A Theory of Physical Vacuum. A New Paradigm, Part II 5-7273-0011-8

The monograph contains the results of the programe of the geometri- zation of the equations of physics advanced by Einst

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A Theory of Physical Vacuum. A New Paradigm, Part II
 5-7273-0011-8

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Table of contents :
Part II: GEOMETRY OF ABSOLUTE PARALLELISM

5 Geometry of absolute parallelism in vector basis
5.1 Object of anholonomicity. Connection of absolute parallelism
5.2 Covariant di erentiation in A4 geometry. Ricci rotation coeficients
5.3 Curvature tensor of A4 space
5.4 Formalism of external forms and the matrix treatment of Cartan's structural equations of the absolute parallelism geometry
5.5 A4 geometry as a group manifold. Killing-Cartan metric
5.6 Structural equations of A4 geometry in the form of expanded, completely geometrized Einstein-Yang-Mills set of equations
5.7 Equations of geodesics of A4 spaces
5.8 Structural equations of right and left A4 geometry

6 The geometry of absolute parallelism in spinor basis
6.1 Three main spinor bases of A4 geometry
6.2 Spinor representation of the structural Cartan equations of A4 geometry
6.3 Splitting of structural Cartan equations into irreducible representations of the group SL(2,C)
6.4 Carmeli matrices
6.5 Component-by-component rendering of structural equations of A4 geometry
6.6 Connection of structural Cartan equations of A4 geometry with the NP formalism
6.7 Variational principle of derivation of the structural Cartan equations and the second Bianchi identities of A4 geometry
6.8 Decomposition of spinor elds of A4 geometry into irreducible parts
6.9 Spinor set of Einstein-Yang-Mills equations
6.10 Formalism of two-component spinors

7 Construction of solutions to structural Cartan equations of the geometry of absolute parallelism
7.1 Selection of a frame of reference and specialization of Newman-Penrose symbols
7.2 Specialization of the spinor components of the Ricci rotation coeficients
7.3 Specialization of the spinor components of the Riemann tensor
7.4 Construction of the asymptotic behavior of insular-type geometries
7.4.1 Radial equations containing derivatives with respect to r
7.4.2 Nonradial equations
7.4.3 U-derivative equations
7.5 Classification of solutions to the structural Cartan equations of the A4 geometry by isometry groups
7.6 A4 geometry with a Schwarzschild-type metric
7.7 Some physically meaningful solutions of the structural Cartan equations of A4 geometry
7.7.1 Solution with a variable source function
7.7.2 Solution with quark interaction
7.7.3 Solution with a short-range (nuclear) interaction
7.7.4 Solution with an electronuclear interaction
7.7.5 Solution with electronuclearquark interaction
7.7.6 Solution with Coulomb-Newton interaction and three-dimensional rotation of a source
7.7.7 Purely torsional solution
7.7.8 Solution with a variable Coulomb-Newton interaction and three-dimensional rotation of a source
7.7.9 Solution with electronuclear interaction and three-dimensional rotation of a source

Citation preview

RUSSIAN ACADEMY OF NATURAL SCIENCES INTERNATIONAL INSTITUTE FOR THEORETICAL AND APPLIED PHYSICS

G. I. Shipov

A THEORY OF PHYSICAL VACUUM A New Paradigm

Moscow 1998

UDC 530.1 Reviewers: Professor of Physics, State University of Moscow R.N.Kysmin, Professor of Physics, Physical Institute of RAN A.A.Ruchadze.

Shipov G.I

A Theory of Physical vacuum: A New Paradigm. This book was originally published in the Russian language by Moscow ST-Center, Russia 1993. { 362 p. English edition { 312 p. ISBN 5-7273-0011-8 The monograph contains the results of the programe of the geometrization of the equations of physics advanced by Einstein at the beginning of the 20th century and developed by the author who gives the idea of the universal principle of relativity and the theory of physical vacuum and analyses theoretical and experimental e ects of the theory. The second part of the monograph develops the mathematical system of the physical vacuum theory. It contains the main characteristics of the absolute parallelism geometry in vector and spinor basis. The book is intended for specialists in theoretical physics, teachers, post-graduates, students and everyone interested in new physical theories.

1604030000 02 5 T O (03) 93

Without announcement

ISBN 5-7273-0011-8

c

c

c

G.I.Shipov, 1998 Intrnational Institute for Theoretical and Applied physics RANS, 1998 Translated into English by A.P.Repiev, 1998

3

Preface This book gives a concise presentation of ideas and methods used by the author to develop the Cli ord-Einstein program of the geometrization of the equations of physics, and also to solve various fundamental problems of modern theoretical physics proceeding from the concept of the universal principle of relativity and the theory of physical vacuum. In his studies the author made an attempt to combine phenomena of seemingly di erent nature and to sketch a coherent picture of modern physics. The author is most grateful to V. Yu. Tatur and all those who, directly or indirectly, made the publication of this book possible. Special thanks are due to my friends and colleagues E. A. Gubarev, A. N. Sidorov, and I. A. Volodin. Many ideas expounded in this book were presented in my rst monograph published in 1979 with a support of M. A. Adamenko and I. S. Lakoba at Moscow University Press. I remember with gratitude my productive talks with V. Skalsky, an Associate Professor at Slovak Polytechnic, who made some valuable points about various vacuum states of matter. Useful comments of A. E. Akimov have been encouraging in many respects for my studies of torsion elds and interactions. The attention and support of all these persons contributed enormously to the publication of this book. Last but not least, the author must record his deep obligation to Elena Turantaeva who was good enough to edit the book and read the proofs.

1993

Gennady Shipov

Preface to English edition The translation of the book into English is shortened as far as it doesn't contain the 5th chapter of the Russian edition. This chapter is dedicated to phenomena of seemingly di erent nature and its absence doesn't in uence the main scienti c results.

1998

Gennady Shipov

4

Conventions

1

Three-dimensional tensor indices are denoted by the Greek letters ; ; ; : : : and take the values 1, 2, 3. Three-dimensional vectors (e.g., linear and angular velocity) are denoted as: ~v and ~! or v and !: Four-dimensional tensor indices are denoted by Latin letters i; j; k : : :; they assume the values 0, 1, 2, 3. Letters from the rst part of the alphabet (a; b; :: :; h) are used as tetrad indices, e.g., ei a ; (a = 0; 1; 2; 3): Spinor indices in the spinor -basis are denoted by Roman capitals A; B : : :; C_ : : : D_ and take the values 0,1 or 0_ ; 1_ . Spinor indices in the -basis are labeled by Greek letters ; ; : : : ; ; _ Æ_ : : :: Symmetrization and antisymmetrization of pairs of indices: 1 S(ij ) = 1 (Sij + Sji); S[ij ] = (Sij Sji ):

2 2 Exclusion of an index from symmetrization or antisymmetrization:

S(ijj jk) = 1 (Sijk + Skji ); S[ijj jk] = 1 (Sijk Skji ): 2 2 Passing over to local (tetrad) indices: S abc = ea i S ijk ej b ek c. External product: ea ^ ec = aa ec ec ea .  The Levi-Chivita pseudotensor: "ijkm ; dual tensor: S ij = 12 "ijkm S km . The matrix representation of (a) tensor quantities : S a bk or, discarding the matrix indices a and b, S abk ! Sk ; (b) spin-tensor quantities: S ABC_ Dk ! Sk . _ A matrix product: [Tm; Tk ] = Tm Tk Tk Tm . Hermitian conjugate matrices: S +B Dkn . _ Derivatives

Partial derivatives with respect to the translational coordinates xi are labeled by a comma in front of an index, i.e., f;k = @f=@xk = @k f ; a covariant derivative with respect to the Christo el symbols i jk is denoted by rk or rk ui = @k ui + i uj . jk A local covariant derivative: raub = @aub + bca uc .  A covariant derivative rk with respect to the connection ijk = eia eaj;k of  the A4 geometry: rk ui = @k ui + ijk uj . 1 The following is a list of only some important notations. All the conventions are explained in the text.

5 An external derivative: d. A spinor derivative: @AB_ . Translational metric and tetrads

Translational coordinates: x0 ; x1 ; x2 ; x3 . The metric signature: (+ ): A translational linear element: ds2 = abeai ebj dxi dxj , ab =  ab = diag(1 1 1 1) The structural equations of the group of translations of the A4 geometry: r[arb] xi = ab: : c rc xi . 1-form of the tetrad: ea = eai dxi . Rotational metric and torsion

Rotational coordinates: '1 ; '2 ; '3 ; 1 ; 2 ; 3 : Rotational metric: d 2 = db adab = T abiT baj dxi dxj , dab = dab : 1 i a a : :i = ei ea The torsion of A4 geometry: jk a [k;j ] = 2 e a (e k;j e j;k ): The contorsion tensor of A4 geometry (the rotational Ricci coeÆcients): : :s + g : :s ) = ei r ea : Tjki = jk: :i + g im(gjs mk ks mj a k j

1-form of contorsion: T ab = T abk dxk = T abc ec ; T(ab) = 0: The structural equations of a rotational group (the matrix indices are discarded): r[k rm] ei = 12 Rkm ei , where Rkm = 2r[mTk] + [Tm; Tk ]: Connection and curvature of A4 geometry

Connection: ijk = i jk + T ijk = ei a eaj;k ; : :i ; i = i + g im(g :: s + g : :s ): = T[ijk] = jk js mk ks mj jk (jk) Curvature:

i[jk]

i Sjkm = 2ij [m;k] + 2is[ksjj jm] = = Rijkm + 2r[k Tjijjm] + 2Tci[k Tjcj jm] = 0

where Rijkm = 2 ij [m;k] + 2 is[k sjj jm] | the Riemann tensor. 1-form of connection: ab = abk dxk = abc ec : The Cartan structural equations: (a) rst structural equations: dea ec ^ T ac = 0; (b) second structural equations: Rab + dT ab + T cb ^ T ac = 0: Spinor

-basis

Newman-Penrose symbols: iAB_ :

6 Translational metric: gij = "AC "B_ D_ iAB_ jC D_ , where "AB is a fundamental spinor   0 1 _D _ AB C " = "AB = " = "C_ D_ = 1 0 : The rotational Ricci coeÆcients:

TABC = Ci D_ rk ABi _ Dk _ _ :

The rotational Ricci coeÆcients in terms of Carmeli matrices: TAB_ with matrix elements (TAB_ )C D : The Riemann curvature in terms of Carmeli matrices: RABC _ D _: The equations of physical vacuum written in Carmeli matrices:

@C D_ Ai B_

+ R_ @AB_ Ci D_ = (TC D_ )AP Pi B_ + Ai B_ (TDC _ _ ) B

(TAB_ )C P Pi D_

RF ED @F E_ TDB_ _ B_ = @DB _ TF E _

+ R_ ; Ci R_ (TBA _ _ ) D

(TDB_ )F S TS B_

(As) + F_ (TED _ ) B_ TF F_ +

+ F_ +(TF E_ )D S TS B_ + (TEF _ ; TDB _ ]: _ ) B_ TDF_ + [TF E + Hermitially conjugate equations.

(B s+ )

Part II GEOMETRY OF ABSOLUTE PARALLELISM

9 Introduction

Geometry with absolute parallelism was rst considered in 1923-24 in the works of Weitzenbock [1, 2] and Vitali [3, 4]. Weitzenbock suggested that there exist in the n-dimensional manifold M with coordinates x1 ; : : : ; xn of Riemannian spaces with a zero Riemann-Christo el tensor

S ijkm = 2ij [m;k] + 2is[ksjj jm] = 0:

(4.1)

Relationship (4.1) was regarded as the condition of parallel displacement of an arbitrary vector in a given space in the absolute (independent of path) sense. In 1924 Vitali introduced the concepts of the connection of absolute parallelism [3] kij = eka eai;j; (4.2)

;j =

@ @xj ;

i; j; k : : : = 0; 1; 2; 3; a; b;c : : : = 0; 1; 2; 3;

where eka and eai are basic vectors de ned at each point of space and translatable in the absolute sense to any point of the space in any direction. Weitzenbock [5] showed that the connection (4.2) can be represented as the sum i ; ijk = ijk + Tjk

where

1 = g im(gjm;k + gkm;j 2 are the Christo el symbols and i jk

(4.3)

gjk;m);

im s ::s Tjki = ::i jk + g (gjs mk + gks mj )

(4.4) (4.5)

are the Ricci rotation coeÆcients [6] for the basis eai . The tensor ::i jk , de ned as 1 i a i a a

::i (4.6) jk = e a e[k;j ] = 2 e a (e k;j e j;k ); came to be known as the anholonomity object [7], therefore the emergence of the geometry of absolute parallelism continued the development of anholonomic di erential geometry [8]. Cartan and Schouten [9, 10], proceeding from the group properties of the space of constant curvature, introduced the connection (4.3), in which the components of the Ricci rotation coeÆcients (4.5) are constants. Cartan and Schouten reasoned as follows. Suppose that in a n - dimensional di erentiable manifold M with the coordinates x1 ; : : :; xn we have n contravariant vector elds aj = aj (xk ); (4.7) where a; b; c : :: = 1 : : : n

10

PART 2.

are vector indices, and

i; j; k : : : = 1 : : :n

are coordinate indices. Suppose that

det(aj ) 6= 0 and that the functions aj satisfy the equations

aj  kb;j

::f  k ; bi  ka;j = Cab f

::f have the following properties: where the constants Cab ::f = C ::f ; Cab ba

(4.8)

::a C ::f + C ::a C ::f + C ::a C ::f = 0: Cfb (4.9) fc db fd bc cd We can then say that we have an n-parametric simple transitive group (group ::f are structural constants of the Tn ) operating in the manifold such that Cab group that obey the Jacobi identity (4.9). The vector eld bj is said to be

in nitesimal generators of the group. Let now the basis ekb , de ned at each point of the manifold M , meet the condition det(eja ) 6= 0: If we suppose that eja (xk0 ) = aj (xk0 ); where xk0 are the coordinates of some arbitrary point P , then we have for the function eja (xk0 ) the equations

ejaekb;j

::f ek : ejb eka;j = Cab f

(4.10)

It follows from the normalization condition for the basis

eai eja = Æij ; eaiei b = Æba;

(4.11)

::i = 2ei ea i ::a b c Cjk a [k;j ] = e aC bc ej ek :

(4.12)

and from (4.10), that Comparing (4.8) and (4.6), we see that 1 ::i

::i jk = 2 Cjk ; i.e., the components of the anholonomity object of a homogeneous space of absolute parallelism are constant. It is easily seen that the connection (4.2) possesses a torsion. In our speci c case 1 ::i k k[ij ] = ::k ij = T[ij ] = 2 Cjk :

11

INTRODUCTION

It was exactly in this manner that Cartan and Schouten introduced connection with torsion [9, 10]. Therefore, the development of the geometry of absolute parallelism brought about the emergence of the Riemann-Cartan geometry with the connection 1 ijk = ijk + (Cijk Cjki Ckij ); (4.13) 2 where Sijk = 12 Cijk is the torsion of space. Further development of the geometry of absolute parallelismin the n-dimensional di erentiable manifold M with coordinates x1 ; : : : ; xn (geometries An ) is described in the works of Bortolotti [11{14], Griss [15], Schouten [16, 17], Eisenhart [18] and other authors [19-25]. Speci cally Bortolotti [12] was the rst to point out that the Cartan-Schouten connection and the Weinzbock-Vitali (4.2) connection is one and the same thing. Besides, Bortolotti showed that the tensor (4.1) can be represented as the sum

S ijkm = Ri jkm + 2r[k Tjij jm] + 2Tci[k Tjcjjm] = 0; where

Ri jkm = 2

i i s j [m;k] + 2 s[k jj jm] i are given quantities Tjk

(4.14) (4.15)

is the Riemann tensor, and the by (4.5). In 1937 Thomas [20, 21] approached absolute parallelism as parallel displacement of vectors "in toto," since the connection of space An (just as that of a at space En) is integratble. Therefore, a vector speci ed at some point An can be speci ed at any other point of space. Lastly, the works [23{25] give a classi cation of spaces with absolute parallelism. Geometry A4 has been rst used by Einstein [26] in applications to problems of theoretical physics. The scientist made an attempt to combine the equations of his theory with the equations of the Maxwell-Lorentz electrodynamics [27]. We note in passing that within the framework of the geometry of absolute parallelism Einstein has written most (all in all 13) works. By developing Einstein's program to construct a uni ed eld theory, this author came to the conclusion that it is necessary to use the A4 geometry as a geometry of space of events in universal relativity theory and the theory of physical vacuum. Unlike Einstein and his following, the author employed Cartan's structural equations of the geometry of absolute parallelism, which are generalizations of Einstein's vacuum equations Rik = 0 for the case where the energy-momentum tensor on the right-hand side of Einstein's equations is geometric in nature. The program of uni ed eld theory put forward by Einstein boils down to solving two strategic problems of modern theoretical physics: (a) the minimum program has it as its goal to geometrize the equations of electromagnetic eld and to combine them with the equations of Einstein's theory of gravitation; (b) the maximum program is aimed at the search for completely geometrized equations of the gravitational and electromagnetic eld (including sources), i.e., the geometrization of the elds that form matter.

12

PART 2.

Although much time was devoted to this search (around 30 years), Einstein failed to solve the problem in a form acceptable to science. Together with many outstanding scientists of the time he wrote a wealth of works relying on various geometries. But all of them failed to meet the above requirements (a) and (b). Also, it was unclear how to geometrize spin elds (e.g., Dirac's eld) that are sources of electromagnetic elds. Wheeler added to the program of uni ed eld theory a further point that required a spinor treatment of the equations of the uni ed eld. The latter condition can be met in the case where the main geometric quantities of the theory are spinors rather than tensors. A spinor treatment of classical geometries was given in the works by Penrose [38, 40, 54], which was of much help to me in my constructing a theory of physical vacuum, a present-day outgrowth of Einstein's program of uni ed eld theory.

Chapter 5

Geometry of absolute parallelism in vector basis 5.1 Object of anholonomicity. Connection of absolute parallelism Consider a four-dimensional di erentiable manifold with coordinates xi (i = 0; 1; 2; 3) such that at each point of the manifold we have a vector eai (i = 0; 1; 2; 3) and a covector ejb (b = 0; 1; 2; 3) with the normalization conditions

eaieja = Æij ; eai ei b = Æba :

(5.1)

For arbitrary coordinate transformations 0 dxi

0

@xi = k dxk @x

(5.2)

in coordinate index i the tetrad eai transforms as a vector

i eai0 = @xi0 eai: (5.3) @x In the process, in the tetrad index a relative to the transformations (5.2) it

behaves as a scalar. Tetrad eai de nes the metric tensor of a space of absolute parallelism

gik = abeai ebk ; ab =  ab = diag(1

1

1

1)

(5.4)

and the Riemannian metric

ds2 = gik dxi dxk :

(5.5)

Using the tensor (5.4) and the normal rule [29], we can construct the Christoffel symbols i = 1 g im(g (5.6) jm;k + gkm;j gjk;m): jk 2 13

14

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

that transform following a nontensor law of transformation [29] k00 0 ji

=

@ 2xk @xk0 @xi @xj @xk0 @xi0 @xj 0 @xk + @xi0 @xj 0 @xk

k ji

(5.7)

with respect to the coordinate transformations (5.2). In the relationship (5.6) and farther on we will denote the partial derivative with respect to the coordinates xi as ;k = @ : (5.8)

@xk

Di erentiating the arbitrary vector eai gives

@xj eai;j0 = @xj 0 eai;j :

(5.9)

Applying the di erentiation operation (5.9) to the relationship (5.3) gives

@ 2xi @xi @xj (5.10) eai0 ;j 0 = @xi0 @xj 0 eai;j + @xi0 @xj 0 eai : Alternating the indices i0 and j 0 and subtracting from (5.10) the resultant

expression, we have

eai0 ;j 0

@xi @xj eaj 0 ;i0 = (eai;j eaj;i) @xi0 @xj 0 :

Considering (5.3), we can rewrite this relationship in the form 0

eka(eai0 ;j 0

i j k0 eaj 0;i0 ) = eka (eai;j eaj;i) @xi0 @xj 0 @x k : @x @x @x

By de nition, the di erential

dsa = eai dxi

(5.11)

is said to be complete, if the following relationship holds:

eai;j eaj;i = 0:

(5.12)

Otherwise, for eai;j eaj;i 6= 0, the di erential (5.11) is not integrable (equality (5.12) is the condition of integration for the relationship (5.11)). We will introduce the following geometric object [30] 1 i a a i a (5.13)

::i jk = e a e[k;j ] = 2 e a (e k;j e j;k ) with a tensor law of transformation relative to the coordinate transformations (5.2) 0

::i j 0 k0

=

@xj

::i jk @xj 0

0

@xk @xi : @xk0 @xi

(5.14)

15

5.2. COVARIANT DIFFERENTIATION. . .

Clearly, if the condition (5.12) is met, this object vanishes. In that case, tetrad eai is holonomic and the metric (5.5) characterizes holonomic di erential geometry. If the object (5.13) is nonzero, we deal with anholonomic di erential geometry, and the object (5.13) itself is called an object of anholonomicity. We will rewrite the relationship (5.10) in the following manner:

@ 2 xi ea + @xi @xj ea = @xi0 @xj 0 i @xi0 @xj 0 i;j  @ 2 xk + @xi @xj k ea ; = @xi0 @xj 0 @xi0 @xj 0 ij k

eai0 ;j 0 =

(5.15)

where we have introduced the notation kij = ekaeai;j

(5.16)

and used the orthogonality condition (5.1). It is seen from the relationships (5.15 ) that the object kij gets transformed relative to the transformations (5.2) as the connection 0

0

0

@ 2 xk @xk + @xi @xj @xk k : @xi0 @xj 0 @xk @xi0 @xj 0 @xk ij

(5.17)

2 k k0 i @xj @xk0 @ x @x @x = j 0 i0 k + j 0 i0 k kji: @x @x @x @x @x @x

(5.18)

ki0 j 0 =

The connection of a space given by (5.16) is called the connection of absolute parallelism [31]. Interchanging in (5.17) the indices i and j gives 0 kj 0i0

Subtracting (5.18) from (5.17) gives 0 k[i0j 0 ]

@xi @xj @xk0 k = j 0 i0 k [ij ]: @x @x @x

(5.19)

It follows from the relationships (5.16) and (5.13) that the connection of absolute parallelism features the torsion k[ij] = ::k ij ; de ned by the object of anholonomity.

(5.20)

5.2 Covariant di erentiation in A4 geometry. Ricci rotation coeÆcients The de nition of the covariant derivative with respect to the connection of the geometry of absolute parallelism (A4 geometry) ijk from a tensor of arbitrary i:::p has the form valence Um:::n  i:::p = U i:::p + i U j:::p + : : : + p U i:::j rk Um:::n jk m:::n m:::n;k jk m:::n (5.21) i:::p : : : j U i:::p : jmk Uj:::n nk m:::j

16

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

This de nition enables some quite useful relationships in A4 geometry to be proved. Proposition 5.1. Parallel displacement of the tetrad eai relative to the connection ijk equals zero identically. Proof. From the de nition (5.21) we have the following equalities:  (5.22) rk ei a = aia;k + ijk eja;  (5.23) rk eaj = eaj;k ijk eai : Since the connection ijk is de ned as ijk = ei a eaj;k ;

(5.24)

we have

ei aeaj;k ijk = 0: Multiplying this equality by eai and taking into consideration the orthogonality conditions (5:1), we get 

rk eaj = eaj;k

ijk eai = 0:

(5.25)

To prove that the relationship (5.22) is zero, we will take a derivative of the convolution eaj ei a = Æji (Æji );k = (eaj eia);k = ei a eaj;k + eaj eia;k = 0: Hence, by (5.24), we have

ijk = eaj ei a;k

or

(5.26)

eaj eia;k + ijk = 0: Multiplying this relationship by eja and using the conditions eaj ei a = Æji , we have



rk ei a = eia;k + ijk eja = 0:

Proposition 5.2.

(5.27)

Connection ijk can be represented as the sum i ; ijk = ijk + Tjk

(5.28)

where ijk are the Christo el symbols given by the relationship (5.6), and im ::s ::s Tjki = ::i jk + g (gjs mk + gks mj )

(5.29)

are the Ricci rotation coeÆcients [30]. Proof. Let us represent the connection (5.28) as the sum of parts symmetrical and skew-symmetrical in indices j; k ijk = i(jk) + i[jk];

(5.30)

17

5.2. COVARIANT DIFFERENTIATION. . .

where

1 1 i(jk) = (ijk + ikj ); i[jk] = (ijk ijk ): 2 2 We now add to and subtract from the right-hand side of (5.30) the same expression ijk = i(jk) + i[jk] + g im(gjss[km] + gks s[jm]) (5.31) g im(gjss[km] + gks s[jm]): We then group the terms on the right-hand side of (5.31) as follows: ijk = i(jk) g im(gjss[km] + gks s[jm]) + + i[jk] + g im(gjss[km] + gks s[jm]):

(5.32)

Since

i[jk] = ::i jk ; it follows from (5.32) and (5.29) that ijk = i(jk)

g im(gjss[km] + gks s[jm]) + Tjki :

(5.33)

g im(gjss[km] + gks s[jm]):

(5.34)

We now show that i jk

= i(jk)

Actually, we have the relationships 1 i(jk) = ei a ea(j;k) = ei a(eaj;k + eak;j ); 2 1 i[jk] = ei a ea[j;k] = ei a (eaj;k eak;j ); 2 gjs = ab eaj ebs; therefore (5.34) become

(5.35)

i jk

= eia ea(j;k) + g im(ab eaj eb[m;k] + ab eak eb[m;j ]) = 1 b a b c =  cd ab eic em d (em e j;k + em e k;j ) + 2  1 + g im ab (eaj ebm;k eaj ebk;m ) + ab(eak ebm;j eak ebj;m) : 2 Regrouping the terms here gives i jk

1 = g im (abeaj ebm );k + (ab eak ebm );j 2



(ab eaj ebk );m :

Hence, by (5.35), we obtain i jk

1 = g im(gjm;k + gkm;j 2

gjk;m);

(5.36)

18

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

or 1 im g (gjm;k + gkm;j gjk;m ) = 2 = i(jk) g im (gjss[km] + gkss[jm]) = ijk :

(5.37)

Substituting (5.37) into (5.33), we get the relationship (5.28). i can be represented Proposition 5.3. The Ricci rotation coeÆcients Tjk in the form Tjki = ei ark eaj ; (5.38)

Tjki = eaj rk ei a ;

(5.39)

where rk stands for a covariant derivative with respect to the Christo el ijk symbols. Proof. We will represent in the relationships (5.25) and (5.27) the connection ijk as the sum (5.28) 

rk eaj = eaj;k 

rk ei a = eia;k +

i ea jk i

Tjki eai = 0;

i ej + T i ej jk a jk a

=0

(5.40) (5.41)

Since, by de nition [29], we can write

rk eaj = eaj;k

i ea ; jk i

rk ei a = eia;k +

i ej ; jk a

then (5.40) and (5.41) can be written as

rk eaj Tjki eai = 0;

(5.42)

rk ei a + Tjki eja = 0:

(5.43)

Multiplying (5.42) by ei a and (5.43) by eaj , respectively, we will obtain (using the orthogonality conditions (5.1)), by (5.42), (5.43), the relationships (5.38) and (5.39).  We will now calculate the covariant derivative rk with respect to the metric tensor g jm, knowing that g jm =  ab eja emb 





rk g jm =rk  ab eja emb =rk eja ema =   = ema rk eja + eja rk ema : From the relationships (5.25) and (5.27), we have  r g jm = 0:

(5.44)

19

5.3. CURVATURE TENSOR. . .

On the other hand, applying the formula (5.21) to the relationship (5.44), we nd that  (5.45) rk g jm = g;kjm + jpk g pm + mpk g jp = 0: Substituting the connection ijk as the sum (5.28), we will write the relationship (5.45) in the form  (5.46) rk g jm = rk g jm + Tpkj g pm + Tpkm g jp = 0: From the equality

rk g jm = g;kjm +

j g pm + m g jp pk pk

= 0;

(5.47)

we have, by (5.46),

Tpkj g pm + Tpkmg jp = Tkjm + Tkmj = 0: This equality establishes the following symmetry properties for the Ricci rotation coeÆcients: Tjmk = Tmjk : (5.48) Therefore, in the A4 geometry the Ricci rotation coeÆcients have 24 independent components.

5.3 Curvature tensor of A4 space The curvature tensor of the space of absolute parallelism S ijkm is de ned in terms of the connection ijk following a conventional rule [18]

S ijkm = 2ij [m;k] + 2is[ksjj jm] = 0;

(5.49)

where the parentheses [ ] signify alternation in appropriate indices, whereas the index within the vertical lines j j is not subject to alternation. Proposition 5.4. The Riemann-Christo el tensor of a space with the connection (5.26) equals zero identically. Proof. From the relationship (5.26) we have

eaj;k = ijk eai :

(5.50)

Di erentiating the relationship (5.50) with respect to m gives

eaj;k;m = (ijk eai );m = ijk;m eai + eai;mijk = = (ijk;m + ei a eas;msjk )eai = (ijk;m + ismsjk )eai: Alternating this relationship in indices k and m we get 2eaj;[k;m] = 2(ij[m;k] + 2is[k sjj jm]) = S ijkm eai :

(5.51)

20

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

Since the operation of di erentiating with respect to indices k and m is symmetrical, we have eaj;[k;m] = 0; From this equality, considering that eai in (5.51) is arbitrary, we will get

S ijkm = 0: Proposition 5.5.

Tensor S ijkm can be represented as the sum

S ijkm = Ri jkm + 2r[k Tjijjm] + 2Tci[k Tjcj jm] = 0; where

(5.52)

Rijkm = 2

i i s j [m;k] + 2 s[k jj jm]

(5.53) (5.54)

is the tensor of the Riemannian space A4 . i into (5.49) gives Proof. Substituting the sum ijk = ijk + Tjk

S ijkm = 2

i i s i i s j [m;k] + 2 s[k jj jm] + 2Tj [m;k] + 2Ts[k Tjj jm] + 2Tsi[k sjj jm] + 2 is[k Tjsjjm] = 0:

(5.55)

Using (5.54), we will write (5.55) as follows:

S ijkm = Ri jkm + 2Tji[m;k] + 2Tsi[k Tjsj jm] + +2 sj[k Tjisjm] + 2 is[k Tjsjjm] = 0:

(5.56)

If now we add to the right-hand side of this relationship the expression 2 s[km] Tsji = 0; and take into consideration that [29] i:::p = U i:::p + rk Um:::n m:::n;k

i U j:::p + : : : + p U i:::j jk m:::n jk m:::n j U i:::p : : : j U i:::p ; mk j:::n nk m:::j

(5.57)

we will obtain from (5.56) the equality (5.53). Let us now rewrite the relationship (5.53) as

Ri jkm = 2Tji[m;k] 2Tsi[k Tjsj jm]: Substituting here (5.38) and (5.39)

Tjki = ei ark eaj ; Tjki = eaj rk eia ; we obtain

2Tji[m;k] = 2ei ar[k rm] eaj 2r[k eijajrm] eaj ; 2Tsi[k Tjsj jm] = 2easr[k eija esajrm] eaj = 2r[k eijaj rm]eaj :

(5.58)

21

5.3. CURVATURE TENSOR. . .

Therefore, it follows from the relationships (5.58) that

Ri jkm = 2ei a r[krm] eaj = 2eia r[mrk] eaj : Proposition 5.6.

equations

(5.59)

The torsion eld ::i jk of the A4 space satis es the 

r k ::i jm

::i + 2 ::s (5.60) [kj m]s = 0: Proof. Alternating the expression (5.49) in indices j; k; m and using the relationship i[jk] = ::i jk , we get [

]

i ::s S[ijkm] = 2 ::i [jm;k] + 2s[k jm] = 0:

(5.61)

If then we add and subtract here the quantity s ::i 2s[kj ::i jsjm] + 2[km j ]s;

we will have i ::s 2 ::i [jm;k] + 2s[k jm]

s ::i 2s[kj ::i jsjm] 2[km j ]s + s ::i 2s[kj ::i jsjm] + 2[km j ]s = 0:

Using the formula (5.21), we can rewrite this relationship as follows:  ::s ::i ::s ::i 2 r[k ::i jm] 2 [kj jsjm] 2 [km j ]s =  ::s ::i = 2 r[k ::i jm] + 4 [kj m]s = 0;

(5.62)

whence we have (5.60). Proposition 5.7.

the equality

The Riemann tensor Ri jkm of the A4 space satis es

Ri[jkm] = 0:

(5.63) Proof. Alternating the relationship (5.54) in indices j; k; m and using the equality T[ijk] = ::i jk ; we have i ::s Ri[jkm] = 2r[k ::i jm] + 2Ts[k jm]: If in the right-hand side of the equality we add and subtract the quantity s ::i 2T[skj ::i jsjm] + 2T[km j ]s;

we obtain i ::s s ::i s ::i Ri[jkm] = 2r[k ::i jm] + 2Ts[k jm] 2T[kj jsjm] 2T[km j ]s +

 s ::i ::i ::s ::i +2T[skj ::i jsjm] + 2T[km j ]s = 2 r[k jm] 2 [kj jsjm]  ::i ::i ::s ::i 2 ::s [km j ]s = 2 r[k jm] + 4 [kj m]s = 0;

which proves the validity of the relationship (5.63).

22

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

5.4 Formalism of external forms and the matrix treatment of Cartan's structural equations of the absolute parallelism geometry Consider the di erentials

dxi = eaei a ;

(5.64)

dei b = ab eia ;

(5.65)

where

ea = eai dxi ; (5.66) a a i a k  b = e i de b =  bk dx (5.67) are di erential 1-forms of tetrad eai and connection of absolute parallelism abk .

Di erentiating the relationships (5.64), (5.65) externally [31], we have, respectively, d(dxi ) = (dea ec ^ ac )ei a = S a ei a ; (5.68)

d(dei a) = (dba ca ^ bc )ei b = S baei b :

(5.69)

Here S a denotes the 2-form of Cartanian torsion [31], and S ba { the 2-form of the curvature tensor. The sign ^ signi es external product, e.g,

ea ^ eb = ea eb

eb ea :

(5.70)

By de nition, a space has a geometry of absolute parallelism, if the 2-form of Cartanian torsion S a and the 2-form of the Riemann-Christo el curvature S ba of this space vanishe S a = 0; (5.71)

S ba = 0:

(5.72)

At the same time, these equalities are the integration conditions for the di erentials (5.64) and (5.65). Equations dea ec ^ ac = S a ; (5.73)

dba ca ^ bc = S ba ;

(5.74)

which follow from (5.68) and (5.69), are Cartan's structural equations for an appropriate geometry. For the geometry of absolute parallelism hold the conditions (5.71) and (5.72), therefore Cartan's structural equations for A4 geometry have the form dea ec ^ ac = 0; (5.75)

dba ca ^ bc = 0:

(5.76)

23

5.4. FORMALISM OF EXTERNAL FORMS. . .

Considering (5.28), we will represent 1-form ab as the sum ab = ab + T ab :

(5.77)

Substituting this relationship into (5.75) and noting that

ec ^ ac = ec ^ T ac ; we get the rst of Cartan's structural equations for A4 space.

dea ec ^ T ac = 0:

(A)

Substituting (5.77) into (5.76) gives the second of Cartan's equations for A4 space. Rab + dT ab T cb ^ T ac = 0; (B ) where Rab is the 2-form of the Riemann tensor

Rab = d

a b

b^

c

a: c

(5.78)

By de nition [31], we always have the relationships

dd(dxi ) = 0;

(5.79)

dd(dei a ) = 0:

(5.80)

In the geometry of absolute parallelism these equalities become

d(dea ec ^ T ac ) = Racfdec ^ ef ^ ed = 0; d(Rab + dT ab T cb ^ T ac ) = dRab + Rfb ^ T af Here

T fb ^ Raf = 0:

(5.81) (5.82)

Racfd = 2T ac[d;f ] 2T ab[f T bjcjd]:

Equalities (5.81) and (5.82) represent the rst and second of Bianchi's identities, respectively, for A4 space. Dropping the indices, we can write Cartan's structural equations and Bianchi's identities for the A4 geometry as

de e ^ T = 0; R + dT T ^ T = 0; R ^ e ^ e ^ e = 0; dR + R ^ T T ^ R = 0:

(A) (B ) (C ) (D )

Proposition 5.8. The matrix treatment of the rst of Cartan's structural equations (A) of the A4 geometry has the form

r k eam [

]

eb[k T ajbjm] = 0:

(5.83)

24

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

Proof. Let us write equations (A) as ec ^ T ac = 0:

dea

(5.84)

Further, by (5.66), we have

dea = d(eam dxm ) = rk eam dxk ^ dxm = 1 (rk eam 2

rmeak )dxk ^ dxm

and, also,

eb ^ T ab = ebk T abm dxk ^ dxm = 1 (ebk T abm ebm T abk )dxk ^ dxm : 2

Substituting these relationships into equations (5.84) we will derive the matrix equations in the form

r k eam [

eb[k T ajbjm] = 0;

]

(A)

where the matrixes eam and T abm in world indices i; j; m; : : : are transformed as vectors m (5.85) eam0 = @xm0 eam ;

@x @xm T abm0 = @xm0 T abm ; and in the matrix indices a; b;c; : : : they are transformed as follows: 0

0

eam = aa eam ; 0

0

(5.86) (5.87)

0

T ab0 k = aa T abk bb0 + aa ab0 ;k : m0

(5.88)

In relationships (5.87) and (5.88) the matrices @x =@xm form a translation i group T4 that is de ned 0 on a manifold of world coordinates x . On the other a hand, the matrices  a form a group of four-dimensional rotations O (3:1) 0

aa

2 O (3:1);

de ned on the manifold of "angular coordinates" ea i . Actually, the tetrad ea i is a mathematical image of an arbitrarily accelerated four-dimensional reference frame. Such a frame has ten degrees of freedom: four translational ones connected with the motion of its origin, and six angular ones describing variations of its orientation. The six independent components of the tetrad ea i represent six direction cosines of six independent angles de ning the orientation of the tetrad in space. Proposition 5.9. The matrix rendering of the second of Cartan's structuring equations (B ) of the A4 geometry has the form

Rabkm + 2r[k Tjabjm] + 2T ac[k T cjbjm] = 0:

(5.89)

25

5.4. FORMALISM OF EXTERNAL FORMS. . .

Proof. We will expand the 2-form Ra d as 1

1

Rab = 2 Rabcd ec ^ ed = 2 Rabkm dxk ^ dxm :

(5.90)

Further, we have

dT ab = d(T abm dxm ) = rk T abm dxk ^ dxm = 1 (r T a 2 k bm

rm T abk )dxk ^ dxm ;

(5.91)

and also

T ac ^ T cb = T ack T cbm dxk ^ dxm =

1 = (T ack T cbm T cbm T ack )dxk ^ dxm : 2 Let us substitute the relationships (5.92){(5.94) into

(5.92)

Rab + dT ab T cb ^ T ac = 0: Simple transformations yield 1 a (R + rk T abm rm T abk + T ack T cbm T cbm T ack )dxk ^ dxm = 0: 2 bkm Since here the factor dxk ^ dxm is arbitrary, we have

Rabkm + rk T abm rm T abk + T ack T cbm T cbm T ack = 0; which is equivalent to the equations (5.89). Proposition 5.10. The matrix form of the Bianchi identity (D ) of A4 geometry is r[nRajbjkm] + Rcb[km T ajcjn] T cb[n Rajcjkm] = 0: (5.93) Proof. The external di erential dRa b in the identities (D ) has the 2-form 1 dRab = 2 rn Rabkm dxn ^ dxk ^ dxm = 1 = (rnRabkm + rm Rabkn + rk Rabmn )dxn ^ dxk ^ dxm : (5.94) 6 In addition, we have 1 Rfb ^ T af = 2 Rfbkm T afndxk ^ dxm ^ dxn = 1 = (Rfbkm T afn + Rfbnk T afm + Rfbmn T afk )dxk ^ dxm ^ dxn ; (5.95) 6 1 2 1 f a f = (T bn R fkm + T bm Rafnk + T fbk Rafmn)dxn ^ dxk ^ dxm : 6

T fb ^ Raf = T fbn Rafkm dxn ^ dxk ^ dxm =

(5.96)

26

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

Substituting relationships (5.94){(5.96) into the identity

dRab + Rfb ^ T af

T fb ^ Raf = 0

and considering that dxn ^ dk ^ dxm is arbitrary, we get

rnRabkm + rm Rabkn + rk Rabmn + Rfbkm T afn + Rfbnk T afm + +Rfbmn T afk

T fbn Rafkm T fbm Rafnk T fbk Rafmn = 0;

which is equivalent to the identity (5.93). The rst of Bianchi's identities (C ) of A4 geometry in indices of the group O (3:1) is written as Ra[bcd] = 0; (5.97) or, which is the same, as  ::f ::a (5.98) r[b ::a cd] + 2 [bc d]f = 0:

5.5 A4 geometry as a group manifold. Killing-Cartan metric The matrix representation of Cartan's structural equations of the geometry of absolute parallelism indicates that, in fact, this space behaves as a manifold, on which the translations group T4 and the rotations group O (3:1) are speci ed. We will consider A4 geometry as a group 10-dimensional manifold formed by four translational coordinates xi (i = 0; 1; 2; 3) and six (by the relationship ea i ej a = Æi j ) angular coordinates eai (a = 0; 1; 2; 3). Suppose that on this manifold a group of four-dimensional translations T4 and a rotations group O (3:1) are de ned. We then introduce the Hayashi invariant derivative [32]

rb = ekb @k ;

(5.99)

whose components are generators of the translations group T4 that is speci ed on the manifold of translational coordinates xi . If then we represent as a sum

ekb = Æ kb + akb ; eld akb

(5.100)

i; j; k : : : = 0; 1; 2; 3; a; b; c;: : : = 0; 1; 2; 3;

then the can be viewed as the potential of the gauge eld of the translations group T4 [32]. In the case where akb = 0, the generators (5.99) coincide with the generators of the translations group of the pseudo-Euclidean space E4 . We know already that in the coordinate index k the nonholonomic tetrad eka transforms as the vector 0 eka

0

@xk = k eka ; @x

5.5.

A4 GEOMETRY . . .

27

whence, by (5.100), we have the law of transformation for the eld aka relative to the translationss 0 0 @xk 0 @xk0 akb = n anb + n Æ nb Æ kb : (5.101)

ei

@x

@x

We de ne the tetrad a as

ei a = ra xi

(5.102)

and write the commutational relationships for the generators (5.99) as

r arb [

]

= ::c ab rc ;

(5.103)

where ::c ab are the structural functions for the translations group of the spaceA4. If then we apply the operator (5.103) to the manifold xi , we will arrive at the structural equations of the group T4 of the space A4 as

r arb xi = [

or

]

i

::c ab rc x

r aei b

i = ::c ab e c : In this relationship the structural functions ::c ab are de ned as [

]

c i

::c ab = e ir[ae b] :

(5.104) (5.105) (5.106)

It is seen from this equality that when the potentials of the gauge eld of translations group akb in the relationship (5.100) vanish, so do the structural functions (5.106). Therefore, we will refer to the eld ::c ab as the gauge eld of the translations group. Considering that T c[ab] = ::c ab , we will rewrite the structural equations (5.106) as r[k eam] eb[k T ajbjm] = 0: (5.107) It is easily seen that the equations (5.107) can be derived by alternating the equations (5.42). What is more, they coincide with the structural Cartan equations (A) of the geometry of absolute parallelism. The structural equations of group T4 , written as (5.106), can be regarded as a de nition for the torsion of space A4 . So the torsion of space A4 coincides with the structural function of the translations group of this space, such that the structural functions obey the generalized Jacobi identity  ::f ::a (5.108) r[b ::a cd] + 2 [bc d]f = 0;  where rb is the covariant derivative with respect to the connection of absolute parallelism abc . Comparing the identity (5.108) with the Bianchi identity (5.98) of the geometry A4 , we see that we deal with the same identity. The Jacobi identity (5:108), which is obeyed by the structural functions of the translations

28

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

group of geometry A4 , coincides with the rst Bianchi identity of the geometry of absolute parallelism . The vectors ei a = raxi ; (5.109) that form the vector strati cation [31] of the A4 geometry, point along the tangents to each point of the manifold xi of the pseudo-Euclidean plane with the metric tensor ab =  ab = diag(1; 1; 1; 1): (5.110) Therefore, the ten-dimensional manifold (four translational coordinates xi and six "rotational" coordinates ei a) of the geometry of absolute parallelism can be regarded as the strati cation with the coordinates of the base xi and the (anholonomic) "coordinates" of the bre ei c : If on the base xi we have the translations group T4 , then in the bre ei c we have the rotation group O (3:1). It follows from (5.109) that the in nitesimal translations in the base xi in the direction a are given by the vector

dsa = eaidxi : (5.111) If from (5.111) and the covariant vector dsa = ei a dxi we form the invariant convolution ds2 , we will obtain the Riemannian metric of A4 space ds2 = gik dxi dxk (5.112) with the metric tensor

gik = abeai ebk :

Therefore, the Riemannian metric (5.112) can be viewed as the metric de ned on the translations group T4 . Since in the bre we have the "angular coordinates" ei a that form a manifold in which group O (3:1) is de ned, then it would be natural to de ne the structural equations for this group, as well as the metric speci ed on the group O (3:1). Let us rewrite the relationships (5.38) and (5.39) in matrix form

T abk = eai T ijk ejb = rk eaj ejb ;

(5.113)

T abk = eaiT ijk ejb = eairk ei b :

(5.114) These relationships enable the dependence between the in nitesimal rotation dab = d ba of the vector eai at in nitesimal translations dsa to be established. In fact, by (5.113) and (5.114), we have

dab = T abk dxk = Deaj ejb ;

(5.115)

dab = T abk dxk = eaiDei b :

(5.116) where D is the absolute di erential [29] with respect to the Christo el symbols i . Using (5.115), we can form the invariant quadratic form d 2 = da db a jk b to arrive at the Killing-Cartan metric

d 2 = dab dba = T abk T bandxk dxn = Deai Dei a

(5.117)

5.5.

A4 GEOMETRY . . .

29

with the metric tensor

Hkn = T abk T ban :

(5.118)

Unlike metric (5.112), the metric (5.117) is speci ed on the rotations group

O (3:1) that acts on the manifold of the "rotational coordinates" eai . Let us now introduce the covariant derivative  rm = rm + Tm ;

(5.119)

where Tm is the matrix T abm with discarded matrix indices. We will regard the components of the derivative as generators of the rotations group O (3:1). Applying this operator to the tetrad ei that forms the manifold of "angular coordinates" of the A4 geometry, we will arrive at  (5.120) rm ei = rm ei + Tm ei = 0; hence

Tm = ei rm ei :

(5.121)

It is interesting to note that, just as in (5.109) we have de ned six "angular coordinates" ei a through the four translational coordinates xi , so in (5:121) we can de ne 24 "supercoordinates" T abm through the six coordinates eia . It follows from (5.120) that

rm ei = Tm ei :

(5.122)

Recall that in the relationships (5.120)-(5.122) we have de ned through rm the covariant derivative with respect to ijk . We will now take the covariant derivative rk of the relationships (5.122)

rk rm ei = rk (Tm ei ) = (rk Tm ei + Tm rk ei ) = = (rk Tm ei + Tm ei ei rk ei ): Using (5.121), we will rewrite this expression as follows

rk rm ei = (rk Tm Tm Tk )ei: Alternating this expression in the indices k and m gives

r k rm ei = 12 Rkm ei ; [

where

]

(5.123)

Rkm = 2r[mTk] + [Tm; Tk ]:

(5.124) Introducing in equations (5.124) the matrix indices (the bre indices), we will obtain the structural equation of the group O (3:1)

Rabkm = 2r[m T ajbjk] + 2T ac[m T cjbjk] :

(B )

30

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

It is easily seen that the structural equations of the rotations group (B ) coincide with the second of Cartan's structural equations (5.124) of the geometry A4 . In this case the quantities T abk and Rabkm vary in the rotations group O (3:1) following the law 0 0 0 T ab0 k = aa T abk bb0 + aa ab0 ;k ; (5.125) and appear as the potentials of the gauge eld Rabkm of the rotations group O (3:1). In the process, the gauge eld of the group O (3:1) obeys the formula 0

0

Rab0 km = aa Rabkm bb0 :

(5.126)

Note that the structural functions of the rotations group of A4 geometry are the components of the curvature tensor Rabkm . It can be shown that the structural functions Rabkm of the rotations group O (3:1) satisfy the Jacobi identity

r nRajbjkm [

]

+ Rcb[km T ajcjn]

T cb[nRajcjkm] = 0;

(D )

which, at it was shown in the previous section, are at the same time the second Bianchi identities of the A4 space. Let us introduce the dual Riemann tensor  1 (5.127) Rijkm = 2 "spkm Rijsp; where "spkm is the completely skew-symmetrical Levi-Chivita tensor. Then the equations (D) can be written as    rn R a b kn + R c b kn T acn T cbn R a c kn = 0 (5.128) or, if we drop the matrix indices, as rn R kn + R kn Tn



Tn R kn = 0:

(5.129)

5.6 Structural equations of A4 geometry in the form of expanded, completely geometrized Einstein-Yang-Mills set of equations Einstein believed that one of the main problems of the uni ed eld theory was the geometrization of the energy-momentum tensor of matter on the righthand side of his equations. This problem can be solved if we use as the space of events the geometry of absolute parallelism and the structural Cartan equations for this geometry. In fact, folding the equations (B ); written as

Rijkm + 2r[k Tjij jm] + 2Tsi[k Tjsj jm] = 0

(5.130)

in indices i and k, gives

Rjm = 2r[i Tjijjm] 2Tsi[iTjsj jm]:

(5.131)

31

5.6. STRUCTURAL EQUATIONS. . .

If then we fold the equations (5.131) with the metric tensor g jm, we have

R = 2g jm (r[iTjij jm] + 2Tsi[iTjsj jm]):

(5.132)

Forming, using (5.131) and (5.132), the Einstein tensor

Gjm = Rjm

1 g R; 2 jm

we obtain the equations 1 g R = Tjm ; (5.133) 2 jm which are similar to Einstein's equations, but with the geometrized right-hand side de ned as Tjm = 2 f(r[iT ijj jm] + T is[iT sjj jm])  1 g g pn (r[iT ijpjn] + T is[iT sjpjn])g (5.134) 2 jm Using the notation

Rjm

Pjm = (r[iT ijj jm] + T is[iT sjj jm]) then, by (5.134), we have

Tjm =

2



(Pjm

1 g g pn Ppn ): 2 jm

(5.135)

Tensor (5.135) has parts that are both symmetrical and skew-symmetrical in indices j and m, i.e., Tjm = T(jm) + T[jm]: (5.136) The left-hand side of the equations (5.133) is always symmetrical in indices j and m, therefore these equations can be written as

Rjm

1 g R = T(jm) ; 2 jm

1

where

T[jm] =  ( ri ::i jm rm Aj Aj = Tjii :

As ::s jm ) = 0;

(5.137) (5.138)

(5.139) Relationship (5.138) can be taken to be the equations obeyed by the torsion elds ::i jm , which form the energy-momentum tensor (5.135). i is skew-symmetrical in all the three indices, In the case where the eld Tjk we get Tijk = Tjik = Tjki = ijk : (5.140)

32

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

For such elds the equations (5.138) become simple, namely

ri ::i jm = 0:

(5.141)

The energy-momentum tensor (5.135) is symmetrical in indices j; m and appears to be given by

Tjm = 1 ( ::i

::s  sm ji

1 g :ji ::s ): 2 jm s ji

(5.142)

By (5.137), we have

Tjm = 1 (Rjm 

1 g R): 2 jm

(5.143)

Using (5.131), (5.140) and (5.142) gives ::s Rjm = ::i sm ji ;

(5.144)

::s :ji ::s R = g jm ::i sm ji = s ji :

(5.145) Substituting (5.144) and (5.145) into (5.143), we arrive at the energy-momentum tensor (5.142). Through the eld (5.140) we can de ne the pseudo-vector hm as follows

ijk = "ijkm hm ; ijk = "ijkm hm ;

(5.146)

where "ijkm is the fully skew-symmetrical Levi-Chivita symbol. In terms of the pseudo-vector hm we can write the tensor (5.142) as follows 1 g hi h ): 2 jm i Substituting the relationships (5.146) into (5.141), we get

Tjm = 1 (hj hm 

hm;j hj;m = 0:

(5.147)

(5.148)

These equations have two solutions: the trivial one, where hm = 0, and

hm =

;m ;

(5.149)

where is a pseudo-scalar. Writing the energy-momentum tensor (5.148) in terms of this pseudo-scalar, we will have Tjm = 1 ( ;j ;m 1 gjm ;i ;i): (5.150)  2 Tensor (5.150) is the energy-momentum tensor of a pseudo-scalar eld. Let us now decompose the Riemann tensor Rijkm into irreducible parts 1 3

Rijkm = Cijkm + gi[kRm]j + gj [k Rm]i + Rgi[mgk]j ;

(5.151)

33

5.6. STRUCTURAL EQUATIONS. . .

where Cijkm is the Weyl tensor; the second and third terms are the traceless part of the Ricci tensor Rjm and R is its trace. Using the equations (5.133), written as 

Rjm =  Tjm



1 g T ; 2 jm

(5.152)

we will rewrite the relationship (5.151) as 1 T gi[mgk]j ; 3

Rijkm = Cijkm + 2g[k(iTj )m]

(5.153)

where T is the tensor trace (5.135). Now we introduce the tensor current

Jijkm = 2g[k(iTj )m]

1 Tg g 3 i[m k]j

(5.154)

and represent the tensor (5.153) as the sum

Rijkm = Cijkm + Jijkm :

(5.155)

Substituting this relationship into the equations (5.130), we will arrive at

Cijkm + 2r[k Tjijjm] + 2Tis[k Tjsj jm] = Jijkm :

(5.156)

Equations (5.156) are the Yang-Mills equations with a geometrized source, which is de ned by the relationship (5.154). In equations (5.156) for the YangMills eld we have the Weyl tensor Cijkm , and the potentials of the Yang-Mills i . eld are the Ricci rotation coeÆcients Tjk We now substitute the relationship (5.155) into the second Bianchi identities (D ) r[nRjij jkm] + Rsj[km Tjisjn] Tjs[nRjisjkm] = 0: (5.157) We thus arrive at the equations of motion

r nCjij jkm + Cjs km Tjisjn Tjs nCjisjkm [

]

[

]

[

]

= Jnijkm

(5.158)

for the Yang-Mills eld Cijkm , such that the source Jnijkm in them is given in terms of the current (5.154) as follows:

Jnijkm = r[nJjij jkm] + Jjs[km Tjisjn] Tjs[nRjisjkm] :

(5.159)

Using the geometrized Einstein equations (5.133) and the Yang-Mills equations (5.156), we can represent the structural Cartan equations (A) and (B ) as an extended set of Einstein-Yang-Mills equations

r keaj + T ikj eai = 0;

(A) [ ] [ ] (B:1) Rjm 12 gjm R = Tjm ; i + 2r T i i s Cjkm J ijkm ; (B:2) [k jj jm] + 2Ts[k Tjj jm] =

(5.160)

34

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

in which the geometrized sources Tjm and Jijkm are given by (5.135) and (5.154). For the case of Einstein's vacuum the equations (5.160) are much simpler

r k eaj + T ikj eai = 0;

(i) Rjm = 0; (ii) C ijkm + 2r[k Tjijjm] + 2Tsi[k Tjsjjm] = 0: (iii) [

]

[

]

(5.161)

The equations of motion (5.158) for the Yang-Mills eld Cijkm will then become r[nCjij jkm] + Cjs[km Tjisjn] Tjs[nCjisjkm] = 0: (5.162) Equations (A) and (B:2) can be written in matrix form

r k eam [

eb[k T ajbjm] = 0;

]

C abkm + 2r[k T ajbjm] + 2T af [k T fjbjm] = J abkm ;

(A) (B:2)

where the current

J abkm = 2g[k (a Tb)m] is given by

1

1 a T g [m gk]b ; 3 1

T am =  (Ram 2 g am R); m = 0; 1; 2; 3; a = 0; 1; 2; 3:

(5.163) (B:1)

By writing the equations (5.158) in matrix form, we have

r nC ajbjkm [

where

]

+ C cb[km T ajcjn]

T cb[nC ajajkm] = J anbkm ;

J anbkm = r[nJ ajbjkm] + J cb[km T ajcjn] T cb[nJ ajcjkm] :

(5.164) (5.165)

Dropping the matrix indices in the matrix equations, we have

r k em [

]

e[k Tm] = 0;

Ckm + 2r[k Tm] [Tk ; Tm] = Jkm ;

rn C kn + [C kn ; Tn] =  J k ;

  where the dual matrices C kn and J k are given by  C kn = "knij Cij ;  J nk = "nkim Jim;    J k = frn J kn + [J kn ; Tn]g:

(A) (B:2) (D )

(5.166) (5.167)

35

5.7. EQUATIONS OF GEODESICS. . .

For the Einstein vacuum we have



Rijkm = Cijkm =R ijkm = Cijkm ;

(5.168)

therefore the equations (B:2) and (D ) become simpler

Ckm + 2r[k Tm] [Tk ; Tm] = 0;

(B:2)

rnC kn + [C kn ; Tn] = 0:

(D ) Using the formalismof external di erential forms, we can write the structural equations (A) and (B:2) as follows:

dea eb ^ T ab = 0; C ab + dT ab T ac ^ T cb = J ab ;

(A) (B:2)

and the equations (D) as

dC ab + C af ^ T fb T fb ^ C af = N ab ;

(D )

where

N ab = dJ ab + J af ^ T fb T fb ^ J af : (5.169) Thus, the structural equations of A4 geometry, written as (5.160), represent

an extended set of Einstein-Yang-Mills equations with the gauge translations group T4 de ned on the base xi with the structural equations (A); and with the gauge rotations group O (3:1), de ned in the bre ei a with the structural equations in the form of the geometrized equations (B:1) and (B:2):

5.7 Equations of geodesics of A4 spaces

The equations of geodesics for the geometry of absolute parallelism can be obtained from the conditions of parallel vector displacement

dxi ui = ds with respect to the connection of A4 geometry i = ei ea : ijk = ij + Tjk a j;k

(5.170) (5.171)

In fact, we specialize the tetrad ei a so that the vector ei 0 would coincide with the tangent to the world line, i.e.,

dxi ei 0 = ui = ds : From the relationships (5:27) for the vector (5.172) we have  rk ui = ui;k + ijk uj = 0

(5.172)

(5.173)

36 or

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

@ui i j i j @xk + jk u + Tjku = 0: Multiplying this by uk = dxk =ds gives dui + ds

i uj uk jk

i uj uk = 0 + Tjk

(5.174)

(5.175)

or, by (5.170),

j k j k d2 xi i dx dx + T i dx dx = 0: + (5.176) jk jk ds2 ds ds ds ds These four equations (i = 0; 1; 2; 3) are the equations of geodesics of A4 space. They are also the equations of motion for the origin O of tetrad ei a. Since in i have both symmetrical the equations (5.176) the Ricci rotation coeÆcients Tjk and skew-symmetrical parts in indices j and k

Tjki = T(ijk) + T[ijk] = im ::s ::s = ::i jk + g (gjs mk + gks mj ); ::s T(ijk) = g im(gjs ::s mk + gks mj ); T[ijk] = ::i jk ;

(5.177) (5.178) (5.179)

we can write the equations (5.176) as

d2 xi ds2 +

j i dx jk ds

dxk dxj dxk + T(ijk) ds ds ds = 0:

(5.180)

Considering the structure of the equality (5.178), we will write it in the form ::s im T(ijk) = g im(gjs ::s mk + gks mj ) = 2g m(jk) ;

(5.181)

hence the equations of geodesics for A4 space can be represented as

d2 xi + ds2

j i dx jk ds

dxk + 2g im

dxj dxk = 0: m (jk) ds ds ds

(5.182)

For the terns in (5.181) we can introduce the following notation: i im ::s

:ik:j = g imgks ::s jm ; :jk = g gks mj ; i for space A will become then the contorsion tensor Tjk 4

Tjki = ::i jk where whence

:ik:j + i:jk ;

(5.183)

:ik:j = i:jk ; i Tjki = ::i jk + 2 :jk :

(5.184)

37

5.7. EQUATIONS OF GEODESICS. . .

The covariant di erential of an arbitrary vector v i with respect to the connection (5.171) for parallel displacement from point xi to point xi + dxi becomes

Æv i = dv i + ijk dxj = 0:

(5.185)

If at an arbitrary point xi of A4 space we have two linear elements Æxi and and make a parallel translation of Æxi along the element dxi , then for the nal point we will have [30]

dxi

xi + dxi + Æxi ijkÆxk dxj = xi + dxi + Æxi + dÆxi :

(5.186)

On the other hand, parallel translation of the vector dxi along the vector

Æxi gives

xi + Æxi + dxi

ijkdxk Æxj = xi + Æxi + dxi + Ædxi :

(5.187)

Subtracting from the relationships (5.186) the equality (5.187), we get

dÆxi

Ædxi = (ijk Æxk dxj + ijkdxk Æxj ) = (ijk ikj )Æxk dxj = 2i[jk]Æxk dxj = k j ::s j k = 2 ::s jk Æx dx = 2 jk Æx dx :

(5.188)

Let us now consider the variation of the integral Z b

a

L(xi ; ui )ds;

(5.189)

where ui is given by the relationship (5.170). We will write (5.188) as j k Ædxi = dÆxi + 2 ::s jk Æx dx :

(5.190)

Then at each point of the extremum we have k dxi d j dx Æui = Æ ds = ds Æxi + 2 ::i jk Æx ds :

(5.191)

Applying a common variational procedure to the integral (5.189), we get Z b Z b

a

a

ÆL(xi; ui )ds = 

L(xi + Æxi ; ui + Æui) L(xi; ui ) ds = =

Z b @L

a

@x

i i Æx +



@L i Æu ds = 0: @ui

Substituting here the relationship (5.191) gives Z b @L

a

@xi

@xi +



@L d i @L ::i j k @ui ds @x + @ui 2 jk @x u ds = 0:

(5.192)

38

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

We now integrate by parts the second term here to obtain Z b @L



d @L @L k i + 2 ::j ik i ds @u @uj u @x = 0

@xi

a

or, since @xi is arbitrary, we arrive at [30]

d @L ds @ui Let now

@L @L k + 2 ::j ki i @x @uj u = 0: L = (gik ui uk )1=2 ;

(5.193) (5.194)

along the extremum L = 1 by the relationship

gik ui uk = ui ui = 1: Substituting the Lagrangian (5.194) into equations (5.193) gives i

gmi du + ds

j uk

mjk u

k j + 2 ::s mj gsk u u = 0:

(5.195)

Multiplying this relationship by g im, we get

or

dui + ds

i uj uk kj

j k + 2g imgks ::s mj u u = 0

dui + ds

i uj uk kj

+ 2g im m(jk) uj uk = 0:

(5.196)

We have thus obtained, using the variational principle, the equations of the geodesics in the form (5.182). Consider now the equations that describe the variation of the orientation of the tetrad ei a as it moves according to the equations of the geodesics (5.196). We will rewrite the equations (5.43) as

@k ei a + ijk eja = 0 or

dei a + ijkeja dxk = 0: Dividing these equations by ds yields dei a + i ej dxk = 0: jk a ds ds

(5.197)

(5.198)

Further, taking the second derivative d2 ei a=ds2 , we will have 







d dei a d @ei a dxk @ 2 ei a dxk dxm @ei a d2 xk = = ds ds ds @xk ds @xm @xk ds ds + @xk ds2 :

(5.199)

39

5.7. EQUATIONS OF GEODESICS. . .

Since

@ 2 eia @ i i j @xm @xk = @xm ( ak ) = jk;m e a isk ( sjmeja ) = ( ijk;m + isk sjm)eja and we have

@eia d2 xk = i s dxk dxm ej ; js km ds ds a @xk ds2 k m d2 ei a i i s i s ) dx dx ej = 0: + (   jk;m sk jm js km ds ds a ds2

(5.200)

Substituting here the sum (5.171), we have

d2 ei a +( ds2 Tski

i jk;m s jm

i + Tjk;m s Tski Tjm

i Ts js km

i sk i js

s ) Tjsi Tkm

s jm s km dxk

ds

i Ts sk jm Tjsi skm dxm j e = ds a

0:

(5.201)

Since independent equations (5.201) (for three Euler's angles and three pseudo-Euclidean angles) describe the variation of the orientation of tetrad ei a as it moves from the origin O according to the equations of geodesics (5.196). In A4 spaces, where the metric is at

gik = ik = diag(1 1 1 1);

(5.202)

the Christo el symbols i js vanish and the equations (5.201) become k m d2 ei a + (T i s s ) dx dx ej = 0; Tski Tjm Tjsi Tkm jk;m 2 ds ds ds a

(5.203)

and the equations of geodesics (5.175) will become j k d2 xi i dx dx = 0: + Tjk 2 ds ds ds

(5.204)

We now introduce the tensor of the four-dimensional angular velocity of rotation tetrads ei a [33]

ij = Tijk

dxk deia a deja a = ds ds e j = ds e i

(5.205)

with the symmetry properties

ij = ji ;

(5.206)

determined by the symmetry (5.48), for which the Ricci rotation coeÆcients hold.

40

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

Using (5.205), we will write the equations (5.204) and (5.203) as

d2 xi + i dxj = 0; j ds ds2 d i j ds

(5.207)

dxk dxm dx T ijk;m ds ds + T ijs sk dsk = 0:

(5.208)

The skew-symmetric matrix (5.206) can be represented as 0

0 01 02 03

10 0 12 13

20 21 0 23

30 31 32 0

B

ij = B @

1 C C A

(5.209)

Let us now give a physical interpretation of the components of the matrix (5.209). We multiply the equations (5.207) by the mass m and rewrite them as j

dx = 0: m d x2i + m ij ds ds 2

(5.210)

If the condition (5.202) holds, there equations can be represented as j m du i + m ij dx = 0; dso dso

(5.211)

where

dso = (ik dxi dxk )1=2 is the pseudo-Euclidean metric and ui = dxi =dso .

(5.212)

We represent the equations (5.211) in the form

du i dxj dxk = mTi(jk) ; (5.213) dso dso dso where the part of T symmetric in indices j and k is given by (5.178). m

Assuming that motion governed by the equations (5.213) is nonrelativistic (v=c  1), we will write the three-dimensional part of these equations as

m

du dxo dxk = mT (ok ) dso dso dso

2mT ( k)

dx dxk dso dso

(5.214)

or, from the relationship (5.205), as o m du = m o dx dso dso

2m

Since in the nonrelativistic approximation

dso = cdt; ua =

v ; c

dx : dso

(5.215)

41

5.7. EQUATIONS OF GEODESICS. . .

and dxo = cdt, the equations (5.215) can be written as

m

dv = mc2 o dt

2mc2

1 dx

c dt

:

(5.216)

It is known from classical mechanics that the nonrelativistic equations of motion of the origin O of a three-dimensional accelerated reference frame under inertia forces alone have the form [34]

d dt (mv) = m ( W + 2[v! ]) ;

(5.217)

where W is the vector of translational acceleration, and ! is the vector of the three-dimensional angular velocity of rotation of the accelerated reference frame. We write these equations as

d (mv ) = m dt





W o + 2! dx dt

;

(5.218)

where W = (W10; W20 ; W30); 0

! = ! =

!3

0

!3 !2

@

0

!1

!2 !1 0

1 A

(5.219)

! = (!1 ; !2 ; !3 ); and comparing these with (5.217), we obtain

10 =

W1 ; = W2 ; = W3 ; 20 30 c2 c2 c2

12 =

! 3 ; = !2 ; = ! 1 : 13 23 c c c

Therefore, the matrix (5.209) in this case has the form 0

ij =

1

c

2

B B @

0

W1 W2 W3

W1 0

c!3 c!2

W2 c!3 0

c!1

W3 c!2 c!1 0

1 C C A

(5.220)

It is seen from this matrix that the four-dimensional rotation of the tetrad

ei a, caused by the torsion of the A4 spaces, gives rise in physics to inertia elds associated with translational and rotational accelerations.

42

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

5.8 Structural equations of right and left A4 geometry We can consider three forms of the geometry of absolute parallelism. (1) A4 geometry, with the nonzero Riemannian tensor Ri jkm and torsion ::i

jk . The structural Cartan equations then become

r k eaj [

]

+ T i[kj ]eai = 0;

(5.221)

Ri jkm + 2r[k Tjij jm] + 2Tsi[k Tjsj jm] = 0: (5.222) (2) A4 geometry, with the zero Riemannian Ri jkm and nonzero torsion ::i jk .

In that case the structural Cartan equations can be written as

r k eaj + T ikj eai = 0; [

]

[

r k Tjij jm + Tsi kTjsj jm [

]

(5.223)

]

[

]

= 0:

(5.224)

(3) A4 geometry, with the zero Riemannian tensor Ri jkm and noncoordinate torsion ::i jk . The structural Cartan equations of the geometry coincide with the structural equations of the pseudo-Euclidean space E4, and they look like Æ Æ (5.225) r[k eÆ aj]+ T i[kj ] eÆ ai = 0; Æ Æ

Æ

Æ

r k T ijj jm + T is k T sjj jm [

]

[

]

= 0;

(5.226)

Æ where the tetrad e ai determines the "coordinate torsion" Æ Æa Æi Æa 1 Æi Æa (5.227)

::i jk =e a e [k;j ] = 2 e a (e k;j e j;k ): Since in the pseudo-Euclidean space the T4 and O (3:1) groups hold globally and its internal geometry is trivial, then, for example, in the Cartesian coordinate x0 = ct, x1 = x, x2 = y , x3 = z the structural equations (5.225) and (5.226) become the identities 0  0; (5.228) 0  0: (5.229) If we now go over to the spherical coordinates

x0 = ct; x1 = r; x2 = ; x3 = '; we will get the equations (5.225){(5.227), which include: (a) components of the \coordinate" tetrad Æe (0) eÆ (1) Æ Æ = 1 = 1; e (2) = r; e (3) = r sin ; 0 2 3 1 Æ3 1 Æe 2 Æe 0 eÆ 1 ; e (3) = ; (2) = (0) = (1) = 1; r r sin 

(5.230)

43

5.8. STRUCTURAL EQUATIONS OF THE RIGHT. . .

(b) components of \coordinate torsion" Æ Æ Æ 1

::212 = ::313 = (2r ) 1 ; ::323 = 2 cot  ; (c) components of the Ricci rotation coeÆcients Æ Æ Æ T 122 = r; T 133 = r sin2 ; T 233 = sin  cos ; Æ Æ Æ 1 T 212 =T 313 = ; T 323 = cot :

(5.231)

(5.232)

r

Using the formulas gÆ ik = ab eÆ ai eÆ bk ; ab =  ab = diag(1

1

1

1);

we nd the components of the metric tensor gÆ 00 =gÆ 11 = 1; gÆ 22 = r; gÆ 33 = r 2 sin2 ; the metric

Æ

ds2o =gij dxi dxj = c2 dt2 dr 2

r 2(de2 + sin2 d'2 )

and the components of the Christo el symbols Æ1 Æ1 Æ2 Æ3 r sin2 ; r; = 13 = 1r ; 33 = 22 = Æ2 Æ 3 12 sin  cos ; 33 = 23 = cot :

(5.233)

Thus, in the pseudo-Euclidean geometry A4, when we deviate from Cartesian coordinates, instead of the identities (5.228) and (5.229) we get the "coordinate structural equations" (5.225) and (5.226). Suppose now that the initial pseudo-Euclidean space A4 is deformed in a continuous manner (e.g., using conformal transformations) into an A4 space with a nonzero dynamic torsion eld and the structural equations (5.223) and (5.224). We can distinguish the right + 1 i a i a

::i jk = r ar [k;j ] = 2 r a(r k;j

and left

r aj;k )

(5.234)

1i a a i a (5.235)

::i jk = l a l [k;j ] = 2 l a (l k;j l j;k ) torsion elds. In these equations rai and lia stand for the right and left tetrads. respectively. + We well take the right tetrad rai to mean a tetrad e i a, such that when the three-dimensional spatial part rotates from the x axis to the y axis the vector of the angular rotational velocity points along the z axis, so that the rotation occurs counterclockwise if looking from the side to which the z vector points.

44

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

For example, the four-dimensional rotation matrix (5.220) for the right tetrad looks like 0 1 0 W1 W2 W3 + 1 B W1 0 c!3 c!2 C C (5.236)

ij = c2 B @ W2 c!3 0 c!1 A W3 c!2 c!1 0 whereas for the left rotations we have 0

ij =

1

c

2

W1

0

W1 W2 W3

B B @

It is seen that

0

c!3 c!2

+

ij = From (5.205) and (5.238), we have +

T

W2 c!3

i jk

W3 c!2 c!1

0

c!1

0

1

C C: A

(5.237)

ij :

(5.238)

T ijk :

(5.239)

=

Since the metric tensor gik is determined both by the right and left tetrad in a similar manner [35]

gik = ab riarkb = ab lai lbk ;

(5.240)

it follows from the de nition im ::s ::s Tjki = ::i jk + g (gjs mk + gks mj )

(5.241)

that the components (5.234) and (5.235) of the right and left torsion elds di er in sign + ::i (5.242)

::i jk = jk : By dividing the torsion elds into left- and right-hand ones, we thereby split + the translations group T4 into the right T 4 and left T 4 translations groups; and the rotations group O (3:1) into the right SO + (3:1) and left SO (3:1) rotations group. We will write the structural Cartan equations of the A4 geometry, which are transformed using continuous transformations in T4 and SO + (3:1) groups, as follows: + r[k +e aj]+ T i[kj ] +e ai = 0; (5.243)

r k T ijj jm + T is k T sjj jm +

+

[

+

]

[

]

= 0:

(5.244)

Accordingly, the equations

r ke aj + T ikj eai = 0; [

]

[

]

(5.245)

45

5.8. STRUCTURAL EQUATIONS OF THE RIGHT. . .

r kT ijj jm + T is k T sjj jm [

]

[

]

=0

(5.246)

are transformed continuously in the T4 and SO + (3:1) groups. It is clear that discrete transformations | inversion transformations | enable us to transform the right equations (5.243) and (5.244) into left equations (5.245) and (5.246), and vice versa. The property (5.242) of the A4 geometry enables an empty pseudo-Euclidean geometry to be "split" into right- and left-hand geometries: Æ + ::i ::i

::i jk = jk + jk = 0;

(5.247)

whose torsion is nonzero. This property appeared to be quite useful for the description of the production of matter from "nothing" in the theory of physical vacuum [36]. If now we split the structural Cartan equations (5.221) and (5.222) into right and left ones, we will get

r k e aj + T ikj e ai = 0; +

+

[

]

+

[

(5.248)

]

i

s

R i jkm + 2r[k T ijj jm] + 2 T s[k T jj jm]= 0;

(5.249)

r k e aj + T ikj e ai = 0;

(5.250)

+

+

[

]

+

[

+

]

R i jkm + 2r[k T ijj jm] + 2 T

i s[k

T sjj jm] = 0:

(5.251) Writing the structural Cartan equations as the extended right and left EinsteinYang-Mills equations, we will arrive at

r k e aj + T ikj e ai = 0;

+

r k e aj + T ikj e ai = 0;

(A)

+

+

+

(A) + + + + 1 (B :1) R jm 2 gjm R=  T jm; + + + + + + C ijkm + 2r[k T ijj jm] + 2 T is[k T sjj jm] =  J ijkm : (B :2) [

[

]

]

[

[

]

]

R jm gjm R=  T jm; i C jkm + 2r[k T ijj jm] + 2 T is[k T sjj jm] =  J 1 2

(B :1) i jkm : (B :2)

(5.252)

(5.253)

In the theory of physical vacuum that is based on the universal relativity principle [37], equations (5.252) and (5.253) describe the right and left matter produced from vacuum.

46

CHAPTER 5. GEOMETRY OF ABSOLUTE. . .

Chapter 6

The geometry of absolute parallelism in spinor basis 6.1 Three main spinor bases of A4 geometry The geometry of absolute parallelism, as laid down in vector basis, enables the structural equations of this geometry of be represented as right (invariant with respect to the T4+ and SO + (3:1)) groups and left (invariant with respect to the T4 and the SO (3:1) groups) groups of the structural equations (A+ ), (B + ) and (A ) and (B ), respectively. Equations (A+), (B + ) (or (A ), (B )) can, in turn, be split by a transition into a group of equations, whose component elds have opposite spins. For this purpose, we have to use spinor basis and some elements of spinor analysis. We will view the spinor geometry A4 as a di erentiable manifold X4 , such that at each point M with the translational coordinates x (i = 0; 1; 2; 3) a twodimensional spinor space C 2 is introduced [38]. There are three possibilities for introducing the spinor basis in the spinor space C 2 : (a) spinor -basis formed by the Infeld-Van der Werden symbols  i _ [39], which satisfy the equality rn  i _ = 0; (6.1) (b) spinor -basis formed by the Newman-Penrose symbols Ai B_ [40], which satisfy the equality  (6.2) rn Ai B_ = 0; (c) spinor dyad basis B , which satis es the equality [41]

"BD  D rk B = 0: (6.3) _ : : : and A; B; _ : : : are spinor inIn relationships (6.1){(6.3) the indices ; ; _ _ dices that take on the values 0,1 and 0; 1. Any local vector Ai that belongs to 47

48

C

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

can be represented as a spin-tensor of the second rank either in the spinor - basis Ai = A _  i _ ; (6.4)

2

or in the spinor -basis

Ai = AAB_ Ai B_ :

(6.5) All the spin-tensors associated with the -basis will have the spinor indices ; ;_ : : :, and the spin-tensors associated with  basis will have spinor indices _ : : :. As to dyad B , it is a connection between - and -basis A; B;

Ai B_ =  i _ A  B : _

Here

(6.6)

 B = B ; _

and the bar on the right-hand side of the equality implies complex conjugation. Spinor -basis is connected with the vector basis eai by

Ai B_ = eia Aa B_ ;

(6.7)

iAB_ = eai aAB_ ;

(6.8)

_ ) matrices, and the matrices where i AB_ are complex Hermitian (i AB_ = i AB _ a A B  AB_ and a have the form

0

 aAB_ = (2)

= B B

1 2

@

0

aAB_ = (2) where

= B B

1 2

@

det(Aa B_ ) = i;

1 0 0 1

0 1

0 1

0

0

1 0 0 1

0 1 1 0

0

i

i

i

i

0

1 0 0 1 1 0 0 1

1

C C; A

(6.9)

1 C C; A

(6.10)

det(aAB ) = i: _

From the orthogonality conditions for the tetrad eia

eaieja = Æi j ; eaiei b = Æ ab

(6.11)

and the relationships (6.7){(6.10) follows the orthogonality conditions for the spinor -basis _ iAB Aj B_ = Æi j ; (6.12)

iAB Ci E_ = Æ AC Æ B_ E_ : _

(6.13)

49

6.1. THREE MAIN SPINOR. . .

For the spinor -basis the following orthogonality conditions hold [54]

i _  j _ = Æi j ;

(6.14)

i _ i _ = Æ  Æ _ _ :

(6.15) Whence, by (6.6) and (6.12)-(6.13), follow the orthogonality conditions for the spinor dyad  o 1 = 1; o o  =  o o = 0; (6.16) 1  1 = 0: In addition, there are the relationships [54]

 o o  1 o = Æ ;  o  1  1  o = " ; where



(6.17) 

0 1 " = = " _ Æ_ = = 1 0 is the fundamental spinor [40] that obeys the following relationships:

"

" _ Æ_

" " = " = " ; " " = Æ  Æ 

(6.18) (6.19)

Æ  Æ  ;

(6.20)

" = 2;

(6.21)

" [ "Æ ] = 0;

(6.22)





1 0 (6.23) 0 1 : The fundamental spinor " increases and decreases the indices on the spintensor associated with the -basis, similar to the metric tensor gik in the vector basis. In the spinor -basis it has the form

" =

"AB = " A B ; so that



(6.24) 

0 1 (6.25) 1 0 : The fundamental spinor "AB increases and decreases indices on the spintensors associated with the -basis. For example, we have

"AB = "AB = "C_ D_ = "C_ D_ =

:::::: AB ::: ::: :::A::: :::A::: ::: ::: "AB = :::B:::; " :::B::: = :::::: ; _ _ B_ ::: ::: _ ::: A::: :::::: A A::: ':::::: "A_ B_ = ':::B::: " ':::B::: = '::: _ _ :::::: :

(6.26)

50

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

If the spinor is skew-symmetric in two indices

:::A:::B::: = :::B:::A:::;

(6.27)

then, using the fundamental spinor "AB , it can be represented as [40] 1 (6.28) 2 The same properties are valid in the spinor -basis for the fundamental spinor " . ::: :::C::: : :::A:::B::: = "AB :::C:::

6.2 Spinor representation of the structural Cartan equations of A4 geometry The relationship (6.28) makes it possible to reduce spinors skew-symmetric in primed and unprimed indices to spinors that are completely (or partially) symmetrical in primed and anprimed indices. In the space of spinors of this type irreducible representations of the groups SL(2:C ) are realized [42]. This group replaces the group SO (3:1) on passing over to the spinor basis. De nition 6.1. We will say that the components of a spinor with r symmetrical lower indices and with s symmetrical lower primed indices are transformed in D (r=2;s=2) irreducible representation of the group SL(2:C ). For example, the spinor FAB = FBA is transformed in D (1:0), and the spinor

FC_ D_ = FD_ C_ in the D (0:1) irreducible representation of the group SL(2:C ). We will write the main relationships of the A4 geometry in the spinor basis. This can be accomplished using the spinor representation of the arbitrary :::i:::::: in the -basis tensor T::::::j::: _ _ j :::AB:::::: T::::::C = iAB T::::::i:::::: _ :::j:::C E_ E:::

(6.29)

or simply replacing the matrix indices by two spinor ones as follows:

eai $ iAB ;

(6.30)

T abm $ T AB_ C Dm ; _

(6.31)

_

Ra bkm $ RAB C Dkm ; _ ab $ ABC _ D _ = "AC "B_ D _; _

and so on.

(6.32) (6.33)

51

6.2. SPINOR REPRESENTATION. . .

In the spinor -basis the metric tensor gij of the A4

Proposition 6.1.

geometry has the form

gij = "AC "B_ D_ iAB_ jC D_ :

Proof. Substituting into

(6.34)

gij = abeai ebj

the relationships (6.7) and (6.8) written as

eai = iAB_  a AB_ ; ebj =  C D_ j  b C D_ ;

(6.35)

gij = ab iAB_ Aa B_ jC D_ Cb D_ :

(6.36)

we have

From the relationships (6.9), (6.10), (6.25) and the de nition

ab =  ab = diag(1

1

1);

1

we obtain the following equality:

abAa B_ Cb D_ = "AC "B_ D_ : Substituting this into (6.36), we arrive at the formula (6.34). We now write the structural Cartan equations in matrix form

r k eam ebk Tjabjm = 0; Rabkm + 2r k Tjabjm + 2Tcak Tjcbjm [

]

[

(A)

]

= 0: (B ) Using the rules (6.30)-(6.32), we write these equations in the spinor -basis [

]

[

r k mAB  CkD T AjBC D jm

]

_

[

_

]

[

_

_

]

= 0;

RAB_ C Dkm + 2r[k T AB_ jC D_ jm] + 2T AB_ E F_ [k T E F_ jC D_ jm] = 0: _ Consequently, the second Bianchi identity of the A4 geometry

r nRajbjkm [

]

+ Rcb[km Tjacjn]

Tbc[nRajcjkm] = 0

(6.37) (6.38) (D )

in the spinor -basis becomes

r nRABjC D jkm _

[

_

]

+ RE FC_ D_ [km T AjBC_ D_ jn]

Proposition 6.2.

the corresponding spinor

T ECF_D_ [nRAEj_E F_ jkm] = 0:

(6.39)

If Fij = Fji is a real skew-symmetrical tensor, then i j FABC _ D _ = Fij AB _ CD _

(6.40)

can be represented in the form 1 2

("B_ D_ FAC + "AC F B_ D_ ); FABC _ D _ =

(6.41)

52

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

where the spinor

FAC = FCA

(6.42) is transformed in the D (1:0) irreducible representation of the group SL(2:C ), and the spinor F BD = FB_ D_ = FD_ B_ (6.43) in the D (0:1) irreducible representation of the same group. Proof. Since the tensor Fij is skew-symmetric, we have, by (6.40),

FABC FC DA _ D _ = _ B _: We rewrite this as 1 FABC (F _ D_ _ D _ = 2 ABC

1 FC DA (FABC _ B _ = _ D _ 2

(6.44)

FC DA _ B_ + FC DA _ B _ FC DA (6.45) _ B _ ):

Using the fundamental spinor (6.25), we can write (6.45) as follows: 1 E_ ): (" F F + "B_ D_ FAEC (6.46) FABC _ _ D _ = 2 AC F B_ D_ E_ , we have, by (6.44), Denoting FAC = (1=2)FAEC _ 1 E_ = 1 F E_ _ = F : FAC = 2 FAEC (6.47) _ CA 2 A CE Further, introducing the notation F B_ D_ = 21 FF B_ F D_ and considering that Fij is real, we nd 1 1 F B_ D_ = 2 FF B_ F D_ = 2 F F_ B F_ D = F BD : (6.48) Substituting the relationships (6.47) and (6.48) into (6.46), we arrive at (6.44). By de nition, the spinor FAC = FCA belongs to the D (1:0) irreducible representation of the groups SL(2:C ); and spinor F B_ D_ = F D_ B_ { to the D (0:1) irreducible representation of the group. Since the quantities TABC and RABC in the equations (6.37) and _ Dm _ _ Dkn _ (6.38) are skew-symmetric in the pair of spinor matrix indices AB_ and C D_ , we can represent them, by (6.27){(6.28), as 1 TABC = ("B_ D_ TACk + "AC TB+_Dk (6.49) _ Dk _ _ ); 2 1 ); (6.50) RABC = ("B_ D_ RACkn + "AC R+B_ Dkn _ Dkn _ _ 2 where 1 1 TACk = 2 "B_ D_ TABC TB+_Dk = "AC TABC (6.51) _ Dk _ ; _ Dk _ ; _ 2 1 1 RACkn = "B_ D_ RABC ; R+B_ Dkn = "AC RABC : (6.52) _ Dkn _ _ Dkn _ _ 2 2 In these relationships the + sign with the spinor matrices implies Hermitian conjugation.

53

6.3. SPLITTING OF THE STRUCTURAL. . .

6.3 Splitting of structural Cartan equations into irreducible representations of the group SL(2:C) Matrices (6.51) and (6.52) can be transformed in the spinor indices as follows: 0

0

0

0

0

T AC 0 k = SAA T ACk S CC 0 + SAA S AC 0 ;k ; 0

(6.53)

B_ +D_ + S +B_ S B_ ; T +D_B0_k = SB+_ B_ T +Dk _ SD _0 D_ 0 ;k B_

(6.54)

RA C 0 kn = SAA RA Ckn SCC0 ;

(6.55)

R+B_ D_ 0 kn = SB+_ B_ R+B_ Dkn SD+_ D0 _ : _

(6.56)

0

0

0

0

0

Matrices of the transformations SAA and SB+_ B form the group SL(2:C ), and the matrices 0 SAA (6.57) form the subgroup

SL+ (2:C )

(6.58)

of the group SL(2:C ), in which the spinors belonging to the irreducible representation D (r=2; 0) are transformed. On the other hand, the matries

S +B_ form the subgroup

0

B_

SL (2:C )

(6.59) (6.60)

of the group SL(2:C ), in which the spinors belonging to the irreducible representation D (0; s=2) are transformed. These properties of the spinors enable the structural Cartan equations to be split into equations that contain spinors transformed in D (r=2; 0) or D (0; s=2) irreducible representations of the group SL(2:C ): Proposition 6.3. The second structural Cartan equations (B ) in the spinor -basis are split into the equations of the form

RACkn + 2r[kTjAC jn] + 2TAE [k TjEC jn] = 0;

(6.61)

R+B_ Dkn + 2r[k Tj+B_ D_ jn] + 2TB+_F_ [k T +jFD__ jn] = 0: _

(6.62)

Proof. We write the second structural Cartan equations (6.38) as EF_ BABC = RABC + 2r[k TjABC _ Dkn _ _ Dkm _ _ D _ jm] + 2TABE _ F_ [k TjC D _ jm] = 0:

(6.63)

54

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Using the fact that the spinor BABC is skew-symmetric in the pair of _ Dkn _ _ , we will write it in the form spinor indices AB_ and CD 1 BABC = ("B_ D_ BACkn + "AC BB+_Dkn ) = 0; _ Dkn _ _ 2

where

(6.64)

1 (6.65) 2 1 BB+_Dkn = "AC BABC = 0: (6.66) _ Dkn _ _ 2 Substituting (6.61) into the equations (6.65) and (6.66) and using the matrices (6.51) and (6.52), we will arrive at the structural equations (6.61) and (6.62) in split form. In the derivation we have used the properties (6.19)-(6.23) of the fundamental spinor "AB . Proposition 6.4. Matrices TACk and TB+_Dk in the dyad basis  C have _ the following form: TACk =  C rk A = TACk ; (6.67)

BACkn = "B_ D_ BABC = 0; _ Dkn _

_

TB+_Dk =  _ D_ rk  B_ = TB+_Dk _ : _

(6.68)

Proof. We write the matrices

Tabk = ei b rk eai in the spinor basis, using the rules (6.30) and (6.31)

TABC = Ci D_ rk ABi _ Dk _ _ :

(6.69)

Substituting this expression into the rst one of (6.51) gives 1

TACk = 2 "B_ D_ Ci D_ rk ABi_ :

(6.70)

Using the formula (6.6), we write ABi _ as

ABi_ =  _ iA  B_ : _

(6.71)

Substituting (6.71) into (6.70), we have _ 1 _ 1 TACk = "B_ D_ Ci D_ rk ( i_ A B_ ) = "B_ D_ Ci D_  i_ rk (A B_ );

2

2

since rk ( i_ ) = 0. Further, considering that

Ci D_  i_ =  _ i C  D__  i_ =

_

= Æ Æ _ _ C  D_ = C  D_ _ ;

(6.72)

55

6.3. SPLITTING OF THE STRUCTURAL. . .

we will write (6.72) as 1

TACk = 2 "B_ D_ C  D_ _ ( B_ rk A + A rk  B_ ): _

_

(6.73)

In the dyad basis we have the equalities

"B_ D_ =  D_ _  B_ ; "B_ D_  D_ _ rk  B_ = 0; _

_

which are conjugates of (6.3) and (6.24). Using these equalities, we can easily obtain (6.67). Similarly, for the conjugate matrix T + B_ Dk _ , we have (6.68). Proposition 6.5. In the spinor -basis the rst structural Cartan equations (A) of the A4 geometry have the form

r k Ci D T kjCE  E D ji jCi F j Tk D F = 0 [

]

[

_

_

[

]

+ _ ] _

_

(6.74)

or, dropping the matrix indices,

r k  i T k  i  iTk ]

[

[

]

[

+ ]

= 0:

(6.75)

Proof. Let us take the derivative rk Ci D_ :

rk Ci D = rk ( i C  D ) =  i ( D rk C + C rk  D ): _

_ _

_

_ _

_

_ _

Using (6.67) and (6.68), we will write this relationship as

rk Ci D =  i (TCEk  E  D + TDFk C  F ): _

_

+ _ _

_

_

_ _

(6.76)

Here we have used the normalization conditions

 E  E = 1;  _ F_  F = 1: _ _

Multiplying the terms on the right-hand side (6.76) we obtain, from (6.71),

rk Ci D TCEk DiE CiF TDFk = 0 _

_

or

_

+ _ _

rk Ci D TkCE DEi Ci F TkDF : +_ _

(6.77)

(6.78) Alternating this relationship in the indices k and i, we obtain the equations (6.74). Proposition 6.6. The second Bianchi (D ) identities of the A4 geometry in the spinor -basis are split into the following equations:    rn R ACkn R ECkn T E An + R EAkn T E C n = 0; (6.79)    _ _ rn R +B_ Dkn TB+_ Fn + R +F_ Bkn TD+_ F n = 0: (6.80) R +F_ Dkn _ _ _ _

_

_

56

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Proof. Increasing and decreasing, using the metric tensors ab and gik , the tensor indices in the identities (6.150), we will write them in the form    rn Rabkn Rcbkn T ca n + Rackn T cb n = 0:

(6.81)

In this equality we now pass over to the spinor indices using (6.31) and (6.32) go get  _ _ n  n rn R ABC T EF T EF (6.82) _ Dkn _ RE FC _ Dkn _ _ AB_ + RE F_ ABkn C D_ = 0: We now write this relationship in the form

DAn BC = 0; _ Dkn _

(6.83)

where by DAn BC we have denoted all the terms on the left-hand side of (6.82). _ Dkn _ Since the relationship (6.83) are skew-symmetrical in the pair of indices AB_ and C D_ , we will write it in the form 1 n + "AC D +n B_ Dkn ) = 0; DAn BC = ("B_ D_ DACkn _ _ Dkn _ 2

(6.84)

where 1

1

n DACkn = "B_ D_ DAn BC = 0; DB+_ nDkn = "AC DAn BC = 0: _ Dkn _ _ Dkn _ _ 2 2

Substituting here (6.82), we will get (6.79) and (6.80). Physically, the spinor splitting of the structural Cartan equations (A) and (B ) implies splitting into the equations of \matter " and \skew-symmetry", just as it has been done by Dirac in his derivation of equations for the electron and the positron. We can now write equations that are transformed in the groups SL+ (2:C ) as r[kCi]D_ T[kjCE DE_ ji]  [i jC F_ j Tk+]D_ F_ = 0; (As)

RACkn + 2r[k TjAC jn] + 2TAE [k TjEC jn] = 0; and in the group SL (2:C ) as

r k  i C D T kjCE DE ji  i jC F j Tk D F [

]

_

[

_

]

[

_

+ _ ] _

= 0;

_ R+B_ Dkn + 2r[k Tj+B_ D_ jn] + 2TB+_F_ [k Tj+D_Fjn] = 0: _

(B s+ ) (As) (B s )

In the numerations of these formulas s implies transformation in a spinor group. Dropping the matrix indices, we will write these relationships as

r k i

T[k  i]  [iTk+] = 0;

(As)

Rkn + 2r[kTn] [Tk ; Tn] = 0;

(B s+ )

[

]

57

6.4. CARMELI MATRICES. . .

r k  i T k  i  iTk = 0; Rkn + 2r k Tn [Tk ; Tn ] = 0: [

+

]

]

[

+ ]

[

[

(As)

+ ]

+

(B s )

+

Correspondingly, discarding the matrix indices in the equations (6.79) and (6.80), we obtain   rn Rkn +[R kn ; T n] = 0; (D s+)   rn R +kn + [R +kn ; T +n ] = 0: (D s )

6.4 Carmeli matrices

Equalities (6.67) and (6.68) can be written in matrix form

Tk =  rk ;

(6.85)

Tk+ =  + rk  + ;

(6.86) where Tk and  are 2  2 complex matrices with elements T A Bk and Aa , respectively. Multiplying Tk by  k AB_ , we can introduce the traceless Carmeli 2  2 matrices [44-46] TAB_ = Ak B_ Tk ; (6.87) _ D_ : : : = 0_ ; 1_ A; C : : : = 0; 1; B; with the components

T00_ = T10_ =

 

"  

 " 

 



; T01_ = ; T11_ =







 

 

; :

(6.88)

Using matrices (6.87), we can de ne the matrix elements

CD AB_ 00 01 10 11 00_ "   " (TAB_ )C D = 01_   ; 10_   11_  

(6.89)

where (TAB_ )C D is the CD element of the matrices TAB_ . Consequently, the complex conjugate matrices T + AB are _

C_ D_

+ D_ (TAB _ ) C_

_ AB 0_ 0_ 0_ 1_ 1_ 0_ 1_ 1_ _00 "   " : = _ 01   _   10 1_ 1  

(6.90)

58

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Proposition 6.7. In the Carmeli matrices the rst structural Cartan equations (A) of the A4 geometry have the form

@C D_ Ai B_

+ R_ @AB_ Ci D_ = (TC D_ )AP Pi B_ + Ai R_ (TDC _ _ ) B P i i + R_ _ : (TAB_ )C P D_ C R_ (TBA _ ) D

(6.91)

Proof. We will write the equations (6.75) as i + i T F rk Ci D rk Ai B = TC E k DE CF D k _

_

_

+ _ _

_

TA C k Ci B_

Ai E_ TB+_Ek_ :

(6.92)

It is easily seen that the equations (6.92) represent the di erence of the two relationships i +  i T +F_ ; rk Ci D_ = TC E k DE (6.93) _ C F_ D_ k

rk Ai B = TAC k Ci B + Ai E TB E k : _

_

_

+ _ _

(6.94)

Multiplying (6.93) by  k AB_ , and (6.94) by  k C D_ , we get i k + i T F k ; rk Ci D Ak B = TC E k DE AB C F D k AB

(6.95)

rk Ai B Ck D = TAP k Pi B Ck D + Ai E TB E k Ck D :

(6.96)

(TAB_ )C E = TC E k Ak B_

(6.97)

@AB_ = Ak B_ rk ;

(6.98)

+ R_ ; @C D_ Ai B_ = (TC D_ )A P Pi B_ + Ai R_ (TDC _ _ ) B

(6.99)

+ R_ : @AB_ Ci D_ = (TAB_ )C P Pi D_ + Ci R_ (TBA _ _ ) D

(6.100)

Rkn + 2r[kTn] [Tk ; Tn] = 0:

(6.101)

_

_

_

_

_

_

_

_

_

_

+ _ _

+ _ _

_

_

We now introduce the notation and

and rewrite the relationships (6.95) and (6.96) as

Subtracting from (6.99) the equality (6.100), we will arrive at the rst structural Cartan equations (6.91) of the A4 geometry, written in terms of Carmeli matrices. Consider now the second structural Cartan equations (B s+ ); written in matrix forms Multiplying the quantity Rkn by  k AB_ and  n C D_ , we will introduce the traceless Carmeli matrix k n RABC _ D _ = Rkn AB_ C D_

(6.102)

59

6.4. CARMELI MATRICES. . .

with the components [44-46] 





1 0 ; R 10 _ 0_ = 100 + 2 2 1    20 3 2 2 12 R111 ; R110 _ 0_ = _ 1_ = 4  3 22   2 + 11 1 01 R110 _ 0_ = +   + 3 21 2 11   2 + 11 +  1 01 R100 _ 1_ = 3 + 21 2 11 

R010 _ 0_ =

00 10 02 12 

 

; ;

;

(6.103)

:

Proposition 6.8. In terms of Carmeli spinor matrices (6.87) and (6.102), the second structural Cartan equations (B s+ ) of the A4 geometry become

RABC _ D _ = @C D _ TAB _

+ F_ T _ + @AB_ TC D_ (TC D_ )A F TF B_ (TDC _ AF _ ) B _ + F F +(TAB_ )C TF D_ + (TBA _ ; TC D _ ]: (6.104) _ TC F_ + [TAB _ ) D

Proof. We write the equations (6.101) as Rkn = 2r[nTk] + [Tk ; Tn ] or

Rkn = rnTk rk Tn + Tk Tn TnTk : Multiplying this by  k AB_  n C D_ , we will have k RABC @AB_ TnCn D_ + TAB_ TC D_ TC D_ TAB_ = _ D _ = @C D _ Tk AB_ = @C D_ TAB_ @AB TC D_ (@C D_ AB_ k @AB_ C D_ k )Tk + +[TAB_ ; TC D_ ]:

(6.105) (6.106)

(6.107)

We have used here the condition that

k AB_ Ck E_ = ÆCAÆEB__ and the notation

@AB_ = Ak B_ rk :

(6.108)

(6.109) If now in (6.107) we use the relationships (6.99) and (6.100), we will get the equations (6.103). Let us write the second Bianchi identities (D s+) of the A4 geometry in matrix form   rn Rkn +[R kn ; T n] = 0: (6.110) n Multiplying these equations by  E F_ , we will render them in terms of Carmeli matrices as follows:  n (rC D_  AB_ )  @ C D_ RE FC _ D _ +EF RABC _ D _ + _ n   _ _ +(rk  kC D ) RE FC [T C D ; R E FC (6.111) _ D _ _ D _ ] = 0:

60

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Using the relationship (6.99), we can rewrite the identities (6.111) as 



@ C D_ RE FC (T C D_ )AE R AFC _ D _ _ D _   _ _ _ D )CP R _ _ + (T +DC )F_ B R E BC _ D _ +(TP E FC D _  _D _  +C Q C D +(TQ_ ) RE FC ; R E FC _ D _ +[T _ D _ ] = 0:

(6.112)

6.5 Component-by-component rendering of structural equations of A4 geometry Let us now write the equations (6.91) component by component. For convenience, we will introduce the following notation: i AiC DA @AB_ Ci D_ = (TC D_ )A P Pi B_ + _ AB_ _ B_ = @C D + i + R_ _ (T _ )C P  i R_ _ : +Ai R_ (TDC _ ) B AB _ ) D P D_ C R_ (TBA

(6.113)

Also, we will denote the components of the spinor derivative as

B_

_ _ @AB_ = A 0 1 ; 0 D Æ 1 Æ 

(6.114)

and the components of the spinor -basis as

Ai B_ =

A

0_

B_

1_

:

(6.115)

i Ai000 @01_ 0i 0_ = (T00_ )0P Pi 1_ + 0i R_ (T00_+ )R_ 1_ _ 1_ = @00_ 01_ _ + R (T01_ )0 P Pi 0_ 0i R_ (T10 _ _ ) 0

(6.116)

0 li = (Y 0 ; V; Y 2 ; Y 3 ) mi = ( 0; !;  2;  3 ) 0 2 3 1 m i = ( ; !;  ;  ) ni = (X o; U; X 2; X 3)

From (6.113), the spinor component Ai000 _ 1_ will be

or 

Ai _ _ = @00_ 0i 1_ @01_ 0i 0_ = (T00_ )00 0i 1_ + (T00_ )0 1 1i 1_ +  0001   + 0_ i (T + )1_ + 0i 0_ (T00 ) +  (T01_ )00 0i 0_ + (T01_ )0 1 1i 0_ _ _ 1 _ 1_ 01_ 00 



+ 0_ i + 1_ 0i 0_ (T10 _ ) 0_ + 01_ (T10 _ ) 0_ :

(6.117)

61

6.5. COMPONENT-BY-COMPONENT RENDERING. . .

Using the notation of (6.89)-(6.90) and (6.114)-(6.115) for the components + R_ _ ; @ _ and  i , we will obtain, by (6.117), (TC D_ )A P ; (TBA _ ) D AB AB_ 

Dmi



Æli = "mi + ( )ni + li + mi ( " )   li + (  )mi li + mi ( ) = = ( +  )li ni + m i + ( + " ")mi :

(6.118)

Since the vectors mi and li have the following components:

li = (Y 0 ; V; Y 2 ; Y 3 ); mi = ( 0; !;  2;  3 ); it follows from (6.118) that

ÆV

D! = ( +

 )V + U

!

( + "

")!;

(6.119)

ÆY D = ( +  )Y + X  ( + " " ) ;

(6.120)

= 0; 2; 3: In a similar manner we nd the following component rendering of the rst structural Cartan equations of the A4 geometry

ÆV

D! = ( +

 )V + U

! ( + " ")!;

ÆY D = ( +  )Y + X  ( + " ") ; Y DX = ( + )Y + (" + ")X ( +  ) ( +  )!; V

(A:1) (A:2) (A:3)

DV = ( + )V + (" + " )U ( +  )! ( +  )!; ÆU ! = V + ( )U + ! + ( + )!;

(A:4)

ÆX  = Y + ( )X +  + ( + ) ; Æ! Æ! = ( )V + ( )U ( )! ( )!;

(A:6)

Æ

(A:8)





Æ = (

)Y + (

)X

( )



(

) ;

(A:5) (A:7)

= 0; 2; 3; and the complex conjugate equations (A:1) (A:8) (all in all 24 independent equations). Let us now look at the equations (6.107) and write them componentwise. For instance, we will derive the R010 _ 0_ component of these equations

R010 = @00_ T01_ @01_ T00_ (T00_ )00 T01_ (T00_ )0 1 T11_ + _ 0_ +(T0+0_ )0_ 1_ T00_ (T0+0_ )1_ 1_ T01_ + (T01_ )0 0 T00_ + (T01_ )0 1T10_ + +(T1+0_ )0_ 0_ T00_ + (T1+0_ )1_ 0_ T01_ + T01_ T00_ T00_ T10_ : (6.121)

62

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Using the matrices (6.89)-(6.90), (6.103) and the spinor derivative (6.114), we can represent (6.121) as 

1 2 + 2

"

+" +

 

+

  " 



0  1



   "



+



+





=D

 

" 



 





 " 

  "



  

  "  +   " 

Æ    "

  

"  "  



 "  "  

  



+

(6.122)

+ 

:

These equations split into the following three independent equations: (D (D

 + ") (Æ +  )" ( +  ) + ( + ) 1 = 0; (D   3" + " ) (Æ  +  3 ) 0 = 0;  + " + " ) (Æ +  + )  +  2 2 = 0:

Similarly, we will obtain the following independent equations (B s+ ): (D

 " ") (Æ 3 +  )

 +  00 = 0; (D   3" + ") (Æ  +  3 ) 0 = 0; (D  " + ") ( 3 )    1 10 = 0; (D  " + 2") (Æ +  )"  +  +   10 = 0; (D + " + ") ( )" ( +  ) ( +  )  +  +  2 11 = 0; (D  + 3" ") (Æ +  + )  +  20 = 0; (D  + ") (Æ +  )" ( +  ) + ( + ) 1 = 0; (D  + " + ") (Æ +  + )  +  2 2 = 0; (D + 3" + " ) ( +  + )  ( +  )

(B s+ :1) (B s+ :2) (B s+ :3) (B s+ :4) (B s+ :5) (B s+ :6) (B s+ :7) (B s+ :8)

6.6. CONNECTION OF STRUCTURAL. . .

63

3 21 = 0; (B s+ :9) ( +  +  + 3 ) (Æ + 3 + +   ) + 4 = 0; (B s+ :10) (Æ  ) (Æ 3 + ) +  ( )+ + 1 01 = 0; (B s+ :11) (Æ + 2 ) (Æ + )  +  ( ) ( )"  + 2 11 = 0; (B s+ :12) (Æ + 3 ) (Æ +  + + ) ( ) + + + 3 21 = 0; (B s+ :13) (Æ  + + ) ( + + )  +  + +"  12 = 0; (B s+ :14) (Æ  + 3 + ) ( +  + + ) + + 22 = 0; (B s+ :15) (Æ  + ) ( +  3 + ) + + 02 = 0; (B s+ :16) ( +  ) (Æ +  ) +   + 2 + 2 = 0; (B s+ :17) ( + ) (Æ +  ) ( + ") + +( + ) + 3 = 0: (B s+ :18) In addition to these equations, the second structural Cartan equations (B ) include the complex conjugate equations

R+kn + 2r[k Tn+] [Tk+; Tn+] = 0:

(B s )

We can write these equations in terms of components by replacing the equations (B s+:1){(B s+:18) by their complex conjugate equations.

6.6 Connection of structural Cartan equations of A4 geometry with the NP formalism

In 1962 Newman and Penrose [40] put forward a system of nonlinear spinor equations, which appeared to be extremely convenient in the search for novel solutions of Einstein's equations. In the work [47] by the author of this book is was shown that the equations of the Newman-Penrose formalism coincide with the structural Cartan equations of the geometry of absolute parallelism. Indeed, with spinor Carmeli matrices TC D_ one can connect the spintensor TFAC D_ using the relationships (TC D_ )A P = TA P k  k C D_ = T P AC D_ = "PF TFAC D_ :

(6.123)

64

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Using the matrix elements (6.162) of the Carmeli matrices and the fundamental spinor   0 1 AB " = "AB = 1 0 ; we will obtain the following notation for the components of the spintensor TABC D_ : C D_ AB 00_ 01_ 10_ 11_ TABC D_ = 00     : (6.124) (01) " 11     Proposition 6.9. First structural Cartan equations of the A4 geometry coincide with the "coordinate equations" [40]

@AB_ Ci D_

i i )+ @C D_ Ai B_ = "PQ (TPAC D_ QB TPCAB_ QD i i T R_ D_ BA +"R_ S_ (T R_ B_ DC _ AS_ _ C S_ )

(6.125)

in the Newman-Penrose formalism. Proof. We will write the structural Cartan equations (A) of the geometry of absolute parallelism as

@C D_ Ai B_

+ _R @AB_ Ci D_ = (TC D_ )AP Pi B_ + Ai R_ (TDC _ _ ) B + P i i R_ : (TAB_ )C P D_ C R_ (TBA _ _ ) D

(6.126)

Using the relationship (6.123), we will represent the equations (6.126) as

@C D_ Ai B_

@AB_ Ci D_ =

i "PQ (TPAC D_ QB i + "R_ S_ (T R_ B_ DC _ AS_

i )+ TPCAB_ QD  i T R_ D_ BA _ C S_ ) :

It is easily seen that these equations are equivalent to (6.125). We now write the well-known decomposition of the Riemannian tensor Rijkm into irreducible representations 1 Rg g ; (6.127) 3 i[m k]j where Cijkm is the Weyl tensor (10 independent coordinates); Rij is the Ricci tensor (nine independent coordinates); R is the scalar curvature. The spinor representation of these quantities using the Newman-Penrose formalism looks like [48] Cijkm $ ABCD "A_ B_ "C_ D_ + "AB "CD A_ B_ C_ D_ ; (6.128) Rij $ 2AB A_ B_ + 6"AB "A_ B_ ; (6.129) R = 24; (6.130)

Rijkm = Cijkm

2g[i[k Rm]j ]

65

6.6. CONNECTION OF STRUCTURAL. . .

where spinors ABCD and AB A_ B_ have the following symmetry properties: ABCD = (ABCD) ; AB A_ B_ = (AB )A_ B_ :

(6.131)

By de nition the spinors ABCD and AB A_ B_ are transformed following the

D (2:0) and D (1:1) irreducible representation of the groups SL+ (2:C ), respectively. If we now put in juxtaposition to the Riemann tensor Rijkm a spintensor following the rule

Rijkm $ RAAB _ BC _ CD _ D _;

then in terms of the spinors (6.128)-(6.130) it can be written as

RAAB _ BC _ CD _ D _ = ABCD "A_ B_ "C_ D _ + "AB "CD A_ B _ C_ D _ + +AB C_ D_ "CD "A_ B_ + CDA_ B_ "AB "C_ D_ + +2("AC "BD "A_ D_ "C_ D_ + +"AB "CD "A_ D_ "B_ C_ ):

(6.132)

This spintensor being skew-symmetric in the pair of indices AA_ and B B_ , we will write it as 1 ("E_ B_ RACDP_ F Q_ + "AC RE_ BD RAEC _ ); _ BD _ PF _ Q _ = _ P_ F Q 2

where

1 2 1 AC RE_ BD " RAEC _ = _ BD _ P_ F Q _: _ P_ F Q 2 Substituting into these relationships the equality (6.132) gives

RACDP_ F Q_ = "E_ B_ RAEC _ BD _ P_ F Q _;

(6.133) (6.134) (6.135)

RACDP_ F Q_ = ACDF "P_ Q_ + AC Q_ P_ "FD + "P_ Q_ ("CD "AF + "AD "CF ); (6.136) RE_ BD = "DP E_ B_ P_ Q_ +  B_ EPD "Q_ P_ + "DP ("B_ P_ "E_ Q_ + "E_ P_ "B_ Q_ ): (6.137) _ _ BPQ _ The second structural Cartan equations (B s+) are equivalent to the equations [40] Proposition 6.10.

ACDF "E_ B_ + AC B_ E_ "FD + "E_ B_ ("CD "AF + +"AD "CF ) @DB_ TACF E_ + @F E_ TACDB_ + PQ +" (TAPDB_ TQCF E_ + TACP B_ TQDF E_ TAPF E_ TQCDB_ TACP E_ TQFDB_ ) + +"R_ S_ (TACDR_ T S_ B_ EF _

TACF R_ T S_ E_ BD _ ) = 0

(6.138)

66

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

in the Newman-Penrose formalism. Proof. We write the equations (B s+ ) in terms of the Carmeli matrices

RF ED @F E_ TDB_ (TDB_ )F S TS B_ _ B_ = @DB _ TF E _ + F_ T _ + (T _ ) S T _ + (TED _ FF _ ) B F E D SB _ + F +(TEF _ ; TDB_ ]: _ TDF_ + [TF E _ ) B

(6.139)

Using the relationships (6.123), we can represent the equations (6.139) as

RACF ED _ B_

@DB_ TACE F_ + @E F_ TACDB_ + T S FDB_ TACS E_ +

+T F E_ BD T S DF E_ TACS B_ T F B_ EF _ TACF F_ _ TACDF_ + PQ +" (TAPDB_ TQCF E_ TAPF E_ TQCDB_ ) = 0; _

_

or as

RACF ED @DB_ TACE F_ + @E F_ TACDB_ + "PQ (TAPDB_ TQCF E_ + _ B_ +TACP B_ TQDF E_ TAPF E_ TQCDB_ TACP E_ TQFDB_ )+ TACF R_ T S_ E_ BD +"R_ S_ (TACDR_ T S_ B_ EF _ _ ) = 0;

(6.140)

where we have introduced the spinor indices in the matrices RABC _ D _ and TAB _ following the rule k n ; RABC _ D _ ! REFABC _ D _ = REFkn AB_ C D _ k TAB_ ! TCDAB_ = TCDk AB_ :

(6.141)

Substituting into (6.140) the relationship (6.136), we will arrive at the equations (6.138). Spintensors ABCE and AB C_ E_ have the following notation for their components [38]:

CE AB 00 01 11

0 1 2 ; 3 4 C_ E_ AB 0_ 0_ 0_ 1_ 1_ 1_ AB C_ E_ = 00 00 01 02 ; 01 10 11 12 11 20 21 22 ABCE = 00 01 11

(6.142)

(6.143)

 = : (6.144) Using the relationships (6.114), (6.115), (6.124), we can expand the equations (6.126) of the Newman-Penrose formalism component by component to arrive at the equations (A:1) (A:8) plus the complex conjugate equations. Using

67

6.6. CONNECTION OF STRUCTURAL. . .

the relationships (6.142)-(6.144) and (6.114) we also can expand the equations (6.138) of the Newman-Penrose formalism componentwise. We will thus end up with the equations (B s+:1){(B s+:18). The spinor counterpart of the dual Riemann tensor 

1

Rijkm = 2 "km sp Rijsp

(6.145)

can be written as 

ABCD "A_ B_ "C_ D_ RAAB _ BC _ CD _ D _ = i "AB "CD A_ B _ C_ D _  CDA_ B_ "AB "C_ D_ + AB C_ D_ "CD "A_ B_ + +2("AC "BD "A_ B_ "D_ C_ + "AB "CD "A_ C_ "B_ D_ )) :

(6.146)

It follows that 

1 i ( "B_ D_ ACEF + = "P_ Q_ RABC _ DE _ PF _ Q _ = RABC _ DEF _ _ 2 +"EF AC B_ D_ + "B_ D_ ("AE "CF + "CE "AF )) ;

(6.147)

also 



1

"EF RABC RABC _ D _ P_ Q _= _ DE _ PF _ Q _ = i "A_ C_ B _D _ P_ Q _ 2



"P_ Q_  AC B_ D_ + "AC ("D_ P_ "B_ Q_ + "B_ P_ "D_ Q_ ) :

(6.148)

 The dual Weyl tensor C ijkm corresponds to the spintensor of the form 



C ijkm $C AAB _ BC _ CD _ D _ = i("AB "CD A_ B _ C_ D _

ABCD "A_ B_ "C_ D_ ):

The self-dual spintensor RABC will be _ DEF _ 

RABC = i RABC = ACEF "B_ D_ ; _ DEF _ _ DEF _

(6.149)

whereas the anti-self-dual tensor is



R ABC _ D _ P_ Q _ = i RABC _ D _ P_ Q _ = "AC E _ B_ P_ Q _:

(6.150)

The second Bianchi identities (D s+) of the A4 geometry in the spinor -basis can be represented as Proposition 6.11.

1 _ HR _ X _ " F EG @P X_ RABGHF ABCR T R F F D_ _ E _ 2i C D_ 3 RPB (A TC ) RP D_ + RB D_ X_ TA R C X_ + X_

X_

+AB X_ E_ T D_ E_ C + AB D_ X_ T E_ E_ C = 0;

(6.151)

68

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

where _ EG _ HR _ X _ "C DF = i("CG "RF "D_ E_ "H_ X_

"CF "GR "D_ H_ "E_ X_ ):

(6.152)

Proof. We will write the equations (6.79) as 





rn RACkn REAkn TC En RAEkn TC En = 0:

(6.153)

Multiplying these equations by Ck D_ gives 





n F E_ + @ F E_ R BAC DF _ E_ + R BAC DF _ E_ r n  k F E_  RS_ + RBARSF _ E_ C D_ @ k



PF E_ PF E_ = 0: RBPC DF _ E_ TA RPAC DF _ E _ TB

(6.154)

Here we have used the relationships (6.94) and (6.133). Substituting into (6.154) the relationship (6.148), we will get 

_ E _ i  P RF iDABC D_ + AF EP _ R_ iC D_ RBA



2i P R RBAC D_ F E = 0; _

_

i where AF EP _ R _ stands for the equations (6.125), rewritten as i i _ AABC _ D _ = @AB _  CD 

"PQ TPAC D_  iQB_ i "R_ S_ T R_ B_ DC _  AS_

@C D_  iAB_

TPCAB_  iQD_





i T R_ D_ BA _  C S_ = 0;

(6.155)

and DABC D_ = 0 de nes the equations (6.151) 1 _ HR _ X_ "C D_ F EG @RX_ RABGHF _ E _ 2i 3 RPB (A TC ) RP D_ + RB D_ X_ TA R C X_ +

DABC D_ = ABCR T RF F D_

+AB X_ E_ T X D_ E_ C + AB D_ X_ T X E_ E_ C = 0; _

_

which proves the Proposition. Proposition 6.12. The second Bianchi identities (6.151) of the A4 geometry coincide with the Bianchi identities in the work by Newman-Penrose [40].

@ P D_ ABPC

@ X (C AB )D_ X_ 3 PR(AB TC ) PR D_ ABCP T P R R D_ + 2T P (AB X_ C )P X_ D_ T X_ D_ V_ (A BC ) X_ V_ T X_ V_ V_ (A BC ) X_ D_ = 0;

(6.156)

69

6.6. CONNECTION OF STRUCTURAL. . .

3@AB_  + @ P X AP B_ X_ _



"V_ W_ AP X_ W_ T B_ X_ V_ P + 

+AP B_ X_ T X_ W_ V_ P + PRB_ X_ TA PR X_ + +AP B_ X_ T P R R X_ = 0

(6.157)

Proof. Using (6.147) and the equality 

1 _ _ _ _ RABC D_ RX = 2 "C D_ F EGHRX RABGH_ F E_ ; we nd that in (6.151) 1 _ HR _ X _ RX_ = "C D_ F EG @P X_ RABGHF _ E_ = @P X _ RABC D _ 2 i   = @P X_ "D_ X_ ABC R "AB C R D_ X_ "D_ X_ ("CA "B R + "BA "C R ) : Substituting this relationship into (6.151) gives

@ P D_ ABPC

@C X_ AB D_ X_ + 2"C (A @B )D_  ABCR T RF F D_ X_

_ 3 RPB (A TC ) RP D_ + RB D_ X_ TA R C X_ + AB X_ E_ T D_ E C + _ +AB D_ X_ T X E_ E_ C = 0:

(6.158)

The part of (6.158) symmetrical in the indices C and B can be written as (6.156); and the part skew-symmetrical in these indices looks like (6.157). By writing the second Bianchi identities (D s+) of the A4 geometry component by component, we obtain [40] (D

4 2") 1 (Æ 4 +  ) 0 + +3 2 + (Æ 2 2 +  )00 (D 2 2")01 211 + +2 10  02 = 0; (D s+ :1) (D

 12

3) 2 (Æ + 2 2 ) 1 + +2 3 +  0 + (Æ 2 + + )10 (D 2)11 21 00 +  01 +  20 D  = 0; (D s+ :2) (D 2 + 2") 3 (Æ + 3 ) 2 + +2 1 +  4 + (Æ 2 + 2 + + )20 (D 2 + 2")21 210 + 2 11  22 2Æ  = 0; (D s+ :3)

70

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

(Æ 4 2 ) 1 ( 4 + ) 0 + +3 2 + (Æ 2 + 2 )01 (D 2" + 2"  )02 212 + 2 11 00 = 0; (D s+:4) (Æ

3 ) 2 ( + 2 2 ) 1 + 2 3 + + 0 + (Æ + 2 )11 (D + 2"  )12 22 01 +  02 + + 21 10 Æ  = 0; (D s+:5)

(Æ + 2

2 ) 3 ( + 3) 2 + 2 1 + + 4 + (Æ + 2 + 2 )21 (D + 2" + 2" )22 211 + 2 12 20 2 = 0; (D s+:6) (D + 4" ) 4 (Æ + 4 + 2 ) 3 + +3 2 + ( + 2 2 + )20 (Æ + 2 2 )21 2 10 + +211  22 = 0; (D s+:7) (Æ + 4

 ) 4

( + 2 + 4) 3 + 3 2 + +( + 2 + 2)21 (Æ + 2 + +2  )22 2 11 + 212  20 = 0; (D s+:8)

(D 2 2)11 (Æ 2 2 + + )10 (Æ 2 2 +  )01 + +( + 2 2 +  + )00 + + 12 + 21  02  20 + 3D  = 0; (D s+:9) (D 2 + 2" )12 (Æ + 2 2 )11 (Æ + 2 2  +  )02 + ( + 2 2 + +)01 + 22  00 10  21 + 3Æ  = 0; (D s+:10)

6.7. VARIATIONAL PRINCIPLE. . .

71

(D + 2" + 2"  )22 (Æ + 2 + 2  )21 (Æ + +2 + 2  )12 + ( + 2 + +2 )11  10  01 + + 20 + 02 + 3 = 0: (D s+ :11) To arrive at the complete set of the second Bianchi (D ) identities of the A4 geometry, we will have to add to these equations their complex conjugate (D s ).

6.7 Variational principle of derivation of the structural Cartan equations and the second Bianchi identities of A4 geometry To begin with, we will consider the derivation of the structural equations (B ) and of the second Bianchi identities (D ) for self-dual and anti-self-dual elds of Riemannian curvature, whose Carmeli matrices obey the conditions  Rkn = i Rkn ;  R+kn = i Rkn ; where Rkn + 2r[k Tn] [Tk ; Tn] = 0; R+kn + 2r[k Tn+] [Tk+; Tn+] = 0; and  1 Rkn = 2 "knps Rps ;  1 R +kn = 2 "knps R+ps : Let us take the Lagrange function in the form L1 = 1 ( g )1=2T r (RknRkn ) + complex conjugate part: (6.159) 4 Varying this expression in Tk and Tk+ , we will arrive at the equations (D ) rn R kn +[R kn ; T n] = 0; (6.160)   rn R +kn + [R +kn ; T +n ] = 0: (6.161) For arbitrary elds of Riemannian curvature the Lagrange function looks like    kn 1 1 1=2 L2 = 2 ( g ) T r R ( 2 Rkn 2r[k Tn] + [Tk ; Tn]) + c.c. part: (6.162)

72

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

  Variation of this Lagrangian in R kn and R+ kn yields the second Bianchi identities (D)   rn R kn +[Rkn ; T n] = 0; (D s+ )   rn R+ kn +[R +kn ; T +n ] = 0: (D s ) On the other hand, variation of the Lagrangian (6.162) in Tk and T + k gives the second structural Cartan equations (B ) of the A4 geometry

Rkn + 2r[kTn] [Tk ; Tn] = 0; R+kn + 2r[kTn+] [Tk+ ; Tn+] = 0; and

(B s+ ) (B s )



1 Rkn = 2 "knps Rps ;



1

R +kn = 2 "knps R+ps:

Independent variables in the Lagrangian (6.162) are the quantities Rkn , R+kn , Tk , and Tk+ . To obtain from them using the variational principle, the rst structural Cartan equations (A) of the A4 geometry

r k i [

]

T[k  i]  [iTk+] = 0;

(As)

we will have to introduce into the Lagrangian (6.162) as independent variables the matrices  i . This can be done by modifying the Lagrangian (6.162) as it has been done in [49]. We now write the equations (A); (B ) and (D ) in spinor form :

Ai ABC _ D _ = 0;

(6.163)

(B ) BF EACD _ B_ = 0 + c.c. equations; (D ) DABC D_ = 0 + c.c. equations;

(6.164) (6.165)

(A)

where i i AiABC @C D_ Ai B_ "PQ (TPAC D_ QB _ C D _ D _ = @AB _ i ) "R_ S_ (T i TPCAB_ QD R_ B_ D_ C AS_ i T R_ D_ BA _ C S_ ) = 0;

@DB_ TACE F_ + @E F_ TACDB_ + +"PQ (TAPDB_ TQCF E_ + +TACP B_ TQDF E_ TAPF E_ TQCDB_ TACP E_ TQFDB_ ) + +"R_ S_ (TACDR_ T S_ B_ EF TACF R_ T S_ E_ BD _ _ ) = 0;

(6.166)

BACF ED _ B _ = RACF ED _ B _

(6.167)

73

6.7. VARIATIONAL PRINCIPLE. . . _ HR _ X _ DABC D_ = 1 "C D_ F EG @RX_ RABGHF _ E_ 2i R F RP ABCR T F D_ 3 RPB (A TC ) D_ +

+RB D_ X_ TAR C X_ + AB X_ E_ T X D_ E_ C + AB D_ X_ T X E_ E_ C = 0 _

_

(6.168)

and consider the Lagrangian 



 _ L3 =R B Q_ AQkn (2rnTABk + 2TPAn TB P k )

1 P R + c.c. part: 4 BPA nk (6.169)  B AQkn _ _ nkjm B A Q nkjm Here R Q_ =" R Q_ jm and " is a completely skew-symmetrical Levi-Chivita symbol. _ If we take RB Q_ AQkn and TPAn to be independent variables and use the conventional variational procedure, we will obtain the following equations: 1 R P_ 2 B P_ A kn

2r[k TjAB jn] + 2TPA[k T P jB jn] = 0;

(6.170) (6.171)

(D s+)

complex conjugate equations;  Pk Q_ nk 2 R Q_ rk R B QA _ P Q_ (A jnkj TB ) = 0;

(D s )

complex conjugate equations:

(6.173)

(B s+ ) (B s )

(6.172)

Multiplying equations (6.170) by C D_ n F E_ n gives

@F E_ TABC D_

P P @C D_ TABF E_ + TPAF E_ TBA D_ TP AC D_ TBF E_

1 n R _ P F EC _ D _ + TABn (@C D _ F E_ 2 B QA

@F E_ C D_ n ) = 0:

(6.174)

Using the notation (6.166) and (6.167), we will write (6.174) as n BACF ED _ B_ + A CDF E_ TABn = 0:

(6.175)

We will now multiply (6.172) by C D_ k to get the relationship 



Q_ _ _ + R _ Q_ _ _ rk  F E_ + @ F E_ R B QA _ k C DE F B QA C DE F

 Q_ _ _  n _ @ F E_  RS_ + RB QA _ n RSE F C D   _ _ _ _ Q _ _ T PF E R _ Q _ _ T PF E = 0 RB QP _ C DE F A P QA C DE F B

(6.176)

or, from (6.166) and (6.167), n iDABC D_ + AF EP _ R _



 1 _ RE _ F_ QP  _ 2 nC D_ RB QA

n

P R_



RB QA _

Q_

E F_ C D_



= 0: (6.177)

74

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Here we have also used the relationship  Q_ _ _ = i (2 RB QA _ BACF "DE_ 2"CF AB D_ E_ + 2"D_ E_ ("BF "AC + C DE F +"BC "AF )) : It is clear that from the Lagrangian (6.169) it is impossible to obtain the rst structural Cartan equations (A) of the A4 geometry, since it does not contain C D_ n. Let us add to the Lagrangian (6.169) the term _ D _ Aj BC AjABC _ D _

(6.178)

_ D _ where the quantities ABC play the role of Lagrange factors _ D _ L4 = L3 + Aj BC AjABC _ D _ + c.c. part:

(6.179)

_ D _ The quantities Aj BC , just like AjABC _ D _ , are Hermitian matrices, which are skew-symmetrical in the pair of indices [49] AB_ and C D_ . Varying the Lagrange density (6.179) in C D_ n gives [49]

AjABC _ D _ = 0 and Since (D s+ )

(6.180)

k X_ (n nAXP DABC D_ = kP R_ nB _ R_ ) _ R _ C D_ = 0: AXP

nAXP _ R _

(6.181) are Hermitian matrices, from (6.181) we have the equations

DABC D_ = 0:

(6.182) Hence varying the complex conjugate part of the Lagrangian (6.179) gives

D A_ B_ CD = 0: _

(6.183)

_ Qkn and of the Lagrangian (6.179) in RB QA gives _

or, from (6.180),

n BACF ED _ B_ + A CDF E_ TABn = 0

(6.184)

BACF ED _ B _ = 0:

(6.185)

_ Variation of the complex conjugate part in R B Q AQkn yields _

B ACF ED _ B_ = 0:

(6.186)

It has thus been shown that from the Lagrangian (6.179) follow the rst and second of the structural Cartan equations of the A4 geometry (equations (6.180), (6.185) and (6.186)), and also the second of the Bianchi identities (equations (6.182) and (6.183)).

75

6.8. DECOMPOSITION OF SPINOR. . .

6.8 Decomposition of spinor elds of A4 geometry into irreducible parts The torsion tensor ::i jk of the A4 space has 24 independent components, and it can be represented as the sum of three irreducible parts as follows: 2 1

i:jk = Æ[ik j ] + "njks ^ s + i:jk ; 3 3

(6.187)

i:jk = g imgks ::s mj ;

(6.188)

where

i and the vector j , pseudovector ^ j and the traceless part of the torsion :jk are given by

j = i:ji; (6.189) 1

^ j = "jins ins; (6.190) 2 s

:js = 0; ijs + jsi + sij = 0: (6.191)

In the spinor basis the spinor representation of the Ricci rotation coeÆcients TABC C_ has the form [40] 



1 1 TABC C_ = AABC C_ + ("AC B C_ + "BC AC_ ) ; 2 3

(6.192)

where the spinor AABC C_ is completely symmetrical in the unprimed indices

AABC C_ = A(ABC )C_ ;

(6.193)

and the spinor B C_ is given by

AC_ = AAB B C_ :

(6.194)

In turn, the spinor AC_ can be decomposed into the Hermitian and antiHermitian parts: AC_ = AC_ iAC_ ; (6.195) where and

1 2

1 2

AC_ = ( AC_ + A_ C ); AC_ = i( AC_

A_ C )

AC_ =  AC_ = C A_ ; AC_ = AC_ = C A_ :

(6.196) (6.197)

The irreducible parts of torsion (6.189)-(6.191) and the spinors (6.193)(6.197) are related by

j ! AC_ ; (6.198)

^ j ! AC_ ; (6.199)

76

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

k

:js

_

(6.200)

ijk = gsk ::s ij ;

(6.201)

AAB ! ijk ; _ BC _ C_ 1

AAB (

" + A_ B_ CC _ BC _ C_ = _ "AB ); 2 ABC C_ A_ B_

ABC C_ = AC (AB )C_ + C_ (A "B )C :

(6.202)

Since we have

! AABC C :

(6.203) (6.204)

By de nition, the spinor AABC C_ is transformed in the D (3=2:1=2) irreducible representation of the group SL(2; C ). Consequently, the spinors AC_ and AC_ are transformed in the D (1=2:1=2) irreducible representation of the group SL(2:C ). Using the relationship (6.124), we can nd the components of the spinors AC_ and AC_ [50]

AC_ =





( + ) 12 (" + " ) ( ) + 12 (  )

1 2 1 2

1 2 1 2

( + ) + 12 (  ) ( + ) 12 ( + )



;



(  ) 12 (" ") 21 ( ) 12 (  ) ( ) + 12 (  ) 12 ( ) 12 ( ) : The Riemann tensor represented in terms of irreducible parts is

AC_ = i

1 2 1 2

1

Rijkm = Cijkm + gi[kRm]j + gj [k Rm]i + 3 Rgi[mgk]j :

(6.205) (6.206)

(6.207)

In the spinor basis this becomes [40]

RAAB _ BC _ CD _ D _ = ABCD "A_ B_ "C_ D _ + +"AB "CD A_ B_ C_ D_ + AB C_ D_ "CD "A_ B_ + +CDA_ B_ "AB "C_ D_ + 2("AC "BD "A_ B_ "C_ D_ + +"AB "CD "A_ D_ "B_ C_ ):

(6.208)

We also have the following connection:

Cijkm ! ABCD "A_ B_ "C_ D_ + "AB "CD A_ B_ C_ D_ ; Rij ! 2AB C_ D_ + 6"AB "C_ D_ ; R ! 24;

(6.209)

where the spinors ABCD , AB C_ D_ and  meet the following symmetry conditions: ABCD = (ABCD) ; AB C_ D_ = (AB )(C_ D_ ) ;  =  and belong to the D (2:0), D (1:1) and D (0:0) irreducible representations of the group SL(2:C ), respectively.

77

6.9. SPINOR REPRESENTATION. . .

6.9 Spinor set of Einstein-Yang-Mills equations In the rst part of the book it was shown that the structural Cartan equations of the geometry of absolute parallelism (A) and (B ) can be represented as an extended set of Einstein-Yang-Mills equations

r k ea j

+ T[ikj ]ea i = 0; (A) Rjm gjm R = Tjm ; (B:1) C i jkm + 2r[k T ijj jm] + 2Tsi[k T sjj jm] = J i jkm : (B:2) [

] 1 2

(6.210)

We will write this set of equations in the spinor basis. To this end, we will make use of the Carmeli matrices and the Newman-Penrose spinor formalism. Suppose now we have the right spin A4 geometry, then its equations (A) and (B ) have the form + R_ +Ai R_ (TDC _ _ )B

+ F_ T _ (TDC _ AF _ )B

@C D_ Ai B_ @AB_ Ci D_ = (TC D_ )PA Pi B_ + + R_ ; (TAB_ )PC Pi D_ Ci R_ (TBA (As+) _ _ )D

RABC @AB_ TC D_ (TC D_ )FA TF B_ _ D _ = @C D _ TAB _ + F_ T _ + [T _ ; T _ ]; + (TAB_ )FC TF D_ + (TBA (B s+ ) _ CF _ )D AB C D

where the components of the matrices Ai B_ , TAB_ and RABC _ D _ are given by (6.115), (6.88) and (6.103), respectively. Proposition 6.13.

Equations (B:1) in the spinor basis are

2AB C_ D_ + "AB "C_ D_ = TACB _ D _:

(6.211)

Proof. In terms of the irreducible spinors (6.209) P Q the components of the spinor matrices RABC _ D _ are given by [51] 

Q=" Q (RABC _ D _ )P D_ B_ CAP



("PC ÆAQ + "PA ÆCQ) + "CA P Q D_ B_ ; (6.212)

where

Q=" Q (CABC (6.213) _ D _ )P D_ B_ CAP are the P Q components of the spinor matrices of the Weyl tensor with the the components 

1 2  2 C110 _ 0_ = 3

C010 _ 0_ =

0 1 1 2





3 4   2 ; C100 _ 1_ = 3

; C111 _ 0_ =

2 3 1 2





; ;

(6.214)

and related with the spinor "D_ B_ ("PC Æ AQ + "PA ÆC Q ) and "CA P QD_ B_ are the trace and traceless parts of the Ricci tensor 1 k   n Rg ; (6.215) "AB "C_ D_ = 4 AC_ B D_ kn

78

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .





1 AB C_ D_ =  k AC_  n B D_ Rkn 2

1 g R : 4 kn

(6.216)

Substituting relationships (6.215) and (6.216) into (6.211) and multiplying the resultant expression by  AC_ k  B D_ n , we arrive at the equations (B:1): We now represent the matrix RABC _ D _ as the sum

RABC _ D _ = CABC _ D _ + JABC _ D _;

(6.217)

where the matrix current JABC _ D _ has the components [52]: 1 J010 _ 0_ = 2



0 1 T 0 6



0



1 J100 _ 0_ = 2



1

J110 _ 1_ = 2

1 J110 _ 0_ = 2 1 J100 _ 1_ = 2

 

1 ; J111 _ 0 _ = 2



T100_ 0_ T101_ 0_

T000 _ 0_ T100 _ 0_

T011_ 1_ T111_ 1_

T010 _ 1_ T011 _ 1_

T110 _ 0_ T101 _ 1_

T010 _ 0_ T110 _ 0_

T110 _ 0_ T101 _ 1_

T010 _ 0_ T110 _ 0_

 

1 2 1 + 2



0 0  

1 6

1 6

0

0

;

;

T 1 6



; (6.218) 0

0



T

1 6

T



T

0

1 6



T

; :

Here

k n T ; TABC (6.219) _ D _ =  AC _ B D_ kn T = g jmTjm; (6.220) and the energy-momentum tensor Tkn is given in terms of the Ricci rotation

coeÆcients by

2n

Tjm =  r[iTjijjm] + Tsi[iTjsj jm]   1 pn i i s g gjm r[ijTjpjn] + Ts[iTjpjn] :

(6.221)

2

In the special case where the eld T ijk is skew-symmetric in all the three indices, the tensor (6.219) is [33] 

Tjm = 1 ^ j ^ m 



1 g ^ i ^ : 2 jm i

(6.222)

Multiplying this by  j AC_  m B D_ and using (6.199), we get 1



TABC _ D _ =  AB_ C D_



1 P Q_ : " " _ AC _D _ P Q B 2

(6.223)

79

6.10. FORMALISM OF TWO-COMPONENT. . .

In addition, we obtain

T = g jmTjm = 1 ^ j ^ j = 1 P Q_ P Q_ :  

(6.224)

Hence the \density of spinor matter" is

=

1

c2

P Q_ P Q_ :

(6.225)

We substitute (6:217) into the spinor equations (B s+) go get (B s+ :1)

2AB C_ D_ + "AB "C_ D_ = TACB _ D _;

CABC _ D _

as

+ F @C D_ TAB_ + @AB_ TC D_ + (TC D_ )FA TF B_ + (TDC _ )B_ TAF_

+ F_ T _ [T _ ; T _ ] = J _ _ : (TAB_ )FC TF D_ (TBA (B s+ :2) _ CF _ )D AB C D ABC D To conclude, we will write the extended set of Einstein-Yang-Mills equations

@C D_ Ai B_

+ R_ @AB_ Ci D_ = (TC D_ )PA Pi B_ + Ai R_ (TDC _ )B _ _ + R; (TAB_ )PC Pi D_ Ci R_ (TBA _ _ )D 2AB C_ D_ + "AB "C_ D_ = TACB _ D _; + F F_ T _ CABC @ T + @ T + ( T ) T + _ D _ _ AF _ )B C D_ AB_ AB_ C D_ C D_ A F B_ (TDC + F_ T _ [T _ ; T _ ] = J _ _ : (TAB_ )FC TF D_ (TBA ) _ AB C D ABC D D_ C F

(As ) s (B + :1) (B s+ :2)

_ B; _ D_ : : : = 0_ ; 1_ . where the spinor indices take on the values A; B; D : : : = 0; 1, A;

6.10 Formalism of two-component spinors

We will introduce the two-component spinors o and i [53], connected with the components of the spinor dyad  as follows:

0 = o ; 1 =  ;  0__ = o _ ;  1__ =  _ ;

(6.226)

; : : : = 0; 1; ; _ _ : : : = 0_ ; 1_ : From the orthogonality condition for the spinor dyad

0  0

 0 1 = 1; =  0 0 = 0; 1  1 = 0:

 0 0  1 0 = Æ ;  0  1  1  0 = " ;

(6.227)

(6.228)

80

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

where





0 1 1 0 ; we derive the normalization condition for the two-component spinors

" = " = " _ Æ_ = " _ Æ_ =

o  =  o = 1; = o o = 0;   = 0;

o o

(6.229)

(6.230)

and also the relationships

" = o 

 o ; " = o 

o  ; " = o 

 o :

Spinors o and i de ne the components of the Newman-Penrose symbols (6.6) _ Ai B_ =  i _ A  B_ (6.231) as follows:

0i 0_ =  i _ o o _ = li ; 1i 1_ =  i _   _ = ni ; 0i 1_ =  i _ o  _ = mi ; 1i 0_ =  i _  o _ = mi :

(6.232)

The vectors li , ni , mi and m i form an isotropic tetrad. The conventional tetrad ei a can be made up of the vectors of an isotropic tetrad using the relationships

ei0 = (2)

(li + ni ) = (2) 1=2  i _ (o o _ +   _ ); ei1 = (2) 1=2 (mi + m i ) = (2) 1=2  i _ (o  _ +  o _ );

ei2 = (2)

=

1 2

ei3 = (2)

=

1 2

i(mi m i ) = (2) =

1 2

(li

ni ) = (2)

=

1 2

=

 i _ (o  _

1 2

 i _ (o o _

 o _ );   _ ):

(6.233)

Using the relationships

TACk = 1 "B_ D_ Ci D_ rk ABi_ ;

(6.234)

r =  i ri

(6.235)

2

_

we nd the following expressions for the [54]:  = o o _ o r _ o ;  =  o _ o r _ o ;  = o  _ o r _ o ;  =   _ o r _ o ;  =   _  r _  ;  = o  _  r _  ;

_

components of the Carmeli matrices

 =  o _  r _  ;  = o o _  r _  ; " = o o _  r _ o ; = o  _  r _ o ;

=   _ o r _  ; =  o _ o r _  ;

(6.236)

81

6.10. FORMALISM OF TWO-COMPONENT. . .

0 = Æ o o o oÆ ; 1 = Æ o o o Æ ; 2 = Æ o o  Æ ; 3 = Æ o   Æ ; 4 = Æ    Æ ; 00 = 00 =  _ Æ_ o o o_ oÆ_ ; 01 = 10 =  _ Æ_ o o o_ Æ_ ; 02 =  2 =  _ Æ_ o o _ Æ_ ; 11 = 11 =  _ Æ_ o  o_ Æ_ ; 12 = 21 =  _ Æ_ o  _ Æ_ ; 22 = 22 =  _ Æ_   _ Æ_ :

(6.237)

(6.238)

It follows from (6.236) that

r o = o o o o o  o  o + "o  

(6.239)

r  = o o o o o  o  o + o  

(6.240)

_

_

_

_

_

 o o_ +  o _ +   o_   _ ; _

_

_

 o o_ +  o _ +   o_ "  _ :

_

_

The components of the spinor derivative (6.114) can be represented in terms of two-component spinors as

D = o o _ r _ ;  =   _ r _ ; Æ = o  _ r _ ; Æ =  o _ r _ :

(6.241)

In the formalism of two-component spinors there exists the so-called modi ed formalism [53] that takes into account the "primed" symmetry of spinor quantities. This symmetry allows the replacement

o ! i ;  ! io ; o _ ! i _ ;  _ ! io _ ;

(6.242)

where the unprimed quantities are replaced by primed ones following the rule (li )0 = ni ; (mi )0 = m i; (mi )0 = mi ; (ni)0 = li ; (6.243)

 = 0 ;  = o0 ;  = 0 ;  =  0 ; = 0 ; = "0 :

(6.244)

This symmetry property makes it possible to replace in (6.236) unprimed quantities by primes ones  = o _ o o r _ o ;  0 =  _ o  r _  ;  =  o _ o r _ o ;  0 = o o _  r _  ;

 = o  _ o r _ o ;  =   _ o r _ o ; 0 =   _  r _  ; 0 = o  _  r _  ;



" = o o _  r _ o ; = o  _  r _ o ; "0 =   _ o r _  ; 0 =  o _ o r _  ;

(6.245)

82

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Here instead of 12 spinor coeÆcients we have only six. The most general transformation under which spinors o , i and the conditions (6.230) are retained is

o ! Co ;  ! C 1  ;

(6.246)

where C is a complex transformation that forms a subgroup of boosts and three-dimensional rotations. The components of the isotropic tetrad (6.232) vary under these transformations as follows:

li ! A 1 li ; ni ! Ani ; mi ! ei mi ; A = CC; ei = CC 1 :

(6.247)

We will now de ne a scalar quantity with the following properties:

 ! C P C q :

(6.248)

This quantity is said to be a spin and boost weight scalar of the type (p; q ) [53]. It follows from (6.246) that the components of the spinors o and i are scalars of types (1; 0) and ( 1; 0), respectively. The components of the isotropic tetrad will be

li : (1; 1); ni : ( 1; 1); mi : (1; 1); mi : ( 1; 1):

(6.249)

In respect of the transformations (6.247) all the spin coeÆcients (6.236) can be divided into two classes: (a) quantities that are transformed in a uniform manner, e.g.,

 ! (Co )(C 1  _ )(Co )r _ (Co ) = C 3 C 1  ;

(6.250)

(b) quantitites that are transformed in a nonuniform manner using derivatives of C , e.g.,

! (Co )(C 1  _ )(C 1  )r _ (Co ) = 1 1 = CC + C o  _ r _ C:

(6.251)

If we take into account "primed" symmetry and spin and boost weights, the main spinor quantities become

 : (3; 1);  : (3; 1);  : (1; 1);  : (1; 1); 0 : ( 3; 1);  0 : ( 3; 1); 0 : ( 1; 1);  0 : ( 1; 1); 0 = 04 : (4; 0); 1 = 03 : (2; 0); 2 = 02 : (0; 0); 3 = 01 : ( 2; 0); 4 = 00 : ( 4; 0);

(6.252)

6.10. FORMALISM OF TWO-COMPONENT. . .

83

01 = 10 = 021 : (2; 0); 00 =  00 = 022 : (2; 2); 0 02 =  20 = 20 : (2; 2); 10 = 01 = 012 : (0; 2); 11 =  11 = 011 : (0; 0); 12 = 21 = 010 : (0; 2); 20 = 02 = 002 : ( 2; +2); 21 = 12 = 001 : ( 2; 0); 22 = 22 = 000 : ( 2; 2);  =  = 0 = R=24 : (0; 0): For weighted quantities we will introduce new di erential operators, such that their action on a scalar  of the type fp; q g is de ned as P  = (D p" q"); P 0  = ( + p"0 + q"0); (6.253) @ = (Æ p + q 0 ); @ 0  = (Æ + p q ): Operators (6.253) have the following spin weights:

P : (1; 1); @ : (1; 1); P 0 : ( 1; 1); @ 0 : ( 1; 1):

(6.254)

In terms of (6.252) and di erential operators (6.253), we can write the spinor equations (B s+) in a simpler form [53]. Shown schematically in Fig. 6.1 are the boost and spin weights of the main spinors of the A4 geometry.

84

CHAPTER 6. GEOMETRY OF ABSOLUTE. . .

Figure 6.1: Boost and spin weights of main spinors of the A4 geometry

Chapter 7

Construction of solutions to structural Cartan equations of the geometry of absolute parallelism 7.1 Selection of a frame of reference and specialization of Newman-Penrose symbols The structural Cartan equations of any geometry describe the general connection between basic geometrical characteristics of a given geometry. A special solution of structural equations determines speci c geometrical quantities, such as the curvature, connection, metric, etc., characteristic of a given speci c solution [55]. For simplicity we will investigate the structural Cartan equations of the A4 geometry r[k eam] eb [k T ajbjm] = 0; (A)

Rabkm + 2r[k T ajbjm] + 2T ac[k T cjbjm] = 0;

(B )

written in the vector basis, for their compatibility. These equations are essentially a system that in the general case includes 44 (24 equations (A) and 20 equations (B )) nonlinear partial di erential equations of the rst order with the following unknown functions: (a) 6 components of anholonomic tetrad

ei a = raxi ;

(7.1)

(b) 24 components of the Ricci rotation coeÆcients

T abk = ej b rk ea j ; 85

(7.2)

86

CHAPTER 7. CONSTRUCTION OF SOLUTIONS. . .

(c) 20 components of the Riemann tensor

Ra bkm :

(7.3)

Thus, in the general case we have 44 equations for 50 unknown functions. This gives us some freedom in choosing a reference frame xi , of the tetrad ei a, and also of the quantities T abk and Ra bkm . Therefore, search for speci c solution to the set of equations (A) and (B ) should rather be referred to as "constructing solutions." When constructing solutions it is convenient to represent the structural Cartan equations of the geometry of absolute parallelism in the spinor -basis in terms of Carmeli matrices

@C D_ Ai B_

+ R_ _ @AB_ Ci D_ = (TC D_ )AP Pi B_ + Ai B_ (TDC _ ) B

(TAB_ )C P Pi D_

RF ED @F E_ TDB_ _ B_ = @DB _ TF E _

+ R_ ; Ci R_ (TBA _ _ ) D

(TDB_ )F S TS B_

(As) + F_ (TED _ ) B_ TF F_ +

+ F_ +(TF E_ )D S TS B_ + (TEF (B s+ ) _ ; TDB _ ]: _ ) B_ TDF_ + [TF E where the components of the traceless 2  2 matrices RF ED _ B_ and TF E _ are found from the relationships (6:88) and (6:103). Let us now nd the Newman-Penrose symbols via the spinor representation of the invariant Haiashi derivative

AB_ i = rAB_ xi = @AB_ xi ;

(7.4)

where the components of the spinor derivative @AB_ are denoted as

B_

_ _ @AB_ = A 0 1 0 D Æ 1 Æ 

(7.5)

From the relationships (7.4)-(7.5) and

 iAB_ = A 0 1

we obtain and also

li

B_

0_

= (Y ; V; Y ; Y ) 0 2 3 m i = ( ; !;  ;  ) 0

2

3

1_

mi = ( 0 ; !;  2;  3 ) ni = (X 0; U;X 2; X 3)

(7.6)

li = Dxi ; ni = xi ; mi = Æxi ; mi = Æxi ;

(7.7)

Y 0 = Dx0 ; X 0 = x0;  0 = Æx0 ;  = Æx0 ; V = Dx1 ; U = x1 ; ! = Æx1 ; ! = Æx1 ; 2 Y 2 = Dx2 ; X 2 = x2;  2 = Æx2 ;  = Æx2 ; Y 3 = Dx3 ; X 3 = x3;  3 = Æx3 ;  3 = Æx3 :

(7.8)

0

87

7.1. SELECTION OF REFERENCE FRAME. . .

From the equality

@AB_ = AB_ iri = AB_ i ;i

(7.9)

D = li ri;  = ni ri; Æ = mi ri; Æ = m iri

(7.10)

and the relationships (7.5) and (7.6) it follows or

D = V @ 1 + Y @ ; @x @x @ @  = V 1 + X ; @x @x Æ = w @1 +  @ ; @x @x @ @ Æ = w 1 + ; @x @x = 0; 2; 3: as

(7.11)

Using these relationships, we write the vectors that make up the matrix (7.6)

li = V Æ1i + Y Æ i ; ni = UÆ1i + X Æ i ; mi = !Æ1i +  Æ i ; mi = !Æ1i +  Æ i :

(7.12)

From the orthogonality condition for the Newman-Penrose symbols

iAB_ Aj B_ = Æij ;

(7.13)

iAB_ Ci E_ = Æ A C Æ B_ E_ :

(7.14)

follow the orthogonality conditions for the vectors (7.12)

li li = mi mi = m im i = ni ni = 0; lini = mi m i = 1; i i li m = li m = nimi = nim i = 0: And from the formulas

gij = "AC "B_ D_ iAB_ jC D_ ; "00 = "11 = 0; "01 = 1; "10 = 1;

we nd

g ij = li nj + lj ni mi mj

mj mi :

(7.15) (7.16)

(7.17) Vectors that meet the orthogonality conditions (7.15) are null vectors and in physics they are normally associated with propagation of radiation (i.e., with

88

CHAPTER 7. CONSTRUCTION OF SOLUTIONS. . .

matter that has no rest mass), where concepts of wave fronts, waves, rays, etc, hold. In the process a family of null hypersurfaces u(xi ) = const is introduced. We will take the vector li to be orthogonal to these hypersurfaces

li = u;i :

(7.18)

Further, we will select the coordinates so that [60]

x0 = u; x1 = r; where r is the aÆne parameter along the null x2 ; x3 ;

geodesics

(7.19)

;

where x assign numbers to rays on each hypersurface and are constant along the rays. 2 3

When selecting the coordinates, the vector li and li look like:

li = u;i = x0;i = Æi0; dxi dxi = Æ1 i li = 1 = dx dr

or

li = (0; 1; 0; 0); li = (1; 0; 0; 0):

From the orthogonality conditions

(7.20) (7.21) (7.22)

li ni = 1; limi = 0 it follows that

ni = (1; U; X 2; X 3); mi = (0; !;  2;  3 );

(7.23)

and the relationships (7.11) become

D = V @1 = @ ; @x @r @ @ @ @ @ @ =U 1 + 0 +X = +U +X ; @x @x @x @u @r @x Æ = ! @1 +  @ = ! @ +  @ ; @x @x @r @x @ @ @ @ Æ=! 1 + =! + ; @x @x @r @x = 2; 3:

(7.24)

Moreover, the vectors (7.12) will be given by

li = Æ1 i ; ni = Æ0 i + UÆ1i + X Æ i ; mi = !Æ1 i +  Æ i ; m i = !Æ1i +  Æ i ;

(7.25)

89

7.2. SPECIALIZATION OF THE SPINOR. . .

= 2; 3: Since

li = g ik lk = g ik Æk0 = g io = Æ1i ;

the metric tensor has the following structure [56]: 0 B

g ik = B @

0 1 0 0 1 g 11 g 12 g 13 0 g 12 g 22 g 23 0 g 13 g 23 g 33

1 C C: A

(7.26)

(a) (b) (c)

; Æ; = 2; 3:

(7.27)

Using the relationship (7.17) and (7.25), we get

g 22 = 2(U !! ); g 2 = X ( ! +  ! ); g Æ = (  Æ +   Æ );

As is seen from the above reasoning, the coordinates (7.19) selected and the specialization of the Newman-Penrose symbols using the relationship (7.18) made it possible to us to derive the dependence (7.27) and the general form (7.26) of the metric tensor g ik of the A4 geometry.

7.2 Specialization of the spinor components of the Ricci rotation coeÆcients The spinor structural Cartan equations (As) and (B s) of the geometry A4 can be viewed as a matrix of possible geometries of absolute parallelism that di er in speci c set of spinor geometrical characteristics. Therefore, we will assume the solution of the spinor structural equations (As) and (B s ) (in this reference frame) to be a set of variables consisting in the general case of: (a) 6 independent components of the Newman-Penrose symbols

Ai B_ ;

(7.28)

(b) 24 independent spinor components of the Ricci rotation coeÆcients + TAB_ ; TBA _ ;

(7.29)

(c) 20 independent spinor components of the independent spinor components of the Riemannian tensor + RABC (7.30) _ D _ ; R _ _ ; BADC

that transform the equations (As) and (B s) into identities when substituted into these equations.

90

CHAPTER 7. CONSTRUCTION OF SOLUTIONS. . .

In our search for solutions to the structural equations (As) and (B s) we will rely on the symmetry conditions, and also on physical arguments, e.g., we will subject the Riemannian tensor to the conditions of Einstein's vacuum

Rij = 0;

(7.31)

which can be represented in terms of Carmeli matrices (7:103), (7:214) and (7:217) as RABC _ D _ = CABC _ D _ = 0: We will now consider the limitations that can be imposed on the components of the matrices (6:88), using physical reasoning. To this end, we will turn to the relationship   + R_  AB_ rk Ci D_ = (TAB_ )PC Pi D_ + Ci R_ (TBA (7.32) _ k _ )D or + R_ = r  i  k : (TAB_ )PC Pi D_ + Ci R_ (TBA (7.33) k C D_ AB_ _ _ )D

91

7.2. SPECIALIZATION OF THE SPINOR. . .

From (7.6), (6:88), and (7.33) we get (7.34)

lk rk li = (" + " )li nk rk li = ( + )li mk rk li = ( + )li m k rk li = ( + )li lk r

ni

mi m i; mi mi ; mi mi ; mi mi ;

(" + ")ni + m

+ m i;

= i k i i n rk n = ( + )n + mi + m i; k

mk rk ni = ( + )ni + mi + mi ; m k rk ni = ( + )ni + mi + mi ; lk r

k

mi

= ("

")mi + l

i

ni ;

nk rk mi = ( )mi + li ni ; mk rk mi = ( )mi + li ni; mk rk mi = ( )mi + li ni ; lk rk mi = (" nk rk m i = ( mk rk mi = ( m k rk mi = (

")mi + li

)m i + li )m i + li )mi + li

ni; ni ; ni ; ni:

(7:34a) (7:34b) (7:34c) (7:34d) (7.35) (7:35a) (7:35b) (7:35c) (7:35d) (7.36) (7:36a) (7:36b) (7:36c) (7:36d) (7.37) (7:37a) (7:37b) (7:37c) (7:37d)

Further, using the orthogonality condition (7.15), we have

 = rk li mi lk ;  = rk nim ink ;  = rk limi m k ; i k  = rk nim m ;  = rk li mi mk ;  = rk ni mi mk ;  = rk li mi nk ;  = rk nim ilk ; (7.38) 1 2 = 1 (rk li nimk 2 1

= (rk li ni nk 2

= (rk lini mk

rk mi mi m k ); rk mi mi mk ); rk mi m ink );

92

CHAPTER 7. CONSTRUCTION OF SOLUTIONS. . .

" = 1 (rk lini nk

rk mi mi lk ): 2 On the other hand, we can write (7.32) as (7.39)

rk lj = ( + )lk lj

lk mj lk mj + (" + ")nk lj nk mj nk m j ( + )mk lj + mk mj + +mk mj ( + )m k lj + +mk mj + m k m j ; rk nj = ( + )lk nj + lk mj + +lk m j (" + ")nk nj + nk mj + +nk m j + ( + )mk nj mk mj mk mj + ( + )mk nj m k mj m k mj ; rk mj = ( )lk mj + lk lj lk nj + (" " )nk mj + nk lj nk nj + ( )m k mj mk lj + mk nj + ( )mk mj m k lj + m k nj ; rk m j = ( )lk m j + lk lj lk nj + +(" ")nk mj nk lj nk nj + ( )mk mj m k lj + mk nj + ( )mk m j mk lj + mk nj ; Alternating these relationships in the indices k and j gives

(7:39a)

(7:39b)

(7:39c)

(7:39d)

(7.40)

r k lj

(

= 2