the time direction). One of the goals of the paper was to unify the relativistic and the non-relativistic approaches to the study of the space-time. We postulate that all real physical processes appear in some valid pseudo-metric {K mn }m,n=0,1,2,3 , describing the generalized gravity field, weakly correlated with some covector of time direction (w0 , w1 , w2 , w3 ) (see definitions in the sequent Sect. 2). Furthermore, we distinguish two types of generalized gravity. The first type is the fictitious gravity which we call inertia. This type of gravity depends only on the flat Geometry of empty space-time via the choice of specific coordinate system, and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_1

1

2

1 Preliminary Introduction

it is independent on the surrounding real matter consisting of gravitational masses or other real physical fields. The second type of the gravity is the genuine (real) gravity, which depends essentially on the real physical matter, especially on gravitational masses. We assume that this type of gravity vanishes away from essential gravitational masses and strong real physical fields. Then we state the First Law of the Newton as the following: • In the parts of the space-time where we observes the absence of genuine gravity, and in particular away from essential real physical bodies and fields, we have K mn = J mn

∀ m, n = 0, 1, 2, 3, 

and (w0 , w1 , w2 , w3 ) =

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

(1.0.3)  ,

(1.0.4)

where the strongly correlating flat Minkowski’s pseudo-metric {J mn }m,n=0,1,2,3 and the fixed scalar field ϕ, called kinematical global time form the standard kinematical Lorentz’s structure with global time on R4 , as defined in the Definition 2.7 of the sequent section. In particular they assumed to satisfy, firstly the following eikonal-type equation 3 3   ∂ϕ ∂ϕ J jm j m = 1 , (1.0.5) ∂x ∂x j=0 m=0 and secondly

   ∂ϕ δj =0 ∂xk J

∀ k, j = 0, 1, 2, 3 .

(1.0.6)

 

we denote the tensor of the covariant derivatives of the where by δ j ∂∂ϕ xk J  ∂ϕ ∂ϕ ∂ϕ ∂ϕ covector ∂ x 0 , ∂ x 1 , ∂ x 2 , ∂ x 3 with respect to the pseudo-metric J mn . Then it easily can be derived that there exists some coordinate system where matrix {J mn }m,n=0,1,2,3 has a form of ⎧ 00 ⎪ ⎨J = 1 ∀ j = 1, 2, 3 J 0 j = J j0 = 0 ⎪ ⎩ jm ∀ j, m = 1, 2, 3 J := −δ jm

(1.0.7)

and at the same coordinate system the covector of time direction for the global time ϕ has a form 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0)

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . (1.0.8)

1 Preliminary Introduction

3

We call this particular system kinematically preferable, and we show that it is unique, up to equivalence. Furthermore, we define the kinematical tensor of threedimensional Geometry {mn }m,n=0,1,2,3 , given by ⎛ mn := ⎝

3  j=0

⎞⎛ ⎞ 3  ∂ϕ ∂ϕ J mj j ⎠ ⎝ J n j j ⎠ − J mn ∂x ∂x j=0

∀ m, n = 0, 1, 2, 3 , (1.0.9)

where ⎛ ⎝

3  j=0

⎞ 3 3 3    ∂ϕ ∂ϕ ∂ϕ ∂ϕ J0j j , J1j j , J2j j , J3j j ⎠ ∂x ∂ x ∂ x ∂x j=0 j=0 j=0

(1.0.10)

is the contravariant vector of inertia. Obviously, we have 3  j=0

m j

∂ϕ =0 ∂x j

∀ m = 0, 1, 2, 3 ,

(1.0.11)

(as in (1.0.1)), forming the standard Galilean structure. In particular, in the kinematically preferable coordinate system, where (1.0.7) and (1.0.8) holds we have ⎧ 00 ⎪ ⎨ = 0 ∀ j = 1, 2, 3 0 j =  j0 = 0 ⎪ ⎩ jm ∀ j, m = 1, 2, 3 .  := δ jm

(1.0.12)

Furthermore, given arbitrary coordinate system, it is called cartesian if in this system we have simultaneously (1.0.12) and (1.0.8) but, we do not necessary have (1.0.7). On the other hand, given arbitrary coordinate system, it is called Lorentzian if in this system we have (1.0.7) but, we do not necessary have (1.0.12) or (1.0.8). Finally, given arbitrary coordinate system, we call it inertial, if we can get it from kinematically preferable coordinate system by a linear transformation. In the sequence we prove that we obtain a coordinate system which is simultaneously cartesian and inertial from another such system by Galilean transformations. On the other hand, we obtain a coordinate system which is Loretzian (and then also inertial) from another such system by Lorentz’s transformations. The unique, up to equivalence, coordinate system which is simultaneously cartesian and Loretzian is a kinematically preferable coordinate system. In Sect. 2.1.1 we define Pseudo-Lorentzian coordinate systems that generalize both cartesian and Lorentzian systems. We also find the group of transformation of such systems including, in particular as subgroups Lorentzian and Galilean transformations. Furthermore, we describe our model of the gravity: given, an arbitrary dynamical four-covector of the dynamical time direction, (w0 , w1 , w2 , w3 ) (formally unrelated to the kinematical global time ϕ), which is weakly correlated with {J mn }m,n=0,1,2,3

4

1 Preliminary Introduction

and an arbitrary four-covector field (S0 , S1 , S2 , S3 ), which we call the four-covector of genuine gravity, consider the two-times covariant tensor {K mn }m,n=0,1,2,3 defined by:   (1.0.13) K jm = J jm + w j Sm + wm S j ∀ 0 ≤ j, m ≤ 3 , and assume that (S0 , S1 , S2 , S3 ) is such that {K mn }m,n=0,1,2,3 in (1.0.13) satisfies     det {K mn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 .

(1.0.14)

Then, one can prove that {K mn }m,n=0,1,2,3 , is a valid pseudo-metric of signature {1, −1, −1, −1}, correlated with the time direction (w0 , w1 , w2 , w3 ). Moreover, we call such a dynamical pseudo-metric {K mn }m,n=0,1,2,3 , with time direction (w0 , w1 , w2 , w3 ), correlated pseudo-metric with time direction (w0 , w1 , w2 , w3 ), corresponding to the covector of genuine gravity (S0 , S1 , S2 , S3 ). In the case of the simplified approximating model we get  (w0 , w1 , w2 , w3 ) ≈

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 ,

(1.0.15)

and (1.0.13) reeds as  K jm =

J jm

∂ϕ ∂ϕ + Sm + m S j j ∂x ∂x

 ∀ 0 ≤ j, m ≤ 3 ,

(1.0.16)

where ϕ is the kinematical global time. Next for the dynamical time direction, (w0 , w1 , w2 , w3 ) and the covector of genuine gravity (S0 , S1 , S2 , S3 ) one can consider Proca-like Lagrangians, see in the sequel. In the following sections we prove, in particular, that in the framework of our simplified model, the gravitational field, generated by some massive body, resting and spherically symmetric in some cartesian and inertial coordinate system, is given by the pseudo-metric {K mn }m,n=0,1,2,3 , such that there exists some curvilinear (noncartesian) coordinate system in R4 , where {K mn }m,n=0,1,2,3 coincides with the wellknown Schwarzschild metric from the general relativity! In particular, all the optical effects that we find in the framework of our model coincide with the effects considered in the framework of general relativity for the Schwarzschild metric. Finally, all the mechanical effects will be the same in the framework of our model like in the case of the general relativity for the Schwarzschild metric, provided that the time does not appear explicitly in this effects. Furthermore, we also prove that gravitational field, governed by our model, generated by a general slowly (non-relativistically) moving massive matter in some cartesian coordinate system, can be well approximated, by the classical model of the Newtonian gravity. Note here about the following advantage of the presented model of gravity with respect to the usual theory of general relativity. The simplified model for the gravity depends only on four-component field (S0 , S1 , S2 , S3 ), which is by Eqs. (1.0.16) and (1.0.14) has only three independent components. Even the full model dependent

References

5

only on (S0 , S1 , S2 , S3 ) and (w0 , w1 , w2 , w3 ), that is by Eqs. (1.0.13) and (1.0.14) has only seven independent components. On the other hand, in the general relativity the symmetric tensor {K mn }m,n=0,1,2,3 has all ten independent components that makes the corresponding system of partial differential equations to be much more complicated. Finally, in Chap. 9 we give the covariant formulation of the Electrodynamics of the moving dielectric and para/diamagnetic continuum mediums in arbitrary dynamical pseudo-metric. The Lorentz’s covariant theory of the moving para/diamagnetic continuum mediums in the flat Lorentz’s pseudo-metric was first introduced in Minkowski [7]. Here we formulate the generally covariant theory in the different alternative way that suite to formulate it in a general pseudo-metric including the presence of the genuine gravity. Note that for the formulation of the concepts of polarization and magnetization and for the mathematical proofs we use in particular the theory of distributions (generalized functions). The next section plays a role of comprehensive introduction. In the sequent sections we give more detailed description of our results. In the end of the paper including Appendix we give detailed prove of all mathematical statements.

References 1. C. Duval, On Galilean isometries. Class. Quantum Grav. 10, 2217 (1993). https://doi.org/10. 1088/0264-9381/10/11/006; arXiv:0903.1641 2. C. Duval, G. Burdet, H.P. Künzle, M. Perrin, Bargmann structures and Newton-Cartan theory. Phys. Rev. D 31, 1841 (1985). https://doi.org/10.1103/PhysRevD.31.1841 3. C. Duval, P.A. Horváthy, Non-relativistic conformal symmetries and Newton-Cartan structures. J. Phys. A: Math. Theor. 42, 465206 (2009). https://doi.org/10.1088/1751-8113/42/46/465206; arXiv:0904.0531 4. C. Duval, H.P. Künzle, Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation. Gen. Relat. Gravit. 16, 333 (1984). https://doi.org/10.1007/ BF00762191 5. K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime. Sci. Post Phys. 5, 011 (2018) 6. H.P. Künzle, Galilei and lorentz structures on space-time?: comparison of the corresponding geometry and physics. Ann. Inst. H. Poincaré Phys. Théor. 17, 337 (1972) 7. H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern. Nachr. Ges. Wiss. Göttn Math.-Phys. Kl. 53 (1908); Reprinted in Math. Ann. 68, 472 (1910)

Chapter 2

Basic Definitions and Statements of the Main Results

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time Definition 2.1 We say that the generalized-Lorentz’s structure on R4 is chosen, if R4 is equipped with symmetric non-degenerate two-times contravariant tensor field {K mn }m,n=0,1,2,3 , such that the matrix K mn has one positive and three negative eigenvalues at every point in R4 . Then, {K mn }m,n=0,1,2,3 is called a contravariant pseudo-metric on R4 . Moreover, the inverse symmetric non-degenerate two-times covariant tensor field {Kmn }m,n=0,1,2,3 which satisfies 3  k=0

K

mk

 1 if m = n Kkn = 0 if m = n

∀ m, n = 0, 1, 2, 3,

(2.1.1)

is called a covariant pseudo-metric on R4 associated with {K mn }m,n=0,1,2,3 . Definition 2.2 We say that a direction of the global time on R4 is chosen, if R4 is equipped with a four-covector field (w0 , w1 , w2 , w3 ) := (w0 , w1 , w2 , w3 )(x0 , x1 , x2 , x3 ), non-vanishing at every point in R4 (the last property is obviously independent on the choice of a coordinate system in R4 ). Then, chosen covector field (w0 , w1 , w2 , w3 ) is called the covector of the time direction. Furthermore, we say that a scalar global time on R4 is chosen, if R4 is equipped with a covariant scalar field ϕ := ϕ(x0 , x1 , x2 , x3 ) such that the four-covector field (w0 , w1 , w2 , w3 ), defined by ∂ϕ wj := j ∀ j = 0, 1, 2, 3, (2.1.2) ∂x does not vanish at any point on R4 . Then ϕ is called a global time in R4 and wj := wj (x0 , x1 , x2 , x3 ), given by (2.1.2) is called the covector of the time direction of the given global time ϕ. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_2

7

2 Basic Definitions and Statements . . .

8

Definition 2.3 We say that a contravariant pseudo-metric {K mn }m,n=0,1,2,3 and a covector of the time direction (w0 , w1 , w2 , w3 ) are weakly correlated, if we have everywhere 3  3  K jm wj wm > 0 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . (2.1.3) j=0 m=0

In that case we say that generalized-Lorentz’s structure with time direction on R4 is chosen. Then, we define the contravariant four-vector field of the potential of generalized gravity (v0 , v1 , v2 , v3 ) by ⎞− 21 ⎛ ⎞ ⎛ 3  3 3   vm := ⎝ K jk wj wk ⎠ ⎝ K mj wj ⎠ ∀ m = 0, 1, 2, 3, j=0 k=0

(2.1.4)

j=0

so that we have 3  3 

Kjm vj vm = 1 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.1.5)

j=0 m=0

Furthermore, we say that a contravariant pseudo-metric {K mn }m,n=0,1,2,3 and a scalar global time ϕ are strongly correlated on R4 if ϕ satisfies the following eikonaltype equation in pseudo-metric {K mn }m,n=0,1,2,3 : 3  3 

K jm

j=0 m=0

∂ϕ ∂ϕ = 1 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . ∂xj ∂xm

(2.1.6)

In that case we say that generalized-Lorentz’s structure with global time on R4 is chosen. Moreover, in the later case we rewrite the definition of the contravariant four-vector field of the potential of generalized gravity (v0 , v1 , v2 , v3 ) in (2.1.4) as: vm :=

3  j=0

K mj

∂ϕ ∀ m = 0, 1, 2, 3. ∂xj

(2.1.7)

Then we prove the following: Proposition 2.1 Given a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 , weakly correlated with a covector of the time direction (w0 , w1 , w2 , w3 ), define the two-times contravariant symmetric tensor field {mn }m,n=0,1,2,3 given by jm := vj vm − K jm ∀ j, m = 0, 1, 2, 3,

(2.1.8)

with (v0 , v1 , v2 , v3 ) defined by (2.1.4). Then, the matrix mn has one vanishing and three positive eigenvalues at every point in R4 , and we call {mn }m,n=0,1,2,3 the contravariant tensor of three-dimensional Geometry on R4 . Moreover, we have

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time 3 

⎛ vj wj = ⎝

j=0

and

3  3 

9

⎞ 21 K jk wj wk ⎠ ,

(2.1.9)

j=0 k=0

3 

mj wj = 0 ∀ m = 0, 1, 2, 3.

(2.1.10)

j=0

Corollary 2.1 Given a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 , strongly correlated with a scalar global time ϕ, define the contravariant tensor field {mn }m,n=0,1,2,3 , given by jm := vj vm − K jm ∀ j, m = 0, 1, 2, 3,

(2.1.11)

with (v0 , v1 , v2 , v3 ), defined by (2.1.7). Then, the matrix mn has one vanishing and three positive eigenvalues at every point in R4 . Moreover, we have 3  j=0

and

3  j=0

mj

vj

∂ϕ = 1, ∂xj

∂ϕ = 0 ∀ m = 0, 1, 2, 3. ∂xj

(2.1.12)

(2.1.13)

Here, it is important to note that in the framework of the Newton-Cartan Theory we have the following definition (see [1–6]): Definition 2.4 The Galilean structure in R4 consists of a fixed two-times contravariant symmetric tensor {mn }m,n=0,1,2,3 in R4 , such that the matrix mn has one vanishing and three positive eigenvalues at every point in R4 , and a fixed covector (w0 , w1 , w2 , w3 ), non-vanishing at every every point in R4 and such that 3 

mj wj = 0 ∀ m = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.1.14)

j=0

Then, we prove the inverse to Proposition 2.1 statement: Proposition 2.2 Let {mn }m,n=0,1,2,3 be a two-times contravariant symmetric tensor, such that the matrix mn has one vanishing and three positive eigenvalues at every point in R4 , and let (w0 , w1 , w2 , w3 ) be a covector, non-vanishing at every every point in R4 and such that

2 Basic Definitions and Statements . . .

10 3 

mj wj = 0 ∀ m = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.15)

j=0

that form together a Galileian structure. Next, given an arbitrary contravariant fourvector field (v0 , v1 , v2 , v3 ) satisfying the covariant relation 3 

vj wj > 0 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.16)

j=0

consider a contravariant symmetric tensor field {K mn }m,n=0,1,2,3 , given by K mn = vj vm − jm ∀ j, m = 0, 1, 2, 3.

(2.1.17)

Then, {K mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric. Moreover, the generalized-Lorentz’s structure, given by {K mn }m,n=0,1,2,3 is weakly correlated with the time direction (w0 , w1 , w2 , w3 ). Finally, we also have ⎞ 21 ⎛ 3  3 3   ⎝ K jm wj wk ⎠ = vj wj ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , j=0 k=0

(2.1.18)

j=0

and ⎛ ⎞− 21 ⎛ ⎞ 3  3 3   vm = ⎝ K jm wj wk ⎠ ⎝ K jm wj ⎠ j=0 k=0

(2.1.19)

j=0

∀m = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . So, {mn }m,n=0,1,2,3 is a contravariant tensor of three-dimensional Geometry and (v , v1 , v2 , v3 ) is a potential of generalized gravity, corresponding to the pseudometric K mn and the time direction (w0 , w1 , w2 , w3 ). 0

Corollary 2.2 Let {mn }m,n=0,1,2,3 be a two-times contravariant symmetric tensor, such that the matrix mn has one vanishing and three positive eigenvalues at every point in R4 , and let ϕ be a covariant scalar field such that the four-covector field (w0 , w1 , w2 , w3 ), defined by wj :=

∂ϕ ∀ j = 0, 1, 2, 3, ∂xj

does not vanish at any point on R4 and such that

(2.1.20)

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time 3 

mj

j=0

∂ϕ = 0 ∀ m = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . ∂xj

11

(2.1.21)

Next, given an arbitrary contravariant four-vector field (v0 , v1 , v2 , v3 ) satisfying the covariant relation 3 

vj

j=0

∂ϕ = 1 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , ∂xj

(2.1.22)

consider a contravariant symmetric tensor field {K mn }m,n=0,1,2,3 , given by K mn = vj vm − jm ∀ j, m = 0, 1, 2, 3.

(2.1.23)

Then, {K mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric. Moreover, the generalized-Lorentz’s structure, given by {K mn }m,n=0,1,2,3 is strongly correlated with the global time ϕ. Finally, we also have vm =

3  j=0

K jm

∂ϕ ∀ m = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . ∂xj

(2.1.24)

So, {mn }m,n=0,1,2,3 is a contravariant tensor of three-dimensional Geometry and (v , v1 , v2 , v3 ) is a potential of generalized gravity, corresponding to the pseudometric K mn and the global time ϕ. 0

Remark 2.1 The above Proposition and Corollary show that the Galilean structure alone incompletely describes the Geometry of the space-time and for complete description, in addition to {mn }m,n=0,1,2,3 and (w0 , w1 , w2 , w3 ), above we need to specify one more contravariant vector field (v0 , v1 , v2 , v3 ) (that we call here the potential of generalized gravity) satisfying either 3 

vj wj > 0 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

j=0

in the case of the weak coupling, or 3 

vj wj = 1 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

j=0

in the case of the strong coupling. Then we can define a Lorentz’s-like contravariant pseudo-metric {K mn }m,n=0,1,2,3 as: K mn = vj vm − jm ∀ j, m = 0, 1, 2, 3,

2 Basic Definitions and Statements . . .

12

that will either weakly or strongly correlate with the time direction (w0 , w1 , w2 , w3 ). Moreover, in the case of strong coupling, the contravariant vector (v0 , v1 , v2 , v3 ) is just a lifted time direction (w0 , w1 , w2 , w3 ) with respect to a pseudo-metric {K mn }m,n=0,1,2,3 . Very similar to (v0 , v1 , v2 , v3 ) contravariant vector field appears implicitly in Eq. (2.1.2) and before it on [5] (it was denoted (ν 0 , ν 1 , ν 2 , ν 3 ) there). Moreover, then it can be easily shown that the covariant degenerate tensor field {hmn }m,n=0,1,2,3 in (2.1.2) on [5] is just the lowering-index of the tensor {hmn }m,n=0,1,2,3 := {mn }m,n=0,1,2,3 , with respect to the covariant pseudo-metric {Kmn }m,n=0,1,2,3 , which is inverse to the contravariant pseudo-metric K mn = vj vm − jm := ν j ν m − hjm ∀ j, m = 0, 1, 2, 3. Definition 2.5 Consider a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 and let {Kmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {K mn }m,n=0,1,2,3 . Then, in every coordinate system define the Christoffel Symbols  kjm by K

 ⎧  ∂Kjn 1 ∂Kjm ⎪ ⎨ j,mn K := 2 ∂xn + ∂xm −  3    j ⎪ K jk k,mn K ⎩ mn := K

∂Kmn ∂xj

 ∀ j, m, n = 0, 1, 2, 3.

(2.1.25)

k=0

Furthermore, given a covariant four-vector field (h0 , h1 , h2 , h3 ), define the covariant derivative of (h0 , h1 , h2 , h3 ) by 

δj hk



∂hk   m kj − hm ∀ k, j = 0, 1, 2, 3. K ∂xj m=0 3

K

:=

(2.1.26)

Then, it is well known that {{δn hm }K }m,n=0,1,2,3 is a two-times covariant tensor. Definition 2.6 Consider a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 and 4 let {Kmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric R , associated with  on {K mn }m,n=0,1,2,3 . Next, consider the Christoffel Symbols kjm , defined by (2.1.25). K

Given a coordinate system in R4 , we call it inertial coordinate system with respect to {K mn }m,n=0,1,2,3 if, in this particular coordinate system we have  j  mn K = 0 ∀ j, m, n = 0, 1, 2, 3.

(2.1.27)

Then, using Lemma A.2 from the Appendix, we deduce that, given coordinate system in R4 is inertial with respect to {K mn }m,n=0,1,2,3 if and only if the tensor {Kmn }m,n=0,1,2,3 is independent on the local coordinates (x0 , x1 , x2 , x3 ) ∈ R4 in this particular coordinate system. Definition 2.7 If there exists some coordinate system where matrix {K mn }m,n=0,1,2,3 has a form of

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time

⎧ 00 ⎪ ⎨K = 1 K 0j = K j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm K := −δjm ∀ j, m = 1, 2, 3

13

(2.1.28)

at every point in R4 and at the same time the covector of time direction for ϕ has a form   ∂ϕ ∂ϕ ∂ϕ ∂ϕ (x0 , x1 , x2 , x3 ) = (1, 0, 0, 0) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , , , ∂x0 ∂x1 ∂x2 ∂x3 (2.1.29) in the same coordinate system, then the contravariant pseudo-metric {K mn }m,n=0,1,2,3 and the global time ϕ are obviously strongly correlated on R4 and we say that the standard kinematical Lorentz’s structure with global time on R4 is chosen. Moreover, in that case we call the contravariant four-vector field of the potential of generalized gravity (v0 , v1 , v2 , v3 ), corresponding to that structure and defined, as in (2.1.7), by vm :=

3  j=0

K mj

∂ϕ ∀ m = 0, 1, 2, 3, ∂xj

(2.1.30)

the contravariant four-vector of the potential of inertia. Furthermore, we call the coordinate system where (2.1.28) and (2.1.29) hold simultaneously, kinematically preferable for the given generalized-Lorentz’s structure with the global time. In particular, it is clear that a kinematically preferable system is inertial with respect to {K mn }m,n=0,1,2,3 (see Definition 2.6) Furthermore, in kinematically preferable coordinate system we obviously have  0 1 2 3 0 1 2 3 v , v , v , v (x , x , x , x ) = (1, 0, 0, 0) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.1.31)

Moreover, in the same coordinate system the contravariant tensor field {mn }m,n=0,1,2,3 , given as in (2.1.11) by jm := vj vm − K jm ∀ j, m = 0, 1, 2, 3, obviously satisfies

⎧ 00 ⎪ ⎨ = 0 0j = j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm  := δjm ∀ j, m = 1, 2, 3.

(2.1.32)

(2.1.33)

Finally, using Lemma A.3 from the Appendix in the end, we deduce that a general coordinate system is inertial, with respect to the above pseudo-metric, if this coordinate system is obtained from kinematically preferable system by a general linear transformation of the form:

2 Basic Definitions and Statements . . .

14

xm =

3 

Cmj xj + cm ∀m = 0, 1, 2, 3,

(2.1.34)

j=0

where {Cmn }n,m=1,2,3 ∈ R4×4 is a constant (independent on (x0 , x1 , x2 , x3 ) ∈ R4 ) non-degenerate matrix, and (c0 , c1 , c2 , c3 ) ∈ R4 is a constant (independent on (x0 , x1 , x2 , x3 ) ∈ R4 ) vector. Definition 2.8 Given a contravariant pseudo-metric {K mn }m,n=0,1,2,3 , we say that a given general coordinate system in R4 is Lorentzian with respect to {K mn }m,n=0,1,2,3 , if {K mn }m,n=0,1,2,3 has the following simple form in the chosen coordinate system: ⎧ 00 0 1 2 3 4 ⎪ ⎨K = 1 ∀(x , x , x , x ) ∈ R mn K = −δmn ∀1 ≤ m, n ≤ 3 ∀(x0 , x1 , x2 , x3 ) ∈ R4 ⎪ ⎩ 0j K = K j0 = 0 ∀1 ≤ j ≤ 3 ∀(x0 , x1 , x2 , x3 ) ∈ R4 .

(2.1.35)

Note that (2.1.35) also implies: ⎧ ⎪ ⎨K00 = 1 Kmn = −δmn ∀1 ≤ m, n ≤ 3 ⎪ ⎩ K0j = Kj0 = 0 ∀1 ≤ j ≤ 3.

(2.1.36)

Moreover, due to Definition 2.6, every Lorentzian coordinate system is obviously inertial with respect to {K mn }m,n=0,1,2,3 . Definition 2.9 Let

⎧ 0 x ⎪ ⎪ ⎪ ⎨x1 ⎪ x2 ⎪ ⎪ ⎩ 3 x

= f (0) (x0 , x1 , x2 , x3 ), = f (1) (x0 , x1 , x2 , x3 ), = f (2) (x0 , x1 , x2 , x3 ), = f (3) (x0 , x1 , x2 , x3 ).

(2.1.37)

be a change of the first general coordinate system to the second coordinate system. We say that transformations (2.1.37) are Lorentz’s transformations if, for arbitrary contravariant pseudo-metric {K mn }m,n=0,1,2,3 , such that in the first coordinate system we have ⎧ 00 0 1 2 3 4 ⎪ ⎨K = 1 ∀(x , x , x , x ) ∈ R mn (2.1.38) K = −δmn ∀1 ≤ m, n ≤ 3 ∀(x0 , x1 , x2 , x3 ) ∈ R4 ⎪ ⎩ 0j j0 0 1 2 3 4 K = K = 0 ∀1 ≤ j ≤ 3 ∀(x , x , x , x ) ∈ R , in the second coordinate system we also have

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time

⎧ 00 0 1 2 3 4 ⎪ ⎨K = 1 ∀(x , x , x , x ) ∈ R K mn = −δmn ∀1 ≤ m, n ≤ 3 ∀(x0 , x1 , x2 , x3 ) ∈ R4 ⎪ ⎩ 0j K = K j0 = 0 ∀1 ≤ j ≤ 3 ∀(x0 , x1 , x2 , x3 ) ∈ R4 .

15

(2.1.39)

In particular, observe that, using Lemma A.3 from the Appendix, we deduce that every Lorentz’s transformation is necessarily linear. Definition 2.10 Consider a standard kinematical Lorentz’s structure with global time ϕ on R4 , together with the corresponding tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 . We say that a given general coordinate system in R4 is cartesian with respect to the tensor of the three-dimensional Geometry {mn }m,n=0,1,2,3 and the global time ϕ, if {mn }m,n=0,1,2,3 has the following simple form in the chosen coordinate system: ⎧ 00 0 1 2 3 4 ⎪ ⎨ = 0 ∀ (x , x , x , x ) ∈ R 0j j0  =  = 0 ∀ j = 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ⎪ ⎩ mn  = δmn ∀ m, n = 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.40)

and at the same time in the same coordinate system we have 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x0 , x1 , x2 , x3 ) = (1, 0, 0, 0) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . (2.1.41)

Definition 2.11 We say that two cartesian coordinate systems in R4 are equivalent if the change of coordinates from one system to another is given by ⎧ 0 0 0 ⎪ ⎨x = x + c , 3  m ⎪ Bmj xj + cm ∀m = 1, 2, 3, ⎩x =

(2.1.42)

j=1

where {Bmn }n,m=1,2,3 ∈ R3×3 is a constant (independent on (x0 , x1 , x2 , x3 ) ∈ R4 ) matrix, satisfying 3 

Bmj Bnj =

j=1

3 

Bjm Bjn = δmn ∀ m, n = 1, 2, 3,

(2.1.43)

j=1

and (c0 , c1 , c2 , c3 ) ∈ R4 is a constant (independent on (x0 , x1 , x2 , x3 ) ∈ R4 ) vector. Then we prove the following: Theorem 2.1 Consider a standard kinematical Lorentz’s structure with global time on R4 together with the corresponding tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 . Next consider the first cartesian, with respect to {mn }m,n=0,1,2,3

2 Basic Definitions and Statements . . .

16

and ϕ, coordinate system in R4 and the second general coordinate system in R4 . Then, the second coordinate system is also cartesian if and only if the change of the first coordinate system to the second one is given by the following relations: ⎧ 0 0 ⎪ ⎨x = x + c, 3  m ⎪ Amj (x0 ) xj + z m (x0 ) ∀m = 1, 2, 3, ⎩x =

(2.1.44)

j=1

where {Amn (x0 )}n,m=1,2,3 ∈ R3×3 is a 3 × 3-matrix, depending on the coordinate x0 only (independent on x := (x1 , x2 , x3 )), and satisfying 3 

Amj (x0 )Anj (x0 ) =

j=1

3 

Ajm (x0 )Ajn (x0 ) = δmn

j=1

∀ m, n = 1, 2, 3 ∀(x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.45)

c ∈ R is a constant (independent on (x0 , x1 , x2 , x3 ) ∈ R4 ) and z(x0 ) := (z 1 (x0 ), z 2 (x0 ), z 3 (x0 )) ∈ R3 is a three-dimensional vector field, depending on the coordinate x0 only (independent on x := (x1 , x2 , x3 )). In particular, up to equivalence of cartesian coordinate systems (see Definition 2.11), (2.1.44) reduces to 

x0 = x0     x = A x 0 · x + z x 0 ,

(2.1.46)

0 0 where  0  A(x ) ∈ SO(3) is a rotation, depending on the coordinate x only0and where z x is a three-dimensional vector field, depending on the coordinate x only.

As a consequence of Theorem 2.1 together with Lemma A.3 from the Appendix, we deduce the following: Corollary 2.3 Consider a standard kinematical Lorentz’s structure with global time on R4 . Next consider the first coordinate system in R4 , which is simultaneously inertial and cartesian, with respect to this structure, and the second general coordinate system in R4 . Then, the second coordinate system is also simultaneously inertial and cartesian if and only if the change of the first coordinate system to the second one is given by the following relations: ⎧ 0 0 0 ⎪ ⎨x = x + c , 3  m ⎪ Bmj xj + wm x0 + cm ∀m = 1, 2, 3, ⎩x = j=1

(2.1.47)

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time

17

where {Bmn }n,m=1,2,3 ∈ R3×3 is a constant 3 × 3-matrix (independent on (x0 , x1 , x2 , x3 ) ∈ R4 ), and satisfying 3  j=1

Bmj Bnj =

3 

Bjm Bjn = δmn ∀ m, n = 1, 2, 3,

(2.1.48)

j=1

4 0 1 2 3 4 (c0 , c1, c2 , c3 ) ∈ R  is a constant vector (independent on (x , x , x , x ) ∈ R ) and w := w1 , w2 , w3 ∈ R3 is a constant three-dimensional vector (independent on (x0 , x1 , x2 , x3 ) ∈ R4 ). In particular, up to equivalence of cartesian coordinate systems (see Definition 2.11), (2.1.47) reduces to the classical Galilean Transformations:  x0 = x0 (2.1.49) xm = xm + wm x0 ∀m = 1, 2, 3,

  where w := w1 , w2 , w3 ∈ R3 is a constant three-dimensional vector field (independent on the point (x0 , x1 , x2 , x3 ) ∈ R4 ). Theorem 2.2 Consider a standard kinematical Lorentz’s structure with global time on R4 . Next consider the first coordinate system in R4 , which is kinematically preferable and the second general coordinate system in R4 . Then, the second coordinate system is also kinematically preferable, if and only if the first and the second coordinate systems are equivalent cartesian systems. Remark 2.2 Consider a standard kinematical Lorentz’s structure with global time on R4 . Next consider a coordinate system in R4 , which is simultaneously cartesian and Lorentzian, with respect to this structure. Then, by Definition 2.7 this system is kinematically preferable, with respect to this structure. Moreover, note that, all kinematically preferable systems are equivalent cartesian and Lorentzian systems. Finally, note that the arbitrary coordinate system is Lorentzian if and only if it can be obtained from the kinematically preferable system by some Lorentz’s transformation.

2.1.1 Pseudo-Lorentzian Coordinate Systems Definition 2.12 Consider a standard kinematical Lorentz’s structure with global time on R4 (see Definition 2.7), consisting of contravariant pseudo-metric {K mn }m,n=0,1,2,3 and the global time ϕ, together with the corresponding tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , defined, as in (2.1.32) by jm := vj vm − K jm ∀ j, m = 0, 1, 2, 3,

(2.1.50)

where (v0 , v1 , v2 , v3 ) is the contravariant four-vector field of the potential of generalized gravity, given, as in (2.1.30), by

2 Basic Definitions and Statements . . .

18

vm :=

3 

K mj

j=0

∂ϕ ∀ m = 0, 1, 2, 3. ∂xj

(2.1.51)

We say that a given general coordinate system in R4 is Pseudo-Lorentzian with respect to the tensor of the three-dimensional Geometry {mn }m,n=0,1,2,3 and the global time ϕ, if {mn }m,n=0,1,2,3 has the following simple form in the chosen coordinate system: ⎧ 00 2 0 1 2 3 4 ⎪ ⎨ = (w0 ) − 1 ∀ (x , x , x , x ) ∈ R 0j j0  =  = −w0 wj ∀ j = 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ⎪ ⎩ mn  = δmn + wm wn ∀ m, n = 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.52)

and at the same time in the same coordinate system we have 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x0 , x1 , x2 , x3 ) = (w0 , w1 , w2 , w3 )

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.53)

where (w0 , w1 , w2 , w3 ) ∈ R4 is some constant (independent on the point (x0 , x1 , x2 , x3 ) ∈ R4 ) vector, satisfying (w0 )2 −

3  (wj )2 = 1,

(2.1.54)

j=1

(Note that by Lemma A.4 the matrix {mn }0≤m,n≤3 in (2.1.52) is degenerate and moreover, it has one vanishing and three positive eigenvalues.) Remark 2.3 Obviously every Lorentzian coordinate system, where ⎧ 00 ⎪ ⎨K = 1 K 0j = K j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ mn K := −δmn ∀ m, n = 1, 2, 3, is a Pseudo-Lorentzian system, since such a system obtained from the kinematically preferable system by a linear Lorentz’s transformation. Moreover, every cartesian coordinate system is also Pseudo-Lorentzian system with 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x0 , x1 , x2 , x3 ) = (1, 0, 0, 0) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

Theorem 2.3 Given coordinate system is Pseudo-Lorentzian, if and only if it obtained from some cartesian coordinate system by some Lorentz’s transformation.

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time

19

Definition 2.13 In every coordinate system (possibly curvilinear) we consider the following matrices ⎧ 00 ⎪ ⎨M := 1 (2.1.55) M 0j = M j0 := 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm M := −δjm ∀ j, m = 1, 2, 3, ⎧ ⎪ ⎨M00 := 1 M0j = Mj0 := 0 ∀ j = 1, 2, 3 ⎪ ⎩ Mjm := −δjm ∀ j, m = 1, 2, 3,

and

(2.1.56)

(note that M kj and Mkj are not tensors). Next for given fixed constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying 3  3 

M jm wj wm = 1,

(2.1.57)

j=0 m=0

we say that the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of the class PL(w0 , w1 , w2 , w3 ) if we have x

m

:=

3  j=0

Bmj

 3 

 wk x

k

x +z j

k=0

m

 3 

 wk x

k

∀ m = 0, 1, 2, 3,

(2.1.58)

k=0

    where B(s) := Bmj (s) m,j=0,1,2,3 : R → R4×4 and z 0 (s), z 1 (s), z 2 (s), z 3 (s) : R → R4 satisfy M mn =

3  3 

Bmj (s)Bnk (s)M jk ∀ m, n = 0, 1, 2, 3, ∀s ∈ R,

(2.1.59)

j=0 k=0 3  j=0

and

   −1 dB B · wj (s) = 0 ∀ m = 0, 1, 2, 3, ∀s ∈ R, ds jm 3  3  m=0 j=0

wj



B−1



(s) jm

 dz m ds

(s) = 0 ∀s ∈ R.

(2.1.60)

(2.1.61)

As a direct consequence of Definition 2.13 we have the following: Corollary 2.4 Given change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is a Lorentz’s transformation, if and only if this transformation is of class PL(w0 , w1 , w2 , w3 ) for every fixed constant vector (w0 , w1 , w2 , w3 ) ∈ R4 satisfying

2 Basic Definitions and Statements . . .

20

    (2.1.57), where B := Bmj m,j=0,1,2,3 ∈ R4×4 and z 0 , z 1 , z 2 , z 3 ∈ R4 in (2.1.58) are independent on the argument s. As the second direct consequence of Definition 2.13 and Theorem 2.1 we have the following: Corollary 2.5 Consider a standard kinematical Lorentz’s structure with global time ϕ on R4 together with the corresponding tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 . Next consider the first cartesian, with respect to {mn }m,n=0,1,2,3 and ϕ, coordinate system in R4 and the second general coordinate system in R4 . Then, the second coordinate system is also cartesian if and only if the change of the the second one is of class PL(1, 0, 0, 0) where B(s) :=  firstcoordinate system to4×4 in (2.1.58) is given by Bmj (s) m,j=0,1,2,3 : R → R ⎧ ⎪ ⎨B00 (s) = 1 ∀s B0j (s) = Bj0 (s) = 0 ∀j = 1, 2, 3 ∀s ⎪ ⎩ Bmn (s) = Amn (s) ∀ m, n = 1, 2, 3 ∀s,

(2.1.62)

where {Amn (s)}n,m=1,2,3 ∈ R3×3 is a 3 × 3-matrix, satisfying 3 

Amj (s)Anj (s) =

j=1

3 

Ajm (s)Ajn (s) = δmn ∀ m, n = 1, 2, 3 ∀s.

(2.1.63)

j=1

Theorem 2.4 Consider a fixed Pseudo-Lorentzian coordinate system, so that ⎧ 00 2 0 1 2 3 4 ⎪ ⎨ = (w0 ) − 1 ∀ (x , x , x , x ) ∈ R 0j j0  =  = −w0 wj ∀ j = 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ⎪ ⎩ mn  = δmn + wm wn ∀ m, n = 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.64)

and at the same time in the same coordinate system we have 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x0 , x1 , x2 , x3 ) = (w0 , w1 , w2 , w3 )

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.1.65)

where (w0 , w1 , w2 , w3 ) ∈ R4 is some constant (independent on the point (x0 , x1 , x2 , x3 ) ∈ R4 ) vector, satisfying (w0 )2 −

3  (wj )2 = 1.

(2.1.66)

j=1

Next, assume that the new coordinate system is obtained from the given above (old) system by the following transformation

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time

(x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ).

21

(2.1.67)

Then the new coordinate system is also Pseudo-Lorentzian if and only if the transformation in (2.1.67) is of class PL ((w0 , w1 , w2 , w3 )). Proposition 2.3 Given a constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying (2.1.57) assume that the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of class PL(w0 , w1 , w2 , w3 ) and consider another constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , defined by wm =

3 

wj



B−1



(0) jm



∀ m = 0, 1, 2, 3, ∀s ∈ R,

(2.1.68)

j=0

  where B(s) := Bmj (s) m,j=0,1,2,3 : R → R4×4 be as in (2.1.58). Then, we have 3  3 

M mn wm wn = 1.

(2.1.69)

m=0 n=0

3 

3        −1 wj B jm (s) = wj B−1 jm (0) = wm

j=0

j=0

∀ m = 0, 1, 2, 3, ∀s ∈ R.

(2.1.70)

Moreover, at every point in R4 we have 3 

wm

m=0

∂xm = wn ∀ n = 0, 1, 2, 3, ∂xn

(2.1.71)

and as a direct consequence of (2.1.71) there exists a constant C ∈ R such that at every point in R4 we have  3

wk xk



k=0

=C+

 3

 wk x

k

.

(2.1.72)

k=0

Definition 2.14 Given two fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 and (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying 3  3  M jm wj wm = 1, (2.1.73) j=0 m=0

2 Basic Definitions and Statements . . .

22

and

3  3 

M jm wj wm = 1,

(2.1.74)

j=0 m=0

we say that the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )) if it belongs to PL(w0 , w1 , w2 , w3 ) and 3     wm = wj B−1 jm (0) ∀ m = 0, 1, 2, 3, ∀s ∈ R, (2.1.75) j=0

  where B(s) := Bmj (s) m,j=0,1,2,3 : R → R4×4 be as in (2.1.58). Furthermore, for given two fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 and  (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying (2.1.73) and (2.1.74) we define the subspace   L (w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )    PL (w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )   of transformations (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), where B := Bmj m,j=0,1,2,3   ∈ R4×4 and z 0 , z 1 , z 2 , z 3 ∈ R4 in (2.1.58) are independent on the argument s (obviously such a coordinate change is necessarily a Lorentz’s transformation). As a direct consequence of Proposition 2.3 we deduce two following Corollaries: Corollary 2.6 • Given fixed constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying (2.1.73), assume that the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of the class PL(w0 , w1 , w2 , w3 ). Then the constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , given by (2.1.75) satisfies (2.1.74) and the above coordinate change (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )). • Given a Lorentz’s transformation (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), there exist two fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 and (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying (2.1.73) and (2.1.74), such that the above of change coordinates (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of the class L((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )). Corollary 2.7 Consider a standard kinematical Lorentz’s structure with global time ϕ on R4 together with the corresponding tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 . Next consider the first cartesian, with respect to {mn }m,n=0,1,2,3 and ϕ, coordinate system in R4 and the second general coordinate system in R4 . Then, the second coordinate system is also cartesian if and only if the change of the first coordinate system to the second one is of class PL((1, 0, 0, 0); (1, 0, 0, 0)).

2.1 Generalized-Lorentz’s Structures with Time Direction and Global Time

23

Proposition 2.4 Given two fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 and (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying 3 3  

M jm wj wm = 1,

(2.1.76)

M jm wj wm = 1,

(2.1.77)

j=0 m=0

and

3  3  j=0 m=0

assume that the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )). Then the inverse change of variables (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is of the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )). As a direct consequence of Lemma 10.1 we deduce: Proposition 2.5 Given three fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 , (w0 , w1 , w2 , w3 ) ∈ R4 and (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying ⎧3 3 jm ⎪ ⎨j=0 m=0 M wj wm = 1 3 3 jm   j=0 m=0 M wj wm = 1 ⎪ ⎩3 3 jm   j=0 m=0 M wj wm = 1,

(2.1.78)

assume that the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) belongs to the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )) and another change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), belonging to the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )). Then, the composition of the above two changes of coordinate systems: (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) = (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) belongs to the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )). As a direct consequence of the Corollary 10.1 and Proposition 2.5 we deduce: Proposition 2.6 Given three fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 , (w0 , w1 , w2 , w3 ) ∈ R4 and (wˆ 0 , wˆ 1 , wˆ 2 , wˆ 3 ) ∈ R4 , satisfying

24

2 Basic Definitions and Statements . . .

⎧3 3 jm ⎪ ⎨j=0 m=0 M wj wm = 1 3 3 jm   j=0 m=0 M wj wm = 1 ⎪ ⎩3 3 jm ˆ j wˆ m = 1, j=0 m=0 M w

(2.1.79)

the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) belongs to the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )) if and only if there exist three other changes of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), belonging to the class L((w0 , w1 , w2 , w3 ); (wˆ 0 , wˆ 1 , wˆ 2 , wˆ 3 )), (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), belonging to the class PL((wˆ 0 , wˆ 1 , wˆ 2 , wˆ 3 ); (wˆ 0 , wˆ 1 , wˆ 2 , wˆ 3 )) and (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), belonging to the class L(((wˆ 0 , wˆ 1 , wˆ 2 , wˆ 3 ); (w0 , w1 , w2 , w3 )), so that the original transformation (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is a composition of the above three changes of coordinate systems: (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) = (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ).

(2.1.80)

As a direct consequence of Proposition 2.6 we deduce the following: Corollary 2.8 Given two fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 , (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying   3 3 jm m=0 M wj wm = 1 (2.1.81) j=0 3 3 jm   j=0 m=0 M wj wm = 1, the change of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) belongs to the class PL((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )) if and only if there exist three other changes of coordinate system (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), belonging to the class L((w0 , w1 , w2 , w3 ); (1, 0, 0, 0)), (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), belonging to the class PL((1, 0, 0, 0); (1, 0, 0, 0)) and (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ), belonging to the class L(((1, 0, 0, 0); (w0 , w1 , w2 , w3 )), so that the original transformation (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) is a composition of the above three changes of coordinate systems: (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) = (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ) → (x0 , x1 , x2 , x3 ). (2.1.82)

2.2 Kinematical Lorentz’s Structure with Global Time

25

2.2 Kinematical Lorentz’s Structure with Global Time Definition 2.15 We say that a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 is flat if the curvature tensor of this pseudo-metric vanishes. In other words, at every point in R4 we have 

j

Rkmn

K

∂  j  ∂  j  kn − n km m K K ∂x ∂x 3 3     d  d   j j kn K dm − km K nd +

:=

d =0

K

d =0

∀ j, k, m, n = 0, 1, 2, 3,

K

= 0 (2.2.1)

 where the Christoffel Symbols kjm are defined by (2.1.25). It is well known that K the equality in (2.2.1) is independent on the chosen coordinate system. Remark 2.4 Obviously, if given a pseudo-metric {K mn }m,n=0,1,2,3 , there exists an inertial coordinate system, with respect to {K mn }m,n=0,1,2,3 , then this pseudo-metric is obviously flat. In particular, if there exists some coordinate system where matrix {K mn }m,n=0,1,2,3 has a form of ⎧ 00 ⎪ ⎨K = 1 K 0j = K j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm K := −δjm ∀ j, m = 1, 2, 3

(2.2.2)

then a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 is flat. Definition 2.16 Given a generalized-Lorentz’s structure with global time on R4 , formed by a strongly correlating contravariant pseudo-metric {K mn }m,n=0,1,2,3 and a global time ϕ, we say that this generalized-Lorentz’s structure with global time is kinematical, if pseudo-metric {K mn }m,n=0,1,2,3  is flat and, at the same time, the tensor ∂ϕ ∂ϕ ∂ϕ ∂ϕ vanishes: of the covariant derivatives of the covector ∂x 0 , ∂x 1 , ∂x 2 , ∂x 3    ∂ϕ δj = 0 ∀ k, j = 0, 1, 2, 3. ∂xk K

(2.2.3)

Remark 2.5 Consider a standard kinematical Lorentz’s structure with global time on R4 , formed by a pseudo-metric {K mn }m,n=0,1,2,3 and a global time ϕ, so that there exists some coordinate system where matrix {K mn }m,n=0,1,2,3 has a form of ⎧ 00 ⎪ ⎨K = 1 K 0j = K j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm K := −δjm ∀ j, m = 1, 2, 3

(2.2.4)

2 Basic Definitions and Statements . . .

26

and at the same time the covector of time direction for ϕ has a form 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x0 , x1 , x2 , x3 ) = (1, 0, 0, 0) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.2.5) Then, we obviously obtain that this standard kinematical Lorentz’s structure with global time is kinematical in the sense of Definition 2.16. On the other hand, given an arbitrary kinematical Lorentz’s structure with global time, one can prove that every point in R4 has at least some neighborhood, in which (2.2.4) and (2.2.5) hold in some coordinate system.

2.3 Kinematical and Dynamical Generalized-Lorentz Structures with Time Direction We postulate that all real physical processes appear in some valid contravariant pseudo-metric {K mn }m,n=0,1,2,3 , describing the generalized gravity field, weakly correlated with some covector of time direction (w0 , w1 , w2 , w3 ). Furthermore, we distinguish two types of generalized gravity. The first type is the fictitious gravity which we call inertia. This type of gravity depends only on the flat Geometry of empty space-time via the choice of specific coordinate system, and it is independent on the surrounding real matter consisting of gravitational masses or other real physical fields. The second type of the gravity is the genuine (real) gravity, which depends essentially on the real physical matter, especially on gravitational masses. We assume that this type of gravity vanishes away from essential gravitational masses and strong real physical fields. Then we state the First Law of the Newton as the following: Proposition 2.7 In the parts of the space-time where we observe the absence of genuine gravity, and in particular away from essential real physical bodies and fields, we have (2.3.1) K mn = J mn ∀ m, n = 0, 1, 2, 3, 

and (w0 , w1 , w2 , w3 ) =

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 ,

(2.3.2)

where the strongly correlated contravariant, pseudo-metric {J mn }m,n=0,1,2,3 and the global time ϕ form the standard kinematical Lorentz’s structure with global time on R4 , as defined in Definition 2.7. Remark 2.6 We remind, that by Definition 2.7, there exists some coordinate system where matrix {J mn }m,n=0,1,2,3 has a form of

2.3 Kinematical and Dynamical Generalized-Lorentz Structures …

⎧ 00 ⎪ ⎨J = 1 J 0j = J j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm J := −δjm ∀ j, m = 1, 2, 3

27

(2.3.3)

and at the same coordinate system the covector of time direction for the global time ϕ has a form 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x0 , x1 , x2 , x3 ) = (1, 0, 0, 0) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.3.4) We remind that this particular system is called kinematically preferable, and it is unique, up to equivalence. Furthermore, we define the kinematical tensor of threedimensional Geometry {mn }m,n=0,1,2,3 , given by mn where

⎛ ⎞⎛ ⎞ 3 3   ∂ϕ ∂ϕ := ⎝ J mj j ⎠ ⎝ J nj j ⎠ − J mn ∀ m, n = 0, 1, 2, 3, ∂x ∂x j=0 j=0 ⎛ ⎝

3  j=0

⎞ 3 3 3    ∂ϕ ∂ϕ ∂ϕ ∂ϕ J 0j j , J 1j j , J 2j j , J 3j j ⎠ ∂x ∂x ∂x ∂x j=0 j=0 j=0

(2.3.5)

(2.3.6)

is the contravariant vector of inertia. In particular, in the kinematically preferable coordinate system, where (2.3.3) and (2.3.4) hold we have ⎧ 00 ⎪ ⎨ = 0 0j = j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm  := δjm ∀ j, m = 1, 2, 3.

(2.3.7)

We also remind that, given arbitrary coordinate system, it is called cartesian if in this system we have simultaneously (2.3.7) and (2.3.4) but, we do not necessarily have (2.3.3). On the other hand, given arbitrary coordinate system, it is called Lorentzian if in this system we have (2.3.3) but, we do not necessarily have (2.3.7) or (2.3.4). Finally, given arbitrary coordinate system, it is inertial, if we can get it from kinematically preferable coordinate system by a linear transformation. We also remind, that we obtain a coordinate system which is simultaneously cartesian and inertial from another such system by Galilean transformations. On the other hand, we obtain a coordinate system which is simultaneously Loretzian and inertial from another such system by Lorentz’s transformations. The unique, up to equivalence, coordinate system which is simultaneously cartesian and Loretzian is a kinematically preferable coordinate system.

2 Basic Definitions and Statements . . .

28

2.4 Lagrangian of the Motion of a Classical Point Particle in a Given Pseudo-metric with Time Direction Definition 2.17 Consider a classical point particle with the inertial mass m and the charge σ moving in the generalized-gravitational field, given by a contravariant pseudo-metric {K mn }m,n=0,1,2,3 , weakly correlated with a four-covector of the time direction (w0 , w1 , w2 , w3 ), and influenced by the electromagnetic field with the four-covector of the electromagnetic  potential (A0 , A1 , A2 , A3 ). Next assume  that χ (s) := χ 0 (s), χ 1 (s), χ 2 (s), χ 3 (s) : [a, b] → R4 ∈ R4 is a four-dimensional space-time trajectory of the particle, parameterized by some general scalar parameter s ∈ [a, b] (including the cases where a = −∞ and/or b = +∞). Then we say that the parameter of the trajectory s is proper with respect to the time direction (w0 , w1 , w2 , w3 ), if we have 3 

wj (χ (s))

j=0

dχ j (s) > 0 ds

∀s ∈ [a, b].

(2.4.1)

Next, given a proper parametrization s, we define the contravariant four-vector of the velocity of the particle (u0 , u1 , u2 , u3 )(s), with respect to the time direction (w0 , w1 , w2 , w3 ), by the following  u (s) = j

3  k=0

so that

dχ k (s) wk (χ (s)) ds 3 

−1

dχ j (s) ∀ j = 0, 1, 2, 3 ∀s ∈ [a, b], ds (2.4.2)

wj (χ (s)) uj (s) = 1

∀s ∈ [a, b]

(2.4.3)

j=0

(It is obvious that the definition of (u0 , u1 , u2 , u3 ) in (2.4.2.) is independent on the choice of the proper parametrization s). Moreover, given an arbitrary contravariant four-vector field (f 0 , f 1 , f 2 , f 3 ), we say that this field is a speed-like four-vector field, with respect to the time direction (w0 , w1 , w2 , w3 ), if we have 3 

wj (x0 , x1 , x2 , x3 ) f j (x0 , x1 , x2 , x3 ) = 1

∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.4.4)

j=0

Remark 2.7 Note here the difference in our notation with respect to the usual special and general relativity: in the usual theory of relativity the notation of the four-velocity of the particle is booked for the normalized contravariant four-vector (ˆu0 , uˆ 1 , uˆ 2 , uˆ 3 )(s), given by

2.4 Lagrangian of the Motion of a Classical Point Particle …



29

− 21 dχ j dχ m dχ n uˆ (s) := Kmn (χ (s)) (s) (s) (s) ds ds ds m=0 n=0 − 21  3 3  Kmn (χ (s)) um (s) un (s) uj (s) ∀ j = 0, 1, 2, 3 ∀s ∈ [a, b], = j

3  3 

m=0 n=0

(2.4.5) while, in the present paper by the name four-velocity we denote the contravariant four-vector (u0 , u1 , u2 , u3 )(s), given by (2.4.2). Furthermore, given a proper parametrization s, consider the Lagrangian of motion of this particle given by the following (in the Gaussian unit system) ⎞⎫ ⎛ 3  3 ⎬  LG (χ ) = Kjk (χ (s)) uj (s) uk (s)⎠ −m G ⎝ ⎭ ⎩ j=0 k=0 a ⎛ ⎞ 3 j  dχ ⎝ ⎠ ds wj (χ (s)) ds j=0 ⎧ ⎫ b ⎨  3 dχ j ⎬ σ Aj (χ (s)) − + ds ⎩ ds ⎭ j=0 a ⎧ ⎫ ⎞ ⎛ b ⎨ 3  3 3 ⎬   Kjk (χ (s)) uj (s) uk (s)⎠ − σ uj (s) Aj (χ (s)) −m G ⎝ = ⎩ ⎭ j=0 k=0 j=0 a ⎛ ⎞ 3  dχ j ⎠ ⎝ wj (χ (s)) ds, (2.4.6) ds j=0 ⎧ b ⎨

where {Kmn }m,n=0,1,2,3 is the inverse to {K mn }m,n=0,1,2,3 covariant pseudo-metric and G(τ ) : R → R is some fixed function. Obviously, the functional in (2.4.6) is independent on the choice of the proper parametrization. We have the following two important particular cases: √ • The case of relativistic particle where G(τ ) := τ − 1 in (2.4.6). • The case of non-relativistic particle where G(τ ) := 21 (τ − 1) in (2.4.6). In both cases G(1) = 0. Moreover, we have   √ 1 τ − 1 = (τ − 1) + O (τ − 1)2 . 2

(2.4.7)

2 Basic Definitions and Statements . . .

30

In the first relativistic case we simplify (2.4.6) as: ⎧  b ⎨ 3  3  L1 (χ ) = Kjk (χ (s)) −m ! ⎩ j=0 k=0 a ⎧ b ⎨  3 σ Aj (χ (s)) − + ⎩ j=0

a

⎫ dχ j dχ k ⎬ ds ds ds ⎭ j

dχ + ds

3 

⎫

mwj (χ (s))

j=0

j⎬

dχ ds. (2.4.8) ds ⎭

On the other hand, in the second, non-relativistic case we simplify (2.4.6) as: ⎫  ⎧ 3  3  ⎪ dχ j dχ k ⎪ ⎪ ⎪ K (χ (s)) ds ds ⎪ ⎪ ⎪ b ⎪ ⎬ ⎨ m j=0 k=0 jk   ds − L2 (χ ) = ⎪ ⎪ 3 2  ⎪ ⎪ j dχ ⎪ ⎪ a ⎪ ⎪ wj (χ (s)) ds ⎭ ⎩ j=0

b + a

⎧ ⎫ 3 3 ⎨  dχ j  m dχ j ⎬ σ Aj (χ (s)) − + wj (χ (s)) ds. (2.4.9) ⎩ ds 2 ds ⎭ j=0 j=0

Definition 2.18 Given two events on the trajectory of the motion of the particle χ (s1 ) ∈ R4 and χ (s2 ) ∈ R4 , with parameters of the chosen proper parametrization s1 and s2 , respectively, we define the interval of time that passed from the event χ (s1 ) to event χ (s2 ), corresponding to the time direction (w0 , w1 , w2 , w3 ), as: ⎛ ⎞ s2  3 j dχ ⎠ ds. τ (χ (s1 ), χ (s2 )) := ⎝ wj (χ (s)) ds j=0

(2.4.10)

s1

Then obviously the interval of time is independent on the proper parametrization of the trajectory. Moreover if we have  (w0 , w1 , w2 , w3 ) =

∂ψ ∂ψ ∂ψ ∂ψ , , , ∂x0 ∂x1 ∂x2 ∂x3

 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.4.11)

where ψ is some global time (which is obviously weakly correlated with {K mn }m,n=0,1,2,3 ), then, inserting (2.4.11) into (2.4.10) we obviously deduce τ (χ (s1 ), χ (s2 )) = ψ (χ (s2 )) − ψ (χ (s1 )) .

(2.4.12)

So, in the later case the interval of time between two events is the difference in global time between the second and the first events, and therefore, this interval is

2.4 Lagrangian of the Motion of a Classical Point Particle …

31

independent on the trajectory itself, but depends only on the two ending points of the trajectory. Note also that in the general case the quantity τ (s) := τ (χ (s), χ (a))

∀ s ∈ [a, b],

(2.4.13)

can be chosen as a preferable parameter of the trajectory of the motion and we rewrite the relativistic Lagrangian in (2.4.8) as: ⎧ ⎨

⎫ ⎞ ⎛ 3 3  3 ⎬ j⎪ j dχ k   dχ dχ ⎟ ⎜ Kjk (χ (τ )) σ Aj (χ (τ )) m − m ⎝! dτ, ⎠− ⎪ dτ dτ dτ ⎪ ⎩ ⎭ j=0 k=0 j=0

τ(b)⎪

L1 (χ ) = 0

(2.4.14) and we rewrite the non-relativistic Lagrangian in (2.4.9) as ⎧ ⎨m

⎫ ⎛ ⎞ 3 3 3 m ⎝  dχ j dχ k ⎠  dχ j ⎬ − Kjk (χ (τ )) σ Aj (χ (τ )) dτ. − ⎩2 2 dτ dτ dτ ⎭

τ(b)

L2 (χ ) =

j=0 k=0

0

j=0

(2.4.15) Moreover, in the case of general G, we rewrite the general Lagrangian in (2.4.6) as: LG (χ ) =

⎧ τ (b)⎨ 0

⎫ ⎛ ⎞ 3  3 3 j k j⎬   dχ dχ ⎠ dχ Kjk (χ (τ )) σ Aj (χ (τ )) −m G ⎝ − dτ, ⎩ dτ dτ dτ ⎭ j=0 j=0 k=0

(2.4.16) and the velocity satisfies uj (τ ) =

dχ j (τ ) ∀ j = 0, 1, 2, 3 ∀s ∈ [0, τ (b)]. dτ

(2.4.17)

Furthermore, we postulate the Second Law of Newton: Proposition 2.8 The motion of a classical point particle with inertial mass m and charge σ is described by the Lagrangian (2.4.6), where we consider either G(τ ) := √ τ − 1 (the relativistic case) or G(τ ) := 21 (τ − 1) (a non-relativistic approximation). As a consequence of the first and the second laws of Newton we deduce the following. √ Corollary 2.9 Assume that either G(τ ) := τ − 1 or G(τ ) := 21 (τ − 1) in (2.4.6). Then, in the absence of genuine gravitational field, the electromagnetical or any other physical field, given an arbitrary inertial coordinate system, the trajectory of the classical point particle in this coordinate system is a direct-line in R4 . Note that the coordinate system must be inertial, but not necessary cartesian or Lorentzian.

2 Basic Definitions and Statements . . .

32

2.5 Lagrangian of the Electromagnetic Field in a Given Pseudo-metric Definition 2.19 Consider a contravariant pseudo-metric K := {K mn }m,n=0,1,2,3 and let A = (A0 , A1 , A2 , A3 ) := (A0 , A1 , A2 , A3 )(x0 , x1 , x2 , x3 ) be the four-covector of the electromagnetic potential and j = (j0 , j 1 , j 2 , j 3 ) := (j 0 , j 1 , j 2 , j 3 )(x0 , x1 , x2 , x3 ) be the contravariant four-vector of the four-current. As usual, we consider the Lagrangian density of the electromagnetic field given by the following (in the Gaussian unit system):   1 Le (A0 , A1 , A2 , A3 ), {K mn }m,n=0,1,2,3 := 4π ⎛ ⎞     3  3 3  3  3  ∂Ak 1 mn pk ∂Ap ∂Am ∂An ⎝− K K − p − k − 4π j k Ak ⎠ , m n 4 ∂x ∂x ∂x ∂x n=0 m=0 p=0 k=0

k=0

(2.5.1) (we use the dynamical pseudo-metric {K mn }m,n=0,1,2,3 in this definition since it is well known that the electromagnetic field is dependent on the genuine gravity). Correspondingly, given a subregion V ⊂ R4 , the Lagrangian of the electromagnetic field in this subregion is given, as usual, by Le (A0 , A1 , A2 , A3 )   $− 21    $$  mn  $ mn Le (A0 , A1 , A2 , A3 ), {K }m,n=0,1,2,3 $det K n,m=0,1,2,3 $ := V

dx0 dx1 dx2 dx3 .

(2.5.2)

In particular, the critical points of Le (A0 , A1 , A2 , A3 ) must satisfy δLe (A0 , A1 , A2 , A3 ) = 0 ∀j = 0, 1, 2, 3, δAj

(2.5.3)

and, thus by the Euler-Lagrange we deduce 3  ∂ j ∂x j=0

 3 3   $ $− 1 km jn ∂An ∂A m p q $det {K }p,q=0,1,2,3 $ 2 K K − n ∂xm ∂x m=0 n=0 $− 1 $ = −4π $det {K p q }p,q=0,1,2,3 $ 2 j k ∀ k = 0, 1, 2, 3.

(2.5.4)

2.6 Correlated Pseudo-metrics

33

2.6 Correlated Pseudo-metrics Definition 2.20 Consider two contravariant pseudo-metrics {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 on R4 and let {Jmn }m,n=0,1,2,3 and {Kmn }m,n=0,1,2,3 be the inverse covariant pseudo-metrics on R4 , associated with {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 , that satisfy 3 

J

mk

Jkn =

k=0

3 

 K

mk

k=0

Kkn =

1 if m = n 0 if m = n

∀ m, n = 0, 1, 2, 3.

(2.6.1)

Then, we say that {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 are two correlated pseudometrics, if there exists a four-covector of the time direction, (w0 , w1 , w2 , w3 ), weakly correlating with {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 simultaneously, so that we have 3  3 

J jm wj wm > 0 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.6.2)

K jm wj wm > 0 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.6.3)

j=0 m=0 3  3  j=0 m=0

and moreover, there exists a four-covector field (S0 , S1 , S2 , S3 ), such that we have   Kjm = Jjm + wj Sm + wm Sj ∀ 0 ≤ j, m ≤ 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.6.4)

Theorem 2.5 Consider a contravariant pseudo-metric {J mn }m,n=0,1,2,3 on R4 and an arbitrary covariant four-covector field (w0 , w1 , w2 , w3 ), served as a time direction, and such that {J mn }m,n=0,1,2,3 and (w0 , w1 , w2 , w3 ) are weakly correlated. Furthermore, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 and let (r 0 , r 1 , r 2 , r 3 ) be the potential of the generalized gravity, corresponding to {J mn }m,n=0,1,2,3 and (w0 , w1 , w2 , w3 ) and satisfying ⎛ r m := ⎝

3  3  j=0 k=0

⎞− 21 ⎛ ⎞ 3  J jk wj wk ⎠ ⎝ J mj wj ⎠ ∀ m = 0, 1, 2, 3.

(2.6.5)

j=0

Furthermore, as before, consider a contravariant tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , given by jm := r j r m − J jm ∀ j, m = 0, 1, 2, 3,

(2.6.6)

Next, given an arbitrary contravariant four-vector field (v0 , v1 , v2 , v3 ), satisfying

2 Basic Definitions and Statements . . .

34 3 

vj wj > 0,

(2.6.7)

j=0

define the contravariant symmetric tensor field {K mn }m,n=0,1,2,3 by the following relations: (2.6.8) K jm := vj vm − jm ∀ j, m = 0, 1, 2, 3. Then, {K mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric. Moreover, the inverse to {K mn }m,n=0,1,2,3 covariant pseudo-metric {Kmn }m,n=0,1,2,3 is given at every point in R4 by the following:   Kjm = Jjm + wj Sm + wm Sj ∀ 0 ≤ j, m ≤ 3,

(2.6.9)

where the covariant four-covector (S0 , S1 , S2 , S3 ) is defined by the following covariant relation: ⎞− 21 ⎛ ⎞ ⎛ 3 3 3  1 ⎝  jk Sm := J wj wk ⎠ ⎝ Kmj r j ⎠ 2 j=0 j=0 k=0 ⎞−1 ⎛ ⎞ ⎛ 3 3 1  j ⎠ ⎝ − ⎝ v wj Jmj vj ⎠ ∀ 0 ≤ m ≤ 3. 2 j=0 j=0

(2.6.10)

Finally, the following covariant relations are valid 3  3 

Kjm vj vm = 1,

(2.6.11)

j=0 m=0 3 

⎛ ⎞ 3  K jm wj = ⎝ vj wj ⎠ vm ∀ m = 0, 1, 2, 3,

j=0

(2.6.12)

j=0

and   3 3      mn − det {J } J w w mn m,n=0,1,2,3 m n 3  3  m=0 n=0    K jm wj wm = − det {Kmn }m,n=0,1,2,3 m=0 j=0 ⎞2 ⎛ 3  vj wj ⎠ . (2.6.13) =⎝ j=0

2.6 Correlated Pseudo-metrics

35

So, {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 are two correlated pseudo-metrics (see Definition 2.20). Moreover, K mn and (w0 , w1 , w2 , w3 ) are weakly correlated and (v0 , v1 , v2 , v3 ) is a potential of generalized gravity, corresponding to the couple K mn and (w0 , w1 , w2 , w3 ). Theorem 2.6 Consider a contravariant pseudo-metric {J mn }m,n=0,1,2,3 on R4 and an arbitrary covariant four-covector field (w0 , w1 , w2 , w3 ), served as a time direction, and such that {J mn }m,n=0,1,2,3 and (w0 , w1 , w2 , w3 ) are weakly correlated. Furthermore, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 and let (r 0 , r 1 , r 2 , r 3 ) be the potential of the generalized gravity, corresponding to {J mn }m,n=0,1,2,3 and (w0 , w1 , w2 , w3 ) and satisfying (2.6.5). Furthermore, as before, consider a contravariant tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , given by (2.6.6). Next consider, an arbitrary covariant four-covector (S0 , S1 , S2 , S3 ) such that a two-times covariant tensor field {Kmn }m,n=0,1,2,3 , defined by   (2.6.14) Kjm = Jjm + wj Sm + wm Sj ∀ 0 ≤ j, m ≤ 3, satisfies the following at every point in R4 :   det {Kmn }m,n=0,1,2,3 < 0.

(2.6.15)

Then, the inverse to {Kmn }m,n=0,1,2,3 two-times contravariant symmetric tensor field {K mn }m,n=0,1,2,3 is given by K mn = vm vn − mn

∀ m, n = 0, 1, 2, 3.

(2.6.16)

where the contravariant vector field (v0 , v1 , v2 , v3 ) is defined by the following covariant relation: % vm :=

⎞⎛ ⎞ ⎞    ⎛⎛ 3 3 3    } {J − det kn k,n=0,1,2,3  ⎝⎝!  J jk wj wk ⎠ ⎝ mj Sj ⎠ + r m ⎠ − det {Kkn }k,n=0,1,2,3 j=0 k=0 j=0 ∀ m = 0, 1, 2, 3.

Furthermore, we also have

⎛ ⎝

3 

(2.6.17)

⎞ vj wj ⎠ > 0,

(2.6.18)

j=0

and {K mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric. Moreover, the covariant relations in (2.6.10), (2.6.11), (2.6.12) and (2.6.13) are also valid. So, {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 are two correlated pseudo-metrics. Moreover, K mn and (w0 , w1 , w2 , w3 ) are weakly correlated and (v0 , v1 , v2 , v3 ) is a potential of generalized gravity, corresponding to K mn and (w0 , w1 , w2 , w3 ).

2 Basic Definitions and Statements . . .

36

Corollary 2.10 Consider a contravariant pseudo-metric {J mn }m,n=0,1,2,3 on R4 and an arbitrary scalar field ϕ : R4 → R, served as a global time, such that 3  3  j=0 m=0

J jm

∂ϕ ∂ϕ = 1 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . ∂xj ∂xm

(2.6.19)

Furthermore, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 and let (r 0 , r 1 , r 2 , r 3 ) be the contravariant fourvector field, defined by r m :=

3 

J mj

j=0

∂ϕ ∀ m = 0, 1, 2, 3. ∂xj

(2.6.20)

Furthermore, as before, consider a contravariant tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , given by jm := r j r m − J jm ∀ j, m = 0, 1, 2, 3.

(2.6.21)

Next, given an arbitrary contravariant four-vector field (v0 , v1 , v2 , v3 ), satisfying 3  j=0

vj

∂ϕ = 1, ∂xj

(2.6.22)

define the contravariant symmetric tensor field {K mn }m,n=0,1,2,3 by the following relations: (2.6.23) K jm := vj vm − jm ∀ j, m = 0, 1, 2, 3. Then, {K mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric and we have 3  3  m=0 j=0

K jm

∂ϕ ∂ϕ = 1. ∂xj ∂xm

(2.6.24)

Moreover, the inverse to {K mn }m,n=0,1,2,3 covariant pseudo-metric {Kmn }m,n=0,1,2,3 is given at every point in R4 by the following:   ∂ϕ ∂ϕ Kjm = Jjm + Sm j + Sj m ∀ 0 ≤ j, m ≤ 3, ∂x ∂x

(2.6.25)

where the covariant four-covector (S0 , S1 , S2 , S3 ) is defined by the following covariant relation:

2.6 Correlated Pseudo-metrics

Sm :=

37

3  1 j=0

2

Kmj r j −

3  1 j=0

2

Jmj vj ∀ 0 ≤ m ≤ 3.

(2.6.26)

Finally, the following covariant relations are valid 3  3 

Kjm vj vm = 1,

(2.6.27)

∂ϕ = vm ∀ m = 0, 1, 2, 3, ∂xj

(2.6.28)

j=0 m=0 3 

K jm

j=0

    det {Jmn }m,n=0,1,2,3 = det {Kmn }m,n=0,1,2,3 .

(2.6.29)

So, {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 are two correlated pseudo-metrics, that both correlated with the same global time ϕ and (v0 , v1 , v2 , v3 ) is a potential of generalized gravity, corresponding to K mn and ϕ. Corollary 2.11 Consider a contravariant pseudo-metric {J mn }m,n=0,1,2,3 on R4 and an arbitrary scalar field ϕ : R4 → R, served as a global time, such that we have (2.6.19). Next, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 and let (r 0 , r 1 , r 2 , r 3 ) be the contravariant four-vector field, defined by (2.6.20). Furthermore, as before, consider a contravariant tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , given by (2.6.21). Next consider, an arbitrary covariant four-covector (S0 , S1 , S2 , S3 ) such that a two-times covariant tensor field {K mn }m,n=0,1,2,3 , defined by   ∂ϕ ∂ϕ ∀ 0 ≤ j, m ≤ 3, Kjm = Jjm + Sm j + Sj m ∂x ∂x

(2.6.30)

satisfies the following covariant relation at every point in R4 :     det {Kmn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 .

(2.6.31)

Then, the inverse to {K mn }m,n=0,1,2,3 two-times contravariant symmetric tensor field {Kmn }m,n=0,1,2,3 is given by K mn = vm vn − mn

∀ m, n = 0, 1, 2, 3.

(2.6.32)

where the contravariant vector field (v0 , v1 , v2 , v3 ) is defined by the following covariant relation: ⎛ ⎞ 3  vm := ⎝ mj Sj + r m ⎠ ∀ m = 0, 1, 2, 3. (2.6.33) j=0

2 Basic Definitions and Statements . . .

38

Moreover, we also have

3  j=0

vj

∂ϕ = 1, ∂xj

(2.6.34)

{K mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric, and 3  3  m=0 j=0

K jm

∂ϕ ∂ϕ = 1. ∂xj ∂xm

(2.6.35)

Moreover, the covariant relations in (2.6.26), (2.6.27) and (2.6.28) are also valid. So, {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 are two correlated pseudo-metrics, that both correlated with the same global time ϕ and (v0 , v1 , v2 , v3 ) is a potential of generalized gravity, corresponding to K mn and ϕ.

2.7 Kinematically Correlated Models of the Genuine Gravity Definition 2.21 Consider the strongly correlated kinematical contravariant, pseudometric {J mn }m,n=0,1,2,3 and the kinematical global time ϕ, forming a standard kinematical Lorentz’s structure with global time on R4 , as in Proposition 2.7 and Remark 2.6. Furthermore, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 . Given, an arbitrary dynamical four-covector of the time direction, (w0 , w1 , w2 , w3 ) (formally unrelated to the kinematical global time ϕ), which is weakly correlated with {J mn }m,n=0,1,2,3 and an arbitrary four-covector field (S0 , S1 , S2 , S3 ), which we call the four-covector of genuine gravity, consider the two-times covariant tensor {Kmn }m,n=0,1,2,3 defined by:   Kjm = Jjm + wj Sm + wm Sj ∀ 0 ≤ j, m ≤ 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.7.1)

and assume that (S0 , S1 , S2 , S3 ) is such that {Kmn }m,n=0,1,2,3 in (2.7.1) satisfies     det {Kmn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4

(2.7.2)

(the last equality is obviously independent on the choice of coordinate system). Then, by Theorem 2.6 the inverse to {Kmn }m,n=0,1,2,3 two-times contravariant tensor {K mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric, weakly correlated with the time direction (w0 , w1 , w2 , w3 ). Moreover, we call such a pseudo-metric {K mn }m,n=0,1,2,3 , with time direction (w0 , w1 , w2 , w3 ), kinematically correlated pseudo-metric with time direction (w0 , w1 , w2 , w3 ), corresponding to the covector of genuine gravity (S0 , S1 , S2 , S3 ).

2.7 Kinematically Correlated Models of the Genuine Gravity

39

Remark 2.8 If {K mn }m,n=0,1,2,3 , with time direction (w0 , w1 , w2 , w3 ), is a kinematically correlated pseudo-metric with time direction (w0 , w1 , w2 , w3 ), corresponding to the covector of genuine gravity (S0 , S1 , S2 , S3 ), then by Theorem 2.6 {J mn }m,n=0,1,2,3 and {K mn }m,n=0,1,2,3 are two correlated pseudo-metrics (see Definition 2.20). Moreover, denoting by (r 0 , r 1 , r 2 , r 3 ) the contravariant four-vector of the generalized gravity, corresponding to {J mn }m,n=0,1,2,3 and (w0 , w1 , w2 , w3 ), which satisfies ⎛ r m := ⎝

3  3 

⎞− 21 ⎛ ⎞ 3  J jk wj wk ⎠ ⎝ J mj wj ⎠ ∀ m = 0, 1, 2, 3,

j=0 k=0

(2.7.3)

j=0

and considering a contravariant {mn }m,n=0,1,2,3 , given by

tensor

of

three-dimensional

mn := r m r n − J mn ∀ m, n = 0, 1, 2, 3,

Geometry

(2.7.4)

using Theorem 2.6, we have K mn = vm vn − mn = J mn + vm vn − r m r n

∀ m, n = 0, 1, 2, 3.

(2.7.5)

where the contravariant vector of generalized gravity (v0 , v1 , v2 , v3 ) is defined by the following covariant relation: ⎛ ⎞⎛ ⎞ 3  3 3   vm := ⎝! J jk wj wk ⎠ ⎝ mj Sj ⎠ + r m j=0 k=0

∀ m = 0, 1, 2, 3,

(2.7.6)

j=0

and both couples {J mn }m,n=0,1,2,3 , (w0 , w1 , w2 , w3 ), and {K mn }m,n=0,1,2,3 , (w0 , w1 , w2 , w3 ) are weakly correlated. Moreover, we also have ⎛ ⎝

3 

⎞ vj wj ⎠ > 0,

(2.7.7)

j=0

and the following covariant relations are valid ⎞− 21 ⎛ ⎞ ⎛ 3 3 3  1 ⎝  jk Sm := J wj wk ⎠ ⎝ Kmj r j ⎠ 2 j=0 j=0 k=0 ⎞−1 ⎛ ⎞ ⎛ 3 3 1  j ⎠ ⎝ − ⎝ v wj Jmj vj ⎠ ∀ 0 ≤ m ≤ 3, 2 j=0 j=0

(2.7.8)

2 Basic Definitions and Statements . . .

40 3  3 

Kjm vj vm = 1,

(2.7.9)

j=0 m=0 3 

⎛ ⎞ 3  K jm wj = ⎝ vj wj ⎠ vm ∀ m = 0, 1, 2, 3,

j=0

and 3  3  m=0 j=0

K jm wj wm =

(2.7.10)

j=0

3  3  m=0 n=0

⎛ ⎞2 3  J mn wm wn = ⎝ vj wj ⎠ > 0.

(2.7.11)

j=0

Note, however, that the dynamical time direction (w0 , w1 , w2 , w3 ) can differ from   ∂ϕ ∂ϕ ∂ϕ ∂ϕ and the dynamical tensor of three, , , kinematical time direction ∂x 0 ∂x1 ∂x2 ∂x3 dimensional Geometry {mn }m,n=0,1,2,3 can differ from the kinematical tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , given by mn

⎛ ⎞⎛ ⎞ 3 3   ∂ϕ ∂ϕ := ⎝ J mj j ⎠ ⎝ J nj j ⎠ − J mn ∀ m, n = 0, 1, 2, 3. ∂x ∂x j=0 j=0

(2.7.12)

Finally, note that if the covector of genuine gravity (S0 , S1 , S2 , S3 ) vanishes at some point, then by (2.7.1) at this point we have Kjm = Jjm ∀ 0 ≤ j, m ≤ 3,

(2.7.13)

correspondingly to the First Law of Newton (Axiom I). Definition 2.22 Consider the strongly correlated kinematical contravariant, pseudometric {J mn }m,n=0,1,2,3 and the kinematical global time ϕ, forming a standard kinematical Lorentz’s structure with global time on R4 , as in Proposition 2.7 and Remark 2.6. Furthermore, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 . Then given a four-covector field (S0 , S1 , S2 , S3 ), which we call the four-covector of genuine gravity, consider the two-times covariant tensor {Kˆ mn }m,n=0,1,2,3 defined by:   ∂ϕ ∂ϕ ∀ 0 ≤ j, m ≤ 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , Kˆ jm = Jjm + Sm j + m Sj ∂x ∂x (2.7.14) and assume that (S0 , S1 , S2 , S3 ) is such that {Kˆ mn }m,n=0,1,2,3 in (2.7.14) satisfies     det {Kˆ mn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . (2.7.15)

2.7 Kinematically Correlated Models of the Genuine Gravity

41

Then, as before, the inverse to {Kˆ mn }m,n=0,1,2,3 two-times contravariant tensor {Kˆ mn }m,n=0,1,2,3 is a valid contravariant pseudo-metric, strongly correlated with the kinematical global time ϕ. Moreover, we call such a pseudo-metric {Kˆ mn }m,n=0,1,2,3 , with global time ϕ, kinematically semi-scalar super-correlated pseudo-metric with kinematical global time ϕ, corresponding to the covector of genuine gravity (S0 , S1 , S2 , S3 ). Remark 2.9 If {K mn }m,n=0,1,2,3 , with kinematical global time ϕ, is a kinematically semi-scalar super-correlated pseudo-metric with kinematical global time ϕ, corresponding to the covector of genuine gravity (S0 , S1 , S2 , S3 ), then, denoting by (r 0 , r 1 , r 2 , r 3 ) the contravariant four-vector of the inertia, corresponding to {J mn }m,n=0,1,2,3 and ϕ, which satisfies r m :=

3 

J mj

j=0

∂ϕ ∀ m = 0, 1, 2, 3, ∂xj

and considering a contravariant {mn }m,n=0,1,2,3 , given by ⎛ mn := r m r n − J mn = ⎝

3  j=0

tensor

of

three-dimensional

(2.7.16)

Geometry

⎞⎛ ⎞ 3  ∂ϕ ∂ϕ J mj j ⎠ ⎝ J nj j ⎠ − J mn ∀ m, n = 0, 1, 2, 3, ∂x ∂x j=0 (2.7.17)

we have K mn = vm vn − mn = J mn + vm vn − r m r n

∀ m, n = 0, 1, 2, 3.

(2.7.18)

where the contravariant vector of generalized gravity (v0 , v1 , v2 , v3 ) is defined by the following covariant relation: ⎛ vm := ⎝

3 

⎞ mj Sj ⎠ + r m

∀ m = 0, 1, 2, 3,

(2.7.19)

j=0

and both couples {J mn }m,n=0,1,2,3 , ϕ, and {K mn }m,n=0,1,2,3 , ϕ are strongly correlated. Moreover, the following covariant relations are valid ⎛ ⎝

3  j=0

⎞ ∂ϕ vj j ⎠ = 1, ∂x

⎞ ⎞ ⎛ ⎛ 3 3  1 1 ⎝ Sm := Kmj r j ⎠ − ⎝ Jmj vj ⎠ ∀ 0 ≤ m ≤ 3, 2 j=0 2 j=0

(2.7.20)

(2.7.21)

2 Basic Definitions and Statements . . .

42 3  3 

Kjm vj vm = 1,

(2.7.22)

∂ϕ = vm ∀ m = 0, 1, 2, 3, ∂xj

(2.7.23)

j=0 m=0 3  j=0

K jm

3  3 

K jm

∂ϕ ∂ϕ = 1, ∂xj ∂xm

(2.7.24)

J jm

∂ϕ ∂ϕ = 1. ∂xj ∂xm

(2.7.25)

j=0 m=0

and

3  3  j=0 m=0

Finally, note that the dynamical global time equals to kinematical time ϕ and the dynamical tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 equals to the kinematical tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , so that ⎛ mn = mn := ⎝

3  j=0

⎞⎛ ⎞ 3  ∂ϕ ∂ϕ J mj j ⎠ ⎝ J nj j ⎠ − J mn ∀ m, n = 0, 1, 2, 3. ∂x ∂x j=0 (2.7.26)

2.8 Lagrangian for Dynamical Time Direction and Its Limiting Case Consider the strongly correlated kinematical covariant pseudo-metric {J mn }m,n=0,1,2,3 and the kinematical global time ϕ, forming a standard kinematical Lorentz’s structure with global time on R4 . Furthermore, consider an arbitrary four-covector of the time direction (w0 , w1 , w2 , w3 ). Next, consider a Lagrangian density LR := LR (w0 , w1 , w2 , w3 ), (x0 , x1 , x2 , x3 ) , defined by   LR (w0 , w1 , w2 , w3 ), (x0 , x1 , x2 , x3 ) ⎛ ⎞    3  3  3  3  ∂wk 1 mn pk ∂wp ∂wm ∂wn J J − − k ⎠ := − λαμ ⎝ m p n 4 ∂x ∂x ∂x ∂x n=0 m=0 p=0 k=0

2  3 3   αν   mn − J wm wn − 1 − G (w0 , w1 , w2 , w3 ), (x0 , x1 , x2 , x3 ) , 4 m=0 n=0 (2.8.1)

2.8 Lagrangian for Dynamical Time Direction and Its Limiting Case

43

where α = 0, λ = 0 are real dimensionless constants, such that |α|  1 and |λ|  1.

(2.8.2)

and μ = 0, ν = 0 are real constants. Here G is some given function and, for simplicity of the notation, we omit the dependence of the function G by the additional physical fields (like dependence on dynamical pseudo-metric {K mn }m,n=0,1,2,3 , mass densities, electromagnetic fields et.al) and express it throughout explicit dependence on (x0 , x1 , x2 , x3 ). Next, given a subregion V ⊂ R4 , we define the Lagrangian on the region V as: LR ((w0 , w1 , w2 , w3 ))   $− 21    $$   $ LR (w0 , w1 , w2 , w3 ), (x0 , x1 , x2 , x3 ) $det J mn n,m=0,1,2,3 $ := V

dx0 dx1 dx2 dx3 . (2.8.3)   Note that det {J mn }n,m=0,1,2,3 is independent on (w0 , w1 , w2 , w3 ). In particular, the critical points of LN must satisfy δLR ((w0 , w1 , w2 , w3 )) = 0 ∀j = 0, 1, 2, 3, δwj

(2.8.4)

  and, since det {J mn }n,m=0,1,2,3 is independent on (w0 , w1 , w2 , w3 ), by the EulerLagrange we deduce 

   3  3 3   ∂wk ∂ ∂wn mn jk J J λαμ − k ∂xm ∂xn ∂x n=0 k=0 m=0  3   3 3   J mn wm wn − 1 J jn wn − αν m=0 n=0

n=0

 ∂G  − (w0 , w1 , w2 , w3 ), (x0 , x1 , x2 , x3 ) = 0 ∂wj ∀j = 0, 1, 2, 3. So we have

(2.8.5)

2 Basic Definitions and Statements . . .

44



   3 3  3   ∂wk ∂ ∂wn mn jk J J μ − k ∂xm ∂xn ∂x n=0 k=0 m=0  3   3 3  ν   mn jn J wm wn − 1 J wn − λ m=0 n=0 n=0 −

 1 ∂G  (w0 , w1 , w2 , w3 ), (x0 , x1 , x2 , x3 ) = 0 αλ ∂wj ∀j = 0, 1, 2, 3.

(2.8.6)

In particular, in the case of (2.8.2), taking |α| → +∞,

(2.8.7)

we approximate (2.8.6) as: 

   3 3  3   ∂wk ∂ ∂wn mn jk J J − k μ ∂xm ∂xn ∂x n=0 k=0 m=0  3   3 3  ν   mn jn J wm wn − 1 J wn = 0 ∀j = 0, 1, 2, 3. − λ m=0 n=0 n=0

(2.8.8)

In particular, by (2.8.8) we obtain: 3  ∂ ∂xj j=0

=

 3 3 

J

mn

 3 &  jn wm wn − 1 J wn

m=0 n=0 3  j=0

n=0

 3 3 3    ∂wk λμ ∂    ∂ ∂wn mn jk J J − k = 0. (2.8.9) ν ∂xj n=0 ∂xm ∂xn ∂x m=0 k=0

Moreover, taking |λ| → +∞,

(2.8.10)

we approximate (2.8.8) as:  3 3 3    ∂  ∂wk ∂wn mn jk J J − k → 0 ∀j = 0, 1, 2, 3. ∂xm ∂xn ∂x n=0 m=0

(2.8.11)

k=0

In particular, the last equation  3 3 3    ∂  ∂w ∂w k n J mn J jk − k = 0 ∀j = 0, 1, 2, 3, m ∂x ∂xn ∂x n=0 m=0 k=0

(2.8.12)

2.8 Lagrangian for Dynamical Time Direction and Its Limiting Case

45

is the same as the classical Maxwell equations in vacuum with vanishing charges and currents, where the four-covector of the electromagnetic potential (A0 , A1 , A2 , A3 ) is replaced by the four-covector (w0 , w1 , w2 , w3 ). Thus, ignoring wave-type solutions of (2.8.12), that assumed to be negligible, we deduce by (2.8.12) that there exists a proper scalar ψ, such that we have  (w0 , w1 , w2 , w3 ) =

∂ψ ∂ψ ∂ψ ∂ψ , , , ∂x0 ∂x1 ∂x2 ∂x3

 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.8.13)

Thus, inserting (2.8.13) into (2.8.9) gives, 3  ∂ ∂xj j=0

 3 3 

J

m=0 n=0

mn

 3 &  ∂ψ ∂ψ ∂ψ jn −1 J = 0. ∂xm ∂xn ∂xn n=0

(2.8.14)

Therefore, using (2.8.14), with the help of Lemma A.1 from the Appendix, we deduce the following eikonal-type equation 3  3  m=0 n=0

J mn

∂ψ ∂ψ =1 ∂xm ∂xn

∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.8.15)

So the global time ψ is correlated with pseudo-metric {J mn }m,n=0,1,2,3 . On the other hand, for the kinematical global time ϕ we also have 3  3  m=0 n=0

J mn

∂ϕ ∂ϕ = 1 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . ∂xm ∂xn

(2.8.16)

Therefore, in the case, where ψ and ϕ coincide in some initial surface, by (2.8.15) and (2.8.16) we deduce that, in the limiting case (2.8.7) we have for the dynamical time direction, that   ∂ϕ ∂ϕ ∂ϕ ∂ϕ ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , (2.8.17) , , , (w0 , w1 , w2 , w3 ) = ∂x0 ∂x1 ∂x2 ∂x3 where ϕ is a kinematical global time. In particular, in the later case we have for the dynamical pseudo-metric {K mn }m,n=0,1,2,3 :   ∂ϕ ∂ϕ ∀ 0 ≤ j, m ≤ 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . Kjm = Jjm + Sm j + m Sj ∂x ∂x (2.8.18) Moreover, in the latter case we have

2 Basic Definitions and Statements . . .

46 3  3  m=0 n=0

 ∂ϕ ∂ϕ ∂ϕ ∂ϕ = J mn m n = 1 m n ∂x ∂x ∂x ∂x m=0 n=0 3

K mn

3

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.8.19) both triples {J mn }m,n=0,1,2,3 , {mn }m,n=0,1,2,3 , ϕ, and {K mn }m,n=0,1,2,3 , {mn }m,n=0,1,2,3 , ϕ are super-correlated, the dynamical global time equals to kinematical global time ϕ and the dynamical tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 equals to the kinematical tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , so that ⎛ mn = mn := ⎝

3  j=0

⎞⎛ ⎞ 3  ∂ϕ ∂ϕ J mj j ⎠ ⎝ J nj j ⎠ − J mn ∀ m, n = 0, 1, 2, 3. ∂x ∂x j=0 (2.8.20)

2.9 Lagrangian of the Genuine Gravity The model of genuine gravity, we present here, is described by kinematically correlated pseudo-metric with time direction (w0 , w1 , w2 , w3 ), corresponding to the covector of genuine gravity (S0 , S1 , S2 , S3 ) (see Definition 2.21). In order to describe it, consider the strongly correlated kinematical contravariant, pseudo-metric {J mn }m,n=0,1,2,3 and the kinematical global time ϕ, forming a standard kinematical Lorentz’s structure with global time on R4 , as in Proposition 2.7 and Remark 2.6. Furthermore, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 . Given, an arbitrary dynamical four-covector of the time direction, (w0 , w1 , w2 , w3 ) (formally unrelated to the kinematical global time ϕ), which is weakly correlated with {J mn }m,n=0,1,2,3 , satisfying 3  3 

J jm wj wm > 0 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.9.1)

j=0 m=0

and a four-covector field (S0 , S1 , S2 , S3 ), which we called the four-covector of genuine gravity, consider the two-times covariant tensor {Kmn }m,n=0,1,2,3 defined by:   Kjm = Jjm + wj Sm + wm Sj ∀ 0 ≤ j, m ≤ 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.9.2)

and assume that (S0 , S1 , S2 , S3 ) is such that {Kmn }m,n=0,1,2,3 in (2.9.2) satisfies     det {Kmn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.9.3)

so that (S0 , S1 , S2 , S3 ) has only three independent components. Then, as before, the inverse to {Kmn }m,n=0,1,2,3 two-times contravariant tensor {K mn }m,n=0,1,2,3 is a kinematically correlated pseudo-metric with time direction (w0 , w1 , w2 , w3 ), cor-

2.9 Lagrangian of the Genuine Gravity

47

responding to the covector of genuine gravity (S0 , S1 , S2 , S3 ). Next, consider a Lagrangian density of the genuine gravitational field as:   Lg (S0 , S1 , S2 , S3 ), {K mn }m,n=0,1,2,3 ⎞ ⎛    3 3 3 3 ∂Sk 1 ⎝    1 mn pk ∂Sp ∂Sm ∂Sn ⎠ K K := − p − k 4π G n=0 4 ∂xm ∂x ∂xn ∂x m=0 p=0

(2.9.4)

k=0

where G is the gravitational constant. Correspondingly, given a subregion V ⊂ R4 , the Lagrangian of the genuine gravitational field in this subregion, as usual, is given by Lg (S0 , S1 , S2 , S3 )   $− 21    $$   $ := Lg (S0 , S1 , S2 , S3 ), {K mn }m,n=0,1,2,3 $det K mn n,m=0,1,2,3 $ V

dx0 dx1 dx2 dx3 .

(2.9.5)

Moreover, we combine this gravitational Lagrangian with the Lagrangian for the time direction, given by LR ((w0 , w1 , w2 , w3 ))   $   $− 21  $ $ mn LR ((w0 , w1 , w2 , w3 )) $det J dx0 dx1 dx2 dx3 , := n,m=0,1,2,3 $ V

(2.9.6) where, the Lagrangian density LR is given, similarly to (2.8.1), by LR ((w0 , w1 , w2 , w3 )) ⎛ ⎞    3  3  3  3  ∂w ∂w 1 ∂w ∂w p m k n := − λαμ ⎝ J mn J pk − − k ⎠ m p n 4 ∂x ∂x ∂x ∂x n=0 m=0 p=0 k=0

2  3 3 αν   mn − J wm wn − 1 , 4 m=0 n=0

(2.9.7)

where μ = 0, ν = 0 are two real constants, and α = 0, λ = 0 are two real dimensionless constants, such that |α|  1 and |λ|  1.

(2.9.8)

2 Basic Definitions and Statements . . .

48

Furthermore, we combine these Lagrangians with the Lagrangian of the Electromagnetical field, given by Le (A0 , A1 , A2 , A3 )   $− 21    $$   $ := Le (A0 , A1 , A2 , A3 ), {K mn }m,n=0,1,2,3 $det K mn n,m=0,1,2,3 $ V

dx0 dx1 dx2 dx3 ,

(2.9.9)

where similarly to (2.5.1) we consider   Le (A0 , A1 , A2 , A3 ), {K mn }m,n=0,1,2,3 ⎞ ⎛    3 3 3 3 ∂Ak 1 ⎝    1 mn pk ∂Ap ∂Am ∂An ⎠ K K := − − p − k . (2.9.10) 4π n=0 4 ∂xm ∂x ∂xn ∂x m=0 p=0 k=0

Finally, we combine all these Lagrangians with the Lagrangian of the real matter, given by   LM {Kmn }m,n=0,1,2,3 := −

     3

1

2

An j

$  $− 21  $ $ $det K mn n,m=0,1,2,3 $



n=0

V 0

 n

3

dx dx dx dx   $− 21      $$  $ LM {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) $det K mn n,m=0,1,2,3 $ + V

dx0 dx1 dx2 dx3 ,

(2.9.11)

where, in the presence of a matter, consisting of N classical point particles ∀ j = 1, 2, . . . , N , with the inertial mass mj and the charge σj for the j-th particle and

having the four-dimensional space-time trajectory χj (sj ) := χj0 (sj ), χj1 (sj ), χj2 (sj ),  χj3 (sj ) : [aj , bj ] → R4 of the j-th particle, parameterized by some proper parametrization sj ∈ [aj , bj ] ∀ j=1, 2, . . . , N , and given for all instances of time from −∞ to +∞ so that 3  3  2  2    j j χk (sk ) = lim+ χk (sk ) = +∞ lim−

sk →ak

j=0

sk →bk

∀ k = 1, 2, . . . , N ,

j=0

(2.9.12) we consider the total density of the Lagrangian of the given matter LM by

2.9 Lagrangian of the Genuine Gravity

49

N     LM {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) :=

⎛ WLG ,k (sk ) ⎝

3 

wj (χk (sk ))

j=0

j dχk

dsk

⎞

bk 

$  pq $ 1 $det K (χk (sk )) $ 2

k=1 a

k

   (sk )⎠ δ x0 − χk0 (sk ), . . . , x3 − χk3 (sk ) dsk ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.9.13)

where WLG ,k (sk ) is given by ⎛ ⎞ 3  3  j Kjm (χk (sk )) uk (sk ) ukm (sk )⎠ WLG ,k (sk ) := −mk G ⎝ j=0 m=0

∀sk ∈ [ak , bk ] ∀ k = 1, 2, . . . , N ,

(2.9.14)

where (uk0 , uk1 , uk2 , uk3 )(sk ) : [ak , bk ] → R4 is the contravariant four-vector of the fourdimensional velocity of the k-th particle, given by  j uk (sk )

=

3 

dχkm wm (χk (sk )) (sk ) dsk m=0 ∀ j = 0, 1, 2, 3

−1

j

dχk (sk ) dsk

∀sk ∈ [ak , bk ] ∀ k = 1, 2, . . . , N ,

(2.9.15)

δ(·) is the delta of Dirac in R4 , and j m (x0 , x1 , x2 , x3 ) =

N  k=1

bk  ak

 m $  pq $ 1   $det K (χk (sk )) $ 2 σk dχk (sk ) (sk ) δ x0 − χ 0 (sk ), . . . , x3 − χ 3 (sk ) dsk k k dsk ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ∀ k = 1, 2, . . . , N .

(2.9.16)

As before, here we consider two cases:

√ • The case of relativistic particles, where G(τ ) := τ − 1. • The case of non-relativistic approximation, where G(τ ) := 21 (τ − 1). Therefore, taking into account (2.9.3), the total Lagrangian of the interaction of the gravitational and Electromagnetical fields with the matter is given by the following:

2 Basic Definitions and Statements . . .

50

  Ltotal Ltotal ((w0 , w1 , w2 , w3 ), (S0 , S1 , S2 , S3 ), (A0 , A1 , A2 , A3 )) := V

$− 21   $$   $ (w0 , . . . , w3 ), (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 ) $det J mn n,m=0,1,2,3 $    0 1 2 3 dx dx dx dx = Ltotal V

$− 21    $$  $ (w0 , . . . , w3 ), (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 ) $det K mn n,m=0,1,2,3 $  dx0 dx1 dx2 dx3 , (2.9.17) where the total Lagrangian density Ltotal ((w0 , . . . , w3 ), (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 )) is given by:   Ltotal (w0 , . . . , w3 ), (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 )

 3     mn k j Ak := LR ((w0 , w1 , w2 , w3 )) + Lg (S0 , S1 , S2 , S3 ), {K }m,n=0,1,2,3 −   +Le (A0 , A1 , A2 , A3 ), {K mn }m,n=0,1,2,3    +LM {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) ,

k=0

(2.9.18)

so that, we have:   Ltotal (w0 , . . . , w3 ), (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 ) ⎛ ⎞    3  3  3  3  ∂wk 1 mn pk ∂wp ∂wm ∂wn J J − − k ⎠ := − λαμ ⎝ m p n 4 ∂x ∂x ∂x ∂x n=0 m=0 p=0 k=0

− +

− −

2  3 3    αν   mn J wm wn − 1 + LM {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) 2 m=0 n=0 ⎞ ⎛    3 3 3 3 ∂Sk 1 ⎝    1 mn pk ∂Sp ∂Sm ∂Sn ⎠ K K − p − k 4π G n=0 4 ∂xm ∂x ∂xn ∂x k=0 m=0 p=0 ⎞ ⎛    3 3 3 3 ∂Ak 1 ⎝    1 mn pk ∂Ap ∂Am ∂An ⎠ K K − p − k 4π n=0 4 ∂xm ∂x ∂xn ∂x k=0 m=0 p=0  3   k j Ak . (2.9.19) k=0

Furthermore, in the case where:

2.9 Lagrangian of the Genuine Gravity

51

|α| → +∞ and |λ| → +∞,

(2.9.20)

we greatly simplify the total Lagrangian, since, then, by (2.8.17) we have  (w0 , w1 , w2 , w3 ) =

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.9.21)

where ϕ is a kinematical global time, satisfying 3  3  m=0 n=0

J mn

∂ϕ ∂ϕ =1 ∂xm ∂xn

∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(2.9.22)

Thus, (w0 , w1 , w2 , w3 ) is fixed and it does not vary in the Lagrangian, and we can write: Ltotal ((S0 , S1 , S2 , S3 ), (A0 , A1 , A2 , A3 ))    := Ltotal (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 ) V

$   $− 21 $ $ $det J mn n,m=0,1,2,3 $ dx0 dx1 dx2 dx3    = Ltotal (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 ) V

$  $− 21  $ $ $det K mn n,m=0,1,2,3 $ dx0 dx1 dx2 dx3 ,

(2.9.23)

  where the total Lagrangian density Ltotal (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 ) is given by:   Ltotal (S0 , . . . , S3 ), (A0 , . . . , A3 ), (x0 , . . . , x3 )    := LM {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) ⎞ ⎛    3 3 3 3 ∂Sk ∂Sm ∂Sn ⎠ 1 ⎝    1 mn pk ∂Sp K K − p − k + 4π G n=0 4 ∂xm ∂x ∂xn ∂x k=0 m=0 p=0 ⎞ ⎛    3 3 3 3 ∂Ak 1 ⎝    1 mn pk ∂Ap ∂Am ∂An ⎠ K K − − p − k 4π n=0 4 ∂xm ∂x ∂xn ∂x k=0 m=0 p=0  3   − j k Ak , (2.9.24) k=0

with

52

2 Basic Definitions and Statements . . .

  ∂ϕ ∂ϕ ∀ 0 ≤ j, m ≤ 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , Kjm = Jjm + Sm j + m Sj ∂x ∂x (2.9.25) and a four-covector field of genuine gravity (S0 , S1 , S2 , S3 ) satisfies the restriction:     det {Kmn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(2.9.26)

so that (S0 , S1 , S2 , S3 ) has only three independent components. Note that, since the full Lagrangian in (2.9.17), (2.9.19) is independent on the kinematical global time ϕ (it depends only on the dynamical time direction (w0 , w1 , w2 , w3 )), then Lorentzian coordinate systems, where we have ⎧ 00 ⎪ ⎨J = 1 J 0j = J j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm J := −δjm ∀ j, m = 1, 2, 3,

(2.9.27)

are the most convenient coordinate systems, to operate with the full Lagrangian. On the other hand, in the limiting case (2.9.20), where we consider the simplified Lagrangian in (2.9.23) (2.9.24), cartesian coordinate systems are often more convenient, than Lorentzian, since in the limiting case we have (w0 , w1 , w2 , w3 ) =   ∂ϕ ∂ϕ ∂ϕ ∂ϕ , and at the same time in the cartesian coordinate systems we , , , ∂x0 ∂x1 ∂x2 ∂x3 always have ϕ = x0 + Const, so that we obtain (w0 , w1 , w2 , w3 ) = (1, 0, 0, 0). Note also that by Lemma A.5 from the Appendix in an arbitrary cartesian coordinate system we still have     det {Kmn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 = −1 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . (2.9.28) In the following sections we prove, in particular, that the gravitational field, governed by the simplified Lagrangian in (2.9.23) (2.9.24), generated by some massive body, resting and spherically symmetric in some cartesian and inertial coordinate system, is given by the pseudo-metric {Kmn }m,n=0,1,2,3 , such that there exists some curvilinear (non-cartesian) coordinate system in R4 , where {Kmn }m,n=0,1,2,3 coincides with the well-known Schwarzschild metric from the general relativity! In particular, all the optical effects that we find in the framework of our model coincide with the effects considered in the framework of General Relativity for the Schwarzschild metric. In particular, the Michelson-Morley experiment and all Sagnac-type effects will lead to the same result in the framework of our model like in the case of the general relativity. Moreover, since the Maxwell equations in both models have the same tensor form, all the electromagnetic effects, where the time does not appear explicitly will be the same. Similarly, the curvature of the light path in the Sun’s gravitational √ field will be the same in both models. Finally, in the particular case of G(τ ) = τ , i.e. in the case of the relativistic Lagrangian of the motion in (2.4.8) all the mechanical effects will be the same in the framework of our model like in the case of the general relativity for the Schwarzschild metric, provided that the time

References

53

does not appear explicitly in this effects. In particular, the movement of the Mercury planet in the Sun’s gravitational field will be the same in both models. Furthermore, we also prove that gravitational field, governed by the simplified Lagrangian in (2.9.23) (2.9.24), generated by a general slowly (non-relativistically) moving massive matter in some cartesian coordinate system, can be well approximated, by the classical model of the Newtonian Gravity. Note here about the following advantage of the presented here model of gravity with respect to the usual theory of general relativity. The simplified Lagrangian in (2.9.23) (2.9.24) for the gravity depends only on four-component field (S0 , S1 , S2 , S3 ), which is by (2.9.3) has only three independent components. Even the full Lagrangian in (2.9.17), (2.9.19), dependent only on (S0 , S1 , S2 , S3 ) and (w0 , w1 , w2 , w3 ), that is by (2.9.3) has only seven independent components. On the other hand, in the general relativity the symmetric tensor {Kmn }m,n=0,1,2,3 has all ten independent components that make the corresponding system of partial differential equations to be much more complicated. Finally, in Chap. 9 we give the covariant formulation of the Electrodynamics of the moving dielectric and para/diamagnetic continuum mediums in arbitrary dynamical pseudo-metric. Lorentz’s covariant theory of the moving para/diamagnetic continuum mediums in the flat Lorentz’s pseudo-metric was first introduced in [7] by H. Minkowski (1908). Here we formulate the generally covariant theory in the different alternative ways, that suits to formulate it in a general pseudo-metric including the presence of the genuine gravity.

References 1. C. Duval, On Galilean Isometries. Class. Quant. Grav. 10, 2217 (1993). https://doi.org/10. 1088/0264-9381/10/11/006; (arXiv:0903.1641) 2. C. Duval, G. Burdet, H.P. Künzle, M. Perrin, Bargmann structures and Newton-Cartan theory. Phys. Rev. D 31, 1841 (1985). https://doi.org/10.1103/PhysRevD.31.1841 3. C. Duval, P.A. Horváthy, Non-relativistic conformal symmetries and Newton-Cartan structures. J. Phys. A: Math. Theor. 42, 465206 (2009). https://doi.org/10.1088/1751-8113/42/46/465206. (arXiv:0904.0531) 4. C. Duval, H.P. Künzle, Minimal gravitational coupling in the Newtonian theory and the covariant Schrödinger equation. Gen. Relat. Gravit. 16, 333 (1984). https://doi.org/10.1007/ BF00762191 5. K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime. SciPost Phys. 5, 011 (2018) 6. H.P. Künzle, Galilei and lorentz structures on space-time?: comparison of the corresponding geometry and physics. Ann. Inst. H. Poincaré Phys. Théor. 17, 337 (1972) 7. H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern. Nachr. Ges. Wiss. Göttn Math.-Phys. Kl. 53 (1908); Reprinted in Math. Ann. 68, 472 (1910)

Chapter 3

Mass, Charge and Lagrangian Densities and Currents of the System of Classical Point Particles

Definition 1 Consider a classical point particle with the inertial mass m and the charge σ moving in the generalized-gravitational field, given by a contravariant pseudo-metric {K mn }m,n=0,1,2,3 , which is correlated with a four-covector of the time direction (w0 , w1 , w2 , w3 ), and influenced by the electromagnetic field with the four-covector of the electromagnetic  potential (A0 , A1 , A2 , A3 ). Next, assume that  χ (s) := χ 0 (s), χ 1 (s), χ 2 (s), χ 3 (s) : [a, b] → R4 is a four-dimensional space-time trajectory of the particle, parameterized by some proper parametrization s ∈ [a, b] (including the cases where a = −∞ and/or b = +∞). Moreover, assume that the infinite trajectory of the motion is considered for all instances of time from −∞ to +∞ so that lim−

s→a

3    j=0

χ j (s)

2 

= lim+ s→b

3   

χ j (s)

2 

= +∞.

(3.0.1)

j=0

Next, given an arbitrary covariant (contravariant) scalar quantity W (s): [a, b] → R, defined across the trajectory  of motion with  the chosen proper parametrization, consider a four-current density IW0 , IW1 , IW2 , IW3 of the quantity W and a scalar density ρW of the quantity W as generalized functions (distributions), defined by

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_3

55

3 Mass, Charge and Lagrangian Densities . . .

56

j IW (x0 , x1 , x2 , x3 )

b    21

  := det K mn (χ (s)) n,m=0,1,2,3  a

  dχ j (s) δ x0 − χ 0 (s), . . . , x3 − χ 3 (s) ds W (s) ds    21

  = det K mn (x0 , x1 , x2 , x3 ) n,m=0,1,2,3  b W (s)

  dχ j (s) δ x0 − χ 0 (s), . . . , x3 − χ 3 (s) ds ds

a

∀ j = 0, 1, 2, 3

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(3.0.2)

and b    21

  ρW (x , x , x , x ) := det K mn (χ (s)) n,m=0,1,2,3  0

1

2

3

a

⎞ ⎛ 3 j  dχ (s)⎠ W (s) ⎝ wj (χ (s)) ds j=0   δ x0 − χ 0 (s), . . . , x3 − χ 3 (s) ds 3    21   

   mn = det K n,m=0,1,2,3  wj x0 , x1 , x2 , x3 j=0

b W (s)

  dχ j (s) δ x0 − χ 0 (s), . . . , x3 − χ 3 (s) ds ds

a

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(3.0.3)

so that ρW (x0 , x1 , x2 , x3 ) :=

3 

j

wj (x0 , x1 , x2 , x3 ) IW (x0 , x1 , x2 , x3 )

j=0

∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(3.0.4)

  Here, given a point (a0 , a1 , a2 , a3 ) ∈ R4 , δ x0 − a0 , . . . , x3 − a3 is a fourdimensional scalar delta function of the point (a0 , a1 , a2 , a3 ) (the  discrete unit measure of the point (a0 , a1 , a2 , a3 ) ∈ R4 ). Note that obviously, IW0 , IW1 , IW2 , IW3 in (3.0.2) is a valid contravariant four-vector field and ρW in (3.0.3) is a valid covariant (contravariant) scalar. Note also that, (3.0.2) and (3.0.3) are independent on the choice of the proper parametrization. Finally, note that by theory of distributions the

3 Mass, Charge and Lagrangian Densities . . .

57

definition (3.0.2) and (3.0.3) mean that for every   smooth scalar classical function with compact support ξ(x0 , x1 , x2 , x3 ) ∈ Cc∞ R4 we have  j

IW (x0 , x1 , x2 , x3 ) ξ(x0 , x1 , x2 , x3 ) dx0 dx1 dx2 dx3 R4

b =

   21

dχ j   (s) ξ (χ (s)) ds det K mn (χ (s)) n,m=0,1,2,3  W (s) ds

a

∀ j = 0, 1, 2, 3,

(3.0.5)

and  ρW (x0 , x1 , x2 , x3 ) ξ(x0 , x1 , x2 , x3 ) dx0 dx1 dx2 dx3 = R4

⎞ ⎛ b   3  21 j  mn

dχ   (s)⎠ ξ (χ (s)) ds. wj (χ (s)) det K (χ (s)) n,m=0,1,2,3  W (s) ⎝ ds j=0 a

(3.0.6) Definition 2 Given an arbitrary contravariant four-vector field (f 0 , f 1 , f 2 , f 3 ) (x0 , x1 , x2 , x3 ), define the covariant divergence of it with respect to the contravariant pseudo-metric K := {K mn }m,n=0,1,2,3 , by the following :=

div (f 0 , f 1 , f 2 , f 3 )

K

(x0 , x1 , x2 , x3 )

3  ∂f j

(x0 , x1 , x2 , x3 ) + j

3 

f j (x0 , x1 , x2 , x3 ) ∂x j=0 j=0 ⎛   − 21  ⎞ ∂ mn  det {K }n,m=0,1,2,3  ⎜ ∂xj ⎟ 0 1 2 3 ⎝    1 ⎠ (x , x , x , x ) det {K mn }n,m=0,1,2,3 − 2

1 =    1 det {K mn }n,m=0,1,2,3 − 2 ⎞ ⎛   3 − 21  

∂   ⎝ det K mn n,m=0,1,2,3  f j (x0 , x1 , x2 , x3 )⎠ j ∂x j=0 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(3.0.7)

3 Mass, Charge and Lagrangian Densities . . .

58

It is well known tensor analysis that if (f 0 , f 1 , f 2 , f 3 ) is a contravariant four

from 0 1 2 3 vector, then div (f , f , f , f ) K is a valid covariant (contravariant) scalar. Proposition 1 Consider a classical point particle with the inertial mass m and the charge σ , moving in the generalized-gravitational field, given by a contravariant pseudo-metric {K mn }m,n=0,1,2,3 , which is correlated with a four-covector of the time direction (w0 , w1 , w2 , w3 ), and influenced by the electromagnetic field with the fourcovector of the electromagnetic  potential (A0 , A1 , A2 , A3 ). Next, assume that χ (s) :=  0 χ (s), χ 1 (s), χ 2 (s), χ 3 (s) : [a, b] → R4 ∈ R4 is a four-dimensional space-time trajectory of the particle, parameterized by some proper parametrization s ∈ [a, b] (including the cases where a = −∞ and/or b = +∞). Moreover, assume that the infinite trajectory of the motion is considered for all instances of time from −∞ to +∞ so that 3  3     j 2   j 2  χ (s) = lim+ χ (s) = +∞. (3.0.8) lim− s→a

s→b

j=0

j=0

Next, consider an arbitrary quantity W (s): [a, b] → R, defined across the trajectory of motion with the chosen proper parametrization, and consider a four-current  density IW0 , IW1 , IW2 , IW3 of the quantity W , given by (3.0.2). Then, in the case when W (s) is a constant across the trajectory of the motion, independent on the parameter s ∈ [a, b], we have the following conservation of the current: 

 div IW0 , IW1 , IW2 , IW3 K (x0 , x1 , x2 , x3 ) = 0

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(3.0.9)

where the covariant divergence {div (·)}K is defined by (3.0.7). Definition 3 Consider a system of N classical point particles ∀ j = 1, 2, . . . , N , with the inertial mass mj and the charge σj for the j-th particle, moving in the generalizedgravitational field given by a contravariant pseudo-metric {K mn }m,n=0,1,2,3 , which is correlated with a four-covector of the time direction (w0 , w1 , w2 , w3 ), and influenced by the electromagnetic field with the four-covector  of the electromagnetic poten tial (A0 , A1 , A2 , A3 ). Next, assume that χj (sj ) := χj0 (sj ), χj1 (sj ), χj2 (sj ), χj3 (sj ) : [aj , bj ] → R4 is a four-dimensional space-time trajectory of the j-th particle, parameterized by some proper parametrization sj ∈ [aj , bj ] for the j-th particle ∀ j = 1, 2, . . . , N (including the cases where aj = −∞ and/or bj = +∞). Moreover, assume that the infinite trajectory of the motion is considered for all instances of time from −∞ to +∞ so that 3  3  2  2    j j χk (sk ) = lim+ χk (sk ) = +∞ lim−

sk →ak

j=0

sk →bk

∀ k = 1, 2, . . . , N .

j=0

(3.0.10)

3 Mass, Charge and Lagrangian Densities . . .

59

Next, for every k = 1, 2, . . . , N consider covariant (contravariant) scalar quantities Wσk (sk ): [ak , bk ] → R and Wmk (s): [ak , bk ] → R, defined across the trajectories of motion with the chosen proper parametrization sk by the following: 

Wσk (sk ) = σk Wmk (s) = mk

∀sk ∈ [ak , bk ] ∀sk ∈ [ak , bk ]

∀ k = 1, 2, . . . , N ∀ k = 1, 2, . . . , N ,

(3.0.11)

where σk is the charge of the given k-th point particle and mk is the mass of the given k-th point particle. Then, obviously, both quantities Wσk (sk ): [ak , bk ] → R and Wmk (s): [ak , bk ] → R are constant across every trajectory of motion of the k-th particle, with the chosen proper parametrization sk . Moreover, for every k = 1, 2, . . . , N consider covariant (contravariant) scalar quantity WLG ,k (s): [ak , bk ] → R, related to the general Lagrangian LG,k of the k-th particle in (2.4.6), and defined across the trajectory of motion with the chosen proper parametrization by the following: ⎛ WLG ,k (sk ) := −mk G ⎝

3  3 

⎞ Kjm (χk (sk )) uk (sk ) ukm (sk )⎠ j

j=0 m=0

−

3 

j

σk uk (s) Aj (χk (sk ))

j=0

∀sk ∈ [ak , bk ] ∀ k = 1, 2, . . . , N ,

(3.0.12)

where (uk0 , uk1 , uk2 , uk3 )(sk ): [ak , bk ] → R4 is the contravariant four-vector of the fourdimensional velocity of the k-th particle, given as in (2.4.2), by  j uk (sk )

=

3 

dχkm wm (χk (sk )) (sk ) dsk m=0 ∀ j = 0, 1, 2, 3

−1

j

dχk (sk ) dsk

∀sk ∈ [ak , bk ]

∀ k = 1, 2, . . . , N .

(3.0.13)

Finally, for every k = 1, 2, . . . , N consider covariant (contravariant) scalar quantity Wˆ LG ,k (s): [ak , bk ] → R, related to the general Lagrangian LG,k of the k-th particle in (2.4.6), and defined across the trajectory of motion with the chosen proper parametrization by the following: ⎛ Wˆ LG ,k (sk ) := −mk G ⎝

3  3 

⎞ Kjm (χk (sk )) uk (sk ) ukm (sk )⎠ j

j=0 m=0

∀sk ∈ [ak , bk ]

∀ k = 1, 2, . . . , N .

(3.0.14)

3 Mass, Charge and Lagrangian Densities . . .

60

Then, define the contravariant four-vector of the total charge four-current density (j 0 , j 1 , j 2 , j 3 ) := (j 0 , j 1 , j 2 , j 3 )(x0 , x1 , x2 , x3 ) and the total scalar charge density ρ := ρ(x0 , x1 , x2 , x3 ) of the system of the charges by: j (x , x , x , x ) := n

0

1

2

3

N 

IWn σ (x0 , x1 , x2 , x3 ) k

k=1

∀ n = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , N  ρ(x0 , x1 , x2 , x3 ) := ρWσk (x0 , x1 , x2 , x3 ) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,(3.0.15) k=1

  where the contravariant four-vector of the four-current density IW0 , IW1 , IW2 , IW3 is given by (3.0.2) and a scalar density ρW is given by (3.0.3), so that N bk    21 

  j (x , x , x , x ) := det K pq (χk (sk )) p,q=0,1,2,3  n

0

1

2

3

k=1 a

k

  dχ n σk k (sk ) δ x0 − χk0 (sk ), . . . , x3 − χk3 (sk ) dsk dsk ∀ n = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , (3.0.16) and ⎞ ⎛ 3 j  1   dχ σk det K pq (χk (sk )}  2 ⎝ wj (χk (sk )) k (sk )⎠ ρ(x0 , x1 , x2 , x3 ) := dsk j=0 k=1 ak   δ x0 − χk0 (sk ), . . . , x3 − χk3 (sk ) dsk N 

bk

∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(3.0.17)

Then, by Proposition 1 we deduce the following conservation of the total charge current:



 div j 0 , j 1 , j 2 , j 3 K (x0 , x1 , x2 , x3 ) = 0

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(3.0.18)

where the covariant divergence {div (·)}K is defined by (3.0.7). Similarly define the contravariant four-vector of the total mass four-current density 0 1 2 3 0 1 2 3 , jM , jM , jM ) := (jM , jM , jM , jM )(x0 , x1 , x2 , x3 ) and the total scalar mass density (jM 0 1 2 3 M := M (x , x , x , x ) of the system of the masses by:

3 Mass, Charge and Lagrangian Densities . . . n jM (x0 , x1 , x2 , x3 ) :=

N 

61

IWn m (x0 , x1 , x2 , x3 ) ∀ n = 0, 1, 2, 3 k

k=1

∀ (x0 , x1 , x2 , x3 ) ∈ R4 , M (x0 , x1 , x2 , x3 ) :=

N 

ρWmk (x0 , x1 , x2 , x3 ) ∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,(3.0.19)

k=1

  where the contravariant four-vector of the four-current density IW0 , IW1 , IW2 , IW3 is given by (3.0.2) and a scalar density ρW is given by (3.0.3), so that n jM (x0 , x1 , x2 , x3 )

N bk    21 

  := det K pq (χk (sk )) p,q=0,1,2,3  k=1 a

k

  dχ n mk k (sk ) δ x0 − χk0 (sk ), . . . , x3 − χk3 (sk ) dsk dsk ∀ n = 0, 1, 2, 3 ∀ (x0 , x1 , x2 , x3 ) ∈ R4 , (3.0.20) and ⎞ ⎛ 3 j  1 

 dχ mk det K pq (χk (sk ))  2 ⎝ wj (χk (sk )) k (sk )⎠ M (x0 , x1 , x2 , x3 ) := dsk j=0 k=1 ak   δ x0 − χk0 (sk ), . . . , x3 − χk3 (sk ) dsk N 

bk

∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(3.0.21)

Then, as before, by Proposition 1 we deduce the following conservation of the total mass current:

 0 1 2 3 

, jM , jM , jM K (x0 , x1 , x2 , x3 ) = 0 div jM

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(3.0.22)

define where the covariant divergence {div(·)}K is defined by  (3.0.7). Furthermore,  the total scalar density LM := LM {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) of the given Lagrangian of the system of particles by: LM

N     {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) := ρWLG ,k (x0 , x1 , x2 , x3 ) k=1

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(3.0.23)

3 Mass, Charge and Lagrangian Densities . . .

62

where WLG ,k (sk ) is given by (3.0.12), so that N     LM {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) :=

⎛ WLG ,k (sk ) ⎝

3  j=0

⎞

bk 

 pq

 1 det K (χk (sk ))  2

k=1 a

k

 j  0  dχk 0 3 3 ⎠ wj (χk (sk )) (sk ) δ x − χk (sk ), . . . , x − χk (sk ) dsk dsk ∀ (x0 , x1 , x2 , x3 ) ∈ R4 . (3.0.24)

In particular, by (3.0.24) and (3.0.12) we have  LM R4

     − 1 {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) det K pq (x0 , x1 , x2 , x3 )  2

⎛ ⎞ 3 j  dχ dx0 dx1 dx2 dx3 = WLG ,k (sk ) ⎝ wj (χk (sk )) k (sk )⎠ dsk dsk j=0 k=1 a k ⎧ ⎛ ⎞⎫ N bk ⎨ 3  3 ⎬   j = Kjm (χk (sk )) uk (sk ) ukm (sk )⎠ −mk G ⎝ ⎭ ⎩ j=0 m=0 k=1 a k ⎛ ⎞ 3 j  dχk ⎝ wj (χk (sk )) (sk )⎠ dsk dsk j=0 ⎫ ⎧ N bk ⎨  3 ⎬  j + σk uk (s) Aj (χk (sk )) − Uk (χk (sk )) − ⎭ ⎩ j=0 k=1 a k ⎛ ⎞ 3 j  dχk ⎝ wj (χk (sk )) (sk )⎠ dsk , (3.0.25) ds k j=0 N  

bk

so that  LM



  {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 )

R4

 pq  0 1 2 3  − 21 0 1 2 3 det K (x , x , x , x )  dx dx dx dx =

N 

LG,k (χk ),

k=1

where, as before in (2.4.6) we denote the Lagrangian of the k-th particle as

(3.0.26)

3 Mass, Charge and Lagrangian Densities . . .



bk

LG,k (χk ) =

ak

⎛

⎧ ⎨ ⎩

⎛ −mk G ⎝

3  3  j=0 m=0

63

⎞⎫ ⎬ j Kjm (χk (sk )) uk (s) ukm (sk )⎠ ⎭

⎞ j dχk ⎝ ⎠ dsk wj (χk (sk )) ds k j=0 ⎫⎛ ⎧ ⎞  bk ⎨  3 3 j ⎬  dχk j ⎠ dsk + σk uk (sk ) Aj (χk (sk )) ⎝ wj (χk (sk )) − ⎭ ds k ak ⎩ j=0 j=0 ⎧ ⎛ ⎞⎫  bk ⎨ 3  3 ⎬  j = Kjm (χk (sk )) uk (s) ukm (sk )⎠ −mk G ⎝ ⎭ ak ⎩ j=0 m=0 ⎛ ⎞ 3 j  dχk ⎝ ⎠ dsk wj (χk (sk )) ds k j=0 ⎧ ⎫  bk ⎨  3 j dχk ⎬ + σk Aj (χk (sk )) − dsk ∀ k = 1, 2, . . . N . (3.0.27) dsk ⎭ ak ⎩ 3 

j=0

   Moreover, define the total scalar residual density Lˆ M := Lˆ M {Kmn }m,n=0,1,2,3 , (x0 ,  1 2 3 x , x , x ) of the given Lagrangian of the system of particles by:

Lˆ M

N     {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 ) := ρWˆ L k=1

G ,k

(x0 , x1 , x2 , x3 )

∀ (x0 , x1 , x2 , x3 ) ∈ R4 ,

(3.0.28)

where Wˆ LG ,k (sk ) is given by (3.0.14), so that Lˆ M







{Kmn }m,n=0,1,2,3 , (x , x , x , x ) := 0

1

2

⎛

 pq

 1 det K (χk (sk ))  2 Wˆ L ,k (sk ) ⎝ G

3

N bk   k=1 a

⎞ j dχk wj (χk (sk )) (sk )⎠ dsk k

3  j=0

  0  0 3 3 δ x − χk (sk ), . . . , x − χk (sk ) dsk ∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(3.0.29)

3 Mass, Charge and Lagrangian Densities . . .

64

Then, by (3.0.16), (3.0.24), (3.0.29), (3.0.12) and (3.0.14) together we deduce:    {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 )    = Lˆ M {Kmn }m,n=0,1,2,3 , (x0 , x1 , x2 , x3 )   3  0 1 2 3 k 0 1 2 3 Ak (x , x , x , x ) j (x , x , x , x ) −

LM

k=0

∀ (x0 , x1 , x2 , x3 ) ∈ R4 .

(3.0.30)

Chapter 4

The Total Simplified Lagrangian of the Gravity in a Cartesian Coordinate System

4.1 The Total Simplified Lagrangian in (2.9.23), (2.9.24), for the Limiting Case of (2.9.20) in a Cartesian Coordinate System Again consider the strongly correlated kinematical contravariant, pseudo-metric {J mn }m,n=0,1,2,3 and the kinematical global time ϕ, forming a standard kinematical Lorentz’s structure with global time on R4 , as in Proposition 2.7 and Remark 2.6. Furthermore, let {Jmn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {J mn }m,n=0,1,2,3 , which satisfies 3 

 J

mk

Jkn =

k=0

1 if m = n 0 if m = n

∀ m, n = 0, 1, 2, 3.

(4.1.1)

We consider the Lagrangian in (2.9.23), (2.9.24), corresponding to the limiting case of (2.9.20), where as before we have  (w0 , w1 , w2 , w3 ) = 3  3  m=0 n=0

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

J mn

 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

∂ϕ ∂ϕ = 1 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , ∂xm ∂xn

(4.1.2)

(4.1.3)

the dynamical global time equals to global kinematical time ϕ and the dynamical tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 equals to the kinematical tensor of three-dimensional Geometry {mn }m,n=0,1,2,3 , so that

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_4

65

66

4 The Total Simplified Lagrangian of the Gravity in a Cartesian …

⎛ mn = mn := r m r n − J mn = ⎝

3  j=0

−J

⎞⎛ ⎞ 3  ∂ϕ ∂ϕ J mj j ⎠ ⎝ J nj j ⎠ ∂x ∂x j=0

∀ m, n = 0, 1, 2, 3,

mn

(4.1.4)

where (r 0 , r 1 , r 2 , r 3 ) is the contravariant four-vector of the inertia, corresponding to {J mn }m,n=0,1,2,3 and ϕ, defined by r m :=

3 

J mj

j=0

∂ϕ ∂x j

∀ m = 0, 1, 2, 3.

(4.1.5)

Note here that by (4.1.5) and (4.1.3) we have 3 

rj

j=0

∂ϕ = 1 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . ∂x j

(4.1.6)

Next, as before, given, a four-covector field of genuine gravity (S0 , S1 , S2 , S3 ) we define the dynamical pseudo-metric {K mn }m,n=0,1,2,3 by the formula 3 

K

mk

k=0

K kn

 1 if m = n = 0 if m = n

∀ m, n = 0, 1, 2, 3,

(4.1.7)

with  K jm :=

J jm + Sm

∂ϕ ∂ϕ + m Sj j ∂x ∂x

 ∀ 0 ≤ j, m ≤ 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(4.1.8) where we assume that the four-covector field of genuine gravity (S0 , S1 , S2 , S3 ) satisfies the calibration: 



det {K mn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , (4.1.9) so that (S0 , S1 , S2 , S3 ) has only three independent components. Moreover, in the latter case we have 3  3 

 ∂ϕ ∂ϕ ∂ϕ ∂ϕ = J mn m n = 1 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , m n ∂x ∂x ∂x ∂x m=0 n=0 m=0 n=0 (4.1.10) and both couples {J mn }m,n=0,1,2,3 , ϕ, and {K mn }m,n=0,1,2,3 , ϕ are strongly correlated, so that, {K mn }m,n=0,1,2,3 , with global time ϕ, is a kinematically semi-scalar supercorrelated pseudo-metric with with global time ϕ, corresponding to the covector of genuine gravity (S0 , S1 , S2 , S3 ). Moreover, as before, we have 3

K mn

3

4.1 The Total Simplified Lagrangian …

67

K mn = v m v n − mn = J mn + v m v n − r m r n ∀ m, n = 0, 1, 2, 3,

(4.1.11)

where the contravariant vector of generalized gravity (v 0 , v 1 , v 2 , v 3 ) is defined by the following covariant relation: ⎛ v m := ⎝

3 

⎞ m j S j ⎠ + r m ∀ m = 0, 1, 2, 3,

(4.1.12)

j=0

and (r 0 , r 1 , r 2 , r 3 ) is given by (4.1.5). Moreover, the following covariant relations are valid ⎛ ⎞ 3  ∂ϕ ⎝ v j j ⎠ = 1, (4.1.13) ∂x j=0 ⎞ ⎞ ⎛ ⎛ 3 3  1 ⎝ 1 Sm := Km j r j ⎠ − ⎝ Jm j v j ⎠ ∀ 0 ≤ m ≤ 3, 2 j=0 2 j=0 3 3  

(4.1.14)

K jm v j v m = 1,

(4.1.15)

∂ϕ = v m ∀ m = 0, 1, 2, 3. ∂x j

(4.1.16)

j=0 m=0

and

3 

K jm

j=0

Finally, as before, given arbitrary contravariant vector (v 0 , v 1 , v 2 , v 3 ), satisfying ⎛

⎞ ∂ϕ ⎝ ⎠ = 1, v j ∂ x j=0 3 

j

(4.1.17)

as in (4.1.13) (we call such a four-vector speed-like four-vector), if we define {K mn }m,n=0,1,2,3 , as in (4.1.11), by K mn = v m v n − mn = J mn + v m v n − r m r n

∀ m, n = 0, 1, 2, 3,

(4.1.18)

then the inverse tensor, {K mn }m,n=0,1,2,3 , given by 3  k=0

 K mk K kn =

1 if m = n 0 if m = n

∀ m, n = 0, 1, 2, 3,

(4.1.19)

68

4 The Total Simplified Lagrangian of the Gravity in a Cartesian …

necessary satisfies  K jm :=

J jm

∂ϕ ∂ϕ + Sm j + m S j ∂x ∂x

 ∀ 0 ≤ j, m ≤ 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , (4.1.20)

and 



det {K mn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , (4.1.21) as in (4.1.8) and (4.1.9), where (S0 , S1 , S2 , S3 ) satisfies ⎞ ⎞ ⎛ ⎛ 3 3  1 ⎝ 1 Sm := Km j r j ⎠ − ⎝ Jm j v j ⎠ ∀ 0 ≤ m ≤ 3, 2 j=0 2 j=0 ⎛

and

v m := ⎝

3 

(4.1.22)

⎞ m j S j ⎠ + r m

∀ m = 0, 1, 2, 3,

(4.1.23)

j=0

as in (4.1.14), and (4.1.12). So we have a one-to-one and onto mappings between contravariant four-vectors, (v 0 , v 1 , v 2 , v 3 ), satisfying the restriction (4.1.17), and four covectors (S0 , S1 , S2 , S3 ), satisfying the restriction (4.1.21), where {K mn }m,n=0,1,2,3 is given by (4.1.20). So the speed-like contravariant four-vectors of generalized gravity (v 0 , v 1 , v 2 , v 3 ), satisfying the restriction (4.1.17), can be considered as an independent argument in the simplified Lagrangian in (2.9.23), (2.9.24), instead of the four-covector of genuine gravity (S0 , S1 , S2 , S3 ), satisfying the restriction (4.1.21). Next, assume that we fix some cartesian coordinate system, so that in this system we have:   ∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , (w0 , w1 , w2 , w3 ) = = (1, 0, 0, 0) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , ∂x0 ∂x1 ∂x2 ∂x3 (4.1.24) and ⎧ 00 00 ⎪ ⎨ =  = 0 0j (4.1.25)  =  j0 = 0 j =  j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm jm  =  = δ jm ∀ j, m = 1, 2, 3. In particular, (4.1.17) reeds: v 0 = 1,

(4.1.26)

so that (v 0 , v 1 , v 2 , v 3 ) = (1, v 1 , v 2 , v 3 ) = (1, v) where v := (v 1 , v 2 , v 3 ) ∈ R3 . (4.1.27)

4.1 The Total Simplified Lagrangian …

69

Therefore, the independent argument of the Lagrangian is actually a threedimensional vector field v. We call v the three-dimensional gravitational vector potential; however, note that it defined only in cartesian coordinate systems. Similarly, by (4.1.6) we have (r 0 , r 1 , r 2 , r 3 ) = (1, r 1 , r 2 , r 3 ) = (1, r) where r := (r 1 , r 2 , r 3 ) ∈ R3 . (4.1.28) We call r the three-dimensional vector potential of inertia, and as before, note that it defined only in cartesian coordinate systems. Furthermore, by (4.1.18), (4.1.25) and (4.1.27) we deduce: ⎧ 00 ⎪ ⎨K = 1 K jm = −δ jm + v j v m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j K = K j0 = v j ∀1 ≤ j ≤ 3,

(4.1.29)

and therefore, as before, we have ⎧ 2 ⎪ ⎨ K 00 = 1 − |v| K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K 0 j = K j0 = v j ∀1 ≤ j ≤ 3,

(4.1.30)

and 



det {K mn }m,n=0,1,2,3 = det {Jmn }m,n=0,1,2,3 = 1 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . (4.1.31) Similarly, ⎧ 00 ⎪ ⎨J = 1 (4.1.32) J jm = −δ jm + r j r m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j j0 j J = J = r ∀1 ≤ j ≤ 3, and therefore, by Lemma A.5 from the Appendix we have ⎧ 2 ⎪ ⎨ J00 = 1 − |r| J jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ J0 j = J j0 = r j ∀1 ≤ j ≤ 3.

(4.1.33)

On the other hand, by (4.1.33), (4.1.24) identity (4.1.20) reads as ⎧ 2 ⎪ ⎨ K 00 = 1 − |r| + 2S0 K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K 0 j = K j0 = S j + r j ∀1 ≤ j ≤ 3,

(4.1.34)

70

4 The Total Simplified Lagrangian of the Gravity in a Cartesian …

and thus, comparing (4.1.30) with (4.1.34) we deduce: 

so that,



1 − |v|2 = 1 − |r|2 + 2S0 v j = S j + r j ∀1 ≤ j ≤ 3,

(4.1.35)

S0 := 21 |r|2 − 21 |v|2 S j := v j − r j ∀1 ≤ j ≤ 3.

(4.1.36)

Furthermore, since ϕ = x 0 + Const, the first coordinate can be chosen as the proper parameter sk on the Lagrangian of every particle, so that we rewrite (2.9.11) as

 L M {K mn }m,n=0,1,2,3 :=





LM



  {K mn }m,n=0,1,2,3 , (x 0 , x 1 , x 2 , x 3 )

V

dx 0 dx 1 dx 2 dx 3 ,

(4.1.37)

and (2.9.13) as:  

 {K mn }m,n=0,1,2,3 , (x 0 , x 1 , x 2 , x 3 ) ⎛ ⎞   N +∞ 3  dχ j      k (χ 0 )⎠ δ x 0 −χ 0 , x 1 −χ 1 (χ 0 ), x 2 −χ 2 (χ 0 ), x 3 −χ 3 (χ 0 ) := w j χk (χ 0 ) dχ 0 W L ,k ((χ 0 )) ⎝ k k k 0 G dχ

LM

k=1−∞

=

N  k=1

j=0



W L ,k (x 0 ) δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) G



∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(4.1.38)

where the δ-function in the last expression is three-dimensional, W L G ,k (x 0 ) is given, as in (3.0.12), by ⎛

⎞ j m 

dχ dχ k W L G ,k (x 0 ) := −m k G ⎝ K jm χk (x 0 ) (x 0 ) k0 (x 0 )⎠ 0 dx dx j=0 m=0 −

3  j=0

3 3  

 dχk 0 (x ) A j χk (x 0 ) dx 0 j

σk

∀s ∈ (−∞, +∞) ∀ k = 1, 2, . . . , N ,

(4.1.39)

4.1 The Total Simplified Lagrangian …

71

and the four-dimensional trajectory of the kth particle, parameterized by the firs coordinate x 0 is given by ⎧     ⎪ ⎨χk (x 0 ) := χk0 (x 0 ), χk1 (x 0 ), χk2 (x 0 ), χk3 (x 0 ) := x 0 , χk1 (x 0 ), χk2 (x 0 ), χk3 (x 0 ) ∀ k = 1, 2, . . . N  0    1 2 3 1 2 3 dχk 0 dχk (x 0 ), dχk (x 0 ), dχk (x 0 ) := 1, dχk (x 0 ), dχk (x 0 ), dχk (x 0 ) ⎪ ∀ k = 1, 2, . . . N . ⎩ 0 (x ), 0 0 0 0 0 0 dx

dx

dx

dx

dx

dx

dx

(4.1.40) Similarly, the four-current and the charge densities satisfy: (j , j , j , j ) = 0

1

2

3

N  k=1

 σk

dχ 1 dχ 2 dχ 3 1, k0 (x 0 ), k0 (x 0 ), k0 (x 0 ) dx dx dx



 δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 )

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(4.1.41)

and ρ=

N 

 σk δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

k=1

(4.1.42) In particular, we have ( j 0 , j 1 , j 2 , j 3 ) = (ρ, j)

where

j := ( j 1 , j 2 , j 3 ) ∈ R3

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(4.1.43)

where j := ( j 1 , j 2 , j 3 ) =

N 

 σk

k=1

 2 3   0 ), dχk (x 0 ), dχk (x 0 ) δ x 1 − χ 1 (χ 0 ), x 2 − χ 2 (χ 0 ), x 3 − χ 3 (χ 0 ) (x k k k dx 0 dx 0 dx 0

dχk1

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(4.1.44)

In the same way, the mass four-current and the mass densities satisfy: 0 , j1 , j2 , j3 ) = ( jM M M M   N    dχ 1 dχ 2 dχ 3 m k 1, k0 (x 0 ), k0 (x 0 ), k0 (x 0 ) δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) dx dx dx k=1

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(4.1.45)

and M=

N 

 m k δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

k=1

(4.1.46)

72

4 The Total Simplified Lagrangian of the Gravity in a Cartesian …

In particular, we have 0 1 2 3 1 2 3 ( jM , jM , jM , jM ) = (M, j M ) where j M := ( j M , jM , jM ) ∈ R3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(4.1.47)

where 1 2 3 j M := ( j M , jM , jM )=   N 1    dχk 0 dχk2 0 dχk3 0 mk (x ), (x ), (x ) δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) 0 0 0 dx dx dx k=1

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(4.1.48)

Note that by (3.0.18), (3.0.22), using (4.1.31), (4.1.43) and (4.1.47) we have the conservation of the charge and the mass current: ∂ρ + divx j = 0 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , ∂x0

(4.1.49)

∂M + divx j M = 0 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . ∂x0

(4.1.50)

Moreover, by (4.1.39), (4.1.30) and (4.1.40) we have: ⎛ W L G ,k (x 0 ) = −m k G ⎝1 −

3  j=1



  dχk 0 j 0 (x ) − v (x ) χ k dx 0 j

2 ⎞ ⎠−

3 

  dχk 0 0 (x ) A (x ) χ j k dx 0 j

σk

j=0

∀s ∈ (−∞, +∞) ∀ k = 1, 2, . . . , N ,

(4.1.51)

so that, by (4.1.51) and (4.1.41) we rewrite (4.1.38) as: LM



3    {K mn }m,n=0,1,2,3 , (x 0 , x 1 , x 2 , x 3 ) := − An (x 0 , x 1 , x 2 , x 3 ) j n (x 0 , x 1 , x 2 , x 3 ) n=0

⎞2 ⎞ j N 3       dχk 0 ⎟ ⎜ ⎝ m k G ⎝1 − (x ) − v j χk (x 0 ) ⎠ ⎠ δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) − 0 dx ⎛

k=1

⎛

j=1

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(4.1.52)

Then, by (4.1.52) and (4.1.37), in the case V = R4 we deduce N 

 L M {K mn }m,n=0,1,2,3 = − mk k=1

−

+∞

G ⎝1 −

3  j=1

−∞

     3 R4

⎛



  dχk 0 (x ) − v j χk (x 0 ) 0 dx j



An (x , x , x , x ) j (x , x , x , x ) 0

1

2 ⎞ ⎠ dx 0

2

3

n

0

1

2

3

dx 0 dx 1 dx 2 dx 3 .

n=0

(4.1.53)

4.1 The Total Simplified Lagrangian …

73

Next we denote (A0 , A1 , A2 , A3 ) = ( , −A) where A0 = and (A1 , A2 , A3 ) = −A, (4.1.54) and (S0 , S1 , S2 , S3 ) := − (, −h) where S0 = −

and

(S1 , S2 , S3 ) = h. (4.1.55)

In particular, by (4.1.36) we have: 

  = 21 |v|2 − |r|2 h = v − r.

(4.1.56)

Moreover, by (4.1.43) and (4.1.54) we have: 3 

An j n = ρ − A · j ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(4.1.57)

n=0

Lemma 1 Let {K mn }n,m=0,1,2,3 be given by (4.1.29) and {K mn }n,m=0,1,2,3 be given by (4.1.30) Next, let {Fmn }n,m=0,1,2,3 be an antisymmetric two-times covariant tensor, defined by ∂ Aj ∂ Am ∀ m, j = 0, 1, 2, 3. (4.1.58) Fm j := m − ∂x ∂x j Then, denoting (x 0 , x 1 , x 2 , x 3 ) = (x 0 , x) where x := (x 1 , x 2 , x 3 ) ∈ R3 ,

(4.1.59)

(A0 , A1 , A2 , A3 ) = ( , −A) where A0 = and (A1 , A2 , A3 ) = −A, (4.1.60) ⎧ B := (B , B , B ) := curl A, ⎪ 1 2 3 x ⎪ ⎪ ⎨E := (E , E , E ) := −∇ − ∂A , 1 2 3 x ∂x0 (4.1.61) ⎪ , D , D ) := E + v × B, D := (D 1 2 3 ⎪ ⎪ ⎩ H := (H1 , H2 , H3 ) := B + v × D, and F

mn

:=

3  3  k=0 j=0

K m j K nk F jk ∀ m, n = 0, 1, 2, 3,

(4.1.62)

74

4 The Total Simplified Lagrangian of the Gravity in a Cartesian …

⎧ F00 ⎪ ⎪ ⎪ ⎪ ⎪ F0 j ⎪ ⎪ ⎪ ⎨F jj ⎪ F12 ⎪ ⎪ ⎪ ⎪ ⎪ F 13 ⎪ ⎪ ⎩ F23

we have

⎧ F 00 ⎪ ⎪ ⎪ ⎪ ⎪ F0j ⎪ ⎪ ⎪ ⎨F j j ⎪ F 12 ⎪ ⎪ ⎪ ⎪ ⎪ F 13 ⎪ ⎪ ⎩ 23 F

=0 = −F j0 = E j ∀ j = 1, 2, 3 = 0 ∀ j = 1, 2, 3 = −F21 = −B3 = −F31 = B2 = −F32 = −B1 ,

(4.1.63)

=0 = −F j0 = −D j ∀ j = 1, 2, 3, = 0 ∀ j = 1, 2, 3, = −F 21 = −H3 = −F 31 = H2 = −F 32 = −H1 ,

(4.1.64)

and ⎞ ⎛ ⎛ ⎞    3 3 3 3 3 3 ∂ Ak 1 ⎝    1 mn pk ∂ A p ∂ Am ∂ An ⎠ 1 ⎝  jk K K − − − F F jk ⎠ =− 4π 4 ∂xm ∂x p ∂xn ∂xk 4π n=0 k=0 m=0 p=0 j=0 k=0    2  1  ∂A 1  − 1 |curlx A|2 . −∇

− + v × curl A (4.1.65) = x x  4π 2  ∂x0 2

Here, for a = (a1 , a2 , a3 ) ∈ R3 and b = (b1 , b2 , b3 ) ∈ R3 we denote by a × b := (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − b2 b1 ) ∈ R3 ,

(4.1.66)

their vector product and, given a vector valued function f(x) = ( f 1 (x), f 2 (x), f 3 (x)) : R3 → R3 we denote by  curlx f(x) :=

∂ f3 ∂ f2 ∂ f1 ∂ f3 ∂ f2 ∂ f1 − 3, 3 − 1, 1 − 2 2 ∂x ∂x ∂x ∂x ∂x ∂x

 (x).

(4.1.67)

Similarly to (4.1.65), by (4.1.55) we have ⎞ ⎛    3 3 3 3 1 ⎝    1 mn pk ∂ S p ∂ Sk ∂ Sm ∂ Sn ⎠ − − k K K 4π G 4 ∂xm ∂x p ∂xn ∂x n=0 k=0 m=0 p=0   2   1 1 1 ∂S |curlx S|2 − −∇x  − 0 + v × curlx S = = 4π G 2 2 ∂x   2     1 1  ∂ 1 2 1 2 1 2 |curlx (v − r)| − − 0 (v − r) − ∇x |v| − |r| + v × curlx (v − r) , 4π G 2 2 ∂x 2 2

(4.1.68)

4.1 The Total Simplified Lagrangian …

75

where we use (4.1.56) in the last equality. Consequently, by (2.9.23) and (4.1.31) we write the simplified total Lagrangian in (2.9.23) and (2.9.24) in the given cartesian coordinate system, as 



Ltotal v, ( , −A), (x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 ,

L total (v, ( , −A)) = V

(4.1.69) where, using (4.1.68), (4.1.65), (4.1.57) and (4.1.52) we can rewrite the total

 Lagrangian density Ltotal v, ( , −A), (x 0 , x 1 , x 2 , x 3 ) in (2.9.24) as:   Ltotal v, ( , −A), (x 0 , x 1 , x 2 , x 3 ) =    2    1 1  ∂A 1 −∇x − 0 + v × curlx A − |curlx A|2 − ρ(x 0 , x 1 , x 2 , x 3 ) − A · j(x 0 , x 1 , x 2 , x 3 )  4π 2 2 ∂x   2     1 1 1 2 1 2 1 ∂ |curlx (v − r)|2 − − + (v − r) − ∇x |v| − |r| + v × curlx (v − r) 0 4π G 2 2 2 2 ∂x ⎛ ⎞2 ⎞ ⎛ j N 3       d χk 0 ⎜ ⎟ j 0 ⎝ m k G ⎝1 − (x ) − v χk (x ) ⎠ ⎠ δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) . − dx 0 k=1

j=1

(4.1.70) Then, by (4.1.70) we rewrite (4.1.69) in the case V = R4 as:  L total (v, ( , −A)) = R4

1 4π



  2  1  ∂A 1 −∇x − 0 + v × curlx A − |curlx A|2 dx 0 . . . dx 3  2 2 ∂x

    −

ρ(x 0 , x 1 , x 2 , x 3 ) − A · j(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 + R4

1 4π G



R4

 2     1 1 2 1 2 1 ∂  dx 0 dx 1 dx 2 dx 3 |curlx (v − r)|2 − − + v × (v − r) − ∇ − curl |v| |r| − r) (v x x  2 2 2 2 ∂x0 ⎫ ⎧ ⎛ ⎞ ⎞ ⎛ 2 ⎪ j  ⎪ N +∞ 3 ⎬ ⎨     dχ ⎜ 0 ⎝ k (x 0 ) − v j χk (x 0 ) ⎠ ⎟ −m k G ⎝1 − + (4.1.71) ⎠ dx . 0 ⎪ ⎪ dx ⎭ ⎩ k=1 j=1 −∞

In particular, if we assume that our coordinate system is inertial, in addition to being cartesian, then since r, given as in (4.1.28) and (4.1.5) by (r 0 , r 1 , r 2 , r 3 ) = (1, r 1 , r 2 , r 3 ) = (1, r) where r := (r 1 , r 2 , r 3 ) ∈ R3 with r m :=

3  j=0

J mj

∂ϕ ∂x j

∀ m = 0, 1, 2, 3,

(4.1.72)

76

4 The Total Simplified Lagrangian of the Gravity in a Cartesian …

is constant in every inertial coordinate system, which is independent on the point (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , we simplify the total Lagrangian density Ltotal (v, ( , −A), (x 0 , x 1 , x 2 , x 3 )) in (4.1.70) as:

Ltotal v, ( , −A), (x 0 , x 1 , x 2 , x 3 )



   2  

1  1 ∂A 1 2  |curl −∇ =

− + v × curl A − A| − ρ(x 0 , x 1 , x 2 , x 3 ) − A · j(x 0 , x 1 , x 2 , x 3 ) x x  x 4π 2  ∂x0 2   2     1 1 2 1  ∂v 1 |curlx v|2 −  0 + ∇x |v| − v × curlx v + 4π G 2 2 ∂x 2 ⎛ 2 ⎞  j N 3    



dχk 0 j 0 ⎠ δ x 1 − χk1 (χ 0 ), x 2 − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) , m k G ⎝1 − (x ) − v χk (x ) − dx 0 k=1

j=1

(4.1.73) and the total Lagrangian in (4.1.71) as:  L total (v, ( , −A)) = −

 

R4

+ R4

   2  1  ∂A 1 2  −∇x − 0 + v × curlx A − |curlx A| dx 0 dx 1 dx 2 dx 3 2 2 ∂x 

ρ(x 0 , x 1 , x 2 , x 3 ) − A · j(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3

R4



1 4π

1 4π G



 2     1 1 2 1  ∂v |curlx v|2 −  0 + ∇x |v| − v × curlx v dx 0 dx 1 dx 2 dx 3 2 2 ∂x 2

⎧ ⎛ ⎞2 ⎞ ⎫ ⎛ ⎪  ⎪ j N +∞ 3 ⎨ ⎬     dχ ⎜ ⎝ k (x 0 ) − v j χk (x 0 ) ⎠ ⎟ −m k G ⎝1 − dx 0 . + ⎠ 0 ⎪ ⎪ dx ⎩ ⎭ k=1 j=1

(4.1.74)

−∞

So, in the coordinate system which is simultaneously inertial and cartesian (we can obtain such systems by the Galilean Transformations from the kinematically preferable coordinate system) neither the total Lagrangian density in (4.1.73) nor the total Lagrangian in (4.1.74) are dependent on the kinematical pseudo-metric {J mn }0≤m,n≤3 and/or on the contravariant four-vector of inertia (r 0 , r 1 , r 2 , r 3 ).

Chapter 5

The Euler-Lagrange for the Lagrangian of the Motion of A Classical Point Particle in a Cartesian Coordinate System

Consider a cartesian coordinate system and a classical point particle with the inertial mass m and the charge σ moving in the generalized-gravitational field given by a contravariant pseudo-metric {K mn }n,m=0,1,2,3 , satisfying (4.1.29), so that ⎧ 00 ⎪ ⎨K = 1 K jm = −δ jm + v j v m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j K = K j0 = v j ∀1 ≤ j ≤ 3,

(5.1)

and the electromagnetic field with the four-covectorof the electromagnetic poten tial (A0 , A1 , A2 , A3 ). Next assume that χ (x 0 ) := x 0 , χ 1 (x 0 ), χ 2 (x 0 ), χ 3 (x 0 ) : [a, b] → R4 ∈ R4 is a four-dimensional space-time trajectory of the particle, parameterized by the first coordinate x 0 in the interval x 0 ∈ [a, b] (including the cases where a = −∞ and/or b = +∞). Then, since ϕ = x 0 + Const, we rewrite the Lagrangian of motion of this particle in (2.4.6) as: ⎧ ⎞ ⎛ b ⎨ 3  3 j k    dχ dχ ⎠ L G (χ ) = K jk χ (x 0 ) −mG ⎝ ⎩ dx 0 dx 0 j=0 k=0 a ⎫ 3    dχ j ⎬ 0 − σ A j χ (x 0 ) dx , dx 0 ⎭

(5.2)

j=0

where {K mn }n,m=0,1,2,3 is given as in (4.1.30) by ⎧ 2 ⎪ ⎨ K 00 = 1 − |v| K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K 0 j = K j0 = v j ∀1 ≤ j ≤ 3, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_5

(5.3)

77

78

5 The Euler-Lagrange for the Lagrangian of the Motion …

with (v 0 , v 1 , v 2 , v 3 ) = (1, v 1 , v 2 , v 3 ) = (1, v) where

v := (v 1 , v 2 , v 3 ) ∈ R3 . (5.4)

Then, since χ 0 (x 0 ) = x 0 , by (5.3) we rewrite (5.2) as:     dz 0  0 2 0 L G (z) = −m G 1 −  0 (x ) − v x , z(x )  dx 0 dx a   b      dz σ  x 0 , z(x 0 ) − A x 0 , z(x 0 ) · 0 (x 0 ) dx 0 . − dx a 

b





(5.5)

where we denote   z(x 0 ) := χ 1 (x 0 ), χ 2 (x 0 ), χ 3 (x 0 ) ∈ R3 ∀x 0 ∈ [a, b],

(5.6)

and (A0 , A1 , A2 , A3 ) := (, −A) where A0 =  and (A1 , A2 , A3 ) = −A. (5.7) Next, we investigate the critical points z(x 0 ) of (5.5). Then, by the Euler-Lagrange we have: δL G (z) = 0. (5.8) δz Therefore,     2  dz    dz 0 0 0 0 0 0   (x ) − v x , z(x ) 1 −  0 (x ) − v x , z(x )  dx dx 0     2   T  dz    dz 0 ) − v x 0 , z(x 0 ) − 2mG  1 −  0 (x 0 ) − v x 0 , z(x 0 )  · (x ∇x v x 0 , z(x 0 ) dx dx 0           d 0 , z(x 0 ) 0 , z(x 0 ) − d A x 0 , z(x 0 ) T · dz (x 0 ) , (5.9) −σ + ∇  x A x x x dx 0 dx 0

d 0 = −m 0 dx





2G 

and, by the following well-known identity from the vector analysis: f × (curlx g) = (dx g)T · f − (dx g) · f ∀ f(x), g(x) : R3 → R3 ,

(5.10)

5 The Euler-Lagrange for the Lagrangian of the Motion …

79

we rewrite (5.9) as:      2    dz dz 0 0 , z(x 0 ) (x ) − v x (5.11) 2G  1 −  0 (x 0 ) − v x 0 , z(x 0 )  dx dx 0     2  dz d2 z − 2m G  1 −  0 (x 0 ) − v x 0 , z(x 0 )  (x 0 ) dx d(x 0 )2        2  ∂   dz dz 1 2 |v| − + 2m G  1 −  0 (x 0 ) − v x 0 , z(x 0 )  v + ∇x × curlx v x 0 , z(x 0 ) 2 dx ∂x0 dx 0    dz ∂A +σ − 0 − ∇x  + 0 (x 0 ) × curlx A x 0 , z(x 0 ) . ∂x dx

0 = −m

d dx 0 



So, denoting



E = −∇x  − B = curlx A,

∂A ∂x0

(5.12)

we rewrite (5.11) as     2  dz d2 z 2m G  1 −  0 (x 0 ) − v x 0 , z(x 0 )  (x 0 ) = dx d(x 0 )2       2    dz d dz 0 (x ) − v x 0 , z(x 0 ) −m 0 2G  1 −  0 (x 0 ) − v x 0 , z(x 0 )  dx dx dx 0        2  ∂   dz dz 1 2 |v| − v + ∇x × curlx v x 0 , z(x 0 ) +2m G  1 −  0 (x 0 ) − v x 0 , z(x 0 )  2 dx ∂x0 dx 0      dz +σ E x 0 , z(x 0 ) + 0 (x 0 ) × B x 0 , z(x 0 ) . (5.13) dx

However, as before, here we consider two cases: • The case of non-relativistic approximation, where√G(τ ) := 21 (τ − 1). • The case of relativistic particles, where G(τ ) := τ − 1. In the first case we have 2G  (τ ) = 1. In the second case we have 2G  (τ ) = τ − 2 = 1 + O(τ − 1) and so 2G  (τ ) ≈ 1 in the non-relativistic approximation where 1

 2  dz     1. − v  dx 0 

(5.14)

Thus, in the latter case we rewrite (5.13) as m

    2    dz 1 ∂  0  v x , z(x 0 ) + ∇x v x 0 , z(x 0 )  − 0 (x 0 ) × curlx v x 0 , z(x 0 ) 2 ∂x0 dx      dz (5.15) + σ E x 0 , z(x 0 ) + 0 (x 0 ) × B x 0 , z(x 0 ) . dx

d2 z (x 0 ) = m d(x 0 )2



Chapter 6

The Euler-Lagrange for the Lagrangian of the Gravitational and Electromagnetic Fields in a Cartesian Coordinate System

6.1 The Euler-Lagrange for the Lagrangian in (4.1.71) in a Cartesian Coordinate System Lemma 6.1 Consider the strongly correlated kinematical contravariant, pseudometric {J mn }m,n=0,1,2,3 and the kinematical global time ϕ, forming a standard kinematical Lorentz’s structure with global time on R4 as in Proposition 2.7 and Remark 2.6. Next, let (r 0 , r 1 , r 2 , r 3 ) be the contravariant four-vector of the inertia, corresponding to {J mn }m,n=0,1,2,3 and ϕ, defined by r m :=

3 

J mj

j=0

∂ϕ ∂x j

∀ m = 0, 1, 2, 3.

(6.1.1)

and denote (r 0 , r 1 , r 2 , r 3 ) := (r 0 , r)

where

r := (r 1 , r 2 , r 3 ) ∈ R3 .

(6.1.2)

Then, in a cartesian coordinate system we have: ⎧ ⎪ r0 = 1 ⎪

⎪  ⎪ ⎪ ⎨d r + {d r}T  := ∂r m ∂r j =0 + ∂xm x x ∂x j m, j=1,2,3 2 m ⎪ ∂ r ⎪ = 0 ∀ j, m, n = 1, 2, 3 ⎪ ⎪ ∂x j ∂xn ⎪ ⎩ divx r = 0 ,

(6.1.3)

where we denote (x 0 , x 1 , x 2 , x 3 ) := (x 0 , x)

with

x := (x 1 , x 2 , x 3 ) ∈ R3 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_6

(6.1.4)

81

82

6 The Euler-Lagrange for the Lagrangian of the Gravitational …

We investigate critical points of the Lagrangian in (4.1.71). As before, we denote (A0 , A1 , A2 , A3 ) = (, −A) where A0 =  and (A1 , A2 , A3 ) = −A, (6.1.5) and (S0 , S1 , S2 , S3 ) := − (, −h)

where S0 = −

and

(S1 , S2 , S3 ) = h. (6.1.6)

In particular, by (4.1.36) we have:

   = 21 |v|2 − |r|2 h = v−r.

(6.1.7)

Furthermore, we denote ⎧ B = curlx A ⎪ ⎪ ⎪ ⎨D = −∇  − ∂A + v × curl A x x ∂x0 ∂A ⎪ E = −∇x  − ∂ x 0 = D − v × B ⎪ ⎪   ⎩ + v × curlx A = B + v × D. H = curlx A + v × −∇x  − ∂∂A x0

(6.1.8)

Moreover, we define

R = −∇x  − Q = curlx h,

∂h ∂x0

+ v × curlx h

(6.1.9)

and by (6.1.7) we rewrite (6.1.9) as:

  R = − ∂∂x 0 (v − r) − ∇x 21 |v|2 − 21 |r|2 + v × curlx (v − r) Q = curlx (v − r).

(6.1.10)

We also can rewrite (6.1.10) as: Q = curlx (v − r) and 1 2 ∂r − r × curl |r| + ∇ r R= x x ∂x0 2 1 2 ∂v − v × curl |v| + ∇ v − (v − r) × curlx r − x x ∂x0 2 ∂v ∂r + d r · r − + d v · v − (v − r) × curlx r, = x x ∂x0 ∂x0

(6.1.11)

where in the last equality we use the following well-known identity from the vector analysis:

6.1 The Euler-Lagrange for the Lagrangian …

f × (curlx g) = (dx g)T · f − (dx g) · f

83

∀ f(x), g(x) : R3 → R3 .

Moreover, in inertial coordinate system where dx r = 0 and (6.1.11) as:

∂r ∂x0

(6.1.12)

= 0 we simplify

    R = − ∂∂vx 0 − ∇x 21 |v|2 + v × curlx v = − ∂∂vx 0 + dx v · v Q = curlx v.

(6.1.13)

Furthermore, by (6.1.8) and (6.1.9) we have: ⎧ curlx E + ∂∂B = curlx D + ∂∂B − curlx (v × B) = 0 ⎪ x0 x0 ⎪ ⎪ ⎨div B = 0 x ⎪ − curlx (v × Q) = 0 curlx R + ∂∂Q ⎪ x0 ⎪ ⎩ divx Q = 0 .

(6.1.14)

Next, taking the divergence by x of (6.1.11) we deduce ⎞ 3 3  m j  ∂ ∂r ∂r ⎠ divx R = ⎝ 0 {divx r} + r · ∇x {divx r} + ∂x ∂x j ∂xm j=1 m=1 ⎛ ⎞ 3 3  m j  ∂ ∂v ∂v ⎠ − ⎝ 0 {divx v} + v · ∇x {divx v} + ∂x ∂x j ∂xm j=1 m=1 ∂ {(v {divx r} + r · ∇x {divx r} − r) × curlx r} = − divx ∂x0 ⎞ 2  2 3  3  3 3  1 ∂r m 1 ∂r m ∂r j ∂r j ⎠ + m − − m + 4 ∂x j ∂x 4 ∂x j ∂x j=1 m=1 j=1 m=1 ∂ {divx v} + v · ∇x {divx v} − ∂x0 ⎞ 2  2 3  3  3 3  1 ∂v m 1 ∂v m ∂v j ∂v j ⎠ + + m − − m 4 ∂x j ∂x 4 ∂x j ∂x j=1 m=1 j=1 m=1 ⎛

− (curlx {(v − r)}) · (curlx r) + (v − r) · (curlx {curlx r}) ,

(6.1.15)

where in the last equality we use the following well-known identity from the vector analysis: divx (f × g) = g · curlx f − f · curlx g

∀ f(x), g(x) : R3 → R3 .

(6.1.16)

84

6 The Euler-Lagrange for the Lagrangian of the Gravitational …

However, we can rewrite (6.1.15) as: ⎛

 ⎞  3  3 m j 2  ∂ ∂r 1 ∂r divx R = ⎝ 0 {divx r} + r · ∇x {divx r} + + m ⎠ 4 ∂x j ∂x ∂x j=1 m=1 ⎛  ⎞  3  3 m j 2  ∂ ∂v 1 ∂v + m ⎠ − ⎝ 0 {divx v} + v · ∇x {divx v} + 4 ∂x j ∂x ∂x j=1 m=1

1 1 |curlx r|2 + |curlx v|2 − (curlx {(v − r)}) · (curlx r) + (v − r) 2 2 · (curlx {curlx r}) ⎞ ⎛   3  3 m j 2  ∂r 1 ∂r ∂ 2 + m − (divx r) ⎠ = ⎝ 0 {divx r} + divx {(divx r) · r} + 4 ∂x j ∂x ∂x j=1 m=1 ⎞ ⎛   3  3 m j 2  ∂ ∂v 1 ∂v 2 + m − (divx v) ⎠ − ⎝ 0 {divx v} + divx {(divx v) · v} + 4 ∂x j ∂x ∂x −

j=1 m=1

1 + |curlx (v − r)|2 + (v − r) · (curlx {curlx r}) . 2

(6.1.17)

Then, since by (6.1.11) we have curlx (v − r) = Q, we rewrite (6.1.17) as: ⎛

⎞ m 3 3  j 2  ∂ ∂v 1 ∂v ⎝ {divx v} + divx {(divx v) · v} + + m − (divx v)2 ⎠ j ∂x0 4 ∂ x ∂ x j=1 m=1 ⎛ ⎞ m 3 3  j 2  ∂ ∂r 1 ∂r = ⎝ 0 {divx r} + divx {(divx r) · r} + + m − (divx r)2 ⎠ j ∂x 4 ∂ x ∂ x j=1 m=1 + (v − r) · (curlx {curlx r}) +

1 |Q|2 − divx R. 2

(6.1.18)

Next, by Lemma 6.1 we have

⎧  ∂r m T ⎪ ⎪ d := r + {d r} + x ⎨ x ∂x j curlx (curlx r) = 0 ⎪ ⎪ ⎩ divx r = 0 .

∂r j ∂xm

 m, j=1,2,3

=0 (6.1.19)

Then, inserting (6.1.19) into (6.1.18) and using the identity Q = curlx (v − r) in (6.1.11) with (6.1.19) we obtain:

6.1 The Euler-Lagrange for the Lagrangian …

⎛

85



 1 ⎝ ∂ {divx v} + divx {(divx v) · v} + ∂x0 4 j=1 m=1 3

=

3

∂v m ∂v j + ∂x j ∂xm

2

⎞ − (divx v)2 ⎠

1 |Q|2 − divx R and curlx (curlx v) = curlx Q. (6.1.20) 2

Next we investigate the Euler-Lagrange of (4.1.71). We denote L 1 (v, (, −h), (, −A))    2   1 1  ∂A 1 2 −∇x  − 0 + v × curlx A − |curlx A| dx 0 dx 1 dx 2 dx 3 = 4π 2  2 ∂x R4

−

 

  ρ(x 0 , x 1 , x 2 , x 3 ) − A · j(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3

R4



1 4π G

+ R4

+



⎧  ⎨ N +∞ 

 2   1 1  ∂h |curlx h|2 − − 0 − ∇x  + v × curlx h dx 0 dx 1 dx 2 dx 3 2 2 ∂x ⎛

−m k G ⎝1 −

k=1−∞

⎩

3  j=1

where



j

dχk

dx 0

(x 0 ) − v j



χk (x 0 )

 ⎞⎫  2 ⎬ ⎠ dx 0 , ⎭

   = 21 |v|2 − |r|2 h = (v − r) ,

(6.1.21)

(6.1.22)

so that we have:  1 2 2 |v| − |r| , − (v − r) , (, −A) = L total (v, (, −A)) L 1 v, 2    2   1 1  ∂A 1 2 −∇x  − 0 + v × curlx A − |curlx A| dx 0 dx 1 dx 2 dx 3 = 4π 2  ∂x 2 R4     ρ(x 0 , x 1 , x 2 , x 3 ) − A · j(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 − R4

 1 2 1 1  ∂ 2 |curlx (v − r)| − − 0 (v − r) − ∇x |v| − + 2 2 ∂x 2 4 R  +v × curlx (v − r)|2 dx 0 dx 1 dx 2 dx 3 ⎧ ⎛ 2 ⎞⎫  j +∞ N  3 ⎬ ⎨     dχk 0 j 0 ⎠ dx 0 . + (x ) − v (x ) χ −m k G ⎝1 − k ⎭ ⎩ dx 0 k=1 j=1 

−∞

1 4π G



1 2 |r| 2



(6.1.23)

86

6 The Euler-Lagrange for the Lagrangian of the Gravitational …

Next by the Euler-Lagrange of (4.1.71) we have ⎧ δL total ⎪ ⎨ δv (v, (, −A)) = 0 δL total (, −A)) = 0 δA (v, ⎪ ⎩ δL total (, −A)) = 0 . δ (v,

(6.1.24)

On the other hand, by chain rule we have  1 2 δL 1 δL total v, |v| − |r|2 , − (v − r) , (, −A) (v, (, −A)) = δv δv 2  δL 1 2 2 +v v, |v| − |r| , − (v − r) , (, −A) δ 2  δL 1 1 2 2 + |v| − |r| , − (v − r) , (, −A) . v, δh 2 (6.1.25) Moreover,

 1  2    v, |v| − |r|2 , − (v − r) , (, −A)   21  2    v, 2 |v| − |r|2 , − (v − r) , (, −A) . (6.1.26) So, by (6.1.24), (6.1.26) and (6.1.25) we obtain δL total δA δL total δ

(v, (, −A)) = (v, (, −A)) =

δL 1 δA δL 1 δ

δL 1 δA δL 1 δ

 1  2    v, |v| − |r|2 , − (v − r) , (, −A) = 0   21  2    v, 2 |v| − |r|2 , − (v − r) , (, −A) = 0 ,

(6.1.27)

and δL 1 δv

 1 2 v, |v| − |r|2 , − (v − r) , (, −A) 2  δL 1 1 2 +v v, |v| − |r|2 , − (v − r) , (, −A) δ 2  δL 1 1 2 + v, |v| − |r|2 , − (v − r) , (, −A) = 0 . δh 2

(6.1.28)

Moreover, by (6.1.21), (6.1.9), (6.1.10) and (6.1.16) we have δL 1 1 1 = curlx Q − δh 4π G 4π G



∂R − curlx {v × R} , ∂x0

δL 1 =− (divx R) . δ 4π G

(6.1.29)

(6.1.30)

6.1 The Euler-Lagrange for the Lagrangian …

and

1 1 δL 1 = − e − Ev + D×B− R×Q δv 4π 4π G

87

(6.1.31)

where (e0 , e1 , e2 , e3 ) := (E, e) with e = (e1 , e2 , e3 ) ∈ R3 ,

(6.1.32)

and the contravariant four-vector field (e0 , e1 , e2 , e3 ) and the scalar field E are given by (e0 , e1 , e2 , e3 )(x 0 , x 1 , x 2 , x 3 ) ⎛ ⎞  j N 3     dχ k := m k 2G  ⎝1 − (x 0 ) − v j χk (x 0 ) ⎠ 0 dx k=1 j=1 j  dχk 0  1 (x )δ x − χk1 (χ 0 ), . . . , x 3 − χk3 (χ 0 ) 0 dx ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(6.1.33)

and E(x 0 , x 1 , x 2 , x 3 ) ⎛ 2 ⎞  j N 3     dχ k := m k 2G  ⎝1 − (x 0 ) − v j χk (x 0 ) ⎠ 0 d x k=1 j=1  1  1 0 2 δ x − χk (χ ), x − χk2 (χ 0 ), x 3 − χk3 (χ 0 ) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(6.1.34)

where G  is the derivative of the function G. Therefore, using (6.1.29), (6.1.30), (6.1.31) in (6.1.28) we deduce: 1 1 D×B− R×Q e − Ev + 4π 4π G ∂R 1 1 curlx Q − = − curlx {v × R} + (divx R) v . 4π G 4π G ∂ x 0

(6.1.35)

Moreover, by (6.1.27) we have 1 divx D − ρ = 0, 4π

(6.1.36)

88

6 The Euler-Lagrange for the Lagrangian of the Gravitational …

and 1 ∂D 1 1 1 1 ∂D − − curlx B − curlx (v × D) = j + curlx H = 0. 0 0 4π ∂ x 4π 4π 4π ∂ x 4π (6.1.37) So, totally we have j+

⎧ ⎪ curlx H = 4π j + ∂∂D ⎪ x0 ⎪ ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎪ ⎪ curlx E + ∂∂B =0 ⎪ x0 ⎪ ⎪ ⎪ div B = 0 ⎪ ⎪ ⎨ x curlx R + ∂∂Q − curlx (v × Q) = 0 x0 ⎪ ⎪ div Q = 0 ⎪ x ⎪ ⎪    1 ⎪ ∂ 1 T 2 2 ⎪ ⎪ ⎪ ∂ x 0 (divx v) + v · ∇x (divx v) + 4 dx v + {dx v}  = 2 |Q| − divx R ⎪ ⎪ ⎪ ⎪ curlx (curlx v) = curlx Q ⎪ ⎪   ⎪   ⎪ 1 ⎩ D × B − 4π1G R × Q = curlx Q − ∂∂R − curlx {v × R} + (divx R) v , 4π G e − Ev + 4π x0

(6.1.38) where D, B, E, H are given by (6.1.8), so that: ⎧ ⎪ ⎪B = curlx A ⎪ ⎨D = −∇  − ∂A + v × curl A x x ∂x0 ⎪E = −∇x  − ∂∂A = D − v × B 0 ⎪ x  ⎪  ⎩ + v × curlx A = B + v × D, H = curlx A + v × −∇x  − ∂∂A x0

(6.1.39)

and moreover, by (6.1.13) in the inertial frame we have: ⎧ ⎪ curlx H = 4π j + ∂∂D ⎪ x0 ⎪ ⎪ ⎪ ⎪divx D = 4πρ ⎪ ⎪ ⎪ ⎪ =0 ⎪curlx E + ∂∂B ⎪ x0 ⎨ divx B = 0  ⎪ ⎪ ⎪ + dx v · v R = − ∂∂v ⎪ x0 ⎪ ⎪ ⎪ ⎪ ⎪ xv ⎪Q = curl ⎪ ⎪ ⎩4π G e − Ev + 1 D × B − 4π

1 4π G R

   × Q = curlx Q − ∂∂R − curlx {v × R} + (divx R) v . x0

(6.1.40) Furthermore, taking divx of the both sides of the last equality in (6.1.38) and using continuum equation (4.1.50): ∂M + divx j M = 0 ∂x0

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(6.1.41)

6.1 The Euler-Lagrange for the Lagrangian …

89

we deduce

−

  ∂M 1 1 + div + div − − M) v + D × B − R × Q − j (Mv) ) (E (e x x M 4π 4π G ∂x0 1 1 D×B− R×Q = divx e − Ev + 4π 4π G ∂ 1 {(div R) + div R) v} . (6.1.42) =− (div x x x 4π G ∂ x 0

Therefore, considering the proper scalar quantity Q 0 , that we call the field mass, which satisfies 1 divx R, (6.1.43) Q 0 := −M + 4π G by (6.1.42) we deduce   1 ∂ Q0 1 − − M) v + − j + div v) = −div D × B − R × Q . (Q ) (E (e x 0 x M ∂x0 4π 4π G

(6.1.44)

Thus, we rewrite (6.1.38) as: ⎧ curlx H = 4π j + ∂D0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎪ ⎪ curlx E + ∂B0 = 0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪ divx B = 0 ⎪ ⎪ ⎪ ⎪ ⎪ curlx R + ∂Q0 − curlx (v × Q) = 0 ⎪ ⎪ ∂x ⎪ ⎨ divx Q = 0    ⎪ 1 1 ∂R ⎪ ⎪4π G e − Ev + 4π D × B − 4π G R × Q = curlx Q − ∂ x 0 − curlx {v × R} + (divx R) v ⎪ ⎪ ⎪ ⎪ ⎪divx R = 4π G(M + Q 0 ), ⎪ ⎪   ⎪ ⎪ ⎪ ∂ (div v) + v · ∇ (div v) + 1 d v + {d v}T 2 = 1 |Q|2 − div R ⎪ x x x x x x ⎪ 0 4 2 ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ curl (curl v) = curl Q x x x ⎪

 ⎪ ⎪ ⎩ ∂ Q 0 + div (Q v) = −div (e − j ) − (E − M) v + 1 D × B − 1 R × Q , x x 0 M 0 4π 4π G ∂x

(6.1.45)

where D, B, E, H are given by (6.1.8), so that: ⎧ B = curlx A ⎪ ⎪ ⎪ ⎨D = −∇  − ∂A + v × curl A x x ∂x0 ∂A ⎪  − = D − v × B E = −∇ x ⎪ ∂x0  ⎪  ⎩ + v × curlx A = B + v × D, H = curlx A + v × −∇x  − ∂∂A x0

(6.1.46)

90

6 The Euler-Lagrange for the Lagrangian of the Gravitational …

and moreover, we rewrite (6.1.40) in the inertial frame as: ⎧ ⎪ curlx H = 4π j + ∂D0 ⎪ ∂x ⎪ ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎪ ⎪curlx E + ∂B0 = 0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪divx B = 0 ⎪ ⎪   ⎨ R = − ∂v0 + dx v · v ∂x ⎪ ⎪ ⎪ Q = curl ⎪ xv ⎪     ⎪ ⎪ 1 D × B − 1 R × Q = curl Q − ∂R − curl {v × R} + (div R) v . ⎪ ⎪ 4π G e − Ev + 4π x x x ⎪ 0 4π G ⎪ ∂ x ⎪ ⎪ ⎪ divx R = 4π G(M + Q 0 ) ⎪ ⎪

 ⎪ ⎪ ⎩ ∂ Q 0 + div (Q v) = −div (e − j ) − (E − M) v + 1 D × B − 1 R × Q . x x 0 M 0 4π 4π G ∂x

(6.1.47)

However, as before, here we consider two cases: • The case of non-relativistic approximation, where√G(τ ) := 21 (τ − 1). • The case of relativistic particles, where G(τ ) := τ − 1. In the first case we have 2G  (τ ) = 1. In the second case we have 2G  (τ ) = τ − 2 = 1 + O(τ − 1). In the first non-relativistic case by (6.1.33) and (6.1.34) we deduce 1

e = jM

E=M

and

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(6.1.48)

In the second relativistic case (6.1.48) is satisfied approximately, provided we have 

j

dχk − v j (χk ) dx0

2 1

∀ k = 1, 2, . . . , N .

(6.1.49)

Thus, by (6.1.48) we rewrite (6.1.45) as: ⎧ curlx H = 4π j + ∂D0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎪ ⎪ curlx E + ∂B0 = 0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪divx B = 0 ⎪ ⎪ ⎪ ⎪ ⎪curlx R + ∂Q − curlx (v × Q) = 0 ⎪ ⎪ ∂x0 ⎪ ⎨ divx Q = 0    ⎪ 1 1 ∂R ⎪ ⎪4π G j M − Mv + 4π D × B − 4π G R × Q = curlx Q − ∂ x 0 − curlx {v × R} + (divx R) v ⎪ ⎪ ⎪ ⎪ ⎪divx R = 4π G(M + Q 0 ), ⎪ ⎪   ⎪ ⎪ ⎪ ∂ (div v) + v · ∇ (div v) + 1 d v + {d v}T 2 = 1 |Q|2 − div R ⎪ x x x x x x ⎪ 0 4 2 ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪curlx (curlx v) = curlx Q ⎪ 

⎪ ⎪ ⎩ ∂ Q 0 + div (Q v) = −div 1 D×B− 1 R×Q , ∂x0

x

0

x

4π

4π G

(6.1.50)

6.1 The Euler-Lagrange for the Lagrangian …

91

where D, B, E, H are given by (6.1.8), so that: ⎧ B = curlx A ⎪ ⎪ ⎪ ⎨D = −∇  − ∂A + v × curl A x x ∂x0 ∂A ⎪ E = −∇x  − ∂ x 0 = D − v × B ⎪ ⎪   ⎩ + v × curlx A = B + v × D, H = curlx A + v × −∇x  − ∂∂A x0

(6.1.51)

and moreover, we rewrite (6.1.47) in the inertial frame as: ⎧ ⎪ curlx H = 4π j + ∂D0 ⎪ ∂x ⎪ ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎪ ⎪ curlx E + ∂B0 = 0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪divx B = 0 ⎪ ⎪   ⎨ R = − ∂v0 + dx v · v ∂x ⎪ ⎪ ⎪ Q = curl ⎪ xv ⎪     ⎪ ⎪ 1 D × B − 1 R × Q = curl Q − ∂R − curl {v × R} + (div R) v . ⎪ ⎪ 4π G j M − Mv + 4π x x x ⎪ 0 4π G ⎪ ∂ x ⎪ ⎪ ⎪ div R = 4π G(M + Q 0 ) ⎪ ⎪

 ⎪ ∂ Qx ⎪ 1 1 ⎩ 0 + div (Q v) = −div x x 4π D × B − 4π G R × Q . 0 0 ∂x

(6.1.52)

Chapter 7

Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate System Which Is Cartesian and Inertial Simultaneously

Consider some coordinate system which is cartesian and inertial simultaneously. Next consider some resting massive spherically symmetric body of radius r0 with center at the point (x 1 , x 2 , x 3 ) = (0, 0, 0) for all x 0 ∈ R. Assume that the electromagnetical fields are negligible with respect to the inertial mass of the body. Then, we rewrite (6.1.47) as: ⎧   ⎪ R = − ∂∂vx 0 + dx v · v ⎪ ⎪ ⎪ ⎪ ⎪ v ⎨Q = curl     x 4π G e − Ev − 4π1G R × Q = curlx Q − ∂∂R 0 − curlx {v × R} + (divx R) v . x ⎪ ⎪ ⎪ divx R = 4π G(M + Q 0 ) ⎪ ⎪ ⎪ ⎩ ∂ Q 0 + div (Q v) = − div (e − j ) − (E − M) v − 1 R × Q . x 0 x M ∂x0 4π G (7.0.1) Moreover, we obviously have M(x 0 , x) = M1 (|x|) and u(x 0 , x) = 0

∀ (x 0 , x) ∈ R4 ,

(7.0.2)

where we denote (x 0 , x) := (x 0 , x 1 , x 2 , x 3 ) ∈ R4 with x := (x 1 , x 2 , x 3 ) ∈ R3 , u is a three-dimensional velocity field of every point of the given massive body and M1 := M1 (|x|) is the inertial mass density of the body which is assumed to be a radial function such that M1 (|x|) = 0 if |x| > r0 ,

(7.0.3)

where r0 is the radius of the body. In particular we have

 2 E(x 0 , x) = M1 (|x|) 2G  1 − v(x 0 , x)

∀ (x 0 , x) := (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . (7.0.4)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_7

93

94

7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate …

Thus we simplify the equations for the gravity in (7.0.1) as:

⎧ R = − ∂v0 + dx v · v ⎪ ⎪ ∂ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Q = curl

xv



4π G −M1 (|x|) 2G  1 − |v|2 v − 4π1G R × Q = curlx Q − ∂R0 − curlx {v × R} + (divx R) v . ∂ x ⎪ ⎪ ⎪ ⎪ divx R = 4π G(M + Q 0 ) ⎪ ⎪ 



⎪∂Q ⎩  2 −1 v− 1 R×Q . 0 0 + divx (Q 0 v) = − divx −M1 (|x|) 2G 1 − |v| 4π G ∂x

(7.0.5) We look for stationary (i.e. x 0 -independent) solutions of (7.0.5). Thus (7.0.5) implies: ⎧ ⎪ R = −dx v · v ⎪ ⎪ ⎪ ⎪ Q = curl ⎪ ⎪

xv

⎨ 4π G −M1 (|x|) 2G  1 − |v|2 v − 4π1G R × Q = curlx Q − (−curlx {v × R} + (divx R) v) . ⎪ ⎪ ⎪ ⎪divx R = 4π G(M + Q ⎪

0)

 ⎪ ⎪ ⎩div (Q v) = − div −M (|x|) 2G  1 − |v|2 − 1 v − 1 R × Q . x

0

x

1

4π G

(7.0.6) On the other hand, by the symmetry considerations of the problem we look for the solution of (7.0.6) that satisfies v(x) = ∇x Z 0 (|x|) where, again by the symmetry of the problem, the scalar function Z 0 (|x|) should be radial. In particular, by (7.0.6) we obtain (7.0.7) Q = curlx v = 0 and thus we simplify (7.0.6) as: ⎧  1 2 ⎪ |v| , R = −d v · v = −∇ x x ⎪ 2 ⎪ ⎪ ⎪ ⎪ v = 0, ⎨Q = curl     x 4π G −M1 (|x|) 2G  1 − |v|2 v = − (−curlx {v × R} + (divx R) v) . ⎪ ⎪ ⎪divx R = 4π G(M + Q 0 ) ⎪ ⎪ ⎪ ⎩div (Q v) = − div −M (|x|) 2G  1 − |v|2  − 1 v . x 0 x 1 (7.0.8)   In particular, since v = ∇x Z 0 (|x|) and R = − ∇x 21 |v|2 are both gradients of radial functions, we have v × R = 0 and thus, we further simplify (7.0.8) as: ⎧   ⎪ R = −∇x 21 |v|2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Q = curlx v = 0,   divx R = 4π G M1 (|x|) 2G  1 − |v|2 , ⎪ ⎪ ⎪ curlx R = 0, ⎪ ⎪ ⎪ ⎩ Q = M (|x|) 2G  1 − |v|2  − 1 . 0 1

(7.0.9)

7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate …

95

However, (7.0.9) is equivalent to the following: ⎧     1 ⎪ |v|2 = 4π G M1 (|x|) 2G  1 − |v|2 ,  − x ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎨curlx v = 0,   R = −∇x 21 |v|2 , ⎪ ⎪ ⎪Q = 0, ⎪ ⎪ ⎪ ⎩ Q = M (|x|) 2G  1 − |v|2  − 1 . 0 1

(7.0.10)

Therefore, denoting 1 1 := − |v|2 2

(7.0.11)

we rewrite (7.0.10) as: ⎧ x 1 = 4π G M1 (|x|) 2G  (1 + 21 ) , ⎪ ⎪ ⎪ ⎪ ⎪ 1 = − 21 |v|2 , ⎪ ⎪ ⎪ ⎨curl v = 0, x   ⎪ R = − ∇x 21 |v|2 , ⎪ ⎪ ⎪ ⎪ ⎪ Q = 0, ⎪ ⎪   ⎩ Q 0 = M1 (|x|) 2G  (1 + 21 ) − 1 ,

(7.0.12)

where the scalar field 1 =  (|x|) is radial, and thus, outside of the Massive body surface, where |x| > r0 it coincides with the following Newtonian potential of the massive body: G M0 , (7.0.13) 1 (|x|) = − |x| where M0 is the total effective gravitational mass of the massive body, defined as  M0 =

M1 (|x|) 2G  (1 + 21 (|x|)) dx

|x|≤r0



=

  M1 (|x|) 2G  1 − |∇x Z 0 (|x|)|2 dx.

(7.0.14)

|x|≤r0

 Note that, for the inertial mass of the Earth m 0 we have m 0 = |x|≤r0 M1 (|x|) dx and thus, in the non-relativistic case, where 2G  (τ ) = 1 we have M0 = m 0 . On the 1 other hand in the relativistic case, where 2G  (τ ) = (τ )− 2 > 1 with τ < 1 we have M0 > m 0 . Next, since there exists a scalar radial field Z 0 (|x|) such that v(x) = ∇x Z 0 (|x|), by (7.0.12) we obtain

96

7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate …

     dZ 0  = −21 (x),  (|x|)   d(|x|) that implies either v(x) =

√ −21 (|x|) x, |x| √

or v(x) = −

−21 (|x|) x. |x|

(7.0.15)

(7.0.16)

(7.0.17)

In particular, on the Earth surface we have:  |v| =

2G M0 , r0

(7.0.18)

where r0 is the massive body radius and M0 is the total effective gravitational mass of the massive body, i.e. the absolute value of the three-dimensional vectorial gravitational potential on a planet surface equals to the escape velocity and its direction is normal to the planet, either downward or upward.

7.1 Certain Curvilinear Coordinate System in the Case of Stationary Radially Symmetric Gravitational Field and Relation to the Schwarzschild Metric Assume that for a given part of the space-time V ⊂ R4 in some inertial or non-inertial cartesian coordinate system (∗) the gravitational field is stationary and radially symmetric that means that, {K mn }n,m=0,1,2,3 is given by (4.1.29) and {K mn }n,m=0,1,2,3 be given by (4.1.30), so that

and

⎧ 00 ⎪ ⎨K = 1 K jm = −δ jm + v j v m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j K = K j0 = v j ∀1 ≤ j ≤ 3,

(7.1.1)

⎧ 2 ⎪ ⎨ K 00 = 1 − |v| K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K 0 j = K j0 = v j ∀1 ≤ j ≤ 3 ,

(7.1.2)

where we denote (v 0 , v 1 , v 2 , v 3 ) := (1, v)

with

v := (v 1 , v 2 , v 3 ) ∈ R3 ,

(7.1.3)

7.1 Certain Curvilinear Coordinate System in the Case of Stationary …

97

and the three-dimensional vectorial gravitational potential v = (v1 , v2 , v3 ) is independent on variable x 0 and having the form v(x) = g (|x|)

x |x|

∀ x,

(7.1.4)

for some scalar function g(τ ) : R → R with (x 0 , x 1 , x 2 , x 3 ) := (x 0 , x) ∈ V where x := (x 1 , x 2 , x 3 ) ∈ R3 .

(7.1.5)

Next, given some differentiable function F(x) : R3 → R, consider the change of variables in the four-dimensional space-time R4 : 

  x 0 = x 0 + F x 1 , x 2 , x 3 x j = x j ∀ j = 1, 2, 3,

(7.1.6)

that transforms the cartesian coordinate system (∗) to the curvilinear coordinate system (∗∗) in the four-dimensional space-time R4 . Then in the terms of the threedimensional space we rewrite (7.1.6) as: 

x 0 = x 0 + F(x) x = x.

(7.1.7)

Next if we define a matrix  A = a ij 0≤i, j≤3 := then



∂ x i ∂x j

 ∈ R4×4 ,

(7.1.8)

0≤i, j≤3

⎧ 0 a0 = 1 ⎪ ⎪ ⎪ ⎨a m = δ mn ∀1 ≤ m, n ≤ 3 n ∂ 0 ⎪a j = ∂ xFj ∀1 ≤ j ≤ 3 ⎪ ⎪ ⎩ j a0 = 0 ∀1 ≤ j ≤ 3.

(7.1.9)

Next, remind that the contravariant pseudo-metric tensor {K mn }0≤m,n≤3 due to (7.1.1) has the form of ⎧ 00 ⎪ ⎨K = 1 (7.1.10) K jm = −δ jm + v j v m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j j0 j K = K = v ∀1 ≤ j ≤ 3, in the cartesian coordinate system (∗). We would like to find the form {K mn }0≤m,n≤3 of this tensor in the curvilinear coordinate system (∗∗). Then by (10.0.6) and (7.1.8) we have:

98

7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate …

K mn

⎛ ⎞ 3 3  3 3    m kj n  ak K a j = = akm ⎝ K k j a nj ⎠ k=0 j=0

k=0

∀0 ≤ m, n ≤ 3. (7.1.11)

j=0

In other words ⎛ K mn = a0m ⎝ K 00 a0n +

+

3 

3 

⎞ K 0 j a nj ⎠

j=1

⎛

akm ⎝ K k0 a0n +

3 

k=1

⎞ K k j a nj ⎠ ∀0 ≤ m, n ≤ 3.

(7.1.12)

j=1

In particular, by (7.1.9) and (7.1.12) together with (7.1.10) we obtain: ⎛ K 00 = a00 ⎝ K 00 a00 +

= a00 ⎝a00 +

3  j=1

= ⎝a00 +

K 0 j a 0j ⎠ +

j=1

⎛

⎛

⎞

3 

3 

⎞

⎛

2v j a 0j ⎠ + ⎝

ak0 ⎝ K k0 a00 +

k=1 3 

v j a 0j ⎠ −

K k j a 0j ⎠

a 0j v j ⎠ −

3   0 2 aj j=1

3   0 2 aj ,

(7.1.13)

j=1

⎛ K 0n = K n0 = a00 ⎝g 00 a0n +

3 

⎞ K 0 j a nj ⎠ +

3 

j=1

= a00 v n − an0 +

⎞

3  j=1

⎞2

j=1

⎞2

j=1

⎛

3 

3 

ak0 v k v n

⎛ ak0 ⎝ K k0 a0n +

k=1

3 

⎞ K k j a nj ⎠

j=1

∀1 ≤ n ≤ 3,

(7.1.14)

k=1

and ⎛ K

mn

=

a0m

⎝ K 00 a0n

+

3  j=1

= v v − δmn m

n

⎞ K 0 j a nj ⎠

+

3  k=1

⎛ akm

⎝ K k0 a0n

+

3 

⎞ K k j a nj ⎠

j=1

∀1 ≤ m, n ≤ 3.

Thus by the last three equations together with (7.1.9) we deduce:

(7.1.15)

7.1 Certain Curvilinear Coordinate System in the Case of Stationary …

⎧ 2   

3 2 00 j ∂F ⎪ K = 1 + v − 3j=1 ∂∂xFj ⎪ j j=1 ⎨

∂x  K 0n = K n0 = v n 1 + 3j=1 v j ∂∂xFj − ∂∂xFn ⎪ ⎪ ⎩ mn = v m v n − δmn ∀1 ≤ m, n ≤ 3. K

99

∀1 ≤ n ≤ 3,

(7.1.16)

Next if v satisfies (7.1.4) then choosing the function F to be defined as: F(x) = ξ (|x|) ∀ x, where we find that:

in other words

g(τ ) dξ (τ ) = dτ 1 − g 2 (τ )

⎞−1 ∂ F ∂F vn = ⎝1 + vj j ⎠ ∂x ∂xn j=1 ⎛

⎛

3 

⎞ ∂ F ∂F vn ⎝1 + vj j ⎠ = n ∂x ∂x j=1 3 

∀ τ,

∀1 ≤ n ≤ 3,

∀1 ≤ n ≤ 3.

(7.1.17)

(7.1.18)

(7.1.19)

Then we rewrite (7.1.16) as: ⎧ 2 

  ⎪ 00 2 ⎪ 1 + 3j=1 v j ∂∂xFj , ⎨ K = 1 − |v| K 0n = K n0 = 0 ∀1 ≤ n ≤ 3, ⎪ ⎪ ⎩ mn = v m v n − δmn ∀1 ≤ m, n ≤ 3. K

(7.1.20)

On the other hand by (7.1.19) we have ⎛ ⎞ 3    ∂F |v|2 = 1 − |v|2 ⎝ vj j ⎠. ∂ x j=1

(7.1.21)

We rewrite (7.1.21) as: ⎛ ⎞ 3    ∂ F 1 = 1 − |v|2 ⎝1 + vj j ⎠ ∂x j=1

(7.1.22)

Therefore, by (7.1.20) and (7.1.22) we deduce: ⎧   00 2 −1 ⎪ , ⎨ K = 1 − |v| 0n n0 ∀1 ≤ n ≤ 3, K =K =0 ⎪ ⎩ mn = v m v n − δmn ∀1 ≤ m, n ≤ 3. K

(7.1.23)

100

7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate …

 Next we find that the covariant pseudo-metric tensor {K mn }0≤m,n≤3 in the curvilinear coordinate system (∗∗) has the following form:

⎧    2 ⎪ ⎪ ⎨ K 00 = 1 − |v| ,   = K n0 ∀1 ≤ n ≤ 3, K 0n

= 0  ⎪ ⎪  2 ⎩ K mn = − 1 − |v| −1 v m v n + δmn

(7.1.24) ∀1 ≤ m, n ≤ 3.

 Indeed, if {K mn }0≤m,n≤3 is defined by (7.1.24), then by (7.1.23) we have: 3 

  K 0k K k0 = K 00 K 00 +

k=0

3 

 K mk K k j

=

3 

 K 0k K k0 = 1,

k=1

 K i0 K 0 j

k=0

+

3 

 K mk K k j

k=1 3

   −1 m k  1 − |v|2 = v v + δmk δk j − v k v j k=1

−1 2 m j −1 m j   = δm j − 1 − |v|2 |v| v v − v m v j + 1 − |v|2 v v = δm j ∀1 ≤ m, j ≤ 3, and

3 

  K mk K k0 = K m0 K 00 +

k=0 3 

 K mk K k0 = 0 ∀1 ≤ m ≤ 3,

k=1

  K 0k K k j = K 00 K 0 j +

k=0

So

3 

3 

 K 0k K k j = 0 ∀1 ≤ j ≤ 3.

k=1

3 

 K mk K k j = δm j ∀0 ≤ m, j ≤ 3,

k=0

and thus equalities (7.1.24) indeed define the inverse to {K mn }m,n=0,1,2,3 matrix. So by (7.1.24) we have: ⎧    2 ⎪ ⎪ ⎨ K 00 = 1 − |v| ,   = K n0 ∀1 ≤ n ≤ 3, K 0n

= 0  ⎪ ⎪  2 ⎩ K mn = − 1 − |v| −1 v m v n + δmn

(7.1.25) ∀1 ≤ m, n ≤ 3.

7.1 Certain Curvilinear Coordinate System in the Case of Stationary …

101

In particular, the quadratic form, induced by the covariant pseudo-metric tensor  {K mn }0≤m,n≤3 in the curvilinear  coordinate system (∗∗), that defined on the tangent  vectors dx 0 , dx 1 , dx 2 , dx 3 ∈ R4 where dx := (dx 1 , dx 2 , dx 3 ) has the following form: 3 3  

 



   dx m dx n = 1 − |v|2 (dx  )2 − |dx |2 + 1 − |v|2 −1 v · dx 2 K mn 0

m=0 n=0

2     v · dx  |dx |2 −  |v|  2  2 

−1  v    v + 1 − |v|2 · dx  +  · dx  = 1 − |v|2 (dx0 )2 |v|2  |v| |v|  2  2  

−1  v      + |dx |2 −  v · dx   · dx − 1 − |v|2 (7.1.26) .  |v|    |v|



= 1 − |v|2 (dx0 )2 −



Thus taking into account (7.1.7) and (7.1.4) we rewrite (7.1.26) as: 3 3  

   dx m dx n = 1 − v(x )2 (dx  )2 K mn 0

m=0 n=0

−



 2  2      2 −1  x    x   2        1 − v(x ) .  |x | · dx  + |dx | −  |x | · dx 

(7.1.27) Next, up to the end of this subsection, assume that our cartesian coordinate system (∗) is inertial and cartesian simultaneously and our gravitational field is formed by the resting spherical symmetric massive body of the effective gravitational mass M0 and radius r0 like the Earth, the Sun et al. with the center at the point (x 1 , x 2 , x 3 ) = 0. Then, as we get before, we have either (7.0.16): √ −21 (|x|) x, v(x) = |x| or (7.0.17):

√ v(x) = −

−21 (|x|) x, |x|

(7.1.28)

(7.1.29)

where outside of the massive body surface we have 1 (|x|) = −

G M0 , |x|

(7.1.30)

102

7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate …

with the total effective gravitational mass of the massive body M0 , defined as in (7.0.14) by  M0 =

M1 (|x|) 2G  (1 + 21 (|x|)) dx

|x|≤r0



=

  M1 (|x|) 2G  1 − |∇x Z 0 (|x|)|2 dx.

(7.1.31)

|v(x)|2 = −21 (|x|),

(7.1.32)

|x|≤r0

Thus in particular,

and outside of the massive body surface we have: 2G M0 . |x|

|v(x)|2 =

(7.1.33)

Both (7.1.28) and (7.1.29) are particular cases of (7.1.4), with  g(τ ) = ± −21 (τ ),

(7.1.34)

and in particular, outside of the massive body surface we have:  g (|x|) = ±

2G M0 , |x|

(7.1.35)

Thus defining the function F(x) as in (7.1.17), that always can be done in the case 2G M0 < 1, we can define the change of variables from coordinate system (∗) to r0 the curvilinear coordinate system (∗∗) in the four-dimensional space-time R4 as in (7.1.7):  x 0 = x 0 + F(x) (7.1.36) x = x. Then by inserting (7.1.28) or (7.1.29) into (7.1.25) we deduce the form of the covariant pseudo-metric tensor in the curvilinear coordinate system (∗∗): ⎧     ⎪ ⎪ ⎨ K 00 = 1 + 21 (|x |) ,   =K ∀1 ≤ n ≤ 3, K 0n

n0 = 0 −1 ⎪ x ⎪  ⎩ K mn = 1 + 21 (|x |) 21 (|x |) m

xn |x | |x |

− δmn



∀1 ≤ m, n ≤ 3. (7.1.37)

7.1 Certain Curvilinear Coordinate System in the Case of Stationary …

103

Moreover, by (7.1.27) we have: 3 3  

 dx m dx n K mn

m=0 n=0

  2  2          −1  x   + |dx |2 −  x · dx   · dx = 1 + 21 (|x |) dx02 − 1 + 21 (|x |) .  |x |    |x |

(7.1.38) In particular, outside of the massive body surface, i.e. when |x  | ≥ r0 we rewrite (7.1.37) and (7.1.38) as:

⎧ 2G M0  ⎪ K , = 1 −  ⎪ 00 |x | ⎪ ⎨   K 0n = K n0 = 0 ∀1 ≤ n ≤ 3,  −1

⎪ ⎪ 2G M0 xm xn ⎪ K  = − 1 − 2G M0 ⎩ + δmn mn |x | |x | |x | |x |

(7.1.39) ∀1 ≤ m, n ≤ 3,

and 3 3  

 dx m dx n K mn

m=0 n=0

 2  2           2G M0 −1  x 2G M0 2 −   + |dx |2 −  x · dx  1 − dx · dx = 1− . 0  |x |    |x | |x | |x |

(7.1.40) Therefore, we get that in coordinate system (∗∗), outside of the massive body, the covariant pseudo-metric tensor in (7.1.39) and (7.1.40) exactly the same as the wellknown Schwarzschild metric from the General Relativity (see [1], pages 180–181). Indeed in the spherical coordinates in R3 we rewrite (7.1.40) as: 3  3 

 K mn dx m dx n

m=0 n=0

    

  2G M0 2G M0 −1   2 2  2  2 2  2 = 1− 1− dr + (r ) (dθ ) + sin θ (dϕ ) dx0 − , r r

(7.1.41) and this is exactly the classical Schwarzschild metric! [1, 2] In particular, all the optical effects that we find in the framework of our model coincide with the effects considered in the framework of general relativity for the Schwarzschild metric. In particular, the Michelson-Morley experiment and all Sagnac-type effects will lead to the same result in the framework of our model like in the case of the general relativity. Moreover, since the Maxwell equations in both models have the same tensor form, all the electromagnetic effects, where the time

104

7 Gravity Field of Spherically Symmetric Massive Resting Body in a Coordinate …

does not appear explicitly, will be the same. Similarly, the curvature of the light path in the Sun’s gravitational √ field will be the same in both models. Finally, in the particular case of G(τ ) = τ , i.e. in the case of the relativistic Lagrangian of the motion in (2.4.8) all the mechanical effects will be the same in the framework of our model like in the case of the general relativity for the Schwarzschild metric, provided that the time does not appear explicitly in this effects. In particular, the movement of the Mercury planet in the Sun’s gravitational field will be the same in both models.

References 1. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, 1972) 2. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields Vol. 2 in Course of Theoretical Physics Book, 4th edn. (1975)

Chapter 8

Newtonian Gravity as an Approximation of Our Model

8.1 Newtonian Gravity as an Approximation of (6.1.52) We now approximate the Euler-Lagrange in (6.1.52). First of all, in the usual circumstances we obviously have for the electromagnetic field:     1  D × B  M .   4π

(8.1)

Thus, by (8.1), we approximate (6.1.50) as: ⎧ ⎪ curlx H = 4π j + ∂D0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪ div D = 4πρ ⎪ x ⎪ ⎪ ⎪ ⎪curlx E + ∂B0 = 0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪divx B = 0 ⎪ ⎪ ⎪ ⎪ curlx R + ∂Q0 − curlx (v × Q) = 0 ⎪ ⎪ ∂x ⎪ ⎨ divx Q =0    ⎪ ⎪ 4π G Mu − Mv − 4π1G R × Q = curlx Q − ∂R0 − curlx {v × R} + (divx R) v ⎪ ∂x ⎪ ⎪ ⎪ ⎪ R = 4π G(M + Q ), ⎪div x 0 ⎪ ⎪  2

⎪ ∂ ⎪   ⎪ v) + v · ∇x (divx v) + 41 dx v + {dx v}T  = 21 |Q|2 − divx R (div ⎪ x 0 ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ curlx (curlx v) = curlx Q  ⎪ ⎪ ⎪ ⎪ 1 R×Q , ⎩ ∂ Q00 + divx (Q 0 v) = divx ∂x

4π G

where u := u(x 0 , x 1 , x 2 , x 3 ) is the field of the velocities of the matter so that j M (x 0 , x 1 , x 2 , x 3 ) = M(x 0 , x 1 , x 2 , x 3 ) u(x 0 , x 1 , x 2 , x 3 )

(8.2)

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(8.3) and where D, B, E, H are given by (6.1.8), so that:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_8

105

106

8 Newtonian Gravity as an Approximation of Our Model

⎧ B = curlx A ⎪ ⎪ ⎪ ⎨D = −∇  − ∂A + v × curl A x x ∂x0 ∂A ⎪  − = D − v × B E = −∇ x ⎪ ∂x0 ⎪  ⎩ + v × curlx A = B + v × D. H = curlx A + v × −∇x  − ∂∂A x0

(8.4)

Furthermore, we approximate (6.1.52) in the inertial frame as: ⎧ curlx H = 4π j + ∂D0 ⎪ ⎪ ∂x ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎪curl E + ∂B = 0 ⎪ ⎪ x ⎪ ∂x0 ⎪ ⎪ ⎪ B = 0 div ⎪ x ⎪   ⎨ R = − ∂v0 + dx v · v ∂x ⎪ ⎪Q = curl v ⎪ ⎪ x ⎪     ⎪ ⎪ ⎪ 4π G Mu − Mv − 4π1G R × Q = curlx Q − ∂R0 − curlx {v × R} + (divx R) v . ⎪ ⎪ ∂ x ⎪ ⎪ ⎪ ⎪divx R = 4π G(M + Q 0 ) ⎪ ⎪  ⎪ ⎩ ∂ Q0 1 0 + divx (Q 0 v) = divx 4π G R × Q . ∂x

(8.5)

On the other hand, we can rewrite (8.5) in the inertial frame as: ⎧ ⎪ curlx H = 4π j + ∂∂D ⎪ x0 ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎪ ⎪curlx E + ∂B0 = 0 ⎪ ∂x ⎪ ⎪ ⎨div B = 0 x  ∂v 

∂v 

⎪ u + − div x v 0 + dx v · v 0 + dx v · v × (curlx v) = curlx (curlx v) ⎪ ∂ x ∂ x ⎪    ∂v 

∂v ⎪ ∂ ⎪ ⎪ ⎪+ ∂ x 0 ∂ x 0 + dx v · v − curlx v × ∂ x 0 + dx v · v + 4π G Q 0 (uv − v) . ⎪ ⎪ ⎪ ⎪divx ∂∂vx 0 + dx v · v = −4π G(M + Q 0 ) ⎪ ⎪ 

  ⎩ ∂ Q0 + divx (Q 0 v) = − divx 4π1G ∂∂vx 0 + dx v · v × (curlx v) , ∂x0 

where we denote uv :=

u if M = 0 v if M = 0 .

(8.6) (8.7)

Furthermore, we assume the non-relativistic approximation and quasistationery nature of the field v, so that  2   2   ∂ v       ∂(x 0 )2   dx v ,

   ∂v 2 2    ∂ x 0   |dx v| ,

  2     dx ∂v   d2 v2 , x   0 ∂x

|u|2  1 and |v|2  1. Thus, by (8.8) and (8.3) we approximate (8.6) in the inertial frame as:

(8.8)

8.1 Newtonian Gravity as an Approximation of (6.1.52)

⎧ curlx H = 4π j + ∂∂D ⎪ x0 ⎪ ⎪ ⎪ ⎪ div D = 4πρ x ⎪ ⎪ ⎪ ∂B ⎪ ⎪ ⎨curlx E + ∂ x 0 = 0 divx B = 0 ⎪   ⎪ ⎪ divx ∂∂vx 0 + dx v · v = −4π G(M + Q 0 ) ⎪ ⎪ ⎪ ⎪ ⎪curlx (curlx v) + 4π G Q 0 (uv − v) + ∇x ζ = 0 ⎪ ⎪ ⎩ ∂ Q0 + divx (Q 0 v) = 0. ∂x0

107

(8.9)

where ζ is some unspecified approximately negligible scalar field. However, the obvious solution of the last two equations in (8.9) are 

curlx v = 0 Q 0 = 0.

(8.10)

So, by (8.10) we rewrite (8.9) as: ⎧ curlx H = 4π j + ∂∂D ⎪ ⎪ x0 ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎨curl E + ∂B = 0 x ∂x0 ⎪divx B = 0 ⎪ ⎪ ⎪ ⎪ ⎪ curlx v = 0 ⎪ ⎪   ⎩ divx ∂∂vx 0 + dx v · v = −4π G M.

(8.11)

We obviously can rewrite (8.11) as: ⎧ curlx H = 4π j + ∂∂D ⎪ ⎪ x0 ⎪ ⎪ ⎪ div D = 4πρ ⎪ x ⎪ ⎪ ⎨curl E + ∂B = 0 x ∂x0 ⎪ B = 0 div x ⎪ ⎪ ⎪ ⎪ ⎪curlx v = 0 ⎪ ⎪ 

 ⎩ divx ∂∂vx 0 + ∇x 21 |v|2 = −4π G M,

(8.12)

where D, B, E, H are given by (6.1.8), so that: ⎧ B = curlx A ⎪ ⎪ ⎪ ⎨D = −∇  − ∂A + v × curl A x x ∂x0 ∂A ⎪ E = −∇x  − ∂ x 0 = D − v × B ⎪ ⎪

 ⎩ + v × curlx A = B + v × D, H = curlx A + v × −∇x  − ∂∂A x0 and furthermore, we rewrite two last equations in (8.12) as:

(8.13)

108

8 Newtonian Gravity as an Approximation of Our Model

⎧ ⎪ ⎨v = ∇x Z where divx {∇x } = 4π G M ⎪ ⎩ ∂Z 2 1 ∂Z 1 |∇ | + Z = + |v|2 = −, x ∂x0 2 ∂x0 2

(8.14)

where Z is some scalar field. So, finally ⎧ ⎪ and ⎨curlx v = 0  ∂v 1 2 = −∇x  + ∇ |v| x 2 0 ∂ x ⎪ ⎩ divx {∇x } = 4π G M.

where

(8.15)

On the other hand, we remind that the motion of the particle with the mass m and the charge σ in the gravitational and electromagnetical fields is governed, in the non-relativistic case by Eq. (5.0.15) so that m

d2 z (x 0 ) d(x 0 )2    

 

1 ∂  0 0) + ∇ 0 , z(x 0 ) 2 − dz (x 0 ) × curl v x 0 , z(x 0 ) x v x , z(x =m  v x x 2 ∂x0 dx 0

    dz (8.16) + σ E x 0 , z(x 0 ) + 0 (x 0 ) × B x 0 , z(x 0 ) . dx

Thus, inserting (8.15) into (8.16) gives that in the coordinate system, which is cartesian and inertial simultaneously, we have m

 d2 z (x 0 ) = −m ∇x  x 0 , z(x 0 ) 0 2 d(x )

0 

0  dz 0 0 0 + σ E x , z(x ) + 0 (x ) × B x , z(x ) , dx

where  is given by

divx {∇x } = 4π G M.

(8.17)

(8.18)

However, (8.17) with (8.18) is obviously exactly the case of the classical Newtonian Gravity! Thus, in the non-relativistic approximation and in the case of quasistationery gravitational field, the Newtonian Gravity is indeed a valid approximation of (6.1.52), provided we deal with the coordinate system, which is cartesian and inertial simultaneously. Similarly, if we do not assume anymore that our cartesian coordinate system is inertial, then in the non-relativistic approximation and in the case of quasistationery gravitational field, by (8.8) we approximate (8.2) as:

8.1 Newtonian Gravity as an Approximation of (6.1.52)

⎧ ∂D ⎪ ⎪curlx H = 4π j + ∂ x 0 ⎪ ⎪ ⎪ ⎪ divx D = 4πρ ⎪ ⎪ ⎪ ⎨curlx E + ∂B0 = 0 ∂x B = 0 div ⎪ x ⎪  ⎪    ⎪ T 2 ∂ 1  ⎪ {d ⎪ d = −4π G M v) + v · ∇ v) + v + v} (div (div x x x x x 0 ⎪ ∂x 4 ⎪ ⎪ ⎩curl (curl v) = 0, x x

109

(8.19)

where D, B, E, H are given by (6.1.8), so that: ⎧ B = curlx A ⎪ ⎪ ⎪ ⎨D = −∇  − ∂A + v × curl A x x ∂x0 ∂A ⎪  − = D − v × B E = −∇ x ⎪ ∂x0 ⎪  ⎩ + v × curlx A = B + v × D. H = curlx A + v × −∇x  − ∂∂A x0

(8.20)

Chapter 9

Polarization and Magnetization

It is well known from tensor analysis that if {ϒ mn }m,n=0,1,2,3 is an antisymmetric twotimes contravariant tensor andif {K mn }m,n=0,1,2,3  is a contravariant pseudo-metric, then the four-component field γ 0 , γ 1 , γ 2 , γ 3 defined by      1 ∂ det {K p q } p,q=0,1,2,3 − 2 3 3 m j j   ∂x ∂S γm : = + Sm j   1  ∂x j det {K p q } p,q=0,1,2,3 − 2 j=0 j=0  3    pq − 1 m j 1 ∂   2 S =  − 1 ∂ x j det {K } p,q=0,1,2,3  j=0 det {K p q } p,q=0,1,2,3  2 ∀ m = 0, 1, 2, 3,

(9.0.1)

is a valid contravariant four-vector. Lemma 9.1 Consider an arbitrary  moving point with four-dimensional trajectory z(s) = z 0 (s), z 1 (s), z 2 (s), z 3 (s) : [a, b] → R4 . Moreover, assume that the infinite trajectory of the motion z(s) is considered for all instances of time from −∞ to +∞ so that 3

3

   j 2   j 2  z (s) = lim+ z (s) = +∞. (9.0.2) lim− s→a

j=0

s→b

j=0

 Next consider a point charge σ with four-dimensional trajectory χ (s) = χ 0 (s),  χ 1 (s), χ 2 (s), χ 3 (s) : [a, b] → R4 , parameterized by the same proper parameter s ∈ [a, b]. Moreover, assume that the infinite trajectory of the motion χ (s) is considered for all instances of time from −∞ to +∞ so that for every τ ∈ [0, 1] we have

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_9

111

112

9 Polarization and Magnetization 3 



lim

s→a −

2 

τ χ j (s) + (1 − τ )z j (s)

= lim

s→b+

j=0

3 



2  τ χ j (s) + (1 − τ )z j (s) = +∞ .

j=0

(9.0.3) If we define the antisymmetric two-times contravariant tensor {ϒσmn }m,n=0,1,2,3 by:  

  ϒσmn (x 0 , x 1 , x 2 , x 3 ) := det K pq x 0 , x 1 , x 2 , x 3

 1 2  p,q=0,1,2,3 

  b 1     dz n dχ n σ χ m (s) − z m (s) τ (s) + (1 − τ ) (s) ds ds a 0

    dz m dχ m (s) + (1 − τ ) (s) − χ n (s) − z n (s) τ ds ds





0 0 0 3 dτ ds. δ x − τ χ (s) + (1 − τ )z (s) , . . . , x − τ χ 3 (s) + (1 − τ )z 3 (s)

(9.0.4)

Then, we have 3  n=0

1

   1 det {K pq } p,q=0,1,2,3 − 2

 − 21 ∂   pq   mn ϒσ det K ( p,q)30  ∂xn

   21 b dχ m      pq = det K (s) δ x 0 − χ 0 (s), . . . , x 3 − χ 3 (s) ds σ p,q=0,1,2,3  ds a

   21 b dz m      pq − det K (s) δ x 0 − z 0 (s), . . . , x 3 − z 3 (s) ds, σ p,q=0,1,2,3  ds a

(9.0.5) in the sense of distributions. Definition  9.1 Consider an arbitrary  moving point with four-dimensional trajectory z(s) = z 0 (s), z 1 (s), z 2 (s), z 3 (s) : [a, b] → R4 . Moreover, assume that the infinite trajectory of the motion z(s) is considered for all instances of time from −∞ to +∞ so that 3

3

   j 2   j 2  z (s) = lim+ z (s) = +∞ . (9.0.6) lim− s→a

j=0

s→b

j=0

Next consider a totally neutral system of N point charges σ1 , σ2 , . . . , σ N satisfying N  k=1

σk = 0,

(9.0.7)

9 Polarization and Magnetization

113

 with four-dimensional trajectory of the k-th particle χk (s) = χk0 (s), χk1 (s), χk2 (s),  χk3 (s) : [a, b] → R4 ∀ k = 1, 2, . . . N , parameterized by the same proper parameter s ∈ [a, b]. Moreover, assume that the infinite trajectory of the motion χk (s) is considered for all instances of time from −∞ to +∞ so that for every τ ∈ [0, 1] we have 3 

3 

2  2    j j j j τ χk (s) + (1 − τ )z (s) = lim+ τ χk (s) + (1 − τ )z (s) lim−

s→a

s→b

j=0

= +∞

j=0

∀ k = 1, 2, . . . , N .

(9.0.8)

Then define the antisymmetric two-times contravariant tensor {ϒ mn }m,n=0,1,2,3 by: ϒ mn (x 0 , x 1 , x 2 , x 3 ) :=

  1 N 

  2  det K pq x 0 , x 1 , x 2 , x 3  p,q=0,1,2,3 

k=1

  b 1   dχ n   dz n (s) σk χkm (s) − z m (s) τ k (s) + (1 − τ ) ds ds a 0

  dχ m   dz m (s) − χkn (s) − z n (s) τ k (s) + (1 − τ ) ds ds





 δ x 0 − τ χk0 (s) + (1 − τ )z 0 (s) , . . . , x 3 − τ χk3 (s) + (1 − τ )z 3 (s) dτ ds,

so that ϒ mn (x 0 , x 1 , x 2 , x 3 ) :=

N 

ϒσmn (x 0 , x 1 , x 2 , x 3 ) . k

(9.0.9)

(9.0.10)

k=1

Then by Lemma 9.1 we have 3 

1

   1 det {K pq } p,q=0,1,2,3 − 2

n=0

=

N  3  k=1 n=0

=

∂ ∂xn



− 1    2  det K pq p,q=0,1,2,3  ϒ mn

1

   1 det {K pq } p,q=0,1,2,3 − 2

∂ ∂xn



− 1  pq   2 mn  K ϒ det σ p,q=0,1,2,3 

N   1 b dχ m 

    2 σk k (s)δ x 0 − χk0 (s), . . . , x 3 − χk3 (s) ds det K pq p,q=0,1,2,3  ds k=1

a

 N      1 b 

 dz m 2  σk − det K pq p,q=0,1,2,3  (s)δ x 0 − z 0 (s), . . . , x 3 − z 3 (s) ds. ds a

k=1

(9.0.11)

114

9 Polarization and Magnetization

Therefore, by inserting (9.0.7) into (9.0.11) we deduce 3  n=0

=

1

   1 det {K pq } p,q=0,1,2,3 − 2

∂ ∂xn



− 1    2  det K pq p,q=0,1,2,3  ϒ mn

N   1 b 

   dχkm  2 σk (s) δ x 0 − χk0 (s), . . . , x 3 − χk3 (s) ds = j m , det K pq p,q=0,1,2,3  ds k=1

a

(9.0.12) in the sense of distributions, where the contravariant four-vector of the charge current ( j 0 , j 1 , j 2 , j 3 ) is given as in (3.16). Definition 9.2 Consider a totally neutral system of N point charges σ1 , σ2 , . . . , σ N satisfying N  σk = 0, (9.0.13) k=1

 with four-dimensional trajectory of the k-th particle χk (s) = χk0 (s), χk1 (s), χk2 (s),  χk3 (s) : [a, b] → R4 ∀ k = 1, 2, . . . N , parameterized by the same proper parameter  s ∈ [a, b]. We define the multipole momentum D(s) = D0 (s), D1 (s), D2 (s), D3 (s) : [a, b] → R4 of this system by: D j (s) :=

N 

j

σk χk (s) =

k=1

N 

 j σk χk (s) − z j (s)

∀ j = 0, 1, 2, 3

∀s ∈ [a, b],

(9.0.14)

k=1

  where z(s) = z 0 (s), z 1 (s), z 2 (s), z 3 (s) : [a, b] → R4 is a four-dimensional trajectory of an arbitrary moving point. Definition  moving point with four-dimensional trajectory  9.3 Consider an arbitrary z(s) = z 0 (s), z 1 (s), z 2 (s), z 3 (s) : [a, b] → R4 . Moreover, consider a totally neutral system of N point charges σ1 , σ2 , . . . , σ N (possibly with infinitesimally large absolute values) satisfying N  σk = 0, (9.0.15) k=1

with four-dimensional trajectory of the k-th particle,   χk (s) := z 0 (s) + lk0 (s), z 1 (s) + lk1 (s), z 2 (s) + lk2 (s), z 3 (s) + lk3 (s) : [a, b] → R4 ∀ k = 1, 2, . . . N , parameterized by the same proper parameter s ∈ [a, b], and such that χk (s) is infinitesimally close to z(s) for every s ∈ [a, b] and every k = 1, 2, . . . , N , so that j lk (s) is infinitesimally small for all j = 0, 1, 2, 3 and all k = 1, 2, . . . N . In particular this system of charges could be a dipole or the limiting cases of the totally

9 Polarization and Magnetization

115

neutral polarized molecule. Then we define the infinitesimal multipole momentum   D (z(s)) = D0 (z(s)) , D1 (z(s)) , D2 (z(s)) , D3 (z(s)) : [a, b] → R4 of this system, with respect to the center z(s), as follows: D (z(s)) : = j

N 

σk



 j z j + lk (s) − z j (s)

k=1

=

N 

j

σk lk (s) ∀ j = 0, 1, 2, 3 ∀s ∈ [a, b].

(9.0.16)

k=1 j

Although in this definition lk (s) is assumed to be infinitesimally small and σk is allowed to be infinitesimally large, the quantity D j (s) in (9.0.16) is always assumed to be finite. Remark 9.1 Obviously, the infinitesimal multipole momentum, defined by (9.0.16) j is a valid contravariant four-vector at the point z(s). Indeed since, lk (s) is infinitesimally small, then under the change of the coordinate system given by a smooth non-degenerate invertible transformation from R4 onto R4 , having the form ⎧ 0 x ⎪ ⎪ ⎪ ⎨x 1 ⎪ x 2 ⎪ ⎪ ⎩ 3 x

= = = =

f (0) (x 0 , x 1 , x 2 , x 3 ), f (1) (x 0 , x 1 , x 2 , x 3 ), f (2) (x 0 , x 1 , x 2 , x 3 ), f (3) (x 0 , x 1 , x 2 , x 3 ),

(9.0.17)

we obviously have  

D j z  =

N 

σk

k=1

→

N  k=1



N

 



  j z j + l k − z j = σk f ( j) z 0 + lk0 , . . . , z 3 + lk3 − f ( j) (z) k=1

 3  3  ∂ f ( j)  ∂ f ( j) n σk (z) lk = (z) Dn (z) n ∂x ∂xn n=0

∀ j = 0, 1, 2, 3.

(9.0.18)

n=0

Definition  moving point with four-dimensional trajectory  9.4 Consider an arbitrary z(s) = z 0 (s), z 1 (s), z 2 (s), z 3 (s) : [a, b] → R4 . Moreover, assume that the infinite trajectory of the motion z(s) is considered for all instances of time from −∞ to +∞ so that 3

3

   j 2   j 2  z (s) = lim+ z (s) = +∞ . (9.0.19) lim− s→a

j=0

s→b

j=0

Next, given an infinitesimalmultipole at the point z(s) with the infinitesimal multi pole momentum D (z(s)) = D0 (z(s)) , D1 (z(s)) , D2 (z(s)) , D3 (z(s)) : [a, b] → R4 , define the two-times contravariant antisymmetric tensor field of polarizationmagnetization of this infinitesimal multipole {P mn }n,m=0,1,2,3 by:

116

9 Polarization and Magnetization

P

mn

b  dz n dz m Dm (z(s)) (s) − Dn (z(s)) (s) (x , x , x , x ) := ds ds 0

1

2

3

a

   21      det K pq (z k (s)) p,q=0,1,2,3  δ x 0 − z 0 (s), . . . , x 3 − z 3 (s) ds ∀ m, n = 0, 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(9.0.20)

Remark 9.2 Consider an arbitrary  moving point with four-dimensional trajectory z(s) = z 0 (s), z 1 (s), z 2 (s), z 3 (s) : [a, b] → R4 . Moreover, consider a totally neutral system of N point charges σ1 , σ2 , . . . , σ N (possibly with infinitesimally large absolute values) satisfying N  σk = 0, (9.0.21) k=1

with four-dimensional trajectory of the k-th particle,   χk (s) := z 0 (s) + lk0 (s), z 1 (s) + lk1 (s), z 2 (s) + lk2 (s), z 3 (s) + lk3 (s) : [a, b] → R4 ∀ k = 1, 2, . . . N , parameterized by the same proper parameter s ∈ [a, b], and such that χk (s) is infinitesimally close to z(s) for every s ∈ [a, b] and every k = 1, 2, . . . , N , so that j lk (s) is infinitesimally small for all j = 0, 1, 2, 3 and all k = 1, 2, . . . N . Then, since j lk (s) is infinitesimally small, by (9.0.9), (9.0.16) and (9.0.20) we deduce ϒ mn (x 0 , x 1 , x 2 , x 3 ) → P mn (x 0 , x 1 , x 2 , x 3 ) ∀ m, n = 0, 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(9.0.22) Therefore, by (9.0.22) we can rewrite (9.0.12) as: 3  n=0

1

   1 det {K pq } p,q=0,1,2,3 − 2

 − 21 ∂   pq   mn = jm , P det K p,q=0,1,2,3  ∂xn (9.0.23)

where ( j , j , j , j ) is the contravariant four-vector of the charge current, created by the infinitesimal multipole. 0

1

2

3

Definition 9.5 Given a union of finite number of infinitesimal multipoles, we define the two-times contravariant antisymmetric tensor field of polarization-magnetization {P mn }n,m=0,1,2,3 of this union as a sum of polarizations of every infinitesimal multipole in the union. Then, by (9.0.23) we have 3  n=0

1

   1 det {K pq } p,q=0,1,2,3 − 2

 − 21 ∂   pq   mn K = j pm , P det p,q=0,1,2,3  ∂xn (9.0.24)

9 Polarization and Magnetization

117

where ( j p0 , j p1 , j p2 , j p3 ) is the contravariant four-vector of the charge current of polarization, created by the union of infinitesimal multipoles. Next, given a moving dielectric and para/diamagnetic continuum medium, by (2.5.4) we have the following Maxwell Equations: 

  3  3   − 1 km jn ∂ An ∂ A m p q det {K } p,q=0,1,2,3  2 K K − ∂xm ∂xn m=0 n=0  − 1   = −4π det {K p q } p,q=0,1,2,3  2 j k + j pk ∀ k = 0, 1, 2, 3 , (9.0.25)

3  ∂ ∂ xj j=0

where ( j p0 , j p1 , j p2 , j p3 ) is the contravariant four-vector of the charge current of polarization-magnetization, created by the dielectric and para/diamagnetic medium (this medium can be represented as the union of infinitesimal multipoles) and ( j 0 , j 1 , j 2 , j 3 ) is the contravariant four-vector of the free (real) charge current. Therefore, by (9.0.24) and (9.0.25) we deduce the following Maxwell equations in the dielectric medium: 3  ∂ ∂x j



  1 det {K p q } p,q=0,1,2,3 − 2

 4π P k j +

3  3 

 K km K jn

m=0 n=0

j=0

− 1  = −4π det {K p q } p,q=0,1,2,3  2 j k

∂ Am ∂ An − ∂xm ∂xn



∀ k = 0, 1, 2, 3 .

(9.0.26)

Next in the case of the simplest isotropic dielectric and para/diamagnetic continuum medium we have P mn =

3  3 

 κ K m j K nk

j=0 j=0

+

3  3 

(γ − κ)

 3 3 

k=0 j=0

=

3  3 

κ K m j K nk

j=0 j=0

+

(γ − κ)

k=0 j=0



uk

 −1 k

K kd u u

d



K m j u n u k + u m u j K nk

k=0 d=0



3  3 

∂Aj ∂ Ak − ∂x j ∂xk

∂Aj ∂ Ak − ∂x j ∂xk

∂Aj ∂ Ak − ∂x j ∂xk

 3 3 

∂A

k ∂x j

−

∂Aj ∂xk



 −1 k

K kd u u

d



un K m j − um K n j



k=0 d=0



∀ m, n = 0, 1, 2, 3.

(9.0.27)

where γ and κ are scalar fields and (u 1 , u 2 , u 3 , u 4 ) is the contravariant four-vector field of velocities of the moving dielectric continuum medium. In other words,

118

9 Polarization and Magnetization

Pkj =

3 3  

 κ K jm K kn

j=0 j=0

∂ An ∂ Am − ∂xm ∂xn



  3  3   ∂ An  ∂ Am (γ − κ) K km u˜ j u˜ n + u˜ k u˜ m K jn − ∂xm ∂xn m=0 n=0   3 3   ∂ An ∂ Am = κ K jm K kn − ∂xm ∂xn j=0 j=0 +

+

  3  3    ∂ An ∂ Am (γ − κ) u˜ j K km − u˜ k K jm u˜ n − ∂xm ∂xn m=0 n=0

∀ k, j = 0, 1, 2, 3,

(9.0.28)

where we denote  u˜ j :=

3 3  

− 21 k

K kd u u

d

uj

∀ j = 0, 1, 2, 3 .

(9.0.29)

k=0 d=0

Thus, inserting (9.0.28) into (9.0.26) implies  3 3 3     1 ∂ det {K p q } p,q=0,1,2,3 − 2 j ∂x m=0 n=0 j=0       4π(γ − κ) k m 4π(γ − κ) j n ∂ Am ∂ An (1 + 4π κ) K km + √ K jn + √ u˜ u˜ u˜ u˜ − ∂xm ∂xn 1 + 4π κ 1 + 4π κ 1   − = −4π det {K p q } p,q=0,1,2,3  2 j k ∀k = 0, 1, 2, 3. (9.0.30)

(Note here that the normalized quantity (u˜ 0 , u˜ 1 , u˜ 2 , u˜ 3 ) is the same as the four-speed on the usual Theory of Relativity). Finally, we consider the generally covariant Ohm’s Law in a moving conducting medium of the form  3 3  3  3   m k r K r k u˜ j u˜ m = εK mr Fr k u˜ k , (9.0.31) j − r =0 k=0

r =0 k=0

where ε is a certain scalar field and Fr k is given by (9.1.6). Moreover, we consider the generally covariant Ohm’s Law, including the Hall effect, of the form

9.1 Polarization and Magnetization in a Cartesian Coordinate System

j − m

 3 3  r =0 k=0

=

3  3 

K

mr

119

 K r k u˜ j

u˜ m

k r



 Fr k εu˜ − ς k

j − k

 3 3 

r =0 k=0

 K r d u˜ j

d r

 u˜

k

,

(9.0.32)

r =0 d=0

where ς is a certain scalar field.

9.1 Polarization and Magnetization in a Cartesian Coordinate System Consider the simplified model of the gravity, governed by the Lagrangian in (2.9.23), (2.9.24) and consider we deal with a cartesian coordinate system. Then, {K mn }n,m=0,1,2,3 is given by (4.1.29) and {K mn }n,m=0,1,2,3 is given by (4.1.30), so that ⎧ 00 ⎪ ⎨K = 1 (9.1.1) K jm = −δ jm + v j v m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j j0 j K = K = v ∀1 ≤ j ≤ 3, ⎧ 2 ⎪ ⎨ K 00 = 1 − |v| K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K 0 j = K j0 = v j ∀1 ≤ j ≤ 3 ,

(9.1.2)

where we denote (v 0 , v 1 , v 2 , v 3 ) := (1, v)

v := (v 1 , v 2 , v 3 ) ∈ R3 .

with

(9.1.3)

Furthermore, consider { mn }n,m=0,1,2,3 as

mn = v m v n − K mn

∀ m, n = 0, 1, 2, 3 .

(9.1.4)

Then, by (9.1.1) { mn }n,m=0,1,2,3 is given by ⎧ 00 ⎪ ⎨ = 0

jm = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j

= j0 = 0 ∀1 ≤ j ≤ 3.

(9.1.5)

Next, let {Fmn }n,m=0,1,2,3 be an antisymmetric two-times covariant tensor, defined by Fm j :=

∂ Aj ∂ Am − ∂xm ∂x j

∀ m, j = 0, 1, 2, 3 .

(9.1.6)

120

9 Polarization and Magnetization

Then, denoting (x 0 , x 1 , x 2 , x 3 ) = (x 0 , x)

where

x := (x 1 , x 2 , x 3 ) ∈ R3 ,

(9.1.7)

(A0 , A1 , A2 , A3 ) = (, −A) where A0 =  and (A1 , A2 , A3 ) = −A, (9.1.8) ⎧ B := (B , B , B ) := curl A, ⎪ 1 2 3 x ⎪ ⎪ ⎨E := (E , E , E ) := −∇  − ∂A , 1 2 3 x ∂x0 (0) (0) (0) (0) ⎪ D := (D1 , D2 , D3 ) := E + v × B, ⎪ ⎪ ⎩ (0) H := (H1(0) , H2(0) , H3(0) ) := B + v × D(0) = B + v × (E + v × B) , (9.1.9) and F mn : =

3 3  

K m j K nk F jk

k=0 j=0

=

3 3  

 K m j K nk

k=0 j=0

∂ Aj ∂ Ak − j ∂x ∂xk

 ∀ m, n = 0, 1, 2, 3,

(9.1.10)

by Lemma 4.1 we have ⎧ F00 ⎪ ⎪ ⎪ ⎪ ⎪ F ⎪ 0j ⎪ ⎪ ⎨F jj ⎪ F12 ⎪ ⎪ ⎪ ⎪ ⎪ F 13 ⎪ ⎪ ⎩ F23 and

⎧ F 00 ⎪ ⎪ ⎪ ⎪ ⎪ F0j ⎪ ⎪ ⎪ ⎨F j j ⎪ F 12 ⎪ ⎪ ⎪ ⎪ ⎪ F 13 ⎪ ⎪ ⎩ 23 F

=0 = −F j0 = E j ∀ j = 1, 2, 3 =0 ∀ j = 1, 2, 3 = −F21 = −B3 = −F31 = B2 = −F32 = −B1 ,

(9.1.11)

=0 = −F j0 = −D (0) ∀ j = 1, 2, 3, j = 0 ∀ j = 1, 2, 3, = −F 21 = −H3(0) = −F 31 = H2(0) = −F 32 = −H1(0) .

(9.1.12)

Next, given a moving dielectric and para/diamagnetic medium, represented by a system of N0 infinitesimal multipoles at the points     z k (x 0 ) :=: z k0 (x 0 ), z k1 (x 0 ), z k2 (x 0 ), z k3 (x 0 ) = x 0 , zk (x 0 ) where   zk (x 0 ) := z k1 (x 0 ), z k2 (x 0 ), z k3 (x 0 ) ∈ R3 ∀ k = 1, 2, . . . , N0 ,

(9.1.13)

9.1 Polarization and Magnetization in a Cartesian Coordinate System

121

0 0 0 parameterized by  the0 first  coordinate z k (z )4 := z , with the infinitesimal multipole momentum Dk z k (z ) : (−∞, +∞) → R , given by























Dk z k (z 0 ) = Dk0 z k (z 0 ) , Dk1 z k (z 0 ) , Dk2 z k (z 0 ) , Dk3 z k (z 0 )





   where d(z 0 ) := Dk1 z k (z 0 ) , Dk2 z k (z 0 ) , Dk3 z k (z 0 ) ∈ R3

 = 0, d(z 0 )

∀ k = 1, 2, . . . , N0 ,

(9.1.14)  0



where obviously we have Dk z k (z 0 ) = 0, consider the two-times contravariant antisymmetric tensor {P mn }n,m=0,1,2,3 , given by Definition 9.5 together with (9.0.20), so that we have

 dz n  dz m k 0 k n 0 0 z (z ) − D (z ) (z ) k k dz 0 dz 0 k=1−∞

 δ x 0 − z k0 (z 0 ), x 1 − z k1 (z 0 ), x 2 − z k2 (z 0 ), x 3 − z k3 (z 0 ) dz 0 P mn (x 0 , x 1 , x 2 , x 3 ) =

+∞ N0  



Dkm z k (z 0 )



 dz n  dz m k k 0 n 0 0 z (x ) − D (x ) (x ) k k dx0 dx0 k=1

 δ x 1 − z k1 (x 0 ), x 2 − z k2 (x 0 ), x 3 − z k3 (x 0 ) =

N0  



Dkm z k (x 0 )

∀m, n = 0, 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(9.1.15)

Thus, by (9.0.24) we have 3  ∂ P mn n=0

∂xn

= j pm ,

(9.1.16)

In particular, by (9.1.15) we have P 0n (x 0 , x 1 , x 2 , x 3 ) = −

N0 







Dkn z k (x 0 ) δ x 1 − z k1 (x 0 ), x 2 − z k2 (x 0 ), x 3 − z k3 (x 0 )

k=1

∀ n = 0, 1, 2, 3

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(9.1.17)

Thus, considering P := (P1 , P2 , P3 ) ∈ R3 and M := (M1 , M2 , M3 ) ∈ R3 , defined by ⎧ P 00 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ P 0 j = −P j0 = −P j ∀ j = 1, 2, 3, ⎪ ⎪ ⎪ ⎨ P j j = 0 ∀ j = 1, 2, 3, (9.1.18) ⎪ P 12 = −P 21 = −M3 ⎪ ⎪ ⎪ ⎪ ⎪ P 13 = −P 31 = M2 ⎪ ⎪ ⎩ 23 P = −P 32 = −M1 , by (9.1.17) and (9.1.15) we have

122

9 Polarization and Magnetization

P(x 0 , x 1 , x 2 , x 3 ) =

N0 

    dk z k (x 0 ) δ x 1 − z k1 (x 0 ), x 2 − z k2 (x 0 ), x 3 − z k3 (x 0 )

k=1

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

∀ m, n = 0, 1, 2, 3

(9.1.19)

and N0    dzk 0 (x ) × dk z k (x 0 ) M(x , x , x , x ) = 0 dx k=1   δ x 1 − z k1 (x 0 ), x 2 − z k2 (x 0 ), x 3 − z k3 (x 0 ) 0

1

2

3

∀ m, n = 0, 1, 2, 3

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(9.1.20)

On the other hand by (9.1.18) and (9.1.16) we have divx P = −

3  ∂ P 0n n=1

and ∂ P1 − ∂x0 ∂ P2 − ∂x0 ∂ P3 − ∂x0







∂ M3 ∂ M2 − ∂x2 ∂x3 ∂ M1 ∂ M3 − ∂x3 ∂x1 ∂ M2 ∂ M1 − ∂x1 ∂x2





= − j p0 = −ρ p ,

(9.1.22)

∂ P 20  ∂ P 2n = + = j p2 , n ∂x0 ∂ x n=1

(9.1.23)

∂ P 30  ∂ P 3n = + = j p3 , n ∂x0 ∂ x n=1

(9.1.24)

3

3

3

 divx P = −ρ p ∂P − curlx M = j p . ∂x0

so that

(9.1.21)

∂ P 10  ∂ P 1n + = j p1 , n ∂x0 ∂ x n=1

= 

∂xn

(9.1.25)

where as before, we denote (x 0 , x 1 , x 2 , x 3 ) := (x 0 , x) with x := (x 1 , x 2 , x 3 ) ∈ R3 and ( j p0 , j p1 , j p2 , j p3 ) := (ρ p , j p ) with j p := ( j p1 , j p2 , j p3 ) ∈ R3 . However, by (9.1.10) and (9.1.1) we rewrite (9.0.25) as 3  ∂ Fkj j=0

∂x j

= −4π



j k + j pk



∀ k = 0, 1, 2, 3 ,

(9.1.26)

where F mn is given by (9.1.12) and ( j 0 , j 1 , j 2 , j 3 ) := (ρ, j) is the contravariant four vector of the free (real) charge current with j := ( j 1 , j 2 , j 3 ) ∈ R3 . Therefore, similarly to that we get in (9.1.25) we rewrite (9.1.26) as:

9.1 Polarization and Magnetization in a Cartesian Coordinate System



  divx D(0) = 4π ρ + ρ p   curlx H(0) = 4π j + j p +

∂D(0) ∂x0

,

123

(9.1.27)

where by (9.1.9) we have  D(0) = E + v × B, H(0) = B + v × D(0) = B + v × (E + v × B) .

(9.1.28)

Thus, by (9.1.25) we rewrite (9.1.27) as: 

  divx D(0) + 4π P = 4π ρ  (0)  curlx H + 4π M = 4π j +

∂ ∂x0

 (0)  D + 4π P .

(9.1.29)

Therefore, denoting:  D := D(0) + 4π P = E + v × B + 4π P H := H(0) + 4π M = B + v × (E + v × B) + 4π M ,

(9.1.30)

we rewrite (9.1.29) or equivalently (9.0.26) in the cartesian coordinate system as: ⎧ divx D = 4πρ ⎪ ⎪ ⎪ ⎨curl H = 4π j + ∂D x ∂x0 ⎪curlx E + ∂∂B = 0 0 ⎪ x ⎪ ⎩ divx B = 0 ,

(9.1.31)

where D is the electric displacement field and H is an auxiliary magnetic field in the dielectric medium, given by 

D := E + v × B + 4π P H := B + v × (E + v × B) + 4π M.

(9.1.32)

Next consider two-times contravariant tensor {G mn }m,n=0,1,2,3 defined by: G mn :=

3  3  

3  3    n mj  K m j u n u k + u m u j K nk F jk = u K − u m K n j u k F jk

k=0 j=0

k=0 j=0

∀ m, n = 0, 1, 2, 3,

(9.1.33)

where (u 1 , u 2 , u 3 , u 4 ) is the contravariant four-vector field of velocities of the moving dielectric continuum medium and we have (u 0 , u 1 , u 2 , u 3 ) := (1, u)

with

u := (u 1 , u 2 , u 3 ) ∈ R3 .

(9.1.34)

124

9 Polarization and Magnetization

In particular, since the tensor {G mn }m,n=0,1,2,3 is antisymmetric, i.e. G mn = −G nm ∀ m, n = 0, 1, 2, 3, then we have ⎧ 00 G =0 ⎪ ⎪ ⎪ ⎨G mm = 0 ∀ m = 1, 2, 3, ⎪ G 0m = −G m0 ⎪ ⎪ ⎩ mn G = −G nm ∀ m = n = 1, 2, 3.

(9.1.35)

Next by (9.1.4), since {F jk } j,k=0,1,2,3 is an antisymmetric matrix, we rewrite (9.1.33) as: G mn =

3  3



 u n (v m v j − m j ) − u m (v n v j − n j ) u k F jk

k=0 j=0

=−

3  3



 u n m j − u m n j u k F jk

k=0 j=0

+

3

3  

 u n v m v j − u m v n v j u k F jk

k=0 j=0

=−

3  3



 u n m j − u m n j u k F jk

k=0 j=0

⎞ ⎛ 3  3    n m m n ⎝ j k + u v −u v v u F jk ⎠ j=0 k=0

=−

3  3



 u n m j − u m n j u k F jk

k=0 j=0

⎛ ⎞ 3  3  3 3     n m + (v j − u j )u k F jk ⎠ ∀ m, n = 0, 1, 2, 3. u (v − u m ) − u m (v n − u n ) ⎝ k=0 j=0

j=0 k=0

(9.1.36) So, G mn = −

3  3



 u n m j − u m n j u k F jk

k=0 j=0

⎛ ⎞ 3  3  n m   m m n n ⎝ j j k + u (u − v ) − u (u − v ) (u − v )u F jk ⎠

∀ m, n = 0, 1, 2, 3.

j=0 k=0

(9.1.37) Next consider two-times contravariant tensor {G˜ mn }m,n=0,1,2,3 defined by: − G˜ mn :=

3  3  3

3

   

m j u n u k + u m u j nk F jk = u n m j − u m n j u k F jk k=0 j=0

k=0 j=0

∀ m, n = 0, 1, 2, 3.

(9.1.38)

9.1 Polarization and Magnetization in a Cartesian Coordinate System

125

In particular, since the tensor {G˜ mn }m,n=0,1,2,3 is antisymmetric, i.e. G˜ mn = −G˜ nm ∀ m, n = 0, 1, 2, 3, then we have ⎧ 00 G˜ = 0 ⎪ ⎪ ⎪ ⎨G˜ mm = 0 ∀ m = 1, 2, 3, ⎪ G˜ 0m = −G˜ m0 ⎪ ⎪ ⎩ ˜ mn G = −G˜ nm ∀ m = n = 1, 2, 3.

(9.1.39)

On the other hand, by (9.1.37) and (9.1.38) together we deduce: G

mn

˜ mn

=G

⎛ ⎞ 3  3  n m   m m n n ⎝ j j k + u (u − v ) − u (u − v ) (u − v )u F jk ⎠ j=0 k=0

∀ m, n = 0, 1, 2, 3.

(9.1.40)

In particular, in the case we have |u − v|2  1 ,

(9.1.41)

by (9.1.40), using (9.1.3) and (9.1.34), we rewrite (9.1.33) as  3 3 

−1 G mn =

K kd u k u d

 3 3 3  3  

k=0 d=0



 u n K m j − u m K n j u k F jk

r =0 d=0

k=0 j=0

≈ G˜ mn = −

−1 Kr d ur u d

3  3



 u n m j − u m n j u k F jk

k=0 j=0

∀ m, n = 0, 1, 2, 3.

(9.1.42)

Next we rewrite (9.1.38) as: −G˜ mn =

  3  3

3 3

     n mj m nj k n mj m nj k u −u u F jk = u −u F j0 + u F jk k=0 j=0



= u

n

= u n

m0

m0

−u m

−u m

n0

n0



j=0

 F00 +

 3  

3 



k

u F0k

 +

k

u F0k

+

j=1

u n

mj

−u m

nj

 3  

u n

mj

−u m

nj

j=1

k=1 3



+

k=1

3



3







k=1

F j0 +

3 

 k

u F jk

k=1

 u n m j − u m n j F j0

j=1



k

u F jk

k=1

So, by (9.1.5) and (9.1.43) we deduce

∀ m, n = 0, 1, 2, 3.

(9.1.43)

126

9 Polarization and Magnetization

G˜ n0 = −G˜ 0n

= u n

00

−

n0

 3  

 k

u F0k

k=1

+

3



  3 3

    u n 0 j − n j F j0 + u n 0 j − n j u k F jk

j=1

=−  =

3 

δn j F j0 −

j=1

E − n

3 

δn j

j=1 3 

 3 

j=1

k=1



u k F jk

= −Fn0 −

k=1



3 

u k Fnk

k=1

∀ n = 1, 2, 3.

k

u Fnk

(9.1.44)

k=1

Therefore, by (9.1.11) and (9.1.44) we have ⎧  1   1  2  10 01 2 3 3 ⎪ ⎨G˜ = −G˜ =  E − u F12 − u F13  =  E + u B3 − u B2  G˜ 20 = −G˜ 02 = E 2 − u 3 F23 − u 1 F21 = E 2 + u 3 B1 − u 1 B3 ⎪      ⎩ ˜ 30 G = −G˜ 03 = E 3 − u 1 F31 − u 2 F32 = E 3 + u 1 B2 − u 2 B1 .

(9.1.45)

On the other hand, by (9.1.5) and (9.1.43) we obtain ˜ mn

−G

=

3  

u δm j − u δn j n

m



 3  3   n   m k F j0 + u δm j − u δn j u F jk

j=1

= u n Fm0 − u m Fn0 + u n  =− u

j=1



u k Fmk

k=1

 n

 3 

E − m

3  k=1



k

u Fmk

− um

−u

k=1



u k Fnk

k=1

 m

 3 

E − n

3 

k

u Fnk

 ∀ m, n = 1, 2, 3.

k=1

(9.1.46) Thus, considering G := (G 1 , G 2 , G 3 ) ∈ R3 and V := (V1 , V2 , V3 ) ∈ R3 , defined by ⎧ G 00 ⎪ ⎪ ⎪ ⎪ ⎪ G0 j ⎪ ⎪ ⎪ ⎨G j j ⎪ G 12 ⎪ ⎪ ⎪ ⎪ ⎪G 13 ⎪ ⎪ ⎩ 23 G

=0 = −G j0 = −G j ∀ j = 1, 2, 3, = 0 ∀ j = 1, 2, 3, = −G 21 = −V3 = −G 31 = V2 = −G 32 = −V1 ,

(9.1.47)

by (9.1.44), (9.1.46), (9.1.45) and (9.1.42), together with (9.1.40), we deduce:

9.1 Polarization and Magnetization in a Cartesian Coordinate System

127

⎧ −1 3  3 ⎪  ⎪ ⎪ K kd u k u d G≈E+u×B ⎨ d=0 k=0  −1  3 3 −1 3  3 ⎪   ⎪ k d k d ⎪ K kd u u V= K kd u u u × G ≈ u × (E + u × B) , ⎩ k=0 d=0

k=0 d=0

(9.1.48) provided, we have (9.1.41). Next in the case of the simplest isotropic continuum medium we rewrite (9.0.27) as: P mn =

3  3 

κ K m j K nk F jk + (γ − κ)

j=0 j=0

 3 3 

−1 K kd u k u d

G mn

∀ m, n = 0, 1, 2, 3 ,

k=0 d=0

(9.1.49) where γ ad κ are scalar fields. Thus, by (9.1.18), (9.1.47) and (9.1.48), together with (9.1.49) and (9.1.12) we have  P ≈ (γ − κ) (E + u × B) + κ (E + v × B) M ≈ (γ − κ) u × (E + u × B) + κ (B + v × (E + v × B)) .

(9.1.50)

We rewrite (9.1.50) as: 

P ≈ γ (E + u × B) − κ (u − v) × B M ≈ u × (γ (E + u × B − κ (u − v) × B)) + κB − κ(u − v) × (E + v × B) . (9.1.51) Thus, by (9.1.51) we rewrite (9.1.32) as     4π γ 4π κ 1 v+ u− D ≈ (1 + 4π γ ) E + (u − v) × B 1 + 4π γ 1 + 4π γ 1 + 4π γ       1 + 4π κ 1 + 4π κ v+ 1− u ×B (9.1.52) = (1 + 4π γ ) E + 1 + 4π γ 1 + 4π γ

and  H ≈ (1 + 4π κ)B + (1 + 4π γ )

4π γ 1 v+ u 1 + 4π γ 1 + 4π γ



× E + v × (v × B) + 4π γ u × (u × B) − 4π κu × ((u − v) × B) − 4π κ(u − v) × (E + v × B)   4π γ 4π κ 1 v+ u− = (1 + 4π κ)B + (1 + 4π γ ) (u − v) 1 + 4π γ 1 + 4π γ 1 + 4π γ × E + v × (v × B) + 4π γ u × (u × B) − 4π κu × ((u − v) × B) − 4π κ(u − v) × (v × B) = (1 + 4π κ)B     1 + 4π κ 1 + 4π κ v+ 1− u + (1 + 4π γ ) 1 + 4π γ 1 + 4π γ       1 + 4π κ 1 + 4π κ v+ 1− u ×B × E+ 1 + 4π γ 1 + 4π γ

128

9 Polarization and Magnetization     1 + 4π κ 1 + 4π κ v+ 1− u − (1 + 4π γ ) 1 + 4π γ 1 + 4π γ      1 + 4π κ 1 + 4π κ v+ 1− u × B + v × (v × B) + 4π γ u × 1 + 4π γ 1 + 4π γ × (u × B) − 4π κu × ((u − v) × B) − 4π κ(u − v) × (v × B) .

(9.1.53)

Thus, by (9.1.52) we rewrite (9.1.53) as: 

   1 + 4π κ 1 + 4π κ v+ 1− u 1 + 4π γ 1 + 4π γ     1 + 4π κ 1 + 4π κ u × D − (1 + 4π γ ) v+ 1− 1 + 4π γ 1 + 4π γ      1 + 4π κ 1 + 4π κ × v+ 1− u × B + v × (v × B) 1 + 4π γ 1 + 4π γ

H ≈ (1 + 4π κ)B +

+ 4π γ u × (u × B) − 4π κu × ((u − v) × B) − 4π κ(u − v)     1 + 4π κ 1 + 4π κ u v+ 1− × (v × B) = (1 + 4π κ)B + 1 + 4π γ 1 + 4π γ (1 + 4π γ ) − (1 + 4π κ) × D + (1 + 4π κ) (u − v) × ((u − v) × B) . 1 + 4π γ

(9.1.54)

Therefore, denoting ⎧ 1 ⎪ ⎪ ⎨γ0 := 1+4πγ κ0 := (1 + 4π κ)



⎪ ⎪ ⎩uˆ := (γ0 κ0 v + (1 − γ0 κ0 ) u) = 1+4πκ v + 1 − 1+4πγ

1+4πκ 1+4πγ

  u ,

(9.1.55)

and using (9.1.41) we rewrite (9.1.52) and (9.1.54) as: 

E = γ0 D − uˆ × B H = κ0 B + uˆ × D + κ0 (1 − γ0 κ0 ) (u − v) × ((u − v) × B) .

(9.1.56)

In particular, if |κ0 | |1 − γ0 κ0 | |u − v|2  1 , we have



E = γ0 D − uˆ × B H = κ0 B + uˆ × D .

(9.1.57)

(9.1.58)

Thus, the case of (9.1.57), by (9.1.58) we rewrite (9.1.31), or equivalently we approximate (9.0.30) in the cartesian coordinate system as:

9.1 Polarization and Magnetization in a Cartesian Coordinate System

⎧ div D = 4πρ ⎪ ⎪ ⎪ x ⎪ ∂D ⎪ ⎪ ⎪curlx H = 4π j + ∂ x 0 ⎪ ⎨curl E + ∂B = 0 x ∂x0 ⎪divx B = 0 ⎪ ⎪ ⎪ ⎪ ⎪ E = γ0 D − (γ0 κ0 v + (1 − γ0 κ0 ) u) × B ⎪ ⎪ ⎩ H = κ0 B + (γ0 κ0 v + (1 − γ0 κ0 ) u) × D ,

129

(9.1.59)

provided we have |κ0 | |1 − γ0 κ0 | |u − v|2  1 .

(9.1.60)

Moreover, we can rewrite (9.1.59) as: ⎧ divx D = 4πρ ⎪ ⎪   ⎪ ⎪ ⎪ − curlx uˆ × D curlx (κ0 B) = 4π j + ∂∂D ⎪ x0 ⎪    ⎪ ∂B ⎪ ⎪ ⎨curlx (γ0 D) = − ∂ x 0 − curlx uˆ × B divx B = 0 ⎪ ⎪ ⎪ E = γ0 D − uˆ × B ⎪ ⎪ ⎪ ⎪ ⎪ H = κ0 B + uˆ × D ⎪ ⎪ ⎩ uˆ := (γ0 κ0 v + (1 − γ0 κ0 ) u) .

(9.1.61)

We call γ0 the dielectric permeability of the dielectric medium, we call κ0 the magnetic permeability of the medium and we call the speed-like vector field uˆ ∈ R3 the optical displacement of the moving dielectric medium. Finally, note that the Maxwell equations in the medium in the form (9.1.61) agree with the Fizeau experiment. Finally, In the case of (9.1.41), by (9.1.2) we have 3  3 

  K r k u˜ k j r ≈ 1 − |v|2 ρ + v · j + ρv · u − u · j = ρ − (u − v) · (j − ρv)

r =0 k=0

(9.1.62)

Then, by (9.1.62) and (9.1.41) we have jm −

 3 3 

 K r k u˜ k j r u˜ m ≈ j m − ρu m + (u − v) · (j − ρv) u m

r =0 k=0

≈ j m − ρu m + (u − v) · (j − ρu) u m ,

(9.1.63)

so that  3 3  ⎧   ⎪ 0− k j r u˜ 0 ≈ (u − v) · (j − ρu) ⎪ K u ˜ j rk ⎨ k=0 r =0  3 3   ⎪ ⎪ ⎩ jm − K r k u˜ k j r u˜ m ≈ j m − ρu m + (u − v) · (j − ρu) u m ∀ m = 1, 2, 3. r =0 k=0

On the other hand, by (9.1.41) and (9.1.11) we have

(9.1.64)

130

9 Polarization and Magnetization

⎧ 3 3   ⎪ ⎪ u˜ k F0k ≈ u · (E + u × B) ⎨ F0k u˜ k = u˜ 0 F00 + k=0

k=1

3 3   ⎪ ⎪ ⎩ Fmk u˜ k = u˜ 0 Fm0 + u˜ k Fmk ≈ − (E + u × B) ∀ m = 1, 2, 3 . k=0

(9.1.65)

k=1

However, we have  3   3  ⎧ 3 3   0r   ⎪ k = K 00 k + K 0r k ⎪ K F u ˜ F u ˜ F u ˜ rk 0k rk ⎨ r =0 k=0 k=0 k=03    3 3 3     ⎪ ⎪ ⎩ ∀ m = 1, 2, 3 . K mr Fr k u˜ k = K m0 F0k u˜ k + K mr Fr k u˜ k r =0 k=0

k=0

(9.1.66)

k=0

Thus, by (9.1.66) and (9.1.65), together with (9.1.2) we have ⎧ 3 3   0r ⎪ ⎪ K Fr k u˜ k ≈ Y0 ⎨ r =0 k=0

3  3  ⎪ ⎪ ⎩ K mr Fr k u˜ k ≈ Y m ∀ m = 1, 2, 3 ,

(9.1.67)

r =0 k=0

where (Y0 , Y1 , Y2 , Y3 ) = ((u − v) · (E + u × B) , ((u − v) · (E + u × B)) v + (E + u × B)) ≈ ((u − v) · (E + u × B) , ((u − v) · (E + u × B)) u + (E + u × B))

(9.1.68)

Thus, in a cartesian coordinate system (9.0.31) in the case (9.1.41) is equivalent to: j − ρ u = ε (E + u × B) ,

(9.1.69)

that is the usual form of the Ohm’s Law in a moving conducting medium. Similarly, we deduce     3 3 ⎧ 3 3   0r  ⎪ k d r ⎪ j u˜ k ≈ Z 0 K F − K u ˜ j ⎨ rk rd r =0 k=0 d=0    r =0 3  3 3 3    ⎪ mr k d r ⎪ ⎩ K Fr k j − K r d u˜ j u˜ k ≈ Z m ∀ m = 1, 2, 3, r =0 k=0

r =0 d=0

(9.1.70)

where (Z 0 , Z 1 , Z 2 , Z 3 ) = ((u − v) · ((j − ρ u) × B) , ((u − v) · ((j − ρ u) × B)) v + ((j − ρu) × B)) ≈ ((u − v) · ((j − ρ u) × B) , ((u − v) · ((j − ρu) × B)) u + ((j − ρu) × B))

(9.1.71)

Thus, in a cartesian coordinate system (9.0.32) in the case (9.1.41) is equivalent to: j − ρ u = ε (E + u × B) − ς (j − ρ u) × B, that is the usual form of the Ohm’s Law, including the Hall effect.

(9.1.72)

Chapter 10

Detailed Proves of the Stated Theorems, Propositions and Lemmas

First of all we would like to remind the definitions of the vectors, covectors and covariant and contravariant tensors of second order in R4 : Definition 10.1 Let S := S(R4 ) be the group of all smooth non-degenerate invertible transformations from R4 onto R4 having the form ⎧ 0 x ⎪ ⎪ ⎪ 1 ⎨ x 2 ⎪ x ⎪ ⎪ ⎩ 3 x

= = = =

f (0) (x 0 , x 1 , x 2 , x 3 ), f (1) (x 0 , x 1 , x 2 , x 3 ), f (2) (x 0 , x 1 , x 2 , x 3 ), f (3) (x 0 , x 1 , x 2 , x 3 ).

(10.1)

We say that a one-component field a := a(x 0 , x 1 , x 2 , x 3 ), defined on R4 , is a covariant (contravariant) scalar field on the group S, if under the coordinate transformation in the group S of the form (10.1) this field transforms as: a  = a.

(10.2)

Next we say that a four-component field (a 0 , a 1 , a 2 , a 3 ), defined on R4 , is a contravariant four-vector field on the group S, if under the coordinate transformation in the group S of the form (10.1) every of four components of this field transforms as: a j =

3  ∂ f ( j) k=0

∂xk

a k ∀ j = 0, 1, 2, 3.

(10.3)

Next we say that a four-component field (a0 , a1 , a2 , a3 ), defined on R4 , is a fourcovector field on the group S, if under the coordinate transformation in the group S of the form (10.1) every of four components of this field transforms as: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1_10

131

132

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

aj =

3  ∂ f (k) k=0

∂x j

ak ∀ j = 0, 1, 2, 3.

(10.4)

Furthermore, we say that a 16-component field {amn }m,n=0,1,2,3 , defined on R4 , is a two-times covariant tensor field on the group S, if under the coordinate transformation in the group S of the form (10.1) every of 16 components of this field transforms as: amn =

3 3   ∂ f (k) ∂ f ( j) j=0 k=0

∂xm ∂xn

ak j ∀ m, n = 0, 1, 2, 3.

(10.5)

Next we say that a 16-component field {a mn }m,n=0,1,2,3 , defined on R4 , is a two-times contravariant tensor field on the group S, if under the coordinate transformation in the group S of the form (10.1) every of 16 components of this field transforms as: a mn =

3 3   ∂ f (m) ∂ f (n) j=0 k=0

∂xk ∂x j

a k j ∀ m, n = 0, 1, 2, 3.

(10.6)

Then it is well known that for every two contravariant four-vectors (a 0 , a 1 , a 2 , a 3 ) and (b0 , b1 , b2 , b3 ) on S, the 16-component field {cmn }m,n=0,1,2,3 , defined in every coordinate system by cmn := a m bn ∀ m, n = 0, 1, 2, 3,

(10.7)

is a two-times contravariant tensor on S. Moreover, for every two four-covectors (a0 , a1 , a2 , a3 ) and (b0 , b1 , b2 , b3 ) on S, the 16-component field {cmn }m,n=0,1,2,3 , defined in every coordinate system by cmn := am bn ∀ m, n = 0, 1, 2, 3,

(10.8)

is a two-times covariant tensor on S. It is also well known that if {a mn }m,n=0,1,2,3 is a two-times contravariant tensor field on the group S and if a 16-component field {bmn }m,n=0,1,2,3 satisfies 3  k=0

a mk bkn

 1 if m = n = 0 if m = n

∀ m, n = 0, 1, 2, 3,

(10.9)

then {bmn }m,n=0,1,2,3 is a two-times covariant tensor on S. Next it is well known that, given a four-covector (a0 , a1 , a2 , a3 ) a four-vector (b0 , b1 , b2 , b3 ), a two-times covariant tensor {cmn }m,n=0,1,2,3 and a two-times contravariant tensor {dmn }m,n=0,1,2,3 on the group S, the quantities

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas 3 

3  3 

ak bk and

cmn d mn

133

(10.10)

m=0 n=0

k=0

are covariant (contravariant) scalars on S, the four-component fields defined by 3 

d mk ak

k=0

and

m=0,1,2,3

3 

cmk bk

k=0

m=0,1,2,3

(10.11)

are covariant four-vector and contravariant four-covector on S, and moreover, 16component fields {cˆmn }m,n=0,1,2,3 and {dˆmn }m,n=0,1,2,3 defined by cˆmn :=

3 3  

d m j d nk c jk and dˆmn :=

k=0 j=0

3 3  

cm j cnk d jk ∀ m, n = 0, 1, 2, 3,

j=0 k=0

(10.12) are two-times contravariant and two-times covariant tensors on S. Finally, it is well known that, if a := a(x 0 , x 1 , x 2 , x 3 ) is a scalar field on the group S, then the fourcomponent field (w0 , w1 , w2 , w3 ) defined by: w j :=

∂a ∂x j

∀ j = 0, 1, 2, 3,

(10.13)

is a four-covector field on the group S. Proof (Proof of Proposition 2.1) By (2.1.8) we obviously deduce that {mn }m,n=0,1,2,3 is a symmetric two-times contravariant tensor field. Next, as in (2.1.5), by (2.1.4) we have 3  3 

K jm v j v m = 1 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.14)

j=0 m=0

Moreover, by (2.1.4) we deduce 3 

⎛ v jwj = ⎝

j=0

3  3 

⎞ 21 K jk w j wk ⎠ ,

(10.15)

j=0 k=0

and thus, by (2.1.8), (10.15) and (2.1.4) we have 3  j=0

m j w j = 0 ∀ m = 0, 1, 2, 3.

(10.16)

134

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

Next, fix a constant point (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . Then, since K mn (x 0 , x 1 , x 2 , x 3 ) has one positive and three negative eigenvalues, by Sylvester’s law of inertia there exists a coordinate system, that we now fix, such that at the fixed point (x 0 , x 1 , x 2 , x 3 ) we have ⎧ 00 0 1 2 3 ⎪ ⎨ K (x , x , x , x ) = 1 (10.17) K jm (x 0 , x 1 , x 2 , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j 0 1 2 3 j0 0 1 2 3 K (x , x , x , x ) = K (x , x , x , x ) = 0 ∀1 ≤ j ≤ 3. Then, we also have ⎧ 0 1 2 3 ⎪ ⎨ K 00 (x , x , x , x ) = 1 K jm (x 0 , x 1 , x 2 , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K 0 j (x 0 , x 1 , x 2 , x 3 ) = K j0 (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3.

(10.18)

In particular, by (2.1.8) and (10.17) we have at the point (x 0 , x 1 , x 2 , x 3 ): ⎧  0 0 1 2 3 2 00 0 1 2 3 ⎪ ⎨ (x , x , x , x ) = v (x , x , x , x ) − 1  jm (x 0 , x 1 , x 2 , x 3 ) = δ jm + v j (x 0 , x 1 , x 2 , x 3 )v m (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j 0 1 2 3  (x , x , x , x ) =  j0 (x 0 , x 1 , x 2 , x 3 ) = v 0 (x 0 , x 1 , x 2 , x 3 )v j (x 0 , x 1 , x 2 , x 3 ).

(10.19) Furthermore, by (10.19), (10.14) and (10.18), using Lemma A.4 from the Appendix, we deduce that the matrix mn (x 0 , x 1 , x 2 , x 3 ) is degenerate, and moreover, it has one vanishing and three positive eigenvalues at the point (x 0 , x 1 , x 2 , x 3 ). Thus, since (x 0 , x 1 , x 2 , x 3 ) ∈ R4 was chosen arbitrary, this completes the proof. Proof (Proof of Proposition 2.2) Consider {K mn }m,n=0,1,2,3 , given as in (2.1.17) by K mn = v j v m −  jm ∀ j, m = 0, 1, 2, 3.

(10.20)

Next, fix a constant point (y 0 , y 1 , y 2 , y 3 ) ∈ R4 . Then, since mn (y 0 , y 1 , y 2 , y 3 ) has one vanishing and three positive eigenvalues, by Sylvester’s law of inertia there exists a coordinate system, that we now fix, such that at the fixed point (y 0 , y 1 , y 2 , y 3 ) we have ⎧ 00 0 1 2 3 ⎪ ⎨ (y , y , y , y ) = 0 jm 0 1 2 (10.21)  (y , y , y , y 3 ) = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j 0 1 2 3 j0 0 1 2 3  (y , y , y , y ) =  (y , y , y , y ) = 0 ∀1 ≤ j ≤ 3. Moreover, by (2.1.15) and by (10.21) in the particular point (y 0 , y 1 , y 2 , y 3 ) we must have (10.22) w j (y 0 , y 1 , y 2 , y 3 ) = 0 ∀ j = 1, 2, 3.

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

135

Then by (10.22) and (2.1.16) we deduce v 0 (y 0 , y 1 , y 2 , y 3 )w0 (y 0 , y 1 , y 2 , y 3 ) > 0,

(10.23)

and in particular, at the point (y 0 , y 1 , y 2 , y 3 ) we must have v 0 (y 0 , y 1 , y 2 , y 3 ) = 0.

(10.24)

Thus, by (10.20), (10.21) and (10.24), using Lemma A.5 from the Appendix we deduce that at the point (y 0 , y 1 , y 2 , y 3 ) we have det



K mn (y 0 , y 1 , y 2 , y 3 )



 m,n=0,1,2,3

= 0,

(10.25)

and moreover, the matrix {K mn (y 0 , y 1 , y 2 , y 3 )}0≤m,n≤3 necessary has one positive and three negative eigenvalues at the point (y 0 , y 1 , y 2 , y 3 ). Thus, since a point (y 0 , y 1 , y 2 , y 3 ) ∈ R4 was chosen arbitrary, we deduce that the tensor field {K mn }m,n=0,1,2,3 , given by (10.20) indeed forms a contravariant pseudo-metric and a valid generalized-Lorentz’s structure on R4 . On the other hand, by (2.1.15) and (10.20), at every point (x 0 , x 1 , x 2 , x 3 ) ∈ R4 we have 3 

⎛ K

jm

wj = v ⎝

j=0

m

3 

⎞ v wj⎠ − j

j=0

3 

⎛ 

jm

wj = v ⎝

j=0

m

3 

⎞ v w j ⎠ ∀ m = 0, 1, 2, 3. j

j=0

(10.26) Thus, by (10.26) we also infer ⎛ ⎛ ⎞ ⎞2 3 3 3    ⎝ K jm w j ⎠ wm = ⎝ v j w j ⎠ > 0 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , (10.27) m=0

j=0

j=0

so, by (10.27) and (2.1.16) we get 3 

⎞ 21 ⎛ 3  3  v jwj = ⎝ K jm w j wk ⎠ > 0 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

j=0

(10.28)

j=0 k=0

and moreover, by the Definition 2.3, pseudo-metric K jm and the time direction (w0 , w1 , w2 , w3 ) are weakly correlated. Finally, by (10.28) and (10.26) we deduce ⎛ vm = ⎝

3  3 

⎞− 1 ⎛ 2

K jm w j wk ⎠

j=0 k=0

⎝

3 

⎞ K jm w j ⎠ ∀ m = 0, 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

j=0

(10.29) This completes the proof.

136

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

Proof (Proof of Theorem 2.1) Let ⎧ 0 x ⎪ ⎪ ⎪ ⎨x 1 ⎪ x 2 ⎪ ⎪ ⎩ 3 x

= = = =

f (0) (x 0 , x 1 , x 2 , x 3 ), f (1) (x 0 , x 1 , x 2 , x 3 ), f (2) (x 0 , x 1 , x 2 , x 3 ), f (3) (x 0 , x 1 , x 2 , x 3 ).

(10.30)

be the corresponding change of coordinates and assume that in the first coordinate system we have ⎧ 00 0 1 2 3 4 ⎪ ⎨ = 0 ∀ (x , x , x , x ) ∈ R 0j j0  =  = 0 ∀ j = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩ mn  := δmn ∀ m, n = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 and 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

(10.31)

 (x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . (10.32)

However, by (10.4) we have  ∂ f (k) ∂ϕ ∂ϕ = ∀ j = 0, 1, 2, 3, ∂x j ∂ x j ∂ x k k=0 3

(10.33)

and by (10.6) we have mn



=

3 3   ∂ f (m) ∂ f (n) j=0 k=0

∂xk ∂x j

k j ∀ m, n = 0, 1, 2, 3.

(10.34)

Thus, by (10.32) and (10.33) we have δ0 j =

3  ∂ f (k) ∂ϕ ∀ j = 0, 1, 2, 3. ∂ x j ∂ x k k=0

(10.35)

Moreover, by (10.31) and (10.34) we deduce mn =

3 3   ∂ f (m) ∂ f (n) j=1 k=1

∂xk ∂x j

δk j =

3  ∂ f (m) ∂ f (n) j=1

∂x j ∂x j

∀ m, n = 0, 1, 2, 3. (10.36)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

137

In particular, by (10.36) we deduce that the following identity ⎧ 00  0 1 2 3 4 ⎪ ⎨ = 0 ∀(x , x , x , x ) ∈ R 0 j  j0  =  = 0 ∀ j = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩  mn  = δmn ∀ m, n = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.37)

is equivalent to identity: ⎧   3  ⎪  ∂ f (0) 2 ⎪ ⎪ =0  ⎪ ∂xk  ⎪ ⎪ k=1 ⎪ ⎨ 3 ∂ f (0) ∂ f ( j) = 0 ∀ j = 1, 2, 3 ∂xk ∂xk ⎪ k=1 ⎪ ⎪ 3 ⎪ ⎪ ∂ f (m) ∂ f (n) ⎪ ⎪ = δmn ∀ m, n = 1, 2, 3. ⎩ ∂x j ∂x j

(10.38)

j=1

In other words, (10.37) is equivalent to the following:  ⎧ (0) ∂f ∂ f (0) ∂ f (0) ⎪ ⎨ ∂x1 , ∂x2 , ∂x3 = 0 3 3   ∂ f ( j) ∂ f ( j) ∂ f (m) ∂ f (n) ⎪ = = δmn ∀ m, n = 1, 2, 3. ⎩ m n ∂x ∂x ∂x j ∂x j j=1

(10.39)

j=1

However, using Corollary .1 from the Appendix, we deduce that (10.39) is equivalent to the following: ⎧ 0 0 ⎪ ⎨x = ζ (x ), 3  m ⎪ Am j (x 0 ) x j + z m (x 0 ) ∀m = 1, 2, 3, ⎩x =

(10.40)

j=1

where {Amn (x 0 )}n,m=1,2,3 ∈ R3×3 is a 3 × 3-matrix, depending on the coordinate x 0 only (independent on x := (x 1 , x 2 , x 3 )), and satisfying 3 

Am j (x 0 )An j (x 0 ) =

j=1

3 

A jm (x 0 )A jn (x 0 ) = δmn ∀ m, n = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

j=1

(10.41) 0 1 2 on the coordinate x only (independent on x := (x , x , x 3 )) ζ (x 0 ) ∈ R is depending  1 0 2 0 3 0  0 3 and z(x ) := z (x ), z (x ), z (x ) ∈ R is a three-dimensional vector field, depending on the coordinate x 0 only (independent on x := (x 1 , x 2 , x 3 )). So, (10.37) is equivalent to (10.40). On the other hand, by (10.35) the following equality 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂ x 0 ∂ x 1 ∂ x 2 ∂ x 3



(x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4

(10.42)

138

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

is equivalent to δ0 j =

3  ∂ f (k) k=0

∂x j

δ0k =

∂ f (0) ∀ j = 0, 1, 2, 3. ∂x j

(10.43)

In other words, (10.42) is equivalent to the following: x 0 = x 0 + c,

(10.44)

where c is a constant independent on (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . Finally, by the above, (10.37) and (10.42) together are equivalent to (10.40) and (10.44) together. In other words, (10.37) and (10.42) together are equivalent to (2.1.44). This completes the proof. Proof (Proof of Theorem 2.2) Let ⎧ 0 x ⎪ ⎪ ⎪ ⎨x 1 ⎪ x 2 ⎪ ⎪ ⎩ 3 x

= = = =

f (0) (x 0 , x 1 , x 2 , x 3 ), f (1) (x 0 , x 1 , x 2 , x 3 ), f (2) (x 0 , x 1 , x 2 , x 3 ), f (3) (x 0 , x 1 , x 2 , x 3 ).

(10.45)

be the corresponding change of coordinates and assume that in the first coordinate system we have

and 

⎧ 00 0 1 2 3 4 ⎪ ⎨ K = 1 ∀ (x , x , x , x ) ∈ R K 0 j = K j0 = 0 ∀ j = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩ jm K := −δ jm ∀ j, m = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.46)

⎧ 00 0 1 2 3 4 ⎪ ⎨ = 0 ∀ (x , x , x , x ) ∈ R 0 j =  j0 = 0 ∀ j = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩ mn  := δmn ∀ m, n = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4

(10.47)

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.48) In particular, denoting a two-times contravariant tensor {mn }m,n=0,1,2,3 , defined by mn := K mn + mn ∀ m, n = 0, 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.49)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

139

by (10.46) and (10.47) we have ⎧ 00 0 1 2 3 4 ⎪ ⎨ = 1 ∀(x , x , x , x ) ∈ R 0j j0  =  = 0 ∀ j = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩ jm  := 0 ∀ j, m = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.50)

However, by (10.6) we have mn =

3 3   ∂ f (m) ∂ f (n) j=0 k=0

∂xk ∂x j

k j ∀ m, n = 0, 1, 2, 3.

(10.51)

Thus, by (10.51) and (10.50) we deduce: mn =

∂ f (m) ∂ f (n) ∀ m, n = 0, 1, 2, 3. ∂x0 ∂x0

(10.52)

In particular, by (10.52) we deduce that the following identity ⎧ 00  0 1 2 3 4 ⎪ ⎨ = 1 ∀(x , x , x , x ) ∈ R  0 j =  j0 = 0 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩  mn  = 0 ∀ m, n = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.53)

is equivalent to identity: ⎧ (m) 2 ∂f ⎪ = 0 ∀ m = 1, 2, 3 ⎪ ∂x0 ⎪ ⎪ ⎪ (m) (n) ⎨ ∂ f ∂ f = 0 ∀ m, n = 1, 2, 3 ∂x0 ∂x0 ∂ f ( j) ∂ f (0) ⎪ ∀ j = 1, 2, 3 ⎪ 0 0 ⎪ ∂ x (0)∂x 2 ⎪ ⎪ ⎩ ∂f = 1.

(10.54)

∂x0

In other words, (10.53) is equivalent to the following: ⎧ (m) ⎨ ∂ f 0 = 0 ∀ m = 1, 2, 3 ∂ x (0) 2 ⎩ ∂f 0 = 1.

(10.55)

∂x

On the other hand, using Theorem 2.1 we deduce that ⎧ 00  0 1 2 3 4 ⎪ ⎨ = 0 ∀(x , x , x , x ) ∈ R  0 j =  j0 = 0 ∀ j = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩  mn  = δmn ∀ m, n = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.56)

140

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

and 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂ x 0 ∂ x 1 ∂ x 2 ∂ x 3



(x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4

(10.57)

together are equivalent to the following relations: ⎧ 0 0 ⎪ ⎨x = x + c, 3  m ⎪ Am j (x 0 ) x j + z m (x 0 ) ∀m = 1, 2, 3, ⎩x =

(10.58)

j=1

where {Amn (x 0 )}n,m=1,2,3 ∈ R3×3 is a 3 × 3-matrix, depending on the coordinate x 0 only (independent on x := (x 1 , x 2 , x 3 )), and satisfying 3 

Am j (x 0 )An j (x 0 ) =

j=1

3 

A jm (x 0 )A jn (x 0 ) = δmn ∀ m, n = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

j=1

(10.59)  c ∈ R is a constant (independent on (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ) and z(x 0 ) := z 1 (x 0 ),  z 2 (x 0 ), z 3 (x 0 ) ∈ R3 is a three-dimensional vector field, depending on the coordinate x 0 only (independent on x := (x 1 , x 2 , x 3 )). Therefore, inserting (10.55) into (10.58) we deduce that, (10.53), (10.56) and (10.57) together are equivalent to (10.58), where Am j and z m are constants (independent on x 0 ). So, we deduced that ⎧ 00  0 1 2 3 4 ⎪ ⎨ K = 1 ∀ (x , x , x , x ) ∈ R K  0 j = K  j0 = 0 ∀ j = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩  jm K := −δ jm ∀ j, m = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.60)

together with (10.56) and (10.57) are equivalent to the following ⎧ 0 0 0 ⎪ ⎨x = x + c , 3  m ⎪ Bm j x j + cm ∀m = 1, 2, 3, ⎩x =

(10.61)

j=1

where {Bmn }n,m=1,2,3 ∈ R3×3 is a constant (independent on (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ) matrix, satisfying 3  j=1

Bm j Bn j =

3 

B jm B jn = δmn ∀ m, n = 1, 2, 3,

(10.62)

j=1

and (c0 , c1 , c2 , c3 ) ∈ R4 is a constant (independent on (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ) vector. This completes the proof.

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

141

Proof (Proof of Theorem 2.5) First, by Proposition 2.1 we obviously obtain 3 

 jm w j = 0 ∀ m = 0, 1, 2, 3,

(10.63)

j=0

and

3 3  

J jm r j r m = 1 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.64)

j=0 m=0

Thus, by (2.6.7), using Proposition 2.2, we deduce that {K mn }m,n=0,1,2,3 , given by (2.6.8), is a symmetric non-degenerate two-times contravariant tensor field, which forms a contravariant pseudo-metric and a valid generalized-Lorentz’s structure on R4 . Moreover, by Proposition 2.2 we also have 3 

⎛ ⎞ 3  K mj w j = ⎝ v j w j ⎠ v m ∀ m = 0, 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

j=0

j=0

3  3 

⎛ K jm w j wk = ⎝

j=0 k=0

3 

(10.65)

⎞2 v jwj⎠

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.66)

j=0

and therefore, ⎛ v =⎝ m

3 3   j=0 k=0

⎞− 1 ⎛ 2

K

jm

w j wk ⎠

⎝

3 

⎞ K

jm

w j ⎠ ∀ m = 0, 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

j=0

(10.67) So, (v 0 , v 1 , v 2 , v 3 ) is a potential of generalized gravity, corresponding to K mn and (w0 , w1 , w2 , w3 ). In particular, we have   det {K mn }m,n=0,1,2,3 < 0,

(10.68)

and there exists the inverse to {K mn }m,n=0,1,2,3 covariant pseudo-metric, which satisfies  3  1 if m = n mk K K kn = ∀ m, n = 0, 1, 2, 3. (10.69) 0 if m = n k=0 Moreover, again, at every point in R4 we also have 3 3   j=0 m=0

K jm v j v m = 1.

(10.70)

142

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

Next, fix a constant point (x 0 , x 1 , x 2 , x 3 )∈R4 . Then, since {J mn }m,n=0,1,2,3 has one positive and three negative eigenvalues, by Sylvester’s law of inertia together with (10.64), using Lemma A.7 from the Appendix, we deduce that there exists a coordinate system, that we now fix, such that in this coordinate system at the fixed point (x 0 , x 1 , x 2 , x 3 ) we have ⎧ 00 0 1 2 3 ⎪ ⎨ J (x , x , x , x ) = 1 jm 0 1 2 J (x , x , x , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j 0 1 2 3 J (x , x , x , x ) = J j0 (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3,

(10.71)

and (r 0 , r 1 , r 2 , r 3 )(x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0).

(10.72)

Then, we also obviously have ⎧ 0 1 2 3 ⎪ ⎨ J00 (x , x , x , x ) = 1 J jm (x 0 , x 1 , x 2 , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ J0 j (x 0 , x 1 , x 2 , x 3 ) = J j0 (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3.

(10.73)

Moreover, by (2.6.6), (10.71) and (10.72) we deduce that at the fixed point (x 0 , x 1 , x 2 , x 3 ) we have ⎧ 00 0 1 2 3 ⎪ ⎨ (x , x , x , x ) = 0 jm 0 1 2  (x , x , x , x 3 ) = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j 0 1 2 3  (x , x , x , x ) =  j0 (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3.

(10.74)

On the other hand, by (2.6.5) and (10.72), in the same coordinate system we deduce w j (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3 and w0 (x 0 , x 1 , x 2 , x 3 ) > 0.

(10.75)

Then, by (10.75) and (2.6.7) we deduce that w0 (x 0 , x 1 , x 2 , x 3 )v 0 (x 0 , x 1 , x 2 , x 3 ) =

3 

v j (x 0 , x 1 , x 2 , x 3 )w j (x 0 , x 1 , x 2 , x 3 ) > 0.

j=0

(10.76) In particular, we have v 0 (x 0 , x 1 , x 2 , x 3 ) > 0.

(10.77)

Furthermore, using Lemma A.5 from the Appendix, we deduce that {K mn }m,n=0,1,2,3 is given at the point (x 0 , x 1 , x 2 , x 3 ), in the coordinate system, where (10.74) hold, by the following:

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

143

  ⎧ |v|2 1 0 1 2 3 ⎪ K (x 0 , x 1 , x 2 , x 3 ) (x , x , x , x ) = 00 2 − 2 ⎪ 0 0 ⎨ (v ) (v ) K jm (x 0 , x 1 , x 2 , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3  ⎪ ⎪ ⎩ K (x 0 , x 1 , x 2 , x 3 ) = K (x 0 , x 1 , x 2 , x 3 ) = v j (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ j ≤ 3, 0j j0 v0 (10.78) with v := (v 1 , v 2 , v 3 ), and moreover, by Lemma A.5 from the Appendix, we have   2  det {K mn (x 0 , x 1 , x 2 , x 3 )}m,n=0,1,2,3 = − v 0 (x 0 , x 1 , x 2 , x 3 ) < 0.

(10.79)

Thus, by (10.79) we obviously have −1 0 1 2 3  2   (x , x , x , x ) = v 0 (x 0 , x 1 , x 2 , x 3 ). − det {K mn }m,n=0,1,2,3

(10.80)

Therefore, by (10.80) and (10.77) we deduce   − 21 0 1 2 3 − det {K mn }m,n=0,1,2,3 (x , x , x , x ) = v 0 (x 0 , x 1 , x 2 , x 3 ).

(10.81)

In particular, by (10.78) and (10.72) we have 3 

K 0 j (x 0 , x 1 , x 2 , x 3 )r j (x 0 , x 1 , x 2 , x 3 )

j=0

= K 00 (x 0 , x 1 , x 2 , x 3 )r 0 (x 0 , x 1 , x 2 , x 3 ) +  =

1

   1 − |v|2 (x 0 , x 1 , x 2 , x 3 )

3 

K 0 j (x 0 , x 1 , x 2 , x 3 )r j (x 0 , x 1 , x 2 , x 3 )

j=1

(v 0 )2 3  and K m j (x 0 , x 1 , x 2 , x 3 )r j (x 0 , x 1 , x 2 , x 3 ) j=0

= K m0 (x 0 , x 1 , x 2 , x 3 )r 0 (x 0 , x 1 , x 2 , x 3 ) +  =

3 

 1 m v (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ m ≤ 3. v0

K m j (x 0 , x 1 , x 2 , x 3 )r j (x 0 , x 1 , x 2 , x 3 )

j=0

(10.82)

Next, if we consider, a covariant four-covector (S0 , S1 , S2 , S3 ), defined by: ⎛ ⎛ ⎞− 1 ⎛ ⎞ ⎞−1 ⎛ ⎞ 2 3 3 3 3 3  1 ⎝  jk 1 ⎝ j ⎠ ⎝ j⎠ j⎠ ⎠ ⎝ Sm := J w j wk Km j r v wj Jm j v − ∀ 0 ≤ m ≤ 3, 2 2 j=0 k=0

j=0

j=0

j=0

(10.83)

144

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

then by (10.82), (10.73), (10.75), (10.77) and (10.72) we deduce     1 1  0 1 2 3 2 S0 (x , x , x , x ) = 1 − |v| − 1 (x 0 , x 1 , x 2 , x 3 ) 2w0 (v 0 )2   1 0 1 2 3 m v (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ m ≤ 3. and Sm (x , x , x , x ) = w0 v 0

(10.84)

On the other hand, by (10.78) and (10.73) we have   ⎧   1 ⎪ 0 1 2 3 2 ⎪ ⎪ ⎨(K 00 − J00 ) (x , x , x , x ) = (v 0 )2 1 − |v| − 1   K − J jm (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j, m ≤ 3 ⎪ ⎪ jm  0 1 2 3  0 1 2 3  ⎪ w0 j ∀1 ≤ j ≤ 3. ⎩ K −J 0j 0 j (x , x , x , x ) = K j0 − J j0 (x , x , x , x ) = 0 v w0 v

(10.85)

Thus, by (10.84) and (10.85) we deduce ⎧ 0 1 2 3 0 1 2 3 ⎪ ⎨(K 00 − J00 ) (x , x , x , x ) = 2(w0 S0 )(x , x , x , x ) 0 1 2 3 K jm − J jm (x , x , x , x ) = 0 ∀1 ≤ j, m ≤ 3 ⎪    ⎩ K 0 j − J0 j (x 0 , x 1 , x 2 , x 3 ) = K j0 − J j0 (x 0 , x 1 , x 2 , x 3 ) = (w0 S j )(x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ j ≤ 3.

(10.86) Therefore, inserting (10.75) into (10.86) we deduce ⎧ 0 1 2 3 0 1 2 3 ⎪ 00 − J00 ) (x , x , x , x ) = 2(w0 S0 )(x , x , x , x ) ⎨(K   0 1 2 3   K jm − J jm (x , x , x , x ) = w j Sm + wm S j ∀1 ≤ j, m ≤ 3 ⎪  0 1 2 3  0 1 2 3    ⎩ K 0 j − J0 j (x , x , x , x ) = K j0 − J j0 (x , x , x , x ) = w0 S j + w j S0 ∀1 ≤ j ≤ 3.

(10.87) We can rewrite (10.87) in the equivalent form:   K jm (x 0 , x 1 , x 2 , x 3 ) = J jm (x 0 , x 1 , x 2 , x 3 ) + w j Sm + wm S j (x 0 , x 1 , x 2 , x 3 ) ∀ 0 ≤ j, m ≤ 3.

(10.88) On the other hand, since by (10.71) and (10.75) we have ⎞ ⎛  3 3   ⎝ J mn wm wn ⎠ (x 0 , x 1 , x 2 , x 3 ) = w0 (x 0 , x 1 , x 2 , x 3 ),

(10.89)

m=0 n=0

by (10.73), (10.89), (10.76) and (10.81) we deduce: ⎛  ⎞  3 3      mn ⎜ − det {Jmn }m,n=0,1,2,3 J wm wn ⎟ ⎜ ⎟ 0 1 2 3 m=0 n=0 ⎜ ⎟ (x , x , x , x )    ⎜ ⎟ − det {K mn }m,n=0,1,2,3 ⎝ ⎠ ⎛ ⎞ 3  =⎝ v j w j ⎠ (x 0 , x 1 , x 2 , x 3 ) > 0. j=0

(10.90)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

145

Next, since (x 0 , x 1 , x 2 , x 3 ) ∈ R4 in (10.88) and (10.90) was chosen arbitrary, and moreover since (10.88) and (10.90) are independent on the coordinate system, we deduce that the inverse to {K mn }m,n=0,1,2,3 covariant pseudo-metric is given at every point in R4 by the following   K jm = J jm + w j Sm + wm S j ∀ 0 ≤ j, m ≤ 3,

(10.91)

where the covariant four-covector (S0 , S1 , S2 , S3 ) is defined by (10.83) and, moreover, we have ⎛  ⎞  3 3      mn ⎛ ⎞ ⎜ J wm wn ⎟ − det {Jmn }m,n=0,1,2,3 3  ⎜ ⎟ m=0 n=0 ⎜ ⎟=⎝    v j w j ⎠ > 0. ⎜ ⎟ } − det {K mn m,n=0,1,2,3 ⎝ ⎠ j=0 (10.92) Finally, by (10.66) and (10.92) we deduce   3 3      mn J wm wn − det {Jmn }m,n=0,1,2,3 3  3  m=0 n=0    K jm w j wm = . (10.93) − det {K mn }m,n=0,1,2,3 m=0 j=0 Proof (Proof of Theorem 2.6) As before, we have ⎛ r m := ⎝

3 3  

⎞− 21 ⎛ ⎞ 3  J jk w j wk ⎠ ⎝ J m j w j ⎠ ∀ m = 0, 1, 2, 3,

j=0 k=0

(10.94)

j=0

and  jm := r j r m − J jm ∀ j, m = 0, 1, 2, 3, so that

3  3 

J jm r j r m = 1 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.95)

(10.96)

j=0 m=0

Next, consider, an arbitrary covariant four-covector (S0 , S1 , S2 , S3 ) such that a twotimes covariant tensor field {K mn }m,n=0,1,2,3 , defined by (2.6.14) satisfies (2.6.15) at every point in R4 . Then, by (2.6.15) there exists the inverse to {K mn }m,n=0,1,2,3 two-times contravariant symmetric tensor field {K mn }m,n=0,1,2,3 , which satisfies 3  k=0

K

mk

K kn

 1 if m = n = 0 if m = n

∀ m, n = 0, 1, 2, 3.

(10.97)

146

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

Next fix a constant point (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . Then, by (10.96), since {J mn }m,n=0,1,2,3 has one positive and three negative eigenvalues, by Sylvester’s law of inertia together with (10.96), using Lemma A.7 from the Appendix, we deduce that there exists a coordinate system, that we now fix, such that in this coordinate system at the fixed point (x 0 , x 1 , x 2 , x 3 ) we have ⎧ 00 0 1 2 3 ⎪ ⎨ J (x , x , x , x ) = 1 jm 0 1 2 J (x , x , x , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j 0 1 2 3 J (x , x , x , x ) = J j0 (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3,

(10.98)

and (r 0 , r 1 , r 2 , r 3 )(x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0).

(10.99)

Then, we also obviously have ⎧ 0 1 2 3 ⎪ ⎨ J00 (x , x , x , x ) = 1 J jm (x 0 , x 1 , x 2 , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ J0 j (x 0 , x 1 , x 2 , x 3 ) = J j0 (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3.

(10.100)

Moreover, by (10.95), (10.98) and (10.99) we deduce that at the fixed point (x 0 , x 1 , x 2 , x 3 ) we have ⎧ 00 0 1 2 3 ⎪ ⎨ (x , x , x , x ) = 0  jm (x 0 , x 1 , x 2 , x 3 ) = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j 0 1 2 3  (x , x , x , x ) =  j0 (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3.

(10.101)

Thus, by (10.94) and (10.99), in the same coordinate system we deduce w j (x 0 , x 1 , x 2 , x 3 ) = 0 ∀1 ≤ j ≤ 3

and

w0 (x 0 , x 1 , x 2 , x 3 ) > 0, (10.102)

and by (10.102) and (10.98) we also have ⎞ ⎛  3 3   ⎝ J mn wm wn ⎠ (x 0 , x 1 , x 2 , x 3 ) = w0 (x 0 , x 1 , x 2 , x 3 ).

(10.103)

m=0 n=0

Next by (10.102), (10.100) and (2.6.14) we deduce that {K mn (x 0 , x 1 , x 2 , x 3 )}m,n=0,1,2,3 is given at the point (x 0 , x 1 , x 2 , x 3 ), in the coordinate system, where (10.100) and (10.99) holds, by the following:

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

147

⎧ 0 1 2 3 0 1 2 3 ⎪ ⎨ K 00 (x , x , x , x ) = 1 + (2w0 S0 ) (x , x , x , x ) 0 1 2 3 K jm (x , x , x , x ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K (x 0 , x 1 , x 2 , x 3 ) = K (x 0 , x 1 , x 2 , x 3 ) = w S  (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ j ≤ 3. 0j j0 0 j

(10.104) We rewrite (10.104) as ⎧   2 2 0 1 2 3  0 1 2 3 2 0 1 2 3 2 ⎪ ⎨ K 00 (x , x , x , x ) = 1 + 2w0 S0 + w0 |S| (x , x , x , x ) − w0 |S| (x , x , x , x ) K jm (x 0 , x 1 , x 2 , x 3 ) = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ K (x 0 , x 1 , x 2 , x 3 ) = K (x 0 , x 1 , x 2 , x 3 ) = w S  (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ j ≤ 3, 0j j0 0 j

(10.105) where we denote S := (S1 , S2 , S3 ). Thus, by (10.105) for every (h 0 , h 1 , h 2 , h 3 ) ∈ R4 , with h := (h 1 , h 2 , h 3 ) ∈ R3 , we have 3 3  

K jm (x 0 , x 1 , x 2 , x 3 )h j h m =

j=0 m=0



 2     1 + 2w0 S0 + w02 |S|2 (x 0 , x 1 , x 2 , x 3 )(h 0 )2 − h − h 0 w0 S (x 0 , x 1 , x 2 , x 3 ). (10.106)

  Thus, if we assume that 1 + 2w0 S0 + w02 |S|2 (x 0 , x 1 , x 2 , x 3 ) < 0, then by (10.106) we deduce that the matrix {−K mn (x 0 , x 1 , x 2 , x 3 )}m,n=0,1,2,3 is positively defined, which contradicts to (2.6.15). On the other hand, if we assume that (1 + 2w0 S0 +  w02 |S|2 (x 0 , x 1 , x 2 , x 3 ) = 0 then, by (10.106) we deduce that (h 0 , h 1 , h 2 , h 3 ) ∈ R4 , defined by  h0 = 1 (10.107) h j = w0 S j ∀1 ≤ j ≤ 3, is an eigenvector of the matrix {K mn (x 0 , x 1 , x 2 , x 3 )}m,n=0,1,2,3 , corresponding to the vanishing eigenvalue, and therefore, the matrix {K mn (x 0 , x 1 , x 2 , x 3 )}m,n=0,1,2,3 is degenerate, which also contradicts to (2.6.15). So, (2.6.15) necessary implies   1 + 2w0 S0 + w02 |S|2 (x 0 , x 1 , x 2 , x 3 ) > 0.

(10.108)

Therefore, by (10.108), using Lemma A.5 from the Appendix, we deduce that the inverse to {K mn (x 0 , x 1 , x 2 , x 3 )}m,n=0,1,2,3 matrix {K mn (x 0 , x 1 , x 2 , x 3 )}m,n=0,1,2,3 is given by the following ⎧ 00 0 1 2 3 K (x , x , x , x ) = 1+2w S1+w2 |S|2 (x 0 , x 1 , x 2 , x 3 ) ⎪ ⎪ ⎪   0 0 0 ⎪ ⎨ jm 0 1 2 3 w2 S S K (x , x , x , x ) = 1+2w 0S j+wm 2 |S|2 (x 0 , x 1 , x 2 , x 3 ) − δ jm ∀1 ≤ j, m ≤ 3 0 0 0 ⎪   ⎪ ⎪ w0 S j ⎪ ⎩ K 0 j (x 0 , x 1 , x 2 , x 3 ) = K j0 (x 0 , x 1 , x 2 , x 3 ) = (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ j ≤ 3, 1+2w S +w2 |S|2 0 0

0

(10.109)

148

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

and moreover, we have −1    − det K mn m,n=0,1,2,3      = − det {K mn }m,n=0,1,2,3 = 1 + 2w0 S0 + w02 |S|2 (x 0 , x 1 , x 2 , x 3 ) > 0. (10.110) Then, by (10.110) and (10.109) we infer ⎧ 1 00 0 1 2 3 0 1 2 3 ⎪ ⎪ ⎪ K (x , x , x , x ) = (− det ({K mn }m,n=0,1,2,3 )) (x , x , x , x ) ⎪ ⎨ 2 w S S K jm (x 0 , x 1 , x 2 , x 3 ) = − det {K 0 }j m (x 0 , x 1 , x 2 , x 3 ) − δ jm ∀1 ≤ j, m ≤ 3 ( ( mn m,n=0,1,2,3 )) ⎪ ⎪  ⎪ w0 S j ⎪ ⎩ K 0 j (x 0 , x 1 , x 2 , x 3 ) = K j0 (x 0 , x 1 , x 2 , x 3 ) = (x 0 , x 1 , x 2 , x 3 ) ∀1 ≤ j ≤ 3. (− det ({K mn }m,n=0,1,2,3 ))

(10.111) Thus, by (10.101), (10.99), (10.103) and (10.100) using (10.111) we deduce K mn (x 0 , x 1 , x 2 , x 3 ) = v m (x 0 , x 1 , x 2 , x 3 )v n (x 0 , x 1 , x 2 , x 3 ) − mn (x 0 , x 1 , x 2 , x 3 ) ∀ m, n = 0, 1, 2, 3, (10.112) where the contravariant vector field (v 0 , v 1 , v 2 , v 3 ) is given by  v m :=

⎞⎛ ⎞ ⎞   ⎛⎛   3 3 3    − det {Jkn }k,n=0,1,2,3  ⎝⎝  J jk w j wk ⎠ ⎝ m j S j ⎠ + r m ⎠ − det {K kn }k,n=0,1,2,3 j=0 k=0 j=0 ∀ m = 0, 1, 2, 3.

(10.113) Moreover, by (10.113), (10.99), (10.101) and (10.102) we obtain ⎛ ⎝

3 

⎞





v w j ⎠ (x , x , x , x ) = w0 j

0

1

2

3

j=0

  − det {Jmn }m,n=0,1,2,3   (x 0 , x 1 , x 2 , x 3 ) > 0. − det {K mn }m,n=0,1,2,3

(10.114) Next, since (x 0 , x 1 , x 2 , x 3 ) ∈ R4 in (10.112) and (10.114) was chosen arbitrary, and moreover since (10.112) and the left-hand side of (10.114) are independent on the coordinate system, we deduce that at every point we have K mn = v m v n − mn

∀ m, n = 0, 1, 2, 3.

(10.115)

where the contravariant vector field (v 0 , v 1 , v 2 , v 3 ) is given by (10.113) and moreover, we have ⎞ ⎛ 3  ⎝ v j w j ⎠ > 0. (10.116) j=0

Thus we can apply Theorem 2.5 to complete the proof.

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

149

Proof (Proof of Proposition 2.3) Differentiating (2.1.58) in Definition 2.13 gives, ∂ x m = ∂xn

⎛

Bmn ⎝

3 

k=0

⎞

⎛

⎞ ⎞⎞ ⎛ ⎛ 3 3 3 m    dB dz m j ⎝ ⎝ wk x k ⎠ + ⎝ wk x k ⎠ x j + wk x k ⎠⎠ wn ∀ m, n = 0, 1, 2, 3. ds ds j=0

k=0

k=0

(10.117) However, differentiating the identity B −1 (s) · B(s) = I d4×4 ∀s ∈ R, gives

(10.118)

dB dB −1 (s) · B(s) + B −1 (s) · (s) = 0 ∀s ∈ R, ds ds

(10.119)

dB dB −1 (s) = −B −1 (s) · (s) · B −1 (s) ∀s ∈ R. ds ds

(10.120)

that implies,

Therefore, by (2.1.60) in Definition 2.13 together with (10.120) we deduce: ⎞ ⎛     3 3 −1 !  dB d ⎝ −1 B wj (s) ⎠ = wj (s) jm ds ds jm j=0 j=0   ! 3  dB =− wj (s) (s) · B −1 (s) B −1 (s) · ds jm j=0   ! 3  3    dB B −1 B −1 · =− wj (s) (s) = 0 km ds jk k=0 j=0

∀ m = 0, 1, 2, 3, ∀s ∈ R.

(10.121)

In particular, 3  j=0

wj



B −1



3      −1 B (s) = w (0) ∀ m = 0, 1, 2, 3, ∀s ∈ R. j jm jm j=0

(10.122) Then, using (10.117) and (10.122), together with (2.1.60) and (2.1.61) in Definition 2.13 we deduce:

150

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas 3 

 wm

m=0

3  3  3 3   ∂ x m    ∂ x m = w j B −1 jm (0) = wj ∂xn ∂xn m=0 j=0

3  3 

=

wj

m=0 j=0

+

3  3 

 

 wj

B −1

 3 



wk x k

jm k=0



B −1

wk x k

jm

m=0 j=0

Bmn

 3 





B −1

 3 

 jm

 wk x k

k=0

∂ x m ∂xn

wk x k

k=0

 3 



m=0 j=0





  3    3  3  dBmr  dz m  wk x k x r + wk x k wn ds ds r =0

k=0

k=0

k=0

(10.123)

= wn ∀ n = 0, 1, 2, 3.

Moreover, by (2.1.57) and (2.1.59) in Definition 2.13 we deduce 3 3  

M

mn

 wm wn

=

m=0 n=0

M

mn

⎝

m=0 n=0

3 3  

=

⎛

3 3  

3 

wj

 B

j=0

−1

jm

⎞   3  (0) ⎠ wk B −1 k=0

M jk w j wk = 1.

kn

(0)





(10.124)

j=0 k=0

This completes the proof. Proof (Proof of Proposition 2.4) By Definition 2.13 we have, x m :=

3 

Bm j

 3 

j=0

 wk x k

 x j + zm

k=0

3 

 wk x k

∀ m = 0, 1, 2, 3,

(10.125)

k=0

    where B(s) := Bm j (s) m, j=0,1,2,3 : R → R4×4 and z 0 (s), z 1 (s), z 2 (s), z 3 (s) : R → R4 satisfy M

mn

=

3  3 

Bm j (s)Bnk (s)M jk ∀ m, n = 0, 1, 2, 3, ∀s ∈ R,

(10.126)

j=0 k=0 3 

 wj

B

−1

j=0

and

3  3 

dB · ds

wj



!

(s) = 0 ∀ m = 0, 1, 2, 3, ∀s ∈ R,

(10.127)

jm



B −1

m=0 j=0

 jm

(s)

 dz m ds

(s) = 0 ∀s ∈ R.

(10.128)

However, by (2.1.72) in Proposition 2.3 there exists a constant C ∈ R such that we have     3 3  k k =C+ (10.129) wk x wk x . k=0

k=0

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

151

Thus, by (10.125) and (10.129) we deduce x m =

3 

Bm j



 3 

j=0

wk x k − C

x j + zm

 3 

k=0

 wk x k − C

∀ m = 0, 1, 2, 3,

k=0

(10.130) that we rewrite as: x

m

3  

=

B

−1



mj

j=0

3 

 wk x k

j

−C x −

k=0

3  

B

−1

 3 

mj

j=0

 wk x k

−C z

j

 3 

k=0

−C

k=0

∀ m = 0, . . . , 3.

(10.131) 

Thus, denoting B  (s) := Bm j (s)  z 3 (s) : R → R4 , given by 

 wk x k

m, j=0,1,2,3

 : R → R4×4 and z 0 (s), z 1 (s), z 2 (s),

B  (s) = B −1 (s − C)    z m (s) = − 3j=0 B −1 m j (s − C) z j (s − C) ∀ m = 0, . . . , 3,

(10.132)

by (10.131) we infer x = m

3 

Bm j

 3 

j=0

 wk x k

j

x +z

m

k=0

 3 

 wk x k

∀ m = 0, . . . , 3.

(10.133)

k=0

However, by (10.126) and (10.132) we deduce M mn =

3  3 

 Bm j (s)Bnk (s)M jk ∀ m, n = 0, 1, 2, 3, ∀s ∈ R,

(10.134)

j=0 k=0

Moreover, by (10.132) and (2.1.70) in Proposition 2.3 we deduce 3 

 w j

B −1 ·

j=0

dB  ds



!

(s) = jm

3  3 

wj

m=0 j=0

d  −1 B (s − C) ∀ m = 0, 1, 2, 3, ∀s ∈ R, jm ds

(10.135) and 3 3  

wj

m=0 j=0

=

3 3   m=0 j=0



B −1



jm

(s)

dz m (s) ds

⎞ 3 d ⎝   −1 B wm − (s − C) z k (s − C)⎠ mk ds ⎛

k=0

152

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

⎛

⎞ 3    d −1 k B = wm ⎝− (s − C) z (s − C)⎠ mk ds m=0 j=0 k=0 ⎛ ⎞ 3  3  3 k   dz −1 B wm ⎝− + (s − C) (s − C)⎠ ∀s ∈ R. (10.136) mk ds 3 3  

m=0 j=0

k=0

On the other hand, as in (10.120) we have: dB −1 dB (s − C) = −B −1 (s − C) · (s − C) · B −1 (s − C) ∀s ∈ R. ds ds

(10.137)

Thus, by (10.135) and (10.137) together, using (10.127) we deduce 3 

 w j

B

−1

j=0

dB  · ds



!

(s) = 0 ∀ m = 0, 1, 2, 3, ∀s ∈ R.

(10.138)

jm

Finally, by (10.136) and (10.137) together, using (10.127) and (10.128) we deduce 3  3 

w j



B −1

 jm

(s)

m=0 j=0

 dz m ds

(s) = 0 ∀s ∈ R.

(10.139)

This completes the proof. Lemma 10.1 Given a constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying (w0 )2 −

3 

(w j )2 = 1,

(10.140)

j=1

assume that that the change of coordinate system (x 0 , x 1 , x 2 , x 3 )→(x 0 , x 1 , x 2 , x 3 ) belongs to the class P L(w0 , w1 , w2 , w3 ). Next, as before, let (w 0 , w1 , w2 , w3 ) ∈ R4 be defined as wm

=

3 

wj



B −1

 jm

(0)



∀ m = 0, 1, 2, 3, ∀s ∈ R,

(10.141)

j=0

  where B(s) := Bm j (s) m, j=0,1,2,3 : R → R4×4 be as in (2.1.58). Furthermore, consider another change of coordinate system (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), belonging to the class P L(w0 , w1 , w2 , w3 ). Then, the composition of the above two changes of coordinate systems: (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) = (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 )

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

153

also belongs to the class P L(w0 , w1 , w2 , w3 ). Proof (Proof of Lemma 10.1) As before, assume that (w0 , w1 , w2, w3 ) ∈ R4 ,  B(s) := Bm j (s) m, j=0,1,2,3 : R → R4×4 and z 0 (s), z 1 (s), z 2 (s), z 3 (s) : R → R4 satisfy 3 3   M jm w j wm = 1, (10.142) j=0 m=0

M mn =

3 3  

Bm j (s)Bnk (s)M jk ∀ m, n = 0, 1, 2, 3, ∀s ∈ R,

(10.143)

j=0 k=0 3 

 wj

j=0

and

dB B −1 · ds

3  3 

wj



!

(s) = 0 ∀ m = 0, 1, 2, 3, ∀s ∈ R,

(10.144)

jm



B −1



(s) jm

 dz m

m=0 j=0

ds

(s) = 0 ∀s ∈ R.

(10.145)

Then, consider x m =

3 

 3 

Bm j

j=0

 wk x k

x j + zm

k=0

 3 

 wk x k

∀ m = 0, 1, 2, 3.

(10.146)

k=0

Furthermore, let (w0 , w1 , w2 , w3 ) ∈ R4 be defined as wm =

3 

wj



B −1



(0) jm



∀ m = 0, 1, 2, 3, ∀s ∈ R,

(10.147)

j=0

  : R → R4×4 and z 0 (s), z 1 (s), z 2 (s), and assume that B  (s) := Bm j (s) m, j=0,1,2,3  z 3 (s) : R → R4 satisfy M mn =

3 3  

 Bm j (s)Bnk (s)M jk ∀ m, n = 0, 1, 2, 3, ∀s ∈ R,

(10.148)

j=0 k=0 3  j=0

 w j

B

−1

dB  · ds



!

(s) = 0 ∀ m = 0, 1, 2, 3, ∀s ∈ R, jm

(10.149)

154

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

and

3  3 

w j



B −1



(s) jm

 dz m ds

m=0 j=0

(s) = 0 ∀s ∈ R.

(10.150)

Then, consider x m =

3 

 3 

Bm j

j=0

 wk x k

x  j + z m

 3 

k=0

 wk x k

∀ m = 0, 1, 2, 3. (10.151)

k=0

However by (2.1.72) in Proposition 2.3 we have  3

wk x k

 =C+

 3

k=0

 wk x . k

(10.152)

k=0

In particular, (2.1.72) in Proposition 2.3 together with (10.146) in the case x j = 0, with the help of (10.147), give us the following: 3  3  

B −1



m=0 r =0

rm

 (0) wr z m (0) = C.

(10.153)

Therefore, by (10.153) we rewrite (10.152) as: 3 

wm x m =

m=0

3 

wjx j +

3  3  

B −1

m=0 r =0

j=0

 rm

 (0) wr z m (0) .

(10.154)

Thus, by (10.151), (10.146) and (10.154) we have x

m

=

3 

 3 

Bm j

j=0

 wk x

k

x +z j

k=0

m, j=0,1,2,3

3 

 3 

 wk x

k

∀ m = 0, 1, 2, 3, (10.155)

k=0

 with B  (s) := Bm j (s) Bm j (s) :=

m

: R → R4×4 , given by

 Bmq (τs ) Bq j (s) ∀ m, j = 0, 1, 2, 3 ∀ s ∈ R,

(10.156)

Bm j (τs ) z j (s) + z m (τs ) ∀ m = 0, 1, 2, 3 ∀ s ∈ R,

(10.157)

q=0

and with z m (s) :=

3  j=0

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

155

where we denote τs := s +

3 3   

B −1

d=0 r =0

 rd

 (0) wr z d (0) .

(10.158)

In particular, by (10.143), (10.148) and (10.156) we deduce M

3 3  

=

mn

 Bm j (s)Bnk (s)M jk ∀ m, n = 0, 1, 2, 3, ∀s ∈ R.

(10.159)

j=0 k=0

Moreover, by (10.156) we have 3 

 wj

B

−1

j=0

+

3 

(s) = jm

 wj

B



B −1 (s) · B −1 (τs ) · B  (τs ) ·

wj

3 

3 

 B −1 (s) · B −1 (τs ) ·

wj

−1

(s) · B

−1

j=0



j=0

=



!

dB  · ds



j=0

dB (s) ds

(τs ) ·

dB  (τs ) · B(s) ds



! jm



!

dB  (τs ) · B(s) ds



jm



!

+ jm

3 

 wj

B −1 (s) ·

j=0

∀ m = 0, 1, 2, 3, ∀s ∈ R.

dB (s) ds



! jm

(10.160)

Thus, by (2.1.68) and (2.1.70) in Proposition 2.3 and (10.144), (10.149), we rewrite (10.160) as: 3 

 wj

dB  B −1 · ds

j=0

=

3 

 wj

=

j=0

 w j

(s) jm

 !  !   3 dB  dB −1 B (s) · (s) wj + (τs ) · B(s) ds ds jm jm j=0    ! 3  dB  B −1 (0) · B −1 (τs ) · wj = (τs ) · B(s) ds jm

B −1 (s) · B −1 (τs ) ·

j=0

3 



!

j=0

B −1 (τs ) ·





!  dB  = 0 ∀ m = 0, 1, 2, 3, ∀s ∈ R. (τs ) · B(s) ds jm

(10.161)

Similarly, by (2.1.68) and (2.1.70) in Proposition 2.3 together with (10.145), (10.150), (10.156), (10.149) and (10.157), using (10.158) we obtain:

156

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas 3  3 

wj



B −1

m=0 j=0



jm

(s)

dz m (s) ds

 3  d   k m B (s) · B (τs ) wj Bmk (τs ) z (s) + z (τs ) = jm ds m=0 j=0 k=0  3   3  3     dBmk −1 −1 k B (s) · B (τs ) wj = (τs ) z (s) jm ds m=0 j=0 k=0  3   3  3    dz k −1 −1  wj Bmk (τs ) B (s) · B (τs ) + (s) jm ds 

3  3 







B −1 (s) · B −1 (τs )







−1

−1

m=0 j=0

3  3 

k=0

dz m (τs ) jm ds m=0 j=0   !  3 3 3  3   dz k   dB  B −1 (s) = z k (s) + w j wj B −1 (τs ) · (τs ) (s) jk ds ds jk

+

wj

j=0 k=0

+

3 

3 

k=0 j=0

w j

B −1 (τs )

m=0 j=0

 jm

dz m ds

(τs ) = 0 ∀s ∈ R.

(10.162)

This completes the proof. Lemma 10.2 For every constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying 3  3 

M jm w j wm = 1,

(10.163)

j=0 m=0

    the classes L (w0 , w1 , w2 , w3 ), (1, 0, 0, 0) and L (1, 0, 0, 0), (w0 , w1 , w2 , w3 )  are not empty. Moreover, for every transformation in L (w0 , w1 , w2 , w3 ), (1, 0,    0, 0) , the inverse transformation belongs to L (1, 0, 0, 0), (w0 , w1 , w2 , w3 ) . Proof (Proof of Lemma 10.2) Follows by Lemma A.7. Corollary 10.1 For every given two fixed constant vectors (w0 , w1 , w2 , w3 ) ∈ R4 , (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying   3 3 M jm w j wm = 1 3j=0 m=0 3 jm   w j wm = 1, j=0 m=0 M

(10.164)

  the class L (w0 , w1 , w2 , w3 ), (w0 , w1 , w2 , w3 ) is not empty. Moreover, for every   transformation in L (w0 , w1 , w2 , w3 ), (w0 , w1 , w2 , w3 ) , the inverse transforma  tion belongs to L (w0 , w1 , w2 , w3 ), (w0 , w1 , w2 , w3 ) .

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

157

Lemma 10.3 Assume that the new coordinate system is obtained from some old Pseudo-Lorentzian system by a Lorentz’s transformation. Then the new coordinate system is also a Pseudo-Lorentzian system. Proof ((Proof of Lemma 10.3) Consider a ancillary contravariant pseudo-metric {L mn }m,n=0,1,2,3 , given in the old coordinate system by ⎧ 00 ⎪ ⎨L = 1 L 0 j = L j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm L := −δ jm ∀ j, m = 1, 2, 3.

(10.165)

Note that in general this pseudo-metric can differ from pseudo-metric {K mn }m,n=0,1,2,3 . Then, since the old system is Pseudo-Lorentzian, there exists a covector, which in this system is equal to some constant (independent on the point (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ) vector (w0 , w1 , w2 , w3 ) ∈ R4 satisfying 3  3 

L m j wm w j = 1,

(10.166)

m=0 j=0

and such that in the old system we have mn = −L mn +

 3

L mk wk

  3

k=0

 L nk wk

∀ 0 ≤ m, n ≤ 3,

(10.167)

k=0

and at the same time in the old coordinate system we have 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x 0 , x 1 , x 2 , x 3 ) = (w0 , w1 , w2 , w3 ) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.168) On the other hand, since the new system is obtained from the old one by some Lorentz’s transformation, that are also necessarily linear, in the new coordinate system we also have ⎧ 00 ⎪ ⎨L = 1 (10.169) L 0 j = L  j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩  jm := −δ jm ∀ j, m = 1, 2, 3, L and moreover, the covector (w0 , w1 , w2 , w3 ) ∈ R4 also in the new system is equal to some constant vector in R4 (independent on the point (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ). Finally, by covariance of (10.166), (10.167) and (10.168), in the new system obviously we have 3  3  L m j wm w j = 1, (10.170) m=0 j=0

158

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

mn = −L mn +

 3

L mk wk

  3

k=0

and 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂ x 0 ∂ x 1 ∂ x 2 ∂ x 3



L nk wk

 ∀ 0 ≤ m, n ≤ 3,

(10.171)

k=0

(x 0 , x 1 , x 2 , x 3 ) = (w0 , w1 , w2 , w3 ) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.172) Thus, using (10.169), we deduce by (10.170), (10.171) and (10.172) that the new system is also Pseudo-Lorentzian.  Proof (Proof of Theorem 2.3) Consider a Pseudo-Lorentzian coordinate system, so that ⎧ 00 2 0 1 2 3 4 ⎪ ⎨ = (w0 ) − 1 ∀ (x , x , x , x ) ∈ R 0j j0 (10.173)  =  = −w0 w j ∀ j = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩ mn 0 1 2 3 4  = δmn + wm wn ∀ m, n = 1, 2, 3 ∀ (x , x , x , x ) ∈ R , and at the same time in the same coordinate system we have 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x 0 , x 1 , x 2 , x 3 ) = (w0 , w1 , w2 , w3 ) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.174) where (w0 , w1 , w2 , w3 ) ∈ R4 is some constant (independent on the point (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ) vector, satisfying (w0 )2 −

3 

(w j )2 = 1.

(10.175)

j=1

Next again let {L mn }m,n=0,1,2,3 be an ancillary contravariant pseudo-metric, satisfying ⎧ 00 ⎪ ⎨L = 1 L 0 j = L j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩ jm L := −δ jm ∀ j, m = 1, 2, 3,

(10.176)

in the given coordinate system. Then, by Lemma A.7, there exists a constant non  degenerate matrix {Amn }0≤m,n≤3 ∈ R4×4 , such that det {Amn }0≤m,n≤3 = 0, and if we consider a matrix {L mn }0≤m,n≤3 ∈ R4×4 , defined by L

mn

:=

3 3   j=0 k=0

Amk An j L k j ∀ m, n = 0, 1, 2, 3,

(10.177)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

159

and a vector (w 0 , w 1 , w 2 , w 3 ) ∈ R4 , defined by j

w :=

3 

A jk w k ∀ j = 0, 1, 2, 3,

(10.178)

k=0

then we have:

and

⎧ 00 ⎪ ⎨L = 1 L  jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0 j L = L  j0 = 0 ∀1 ≤ j ≤ 3,

(10.179)

(w 0 , w 1 , w 2 , w 3 ) = (1, 0, 0, 0) .

(10.180)

Obviously matrix {Amn }0≤m,n≤3 ∈ R4×4 represents some Lorentz’s transformation leading to a new system. However, by (10.173), (10.174) and (10.175) we have 3  3 

L m j wm w j = 1,

(10.181)

m=0 j=0

mn = −L mn +

 3

L mk wk

  3

k=0

 ∀ 0 ≤ m, n ≤ 3.

L nk wk

(10.182)

k=0

and 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

 (x 0 , x 1 , x 2 , x 3 ) = (w0 , w1 , w2 , w3 ) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(10.183)

Thus, again by a covariance in the new system we also observe: 3  3 

L m j wm w j = 1,

(10.184)

m=0 j=0

mn



= −L

mn

+

 3 k=0

L

mk

wk

  3

L

nk

wk

 ∀ 0 ≤ m, n ≤ 3.

(10.185)

k=0

and 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂ x 0 ∂ x 1 ∂ x 2 ∂ x 3



(x 0 , x 1 , x 2 , x 3 ) = (w0 , w1 , w2 , w3 ) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

Therefore, inserting (10.180) into (10.186) gives

(10.186)

160

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂ x 0 ∂ x 1 ∂ x 2 ∂ x 3



(x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.187)

and inserting (10.180) together with (10.179) into (10.185) gives, ⎧ 00 ⎪ ⎨ = 0 0 j =  j0 = 0 ∀ j = 1, 2, 3 ⎪ ⎩  jm  = δ jm ∀ j, m = 1, 2, 3.

(10.188)

Therefore, by (10.187) and (10.188) the new system is cartesian and obtained from the old system by some Lorentz’s transformation. Thus, since the inverse transform to Lorentz’s transformation is also a Lorentz’s transformation, we deduce that the original system (old) is obtained from the cartesian system (new) by some Lorentz’s transformation. Conversely, if any new system is obtained from the old cartesian coordinate system by Lorentz’s transformation, then since every cartesian system is also a PseudoLorentzian coordinate system, by Lemma 10.3 we deduce that the new system is also a Pseudo-Lorentzian coordinate system. Proof (Proof of Theorem 2.4) Indeed, assume that the transformation in (2.1.67) is of class P L ((w0 , w1 , w2 , w3 )). Then, obviously there exists another constant vector (w0 , w1 , w2 , w3 ) ∈ R4 , satisfying (w0 )2 −

3 

(w j )2 = 1,

(10.189)

j=1

and such that the transformation in (2.1.67) is of class P L((w0 , w1 , w2 , w3 ); (w0 , w1 , w2 , w3 )). Then by Corollary 2.8 there exists three other changes of coordinate system (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), belonging to the class L((w0 , w1 , w2 , w3 ); (1, 0, 0, 0)), (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), belonging to the class P L((1, 0, 0, 0); (1, 0, 0, 0)) and (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), belonging to the class L(((1, 0, 0, 0); (w0 , w1 , w2 , w3 )), so that the original transformation (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) is a composition of the above three changes of coordinate systems: (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) = (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ).

(10.190)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

161

However, since the original system with coordinates (x 0 , x 1 , x 2 , x 3 ) is PseudoLorentzian and since, the change (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), belongs to the class L((w0 , w1 , w2 , w3 ); (1, 0, 0, 0)), by Lemma 10.3 the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is Pseudo-Lorentzian, and moreover, we have 

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂ x 0 ∂ x 1 ∂ x 2 ∂ x 3



(x 0 , x 1 , x 2 , x 3 ) = (1, 0, 0, 0) ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

and thus also ⎧ 00 0 1 2 3 4 ⎪ ⎨ = 0 ∀ (x , x , x , x ) ∈ R 0 j  j0  =  = 0 ∀ j = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ⎪ ⎩ mn  = δmn ∀ m, n = 1, 2, 3 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

(10.191)

(10.192)

so that the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is cartesian. Therefore, since the change (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), belongs to the class P L((1, 0, 0, 0); (1, 0, 0, 0)), by Corollary 2.7 we deduce that the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is also cartesian, and thus in particular, the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is Pseudo-Lorentzian. Finally, since the change (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) is a Lorentian transformation, and since the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is Pseudo-Lorentzian, by Lemma 10.3 we deduce that the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is also PseudoLorentzian. Conversely, assume that the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is Pseudo-Lorentzian. Then, by Theorem 2.3, there exists two Lorentz’s transformations (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) and (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), so that both systems with coordinates (x 0 , x 1 , x 2 , x 3 ) and (x 0 , x 1 , x 2 , x 3 ) are cartesian. Then, again by Corollary 2.7, the change (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ), belongs to the class P L((1, 0, 0, 0); (1, 0, 0, 0)). However, since the system with coordinates (x 0 , x 1 , x 2 , x 3 ) is cartesian, by (2.1.65) we deduce that the transformation (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) also belongs to the class P L((w0 , w1 , w2 , w3 ); (1, 0, 0, 0)). Therefore, by Proposition 2.5 we obtain that the transformation (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) also belongs to the class P L((w0 , w1 , w2 , w3 ); (1, 0, 0, 0)). On the other hand, since the change (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) is a Lorentz’s transformation, it also belongs to the class P L((1, 0, 0, 0)). Then, by Corollary 2.6 together with Proposition 2.5, we finally deduce that the composed transformation (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) given through: (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) → (x 0 , x 1 , x 2 , x 3 ) belongs to the class P L ((w0 , w1 , w2 , w3 )).

162

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

Proof (Proof of Proposition 3.1) For every  smooth scalar classical function with compact support ξ(x 0 , x 1 , x 2 , x 3 ) ∈ Cc∞ R4 , by (3.0.7) we have """" 

    − 1    2 0 1 2 3 , IW , IW , IW ξ(x 0 , x 1 , x 2 , x 3 ) det K mn n,m=0,1,2,3  dx 0 dx 1 dx 2 dx 3 = div IW K

R4

""""  j 3   − 1  ∂ IW 0 1 2 3  2 0 1 2 3  mn (x , x , x , x ) ξ(x , x , x , x ) K dx 0 dx 1 dx 2 dx 3 + det  n,m=0,1,2,3 ∂x j j=0

R4

""""  3

j

IW (x 0 , x 1 , x 2 , x 3 )ξ(x 0 , x 1 , x 2 , x 3 )

j=0

R4

∂ ∂x j

  − 1     2  dx 0 dx 1 dx 2 dx 3 . det K mn n,m=0,1,2,3 

(10.193) However, by the definition of the derivative of the distribution we have,  """"  j 3 − 1  ∂ IW 0 1 2 3   mn   2 (x , x , x , x ) K ξ (x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 det n,m=0,1,2,3  ∂x j j=0

R4

=−

""""  3 j=0

R4

=−

""""  3

j

IW

j=0

R4

−

j

IW

""""  3 R4

j=0

∂ ∂x j ∂ ∂x j

  − 1      2 det K mn n,m=0,1,2,3  ξ (x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3   − 1     2  ξ(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 det K mn n,m=0,1,2,3 

  − 1 ∂ξ   2 j  IW det K mn n,m=0,1,2,3  (x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 . ∂x j

(10.194) Thus, inserting (10.194) into (10.193) gives """" 

    − 1    2 0 1 2 3 div IW , IW , IW , IW ξ(x 0 , x 1 , x 2 , x 3 ) det K mn n,m=0,1,2,3  dx 0 dx 1 dx 2 dx 3 = K

R4

−

""""  3 R4

j=0

  − 1 ∂ξ   2 j  IW det K mn n,m=0,1,2,3  (x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 . ∂x j

(10.195) On the other hand, by (3.0.5) we deduce: """"  3

j

IW (x 0 , x 1 , x 2 , x 3 )

j=0

R4

"b = a

  − 1 ∂ξ      2 (x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 det K mn n,m=0,1,2,3  ∂x j

⎞ "b 3 j  d ∂ξ dχ ⎠ ds = (s) W (s) ⎝ W (s) (s)) (ξ (χ(s))) ds. (χ ∂x j ds ds ⎛

j=0

a

(10.196)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

163

Thus, if we assume that W (s) is a constant across the trajectory of the motion, so that W (s) = W0 where W0 is independent on s, then by (10.195) and (10.196), using the Newton-Leibnitz formula, we deduce: """" 

    − 1   2  0 1 2 3 div IW , IW , IW , IW ξ(x 0 , x 1 , x 2 , x 3 ) det K mn n,m=0,1,2,3  dx 0 dx 1 dx 2 dx 3 K

R4

"b =−

W (s)

d (ξ (χ (s))) ds = −W0 ds

a

"b

d (ξ (χ (s))) ds = W0 (ξ (χ(a)) − ξ (χ(b))) . ds

(10.197)

a

  However, since ξ(x 0 , x 1 , x 2 , x 3 ) ∈ Cc∞ R4 has a compact support, by (3.0.8) we obtain ξ (χ (b)) = ξ (χ (a)) = 0, (10.198) and thus, inserting (10.198) into (10.197) gives: """" 

    − 1    0 , I1 , I2 , I3 ξ(x 0 , x 1 , x 2 , x 3 ) det K mn n,m=0,1,2,3  2 dx 0 dx 1 dx 2 dx 3 div I W W W W K

R4

=0

  ∀ ξ ∈ Cc∞ R4 .

(10.199)   Therefore, since the test function ξ(x 0 , x 1 , x 2 , x 3 ) ∈ Cc∞ R4 in (10.199) is arbitrary, by (10.199), using the basic properties of distributions, we finally deduce (3.0.9). This completes the proof. Proof (Proof of Lemma 4.1) By inserting (4.1.60) into (4.1.58) we deduce: ⎧ F00 = 0 ⎪ ⎪ ⎪ ⎨ F = −F = − ∂(−A j ) − ∂ ∀ j = 1, 2, 3 0j j0 ∂x0 ∂x j ⎪ = 0 ∀ j = 1, 2, 3 F jj ⎪ ⎪ ⎩ ∂(−A ) m) − ∂ x m j ∀ m = j = 1, 2, 3, Fm j = −F jm = ∂(−A ∂x j

(10.200)

Thus, if we define, as usual the magnetic and the electric field:  B := curlx A, E := −∇x  −

∂A , ∂x0

(10.201)

then denoting E := (E 1 , E 2 , E 3 ) and B := (B1 , B2 , B3 ), by (10.201) we rewrite (10.200) as:

164

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas ⎧ ⎪ F00 ⎪ ⎪ ⎪ ⎪ F0 j ⎪ ⎪ ⎪ ⎨F jj ⎪ F12 ⎪ ⎪ ⎪ ⎪ ⎪ F13 ⎪ ⎪ ⎩ F23

=0 = −F j0 = E j ∀ j = 1, 2, 3 = 0 ∀ j = 1, 2, 3 = −F21 = −B3 = −F31 = B2 = −F32 = −B1 ,

(10.202)

so that we get (4.1.63). Next consider two-times contravariant tensor {F mn }m,n=0,1,2,3 defined by: F mn :=

3 3  

K m j K nk F jk ∀ m, n = 0, 1, 2, 3.

(10.203)

k=0 j=0

We rewrite (10.203) as: F mn = K m0 K n0 F00 +

3 

K m0 K nk F0k +

k=1

+

3  3 

3 

K m j K n0 F j0

j=1

K m j K nk F jk ∀ m, n = 0, 1, 2, 3.

(10.204)

k=1 j=1

In particular, by inserting (4.1.29) into (10.204) we deduce: ⎧  3   00 k ⎪ + 3j=1 v j F j0 + 3k=1 3j=1 v j v k F jk ⎪ ⎪ F = F00 + k=1 v F0k  ⎪ ⎨ F m0 = v m F 00 − F − 3 v k F m0 mk ∀ m = 1, 2, 3, 3 k=1 j 0n = v n F 00 − F − ⎪ F v F ∀ n = 1, 2, 3, ⎪ 0n jn j=1 ⎪ ⎪ ⎩ F mn = v m v n F 00 − 3 v n v k F − 3 v v j F − v m F − v n F k=1

mk

j=1 m

jn

0n

m0

+ Fmn ∀ m, n = 1, 2, 3.

(10.205) We rewrite (10.205) as: ⎧ 00 3 3 3 3 k j j k ⎪ ⎪ F = F00 + k=1 v F0k+ j=1 v F j0 + k=1 j=1 v v F jk ⎪ ⎨ F m0 = v m F 00 − F − 3 v k F m0 mk ∀ m = 1, 2, 3, 3 k=1 j 0n n 00 ⎪ F = v F − F0n − j=1 v F jn ∀ n = 1, 2, 3, ⎪ ⎪ ⎩ mn F = v m F 0n + v n F m0 − v m v n F 00 + Fmn ∀ m, n = 1, 2, 3.

(10.206)

In particular, since the tensor {Fmn }m,n=0,1,2,3 is antisymmetric, i.e., Fmn = −Fnm ∀ m, n = 0, 1, 2, 3, then we simplify (10.206) as ⎧ 00 F =0 ⎪ ⎪ ⎪ ⎨ F mm = 0 ∀ m = 1, 2, 3,  ⎪ F 0m = −F m0 = −F0m + 3k=1 v k Fmk ∀ m = 1, 2, 3, ⎪ ⎪ ⎩ mn F = v m F 0n − v n F 0m + Fmn ∀ m, n = 1, 2, 3.

(10.207)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

165

In particular, since {Fm j }0≤m, j≤3 is the antisymmetric two-times covariant tensor field, then by inserting (10.202) into (10.207) we deduce: ⎧ 00 ⎪ ⎪F ⎪ ⎪ ⎪ F jj ⎪ ⎪ ⎪ ⎪ F 01 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ F 02 F 03 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ F 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ F 13 ⎪ ⎪ ⎪ ⎩ 23 F

=0 = 0 ∀ j = 1, 2, 3,    = −F 10 = −F01 + v 2 F12 + v 3 F13 = − E 1 + v 2 B3 − v 3 B2    = −F 20 = −F02 + v 1 F21 + v 3 F23 = − E 2 + v 3 B1 − v 1 B3    = −F 30 = −F03 + v 1 F31 + v 2 F32 = − E 3 + v 1 B2 − v 2 B1    = −F 21 = v 1 F 02 − v 2 F 01 + F12 = − B3 + v 1 F 20 − v 2 F 10   = −F 31 = v 1 F 03 − v 3 F 01 + F13 = B2 + v 3 F 10 − v 1 F 30    = −F 32 = v 2 F 03 − v 3 F 02 + F23 = − B1 + v 2 F 30 − v 3 F 20 .

Thus, defining:



D := E + v × B H := B + v × D,

(10.208)

(10.209)

and denoting D := (D1 , D2 , D3 ) and H := (H1 , H2 , H3 ) we rewrite (10.208) as: ⎧ F 00 ⎪ ⎪ ⎪ ⎪ ⎪ F0j ⎪ ⎪ ⎪ ⎨F j j ⎪ F 12 ⎪ ⎪ ⎪ ⎪ ⎪ F 13 ⎪ ⎪ ⎩ 23 F

=0 = −F j0 = −D j ∀ j = 1, 2, 3, = 0 ∀ j = 1, 2, 3, = −F 21 = −H3 = −F 31 = H2 = −F 32 = −H1 ,

(10.210)

so that we get (4.1.64). In particular, by (10.202) and (10.210), using (10.209) and using the definition (4.1.58), we deduce that ⎞ ⎛    3  3  3  3 3  3   ∂ Ap ∂ A ∂ A ∂ A m k n mn pk ⎠= ⎝ K K − − F jk F jk ∂xm ∂x p ∂xn ∂xk n=0 k=0 m=0 p=0

= F 00 F00 +

j=0 k=0

3  k=1

F 0k F0k +

3  j=1

F j0 F j0 +

3  3 

F jk F jk = −2E · D + 2B · H

j=1 k=1

  = −2 ((D − v × B) · D − B · (B + v × D)) = −2 |D|2 − |B|2 ,

(10.211)

and so by (10.211), (10.209) and (10.201) we finally obtain (4.1.65). Proof (Proof of Lemma 6.1) Since the current coordinate system is cartesian, by Theorem 2.1 there exists a change of variables from the kinematically preferable coordinate system to the current cartesian coordinate system of the form:

166

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas

⎧ 0 0 ⎪ ⎨x = x + c, 3  m ⎪ Am j (x 0 ) x  j + z m (x 0 ) ∀m = 1, 2, 3, ⎩x =

(10.212)

j=1

where {Amn (x 0 )}n,m=1,2,3 ∈ R3×3 is a 3 × 3-matrix, depending on the coordinate x 0 only (independent on x := (x 1 , x 2 , x 3 )), and satisfying 3 

Am j (x 0 )An j (x 0 ) =

j=1

3 

A jm (x 0 )A jn (x 0 ) = δmn ∀ m, n = 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 ,

j=1

(10.213)  c ∈ R is a constant (independent on (x 0 , x 1 , x 2 , x 3 ) ∈ R4 ) and z(x 0 ) := z 1 (x 0 ),  z 2 (x 0 ), z 3 (x 0 ) ∈ R3 is a three-dimensional vector field, depending on the coordinate x 0 only (independent on x := (x 1 , x 2 , x 3 )). On the other hand, since the kinematically preferable system is simultaneously Lorentzian and cartesian, in this system we have (r 0 , r 1 , r 2 , r 3 ) = (1, 0, 0, 0) so that

r := (r 1 , r 2 , r 3 ) = (0, 0, 0). (10.214) On the other hand, by the rule of transformations of contravariant vector in (10.3), using (10.212), (10.214) and (10.213) we deduce r 0 = 1,

(10.215)

and rm =

3  d Am j 0  j dz m (x ) x + 0 (x 0 ) 0 dx dx j=1

=

3  3   dz m  d Am j 0 0 n n 0 (x ) A (x ) x − z (x ) + 0 (x 0 ) n j dx 0 dx j=1 n=1

∀m = 1, 2, 3.

(10.216) Therefore, differentiating (10.216) gives:  d Am j ∂r m = (x 0 ) An j (x 0 ) ∀m, n = 1, 2, 3. n 0 ∂x dx j=1 3

Thus, by (10.217) and (10.213) we deduce

(10.217)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas



∂r m ∂r n + m n ∂x ∂x

 =

3   d Am j j=1

 d An j 0 0 (x ) A (x ) mj dx 0 dx 0 ⎫ ⎧ 3 ⎬ d d ⎨ Am j (x 0 ) An j (x 0 ) = 0 {δmn } = 0 = 0 ⎭ dx dx ⎩ j=1 (x 0 ) An j (x 0 ) +

∀m, n = 1, 2, 3. In particular,

167

⎧ ∂r m ⎨ ∂xn +



∂r n ∂xm 3 

⎩divx r =

n=1

(10.218)

= 0 ∀m, n = 1, 2, 3 ∂r n ∂xn

= 0.

(10.219)

Moreover, differentiating (10.217) one more time by x j gives ∂ 2r m = 0 ∀ j, m, n = 1, 2, 3. ∂x j ∂xn

(10.220)

So, by (10.219) and (10.220) we deduce (6.1.3). Proof (Proof of Lemma 9.1) By the theory of distribution, the definition in (9.0.4) means that for every   smooth scalar classical function with compact support ξ(x 0 , x 1 , x 2 , x 3 ) ∈ Cc∞ R4 we have """"   "1 "b − 1    2 mn 0 1 2 3  ϒσ (x , x , x , x ), ξ(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 = σ det K pq p,q=0,1,2,3  0 a

R4

        m dz n dz m dχ n dχ m (s) + (1 − τ ) (s) − χ n (s) − z n (s) τ (s) + (1 − τ ) (s) χ (s) − z m (s) τ ds ds ds ds ! ξ (τ χ(s) + (1 − τ )z(s)) ds dτ ∀ m = 0, 1, 2, 3. (10.221)

Next, since by the theory of distributions we have 3 """"  n=0

R4

=−

∂ ∂xn

!   − 1    2 mn ϒσ ξ(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 det K pq p,q=0,1,2,3 

3 """"  − 1     2 mn ∂ξ 0 1 2 3  ϒσ (x , x , x , x ) dx 0 dx 1 dx 2 dx 3 det K pq p,q=0,1,2,3  ∂xn n=0

R4

∀ m = 0, 1, 2, 3,

by (10.221) we can proceed in (10.222) as:

(10.222)

168

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas 3 """"  n=0

R4

!   − 1    2 mn ϒσ ξ(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 det K pq p,q=0,1,2,3 

∂ ∂xn

3 "1 "b 

=−

σ





χ m (s) − z m (s)

n=0 0 a

  dz n dχ n (s) + (1 − τ ) (s) τ ds ds

 !   dχ m dz m ∂ξ − χ n (s) − z n (s) τ (s) + (1 − τ ) (s) (τ χ(s) + (1 − τ )z(s)) dsdτ ds ds ∂xn ∀m = 0, 1, 2, 3.

(10.223)

Furthermore, by the chain rule we can rewrite (10.223) as: 3 """"  n=0

R4

"1 "b

  ∂ξ σ χ m (s) − z m (s) (τ χ(s) + (1 − τ )z(s)) ds dτ ∂s

=− 0 a

"1 "b +

!   − 1    2 mn ϒσ ξ(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3 det K pq p,q=0,1,2,3 

∂ ∂xn

σ 0 a

  dχ m dz m ∂ξ τ (s) + (1 − τ ) (s) (τ χ(s) + (1 − τ )z(s)) ds dτ ∀ m = 0, 1, 2, 3. ds ds ∂τ

(10.224) However, taking into account (9.0.3), integration by partes by s gives: "1 "b − 0 a

  ∂ξ σ χ m (s) − z m (s) (τ χ(s) + (1 − τ )z(s)) ds dτ ∂s 

"1 "b =

σ

dz m dχ m (s) − (s) ds ds

 ξ (τ χ(s) + (1 − τ )z(s)) ds dτ ∀ m = 0, 1, 2, 3.

0 a

(10.225) On the other hand, integration by partes by τ gives: "1 "b σ 0 a

  dχ m dz m ∂ξ τ (s) + (1 − τ ) (s) (τ χ(s) + (1 − τ )z(s)) ds dτ ds ds ∂τ "b +

σ

dχ m (s) ξ (χ(s))) ds − ds

a



"1 "b −

σ

dχ m dz m (s) − (s) ds ds

"b σ

dz m (s) ξ (z(s))) ds ds

a

 ξ (τ χ(s) + (1 − τ )z(s)) ds dτ ∀ m = 0, 1, 2, 3.

0 a

Thus, inserting (10.225) and (10.226) into (10.224) gives:

(10.226)

10 Detailed Proves of the Stated Theorems, Propositions and Lemmas 3 """"  n=0

R4

∂ ∂xn



"b =

σ a

  − 1    2 mn ϒσ det K pq p,q=0,1,2,3 

dχ m (s) ξ (χ (s))) ds − ds

"b σ a

169

& ξ(x 0 , x 1 , x 2 , x 3 ) dx 0 dx 1 dx 2 dx 3

dz m (s) ξ (z(s))) ds ∀ m = 0, 1, 2, 3. ds

(10.227)

  Thus, since ξ(x 0 , x 1 , x 2 , x 3 ) ∈ Cc∞ R4 was chosen arbitrary, by the theory of distributions, using (10.227) we deduce 3  ∂ ∂xn



  − 1    2 mn ϒσ det K pq p,q=0,1,2,3 

& =

n=0

"b σ a

"b σ

− a

  dχ m (s) δ x 0 − χ 0 (s), . . . , x 3 − χ 3 (s) ds ds

  dz m (s) δ x 0 − z 0 (s), . . . , x 3 − z 3 (s) ds. ds

(10.228) Finally, (10.228) obviously implies: 3  n=0

1

   1 det {K pq } p,q=0,1,2,3 − 2

∂ ∂xn

!   − 1   2  det K pq ( p,q)3  ϒσmn 0

   1 "b dχ m     2 = det K pq p,q=0,1,2,3  σ (s) δ x 0 − χ 0 (s), . . . , x 3 − χ 3 (s) ds ds a

    1 "b dz m   2  σ (s) δ x 0 − z 0 (s), . . . , x 3 − z 3 (s) ds. − det K pq p,q=0,1,2,3  ds a

(10.229)

Appendix

Some Technical Statements

Lemma A.1 Assume that a contravariant pseudo-metric {K mn }m,n=0,1,2,3 and a covector of the time-direction (w0 , w1 , w2 , w3 ) are weakly correlated (see Definition 2.3), so that we have everywhere 3  3 

K jm w j wm > 0

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(A.1)

j=0 m=0

Moreover, assume that there exists a scalar field ϕ such that ⎛ ⎞ 3 3   ∂ϕ ∂ϕ ⎝ lim K jm j m ⎠ = 1 ,   ∂ x ∂x 0 2 1 2 2 2 3 2 (x ) +(x ) +(x ) +(x ) →+∞ j=0 m=0 (w0 , w1 , w2 , w3 ) =

∂ϕ ∂ϕ ∂ϕ ∂ϕ , , , ∂x0 ∂x1 ∂x2 ∂x3

(A.2)

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , (A.3)

and ⎧⎛ ⎞⎛ ⎞⎫ 3 3 3 3   ∂ ⎨⎝   mn ∂ϕ ∂ϕ ∂ϕ ⎠⎬ jn ⎠ ⎝ K −1 K = 0 ∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . ∂xm ∂xn ∂xn ⎭ ∂x j ⎩ j=0

m=0 n=0

n=0

(A.4) Then, scalar global time ϕ and pseudo-metric {K mn }m,n=0,1,2,3 are strongly correlated on R4 , i.e. ϕ satisfies the following eikonal-type equation: 3  3  j=0 m=0

K jm

∂ϕ ∂ϕ =1 ∂x j ∂xm

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Poliakovsky, Lorentzian Geometrical Structures with Global Time, Gravity and Electrodynamics, https://doi.org/10.1007/978-3-031-23762-1

(A.5)

171

172

Appendix: Some Technical Statements

Proof Define ⎧ ⎛ 3  3 ⎨  ∂ϕ A1 = (x 0 , x 1 , x 2 , x 3 ) ∈ R4 : ⎝ K jm j ⎩ ∂x j=0 m=0 ⎧ ⎛ 3  3 ⎨  ∂ϕ A2 = (x 0 , x 1 , x 2 , x 3 ) ∈ R4 : ⎝ K jm j ⎩ ∂x j=0 m=0

⎫ ⎞ ⎬ ∂ϕ ⎠ 0 1 2 3 (x , x , x , x ) − 1 < 0 , (A.6) m ⎭ ∂x ⎫ ⎞ ⎬ ∂ϕ ⎠ 0 1 2 3 (x , x , x , x ) − 1 > 0 . (A.7) m ⎭ ∂x

Then, we obviously have 3 3  

K jm

j=0 m=0

∂ϕ ∂ϕ −1=0 ∂x j ∂xm

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ ∂A1 ∪ ∂A2 .

(A.8)

where by ∂A we denote the boundary of the set A. Therefore, by (A.4), (A.2) and (A.8), the Gauss-Green formula gives ⎧     ⎨  3  3 ⎩

Ak

m=0 n=0

 =− Ak

K

mn

⎛

ϕ⎝

3  j=0

⎞⎫ ⎛ 3 3 ⎬  ∂ϕ ∂ϕ jn ∂ϕ ∂ϕ ⎠ ⎝ − 1 K d(x 0 , x 1 , x 2 , x 3 ) ∂xm ∂xn ∂x j ∂xn ⎭ j=0 n=0

∂ ∂x j



3 

3 

K

mn

m=0 n=0

 3 ⎞  ∂ϕ ∂ϕ jn ∂ϕ ⎠ d(x 0 , x 1 , x 2 , x 3 ) −1 K ∂xm ∂xn ∂xn n=0

= 0 ∀ k = 1, 2 .

(A.9)

However, by (A.1) and (A.3) we have 3  3  j=0 m=0

K jm

∂ϕ ∂ϕ >0 ∂x j ∂xn

∀ (x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(A.10)

Thus, inserting (A.6), (A.7) and (A.10) into (A.9) gives  ⎞ ⎛ 3 3        3    3  ∂ϕ ∂ϕ ∂ϕ ∂ϕ mn jn  ⎠ d(x 0 , x 1 , x 2 , x 3 ) K −1 ⎝ K  m n j n ∂x ∂x ∂ x ∂ x   m=0 n=0 j=0 n=0 Ak

=0

∀ k = 1, 2 .

(A.11)

So, by the definition of A1 , A2 we must have  ⎞ ⎛ 3 3        3    3  ∂ϕ ∂ϕ ∂ϕ ∂ϕ mn jn  ⎝ ⎠ d(x 0 , x 1 , x 2 , x 3 ) = 0 , K − 1 K  ∂xm ∂xn ∂ x j ∂ x n   m=0 n=0 j=0 n=0 4 R

which, together with (A.10), finally implies (A.5).

(A.12) 

Appendix: Some Technical Statements

173

Lemma A.2 Consider a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 and on R4 , associated with let {K mn }m,n=0,1,2,3 be the inverse covariant pseudo-metric   {K mn }m,n=0,1,2,3 . Next, consider the Christoffel Symbols kmj , defined by (2.1.25). K Then, in some coordinate system we have 

j mn

 K

=0

∀ j, m, n = 0, 1, 2, 3 ,

(A.13)

if and only if the tensor {K mn }m,n=0,1,2,3 is independent on the local coordinates (x 0 , x 1 , x 2 , x 3 ) ∈ R4 in the given coordinate system. Proof If in some fixed coordinate system {K mn }m,n=0,1,2,3 is independent on the local coordinates (x 0 , x 1 , x 2 , x 3 ) ∈ R4 , then by (2.1.25) we obviously deduce (A.13). Conversely, if in some fixed coordinate system we have (A.13), then we obviously have   ∂ K mn = δ j K mn K = 0 ∀ j, m, n = 0, 1, 2, 3 , (A.14) j ∂x   where δ j K mn K is the covariant derivative of the pseudo-metric K with respect to the same pseudo-metric K , which is vanishes, due to the well-known rule of the tensor analysis. So, by (A.14) we deduce that {K mn }m,n=0,1,2,3 is indeed independent  on the local coordinates (x 0 , x 1 , x 2 , x 3 ) ∈ R4 . Lemma A.3 Consider a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 and let {K mn }m,n=0,1,2,3 be the inverse covariant pseudo-metric on R4 , associated with {K mn }m,n=0,1,2,3 . Next, consider two coordinate systems in R4 , so that the change of coordinates from the first to the second coordinate system is given by: ⎧ 0 x ⎪ ⎪ ⎪ 1 ⎨ x 2 ⎪ x ⎪ ⎪ ⎩ 3 x

= = = =

f (0) (x 0 , x 1 , x 2 , x 3 ), f (1) (x 0 , x 1 , x 2 , x 3 ), f (2) (x 0 , x 1 , x 2 , x 3 ), f (3) (x 0 , x 1 , x 2 , x 3 ).

(A.15)

Finally, assume that the tensor {K mn }m,n=0,1,2,3 is independent on the local coordinates in both given coordinate systems, i.e. {K mn }m,n=0,1,2,3 is independent on the  }m,n=0,1,2,3 is independent on the coorcoordinates (x 0 , x 1 , x 2 , x 3 ) ∈ R4 and {K mn 0 1 2 3 4 dinates (x , x , x , x ) ∈ R . Then, the transformations in (A.15) are linear, i.e. ∂ 2 f ( j) 0 1 2 3 (x , x , x , x ) = 0 ∂ xn∂ xk

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4

∀ j, k, n = 0, 1, 2, 3 . (A.16)

174

Appendix: Some Technical Statements

Proof First of all, observe that since {K mn }m,n=0,1,2,3 is independent on the coordi }m,n=0,1,2,3 is independent on the coordinates nates (x 0 , x 1 , x 2 , x 3 ) ∈ R4 and {K mn 0 1 2 3 4 (x , x , x , x ) ∈ R , then, by (2.1.25) in both coordinate systems we have  and



j mn

j mn

 K

=0

∀ j, m, n = 0, 1, 2, 3

(A.17)

K

=0

∀ j, m, n = 0, 1, 2, 3 .

(A.18)



Next fix an index j ∈ {0, 1, 2, 3} and define the proper scalar field ψ  (x 0 , x 1 , x 2 , x 3 ) = ψ(x 0 , x 1 , x 2 , x 3 ) by the following ψ(x 0 , x 1 , x 2 , x 3 ) = f ( j) (x 0 , x 1 , x 2 , x 3 )

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4

(A.19)

so that we have ψ  (x 0 , x 1 , x 2 , x 3 ) = x  j

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(A.20)

  ∂ψ ∂ψ ∂ψ instead of On the other hand, by (2.1.26) with the covector ∂∂ψ , , , 0 1 2 3 x ∂x ∂x ∂x (h 0 , h 1 , h 2 , h 3 ), using (A.17) and (A.18), in both coordinate systems we deduce 

δn

∂ψ ∂xk

 = K

∂ 2ψ ∂ xn∂ xk

and    ∂ψ ∂ 2ψ  δn =    ∂ xk ∂ xn ∂ xk K

∀ n, k = 0, 1, 2, 3

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 , (A.21)

∀ n, k = 0, 1, 2, 3

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 . (A.22)

However, by (A.20) and (A.22) together we deduce    ∂ψ ∂2ψ  ∂2x j = = = 0 ∀ n, k = 0, 1, 2, 3 ∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 . δn    ∂ xk ∂ xn ∂ xk ∂ xn ∂ xk K

    Therefore, since δn ∂∂ψ xk

K n,k=0,1,2,3

(A.23) is a proper two-times covariant tensor, by

(A.23) we deduce 

 ∂ψ δn =0 ∂xk K

∀ n, k = 0, 1, 2, 3

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(A.24)

Appendix: Some Technical Statements

175

Thus, by (A.24) and (A.21) we deduce ∂ 2ψ =0 ∂ xn∂ xk

∀ n, k = 0, 1, 2, 3

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4 .

(A.25)

Finally, by (A.25) and (A.19) we obtain ∂ 2 f ( j) 0 1 2 3 (x , x , x , x ) = 0 ∂ xn∂ xk

∀(x 0 , x 1 , x 2 , x 3 ) ∈ R4

∀k, n = 0, 1, 2, 3 , (A.26) and since the fixed index j ∈ {0, 1, 2, 3} was arbitrary, we deduce (A.16).  3 Corollary .1 Let f(x) := f(x 1 , x 2 , x 3 ) : R3→ R  be a smooth mapping. Next assume ∂ fm that the Jacoby’s derivatives matrix dx f := ∂ x n of f satisfies (everywhere) 1≤m,n≤3

{dx f(x)}T · {dx f(x)} = I

∀x ∈ R3 ,

(A.27)

where I ∈ R3×3 is the identity matrix. Then dx f is a constant matrix (independent on x) in R3 and so f is a linear orthogonal mapping; in other word there exists a constant matrix A ∈ R3×3 and a constant vector w ∈ R3 such that AT · A = I ,

(A.28)

and we have f(x) = A · x + w

∀x ∈ R3 .

(A.29)

We give here the full proof, although the result is well known. Proof (Proof of Corollary .1) Consider two coordinate systems in R4 , so that the change of coordinates from the first to the second coordinate system is given by: ⎧ 0 ⎪ ⎪x ⎪ ⎨x 1 ⎪x 2 ⎪ ⎪ ⎩ 3 x

= = = =

x 0, f 1 (x 1 , x 2 , x 3 ), f 2 (x 1 , x 2 , x 3 ), f 3 (x 1 , x 2 , x 3 ),

(A.30)

  where f 1 (x 1 , x 2 , x 3 ), f 2 (x 1 , x 2 , x 3 ), f 3 (x 1 , x 2 , x 3 ) := f(x 1 , x 2 , x 3 ). Next, consider a contravariant pseudo-metric {K mn }m,n=0,1,2,3 on R4 and the corresponding inverse covariant pseudo-metric {K mn }m,n=0,1,2,3 , such that in the first coordinate system we have: ⎧ 00 ⎪ ⎨K = 1 (A.31) ∀ j = 1, 2, 3 K 0 j = K j0 = 0 ⎪ ⎩ jm ∀ j, m = 1, 2, 3 K = −δ jm at every point in R4 , so that in the same coordinate system we have:

176

Appendix: Some Technical Statements

⎧ ⎪ ⎨ K 00 = 1 ∀ j = 1, 2, 3 K 0 j = K j0 = 0 ⎪ ⎩ ∀ j, m = 1, 2, 3 K jm = −δ jm

(A.32)

at every point in R4 . On the other hand, by (A.30) and (10.5) we have  K 00 = K 00 ,

K m0 = K 0m =

3  ∂ f (k) k=1

and K mn =

∂xm

 K k0 =

3  ∂ f (k) k=1

3 3   ∂ f (k) ∂ f ( j) j=1 k=1

∂xm ∂xn

∂xm

(A.33)  K 0k

K k j

∀ m = 1, 2, 3.

∀ m, n = 1, 2, 3.

(A.34)

(A.35)

Therefore, using (A.27) and (A.32), by (A.33), (A.34) and (A.35) we deduce ⎧  ⎪ ⎨ K 00 = 1 ∀ j = 1, 2, 3 K 0 j = K j0 = 0 ⎪ ⎩  ∀ j, m = 1, 2, 3 . K jm := −δ jm

(A.36)

Then, by (A.32) and (A.36) we can apply Lemma A.3 to deduce ∂ 2 f ( j) 1 2 3 (x , x , x ) = 0 ∂ xn∂ xk

∀(x 1 , x 2 , x 3 ) ∈ R3

∀ j, k, n = 1, 2, 3 .

(A.37)

Therefore, there exists a constant (independent on x) matrix A ∈ R3×3 such that AT · A = I ,

(A.38)

and dx f(x) = A

∀ x ∈ R3 .

(A.39)

So there exists a constant vector w ∈ R3 , such that f(x) = A · x + w

∀x ∈ R3 .

(A.40) 

This completes the proof. Lemma A.4 Given an arbitrary (w 0 , w 1 , w 2 , w 3 ) ∈ R4 such that (w 0 )2 − |w|2 = 1 ,

(A.41)

Appendix: Some Technical Statements

177

where w := (w 1 , w 2 , w 3 ) ∈ R3 , consider a matrix {mn }0≤m,n≤3 ∈ R4×4 , defined by ⎧ 00 0 2 ⎪ ⎨ = (w ) − 1  jm = δ jm + w j w m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j  =  j0 = w 0 w j ∀1 ≤ j ≤ 3.

(A.42)

Then the matrix {mn }0≤m,n≤3 is degenerate, and moreover, it has one vanishing and three positive eigenvalues. Proof (Proof of Lemma A.4) By (A.42) and (A.41) we have ⎧ 00 2 ⎪ ⎨ = |w|  jm = δ jm + w j w m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j  =  j0 = w 0 w j ∀1 ≤ j ≤ 3,

(A.43)

Thus, in the case w = 0 we obviously have ⎧ 00 ⎪ ⎨ = 0  jm = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j  =  j0 = 0 ∀1 ≤ j ≤ 3,

(A.44)

and so, by (A.44) we infer that {mn }0≤m,n≤3 is degenerate and moreover, it indeed has one vanishing and three positive eigenvalues. Next we assume from now that w = 0. Furthermore, observe that by (A.42), for every (z 0 , z 1 , z 2 , z 3 ) ∈ R4 , with z := (z 1 , z 2 , z 3 ) ∈ R3 , we have 3 

0 j z j = 00 z 0 +

3 

j=0

  2 0 j z j = (w 0 ) − 1 z 0 + w 0 (w · z)

(A.45)

j=1

and 3 

 z j =  z0 + mj

m0

j=0

3 

3    δ jm + w j w m z j  z j = z0 w w + mj

0

j=1

= z 0 w w + z m + (w · z)w 0

m

m

j=1 m

∀ m = 1, 2, 3.

(A.46)

In other words, ⎧ 3 ⎪ ⎪ ⎪ 0 j z j = −z 0 + (w 0 z 0 + w · z)w 0 ⎨ ⎪ ⎪ ⎪ ⎩

j=0 3 j=0

(A.47) m j z j = z m + (w 0 z 0 + w · z)w m

∀ m = 1, 2, 3.

178

Appendix: Some Technical Statements

Furthermore, assume that (z 0 , z 1 , z 2 , z 3 ) ∈ R4 , with z := (z 1 , z 2 , z 3 ) ∈ R3 , satisfies 

where λ solves

1 z 0 = λ+1 w0 1 z = λ−1 w .

(A.48)

   λ λ − 1 + 2|w|2 = 0 ,

(A.49)

or in other words, λ satisfies     either λ = 0 or λ = 2(w 0 )2 − 1 = 1 + 2|w|2 > 1 .

(A.50)

Then, in particular, by (A.48) we have 

 w0 z 0 + w · z =

1 1 (w 0 )2 + |w|2 . λ+1 λ−1

(A.51)

Then by (A.41) and (A.51) we have   0 w z0 + w · z =



1 1 + λ+1 λ−1 1 − 2 . λ −1

(w 0 )2 −

 1 λ  = 2 2(w 0 )2 − 1 λ−1 λ −1 (A.52)

If λ = 0 then, by (A.52) we have clearly w 0 z 0 + w · z = 1.

(A.53)

    On the other hand, if λ = 1 + 2|w|2 = 2(w 0 )2 − 1 then by (A.52) we also have  0  w z0 + w · z =

λ2

λ 1 λ− 2 = 1. −1 λ −1

(A.54)

Thus, if λ solves (A.49) then in both cases w 0 z 0 + w · z = 1, and therefore, by (A.47) we deduce ⎧ 3 ⎪ 0j 0 ⎪ ⎪ ⎨  z j = −z 0 + w j=0 (A.55) 3 ⎪ ⎪ ⎪ m j z j = z m + w m ∀ m = 1, 2, 3. ⎩ j=0

However, by (A.48) we have 

w 0 = (λ + 1)z 0 w = (λ − 1)z .

(A.56)

Appendix: Some Technical Statements

179

Thus by (A.55) and (A.56) we deduce ⎧ ⎪ ⎪ ⎪ ⎨

3

j=0 ⎪ 3

⎪ ⎪ ⎩

0 j z j = λz 0 (A.57)  z j = λ zm mj

∀ m = 1, 2, 3.

j=0

Thus, in the case where λ solves (A.49) and (z 0 , z 1 , z 2 , z 3 ) ∈ R4 , with z := (z 1 , z 2 , z 3 ) ∈ R3 , satisfies (A.48), by (A.57) we deduce that (z 0 , z 1 , z 2 , z 3 ) is an eigenvector of the matrix {mn }0≤m,n≤3 . So,   λ0 = 0 and λ1 = 1 + 2|w|2 > 1 are two eigenvalues of the matrix {mn }0≤m,n≤3 .

(A.58) Furthermore if λ ∈ R and (z 0 , z 1 , z 2 , z 3 ) ∈ R4 with (z 1 , z 2 , z 3 ) := z ∈ R3 satisfy ⎧ ⎪ ⎨λ = 1 z·w =0 ⎪ ⎩ z0 = 0 ,

(A.59)

then by (A.59) and (A.47) we have ⎧ ⎪ ⎪ ⎪ ⎨

3

j=0 ⎪ 3

⎪ ⎪ ⎩

0 j z j = 0 = λz 0 (A.60) m j z j = z m = λz m

∀ m = 1, 2, 3 .

j=0

However, obviously there exists two orthogonal unit vectors z1 ∈ R3 and z2 ∈ R3 such that z1 · w = z2 · w = z1 · z2 = 0 . Therefore, in the case λ = 1, if (z 0 , z 1 , z 2 , z 3 ) ∈ R4 satisfies z 0 = 0 and either (z 1 , z 2 , z 3 ) = z1 or (z 1 , z 2 , z 3 ) = z2 , then by (A.60) we deduce that in both cases (z 0 , z 1 , z 2 , z 3 ) is an eigenvector of of the matrix {mn }0≤m,n≤3 . So, λ2 = 1 and λ3 = 1 are two coinciding eigenvalues of the matrix {mn }0≤m,n≤3 .

(A.61) Therefore, by (A.58) and (A.61) we deduce that the matrix {mn }0≤m,n≤3 is degenerate, and moreover it has one vanishing and three positive eigenvalues. 

180

Appendix: Some Technical Statements

Lemma A.5 Consider an arbitrary (w0 , w 1 , w 2 , w 3 ) ∈ R4 and consider a matrix {K mn }0≤m,n≤3 ∈ R4×4 , defined by: ⎧ 00 0 2 ⎪ ⎨ K = (w ) K jm = −δ jm + w j w m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j K = K j0 = w 0 w j ∀1 ≤ j ≤ 3 . Then,

  in the case w 0 = 0 we have det {K mn }0≤m,n≤3 = 0 .

(A.62)

(A.63)

On the other hand, in the case w 0 = 0 the matrix {K mn }0≤m,n≤3 is invertible, its reverse matrix {K mn }0≤m,n≤3 ∈ R4×4 is defined by the following: ⎧ |w|2 1 ⎪ ⎨ K 00 = (w0 )2 − (w0 )2 K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ j ⎩ ∀1 ≤ j ≤ 3 , K 0 j = K j0 = w w0

(A.64)

where w := (w 1 , w 2 , w 3 ) ∈ R3 . Moreover, the matrix {K mn }0≤m,n≤3 necessary has one positive and three negative eigenvalues and   det {K mn }0≤m,n≤3 = −(w 0 )2 .

(A.65)

Proof (Proof of Lemma A.5) First of all, observe that in the case w 0 = 0 the first row of the matrix, given by (A.62), vanishes and thus we obviously deduce (A.63). Next, assume from now that w0 = 0. Then, consider a matrix {K mn }0≤m,n≤3 ∈ 4×4 defined by (A.64). Thus, by (A.62) and (A.64) we obtain R 3 

K 0k K k0 = K 00 K 00 +

k=0

3 

3 

K 0k K k0 = 1 − |w|2 + |w|2 = 1,

k=1

K mk K k j = K m0 K 0 j +

k=0

3 

K mk K k j = w m w j + δm j − w m w j

k=1

= δm j ∀1 ≤ m, j ≤ 3, and 3  k=0

K mk K k0 = K m0 K 00 +

3  k=1

K mk K k0 = w m w 0 − w m w 0 = 0 ∀1 ≤ m ≤ 3,

Appendix: Some Technical Statements 3 

181

K 0k K k j = K 00 K 0 j +

k=0

K 0k K k j =

k=1

−

3  k=1

So,

3 

3 

 K

mk

Kk j =

k=0

 1  1 − |w|2 w j 0 w

 w  δk j − w k w j = 0 ∀1 ≤ j ≤ 3. 0 w k

1 if m = j 0 if i = j

∀ m, j = 0, 1, 2, 3.

(A.66)

Therefore, we deduce that {K mn }0≤m,n≤3 is an inverse matrix to {K mn }0≤m,n≤3 ∈ R4×4 and, in particular, in the case w 0 = 0 matrix {K mn }0≤m,n≤3 ∈ R4×4 is invertible. Next, in the case w := (w1 , w 2 , w 3 ) = 0 we are done, since then matrix {K mn }0≤m,n≤3 ∈ R4×4 obviously satisfies, ⎧ 00 0 2 ⎪ ⎨ K = (w ) K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j K = K j0 = 0 ∀1 ≤ j ≤ 3 ,

(A.67)

2

and thus it has one positive eigenvalue λ0 = (w 0 ) , three negative eigenvalues λ1 = λ2 = λ3 = −1 and we have (A.65). Thus, we assume from now that w 0 = 0 and w := (w 1 , w 2 , w 3 ) = 0. Furthermore, observe that, by (A.62) for every (z 0 , z 1 , z 2 , z 3 ) ∈ R4 , with z := (z 1 , z 2 , z 3 ) ∈ R3 , we have 3 

K z j = K z0 + 0j

00

j=0

3 

2

K 0 j z j = (w 0 ) z 0 + w 0 (w · z)

(A.68)

j=1

and 3 

K m j z j = K m0 z 0 +

j=0

3 

K m j z j = z 0 w0 wm +

j=1

= z 0 w w − z m + (w · z)w 0

m

3    −δ jm + w j w m z j j=1

m

∀ m = 1, 2, 3.

(A.69)

In other words, ⎧ 3 ⎪ ⎪ ⎪ K 0 j z j = (w 0 z 0 + w · z)w 0 ⎨ ⎪ ⎪ ⎪ ⎩

j=0 3 j=0

(A.70) K m j z j = −z m + (w 0 z 0 + w · z)w m

∀ m = 1, 2, 3.

182

Appendix: Some Technical Statements

Next, if (z 0 , z 1 , z 2 , z 3 ) ∈ R4 , with z := (z 1 , z 2 , z 3 ) ∈ R3 , satisfies 

1 z = λ+1 w 1 z 0 = λ w0 ,

(A.71)

where λ ∈ R is a solution of the following quadratic equation   λ2 + 1 − (w 0 )2 − |w|2 λ − (w 0 )2 = 0

(A.72)

(obviously that if w 0 = 0 and w = 0, then λ ∈ {−1, 0} does not satisfies (A.72)). Thus in particular, by (A.71) and (A.72) we have: w0 z 0 + w · z =

1 02 1 (λ + 1)|w 0 |2 + λ|w|2 |w | + |w|2 = = 1. λ λ+1 λ(λ + 1)

(A.73)

Therefore, by (A.73) and (A.70) we deduce: ⎧ ⎪ ⎪ ⎪ ⎨

3

K 0 j z j = w0

j=0 ⎪ 3

⎪ ⎪ ⎩

(A.74) K m j z j = −z m + w m

∀ m = 1, 2, 3.

j=0

However, by (A.71) we have 

w = (λ + 1)z w0 = λ z 0 .

(A.75)

Therefore, by (A.75) and (A.74) we deduce: 3 

K m j z j := λz m

∀ m = 0, 1, 2, 3.

(A.76)

j=0

So, we obtain that, if λ is a root of the quadratic equation (A.72) then, by (A.76) (z 0 , z 1 , z 2 , z 3 ) ∈ R4 , with z := (z 1 , z 2 , z 3 ) ∈ R3 , given by (A.71), is an eigenvector of the matrix {K mn }0≤m,n≤3 . In other words, every root of the quadratic equation (A.72) is an eigenvalue of the matrix {K mn }0≤m,n≤3 . However, by Vieta’s formulas the quadratic equation (A.72) has two distinct real roots λ0 > 0 and λ1 < 0 and λ0 λ1 = −(w 0 )2 . Moreover, if w 0 = 0 and w = 0 then −1 does not satisfies (A.72), and so, λ0 > 0 and − 1 = λ1 < 0 are two eigenvalues of the matrix {K mn }0≤m,n≤3 and λ0 λ1 = −(w 0 )2 . (A.77)

Appendix: Some Technical Statements

183

Furthermore, if λ ∈ R and (z 0 , z 1 , z 2 , z 3 ) ∈ R4 , with z := (z 1 , z 2 , z 3 ) ∈ R3 , satisfy ⎧ ⎪ ⎨λ = −1 z·w =0 ⎪ ⎩ z0 = 0 ,

(A.78)

then by (A.78) and (A.70) we have ⎧ ⎪ ⎪ ⎪ ⎨

3

j=0 ⎪ 3

⎪ ⎪ ⎩

K 0 j z j = 0 = λz 0 (A.79) K m j z j = −z m = λz m

∀ m = 1, 2, 3.

j=0

However, obviously there exists two orthogonal unit vectors z1 ∈ R3 and z2 ∈ R3 such that z1 · w = z2 · w = z1 · z2 = 0 . Therefore, in the case λ = −1, if (z 0 , z 1 , z 2 , z 3 ) ∈ R4 satisfies z 0 = 0 and either (z 1 , z 2 , z 3 ) = z1 or (z 1 , z 2 , z 3 ) = z2 , then by (A.79) we deduce that in both cases (z 0 , z 1 , z 2 , z 3 ) is an eigenvector of of the matrix {K mn }0≤m,n≤3 . So, λ2 = −1 and λ3 = −1 are two coinciding eigenvalues of the matrix {K mn }0≤m,n≤3 .

(A.80)

Therefore, by (A.77) and (A.80) we deduce that the matrix {K mn }0≤m,n≤3 has one positive and three negative eigenvalues and moreover, λ0 λ1 λ2 λ3 = −(w 0 )2 .

(A.81)

However, it is well known from the linear algebra that   λ0 λ1 λ2 λ3 = det {K mn }0≤m,n≤3 and thus, by (A.81) we finally deduce (A.65).



Lemma A.6 Let {mn }0≤m,n≤3 ∈ R4×4 be a degenerate symmetric matrix, given by ⎧ 00 ⎪ ⎨ = 0  jm = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j  =  j0 = 0 ∀1 ≤ j ≤ 3 , with



δjj = 1 δ jm = 0

∀ j = 1, 2, 3 ∀ j = m = 1, 2, 3 ,

(A.82)

(A.83)

184

Appendix: Some Technical Statements

and let (w 0 , w 1 , w 2 , w 3 ) ∈ R4 be such that w0 = 0 .

(A.84)

Then, ∈ R4×4 , such that  a non-degenerate matrix {Amn }0≤m,n≤3  there exists mn det {Amn }0≤m,n≤3 = 0, and if we consider a matrix { }0≤m,n≤3 ∈ R4×4 , defined by 3 3   mn := Amk An j k j ∀ m, n = 0, 1, 2, 3, (A.85) j=0 k=0

and a vector (w 0 , w 1 , w 2 , w 3 ) ∈ R4 , defined by w  j :=

3 

A jk w k

∀ j = 0, 1, 2, 3,

(A.86)

k=0

then we have:

and

⎧ 00 ⎪ ⎨ = 0  jm = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0 j  =  j0 = 0 ∀1 ≤ j ≤ 3 ,

(A.87)

(w 0 , w 1 , w 2 , w 3 ) = (1, 0, 0, 0).

(A.88)

Proof (Proof of Lemma A.6) Consider a a non-degenerate matrix {Amn }0≤m,n≤3 ∈ R4×4 defined by: ⎧ A00 = w10 ⎪ ⎪ ⎪ ⎨A = δ jm jm ∀1 ≤ j, m ≤ 3 (A.89) wj ⎪ = − ∀1 ≤ j ≤ 3 A j0 0 ⎪ w ⎪ ⎩ A0 j = 0 ∀1 ≤ j ≤ 3 . Then, by (A.86) and (A.89) we obtain w  j = A j0 w 0 +

3  k=1 j

3  wj A jk w k = − 0 w 0 + δ jk w k w k=1

= −w + w = 0 ∀ j = 1, 2, 3, j

and w 0 = A00 w 0 +

3  k=1

A0k w k =

1 w0

(A.90)

w 0 + 0 = 1.

(A.91)

Appendix: Some Technical Statements

185

So we deduce (A.88). On the other hand, by (A.85) and (A.82) we deuce: mn



=

3 3  

Amk An j  = kj

3 3  

j=0 k=0

=

3 

Amk An j δk j

j=1 k=1

∀ m, n = 0, 1, 2, 3.

Am j An j

(A.92)

j=1

Therefore, by (A.92) and (A.89) we infer 00



=

3 

A20 j = 0 ,

(A.93)

j=1

m0 = 0m =

3 

Am j A0 j = 0

∀ m = 1, 2, 3,

(A.94)

∀ m, n = 1, 2, 3.

(A.95)

j=1

and mn =

3  j=1

Am j An j =

3 

δm j δn j = δmn

j=1

So, by (A.93), (A.94) and (A.95) we also deduce (A.87). This completes the proof.  Lemma A.7 Let {K mn }0≤m,n≤3 ∈ R4×4 be a non-degenerate symmetric matrix, given by ⎧ 00 ⎪ ⎨K = 1 (A.96) K jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j j0 K = K = 0 ∀1 ≤ j ≤ 3 , and let (w 0 , w 1 , w 2 , w 3 ) ∈ R4 be such that the real number M, given by 3  2   j 2 w  , M := w 0  −

(A.97)

j=1

satisfies M > 0.

(A.98)

186

Appendix: Some Technical Statements

Then, ∈ R4×4 , such that  a non-degenerate matrix {Amn }0≤m,n≤3  there exists mn det {Amn }0≤m,n≤3 = 0, and if we consider a matrix {K }0≤m,n≤3 ∈ R4×4 , defined by 3 3   K mn := Amk An j K k j ∀ m, n = 0, 1, 2, 3, (A.99) j=0 k=0

and a vector (w 0 , w 1 , w 2 , w 3 ) ∈ R4 , defined by w  j :=

3 

A jk w k

∀ j = 0, 1, 2, 3,

(A.100)

k=0

then we have:

and

⎧ 00 ⎪ ⎨K = 1 K  jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0 j K = K  j0 = 0 ∀1 ≤ j ≤ 3 , (w 0 , w 1 , w 2 , w 3 ) =

√

 M, 0, 0, 0 .

(A.101)

(A.102)

Proof (Proof of Lemma A.7) Without any loss of generality may assume M= 1  we 0 1 2 3 w 0 1 2 3 in (A.97), otherwise we just replace (w , w , w , w ) by √ M , √wM , √wM , √wM . So consider from now that M = 1. In other words, we have 3  0 2   j 2 w  − w  = 1 .

(A.103)

j=1

Then, by (A.103), using Lemma A.4 we deduce that a matrix {mn }0≤m,n≤3 ∈ R4×4 , defined by ⎧ 00 0 2 ⎪ ⎨ = (w ) − 1 (A.104)  jm = δ jm + w j w m ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0j  =  j0 = w 0 w j ∀1 ≤ j ≤ 3, has one vanishing and three positive eigenvalues. Therefore, by the Sylvester’s }0≤m,n≤3 ∈ R4×4 , such that lawof inertia, there  exists a non-degenerate matrix {Bmn mn det {Bmn }0≤m,n≤3 = 0, and if we consider a matrix { }0≤m,n≤3 ∈ R4×4 , defined by 3  3  mn  := Bmk Bn j k j ∀ m, n = 0, 1, 2, 3, (A.105) j=0 k=0

Appendix: Some Technical Statements

187

⎧ 00 ⎪ ⎨ = 0  jm = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0 j  =  j0 = 0 ∀1 ≤ j ≤ 3 .

then we have

(A.106)

However, by (A.101) and (A.104) we have K mn = w m w n − mn :

∀ m, n = 0, 1, 2, 3.

(A.107)

Thus, if we consider a vector (w 0 , w 1 , w 2 , w 3 ) ∈ R4 , defined by w  j :=

3 

B jk w k

∀ j = 0, 1, 2, 3,

(A.108)

k=0

and a matrix {K mn }0≤m,n≤3 ∈ R4×4 , defined by K mn :=

3 3  

Bmk Bn j K k j

∀ m, n = 0, 1, 2, 3,

(A.109)

j=0 k=0

then, by (A.107) and (A.109) we deduce K mn = w m w n − mn :

∀ m, n = 0, 1, 2, 3.

(A.110)

Then, by (A.110) and (A.106) we obtain ⎧ 2 00 ⎪ = (w 0 ) ⎨K K  jm = w  j w m − δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0 j = K  j0 = w 0 w  j ∀1 ≤ j ≤ 3. K

(A.111)

In particular, since the matrix {K mn }0≤m,n≤3 is non-degenerate (follows by the fact that {K mn }0≤m,n≤3 is non-degenerate), we deduce from (A.111) that we necessary have w 0 = 0 . (A.112) Therefore, we can apply Lemma A.6 to deduce that, there exists a non-degenerate  matrix {Amn }0≤m,n≤3 ∈ R4×4 , such that det {Amn }0≤m,n≤3 = 0, and if we consider a matrix {mn }0≤m,n≤3 ∈ R4×4 , defined by mn :=

3 3   j=0 k=0

Amk An j k j

∀ m, n = 0, 1, 2, 3,

(A.113)

188

Appendix: Some Technical Statements

and a vector (w 0 , w 1 , w 2 , w 3 ) ∈ R4 , defined by j

w :=

3 

Ajk w k

∀ j = 0, 1, 2, 3,

(A.114)

k=0

⎧ 00 ⎪ ⎨ = 0  jm = δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0 j  =  j0 = 0 ∀1 ≤ j ≤ 3 ,

then we have:

and

(w 0 , w 1 , w 2 , w 3 ) = (1, 0, 0, 0).

(A.115)

(A.116)

Therefore, considering a non-degenerate matrix {Amn }0≤m,n≤3 ∈ R4×4 , defined by Amn :=

3 

Amk Bkn

∀ j = 0, 1, 2, 3,

(A.117)

k=0

  we obviously deduce, det {Amn }0≤m,n≤3 = 0 and moreover, by (A.105) and (A.113) we have 3  3  mn := Amk An j k j ∀ m, n = 0, 1, 2, 3, (A.118) j=0 k=0

and by (A.108) and (A.114) we have w  j :=

3 

A jk w k

∀ j = 0, 1, 2, 3.

(A.119)

k=0

Thus, if we consider a matrix {K mn }0≤m,n≤3 ∈ R4×4 , defined by K mn :=

3 3  

Amk An j K k j

∀ m, n = 0, 1, 2, 3,

(A.120)

j=0 k=0

then, by (A.118), (A.119) and (A.107) we deduce K mn = w m w n − mn

∀ m, n = 0, 1, 2, 3.

(A.121)

Appendix: Some Technical Statements

189

Finally, inserting (A.115) and (A.116) into (A.121) gives ⎧ 00 ⎪ ⎨K = 1 K  jm = −δ jm ∀1 ≤ j, m ≤ 3 ⎪ ⎩ 0 j K = K  j0 = 0 ∀1 ≤ j ≤ 3 , and

(w 0 , w 1 , w 2 , w 3 ) = (1, 0, 0, 0),

(A.122)

(A.123)

where {Kmn }0≤m,n≤3 is given by (A.120) and (w 0 , w 1 , w 2 , w 3 ) is given by (A.119)  with det {Amn }0≤m,n≤3 = 0. This completes the proof.