Spin-Curvature and the Unification of Fields in a Twisted Space

7/31/2019 Spin-Curvature and the Unification of Fields in a Twisted Space

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INDRANU SUHENDRO

SPIN-CURVATURE

AND THE UNIFICATION

OF FIELDS

IN A TWISTED SPACESVENSKA

FYSIKARKIVET

2008


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Indranu Suhendro

Spin-Curvature

and the Unification of Fields

in a Twisted Space

Spinn-krokning

och foreningen av falt

i ett tvistat rum

2008Swedish physics archive

Svenska fysikarkivet


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Svenska fysikarkivet, that means the Swedish physics archive, is a publisher regis-tered with the National Library of Sweden (Kungliga biblioteket), Stockholm.

Please send all comments on this book or requests to reprint to the Author.E-mails: [email protected]; spherical [email protected]

Edited by Stephen J. Crothers and Dmitri Rabounski

Copyright c Indranu Suhendro, 2008Copyright c Typesetting and design by Dmitri Rabounski, 2008Copyright c Publication by Svenska fysikarkivet, 2008

Copyright Agreement: All rights reserved. The Author does hereby grant Sven-ska fysikarkivet non-exclusive, worldwide, royalty-free license to publish and dis-tribute this book in accordance with the Budapest Open Initiative: this means thatelectronic copying, print copying and distribution of this book for non-commercial,academic or individual use can be made by any user without permission or charge.Any part of this book being cited or used howsoever in other publications must ac-knowledge this publication. No part of this book may b e reproduced in any formwhatsoever (including storage in any media) for commercial use without the priorpermission of the copyright holder. Requests for permission to reproduce any part ofthis book for commercial use must be addressed to the Author. The Author retainshis rights to use this book as a whole or any part of it in any other publications andin any way he sees fit. This Copyright Agreement among the Author and Svenskafysikarkivet shall remain valid even if the Author transfers copyright of the bookto another party. The Author hereby agrees to indemnify and hold harmless Sven-ska fysikarkivet for any third party claims whatsoever and howsoever made againstSvenska fysikarkivet concerning authorship or publication of the bo ok.

Cover photo: Spiral Galaxy M74. This image is a courtesy of the Hubble SpaceTelescope Science Institute (STScI) and NASA. This image is a public domain prod-uct; see http://hubblesite.org/copyright for details. We are thankful to STScI andNASA for the image. Acknowledgement: R. Chandar (University of Toledo) andJ. Miller (University of Michigan).

This book was typeset using teTEX typesetting system and Kile, a TEX/LATEX editorfor the KDE desktop. Powered by Ubuntu Linux.

ISBN: 978-91-85917-01-3

Printed in Sweden and the USA


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Contents

P r e f a c e o f t h e E d i t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

A c k n o w l e d g e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 1 A four-manifold possessing an internal spin

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Geometric construction of a mixed, metric-compatiblefour-manifold R4 possessing an internal spin space Sp . . 11

Chapter 2 The unified field theory

2.1 Generalization of Kaluzas projective theory. . . . . . . . . . . . 202.2 Fundamental field equations of our unified field theory.

Geometrization of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 3 Spin-curvature

3.1 Dynamics in the microscopic limit. . . . . . . . . . . . . . . . . . . . . .443.2 Spin-curvature tensor ofSp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5

3.3 Wave equation describing the geometry ofR

4. . . . . . . . . . 48

Chapter 4 Additional considerations

4.1 Embedding of generalized Riemannian manifolds(with twist) in N= n +p dimensions . . . . . . . . . . . . . . . . . . . 53

4.2 Formulation of our gravoelectrodynamics by meansof the theory of distributions. Massive quantum elec-tromagnetic fi eld tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 On the conservation of currents . . . . . . . . . . . . . . . . . . . . . . . . 644.4 On the wave equations of our unified field theory. . . . . . . 654.5 A more compact form of the generalized Gauss-

Codazzi equations in IR5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9

Appendix The fundamental geometric properties of a curvedmanifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7

B i b l i o g r a p h y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 8


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Preface of the Editor

I am honoured to present this book by Indranu Suhendro, in which headduces theoretical solutions to the problems of spin-curvature and theunification of fields in a twisted space. A twist of space is given hereinthrough the appropriate formalism, and is related to the anti-symmetricmetric tensor. Kaluzas theory is extended and given an appropriate

integrability condition. Both matter and the isotropic electromagneticfield are geometrized through common field equations: trace-free fieldequations, giving the energy-momentum tensor for the electromagneticfield via only the generalized Ricci curvature tensor and scalar, areobtained. In the absence of the electromagnetic field the theory goes toEinsteins 1928 theory of distant parallelism where only the matter fieldis geometrized (through the twist of space-time). Therefore the aboveresults, in common with respective wave equations, are joined into aunified theory of semi-classical gravoelectrodynamics.

There have been few attempts to introduce spin-particles into thetheory of relativity (which is the theory of fields, in the sense pro-pounded by Landau and Lifshitz). Frankly speaking, only two of the

attempts were complete. The first attempt, by A. Papapetrou (1951),was a frontal approach to this problem: he introduced a spin-particleas a swiftly rotating gyro (Proc. Roy. Soc. A, 1951, v. 209, 248258 and259268). This approach however doesnt match experimental data dueto that fact that, considering an electron as a solid ball, the linearvelocity of its rotation at its surface should be 70 times greater thanthe velocity of light. The second attempt, by me and L. Borissova(2001), introduced a spin-particle through the variational principle andLagranges function for such a system (see Chapter 4 of the book Fields,Vacuum, and the Mirror Universe, Editorial URSS, Moscow, 2001). De-spite some success related to its immediate application to the theory ofelementary particles, this method however could not be considered aspurely geometric: spin, the fundamental property of a particle, wasnt

expressed through the geometric properties of the basic space, but in-stead still remained a non-geometrized fundamental characteristic ofmatter.

Insofar as the geometrization of distributed matter is concerned,just one solution was successful before this book. It was given foran isotropic electromagnetic field. Such fields are geometrized via the


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Preface of the Editor 5

well-known Rainich condition and the Nortvedt-Pagels condition: theenergy-momentum tensor of such a field expresses itself through thecomponents of the fundamental metric tensor, so Einsteins equationscontain only the geometric left-hand-side by moving all the right-hand-side terms to the left-hand-side so the right-hand-side becomeszero. Various solutions given by the other authors are particular to theRainich/Nortvedt-Pagels condition, or express an electromagnetic fieldunder a very particular condition.

Both problems are successfully resolved in this book with the useof only geometric concepts on a common basis (a twisted space), which

is the great advantage of this work. This fact places this book in thesame class of great ideas in the theory of fields, produced during the lastcentury, commencing with the time Albert Einstein first formulated hisfield equations.

I am therefore very pleased to bring this book to the reader. I recom-mend it to anyone who is seriously interested in the Theory of Relativityand the geometric approach to physics in particular.

February, 2008 Dmitri Rabounski


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Foreword

In the present research, I consider a unification of gravity and electro-magnetism in which electromagnetic interaction is seen to produce agravitational field. The field equations of gravity and electromagnetismare therefore completely determined by the fundamental electromag-netic laws. Insight into this unification is that although gravity and

electromagnetism have different physical characteristics (e.g., they dif-fer in strength), it can be shown through the algebraic properties of thecurvature and the electromagnetic field tensors that they are just different aspects of the geometry of space-time. Another hint comes fromthe known speed of interaction of gravity and electromagnetism: elec-tromagnetic and gravitational waves both travel at the speed of light.This means that they must somehow obey the same wave equation.This indeed is unity. Consequently, many different gravitational andelectromagnetic phenomena may be described by a single wave equationreminiscent of the scalar Klein-Gordon equation in quantum mechan-ics. Light is understood to be a gravoelectromagnetic wave generatedby a current-producing oscillating charge. The charge itself is gener-ated by the torsion of space-time. This electric (or more generally,electric-magnetic) charge in turn is responsible for the creation of mat-ter, hence also the transformation of matter into energy and vice versa.Externally, the gravitational field manifests itself as the final outcomeof the entire process. Hence gravity and electromagnetism obey thesame set of field equations, i.e., they derive from a common origin. Asa result, the charge produces the so-called gravitational mass. Albeitthe geometric non-linearity of gravity, the linearity of electromagnetismis undisturbed: an idea which is central also in quantum mechanics.Therefore I preserve the most basic properties of matter such as energy,momentum, mass, charge and spin through this linearity. It is a mod-est attempt to once again achieve a comprehensive unification whichexplains that gravity, electromagnetism, matter and light are only dif-

ferent aspects of a single theory.

Karlstad, Sweden, 2004 Indranu Suhendro


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Acknowledgements

This work was originally intended to be my Ph.D. thesis in 2003. How-ever, life has led me through its own existential twists and turns. Inthe process of publishing this book, many individuals have made theirsplendid contributions. It would be rather difficult to give credit with-out overlooking some of these sincere, wonderful persons. I would like

especially to single out my dear colleagues, Dmitri Rabounski, StephenCrothers, and Augustina Budai, who most passionately and sincerelyhelped prepare the entire manuscript in the present form. I would alsolike to express my most sincere gratitude to Bengt and Colette Johnsson,who gave me shelter and showered me with tenderness during my mostcreative years in Sweden. I would also like to thank my dear friends,Barbara Smilgin, Dima Shaheen, Quinton Westrich, Liw Raskog, Vi-taliy Kazymyrovych, and Sana Rafiq, with whom I share the subtlepassions that normally lie hidden in the solitary, silent depths of life.Amidst lifes high tides and despite all human frailties, in our togeth-erness, I believe, we have done enough to motivate each other to reachthe height of existence. I must also recognize the most beautiful love,patience, and understanding of my father, mother, and the rest of myfamily throughout my life of seasons. Finally, this little scientific cre-ation is dedicated with infinite love, yearning, gratitude, and humilityto A. S. and M. H. for always being there for me somewhere betweenmy slumber and wakefulness in this sojourn called life.

February, 2008 Indranu Suhendro


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Chapter 1

A FOUR-MANIFOLD POSSESSING

AN INTERNAL SPIN

1.1 Introduction

Attempts at a consistent unified field theory of the classical fields of

gravitation and electromagnetism and perhaps also chromodynamicshave been made by many great past authors since the field conceptitself was introduced by the highly original physicist, M. Faraday inthe 19th century. These attempts temporarily ended in the 1950s:in fact Einsteins definitive version of his unified field theory as wellas other parallel constructions never comprehensively and compellinglyshed new light on the relation between gravity and electromagnetism.Indeed, they were biased by various possible ways of constructing a uni-fied field theory via different geometric approaches and interpretationsof the basic geometric quantities to represent the field tensors, e.g., theelectromagnetic field tensor in addition to the gravitational field ten-sor (for further modern reference of such attempts, especially the last

version of Einsteins gravoelectrodynamics see, e.g., various works ofS. Antoci). This is put nicely in the words of Infeld: . . . the problem ofgeneralizing the theory of relativity cannot be solved along a purely for-mal way. At first, one does not see how a choice can be made among thevarious non-Riemannian geometries providing us with the gravitationaland Maxwells equations. The proper world-geometry which should leadto a unified theory of gravitation and electricity can only be found byan investigation of its physical content. In my view, one way to justifywhether a unified field theory of gravitation and electromagnetism isreally true (comprehensive) or really refers to physical reality is tosee if one can derive the equation of motion of a charged particle, i.e.,the (generalized) Lorentz equation, if necessary, effortlessly or directlyfrom the basic assumptions of the theory. It is also important to be ableto show that while gravity is in general non-linear, electromagnetism islinear. At last, it is always our modest aim to prove that gravity andelectromagnetism derive from a common source. In view of this onemust be able to show that the electromagnetic field is the sole ingre-dient responsible for the creation of matter which in turn generates agravitational field. Hence the two fields are inseparable.


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Chapter 1 A Four-Manifold Possessing an Internal Spin 9

Furthermore, I find that most of the past theories were based on theLagrangian formulation which despite its versatility and flexibility mayalso cause some uneasiness due to the often excessive freedom of choos-ing the field Lagrangian. This strictly formal action-method looks likea short cut which does not lead along the direct route of true physicalprogress. In the present work we shall follow a more fundamental (nat-ural) method and at the same time bring up again many useful classicalideas such as the notion of a mixed geometry and Kaluzas cylinder con-dition and five-dimensional formulation. Concerning higher-dimensionalformulations of unified field theories, we must remember that there is

always a stage in physics at which direct but narrow physical argumentscan hardly impinge upon many hidden properties of Nature. In fact theuse of projective geometries also has deep physical reason and displays acertain degree of freedom of creativity: in this sense science is an art, acreative art. But this should never exclude the elements of mathemati-cal simplicity so as to provide us with the very conditions that Naturesmanifest four-dimensional laws of physics seemingly take.

For instance, Kaluzas cylinder condition certainly meets such a re-quirement and as far as we speak of the physical evidence (i.e., thereshould possibly be no intrusion of a particular dependence upon thehigher dimension(s)), such a notion must be regarded as importantif not necessary. An arbitrary affine (n + 1)-space can be represented

by a projective n-space. Such a pure higher-dimensional mathematicalspace should not strictly be regarded as representing a real higher-dimensional world space. Physically saying, in our case, the five-dimensional space only serves as a mathematical device to representthe events of the ordinary four-dimensional space-time by a collectionof congruence curves. It in no way points to the factual, exact numberof dimensions of the Universe with respect to which the physical four-dimensional world is only a sub-space. In this work we shall employ afive-dimensional background space simply for the sake of convenienceand simplicity.

On the microscopic scales, as we know, matter and space-time itselfappear to be discontinuous. Furthermore, matter arguably consists ofmolecules, atoms and smaller elements. A physical theory based on acontinuous field may well describe pieces of matter which are so large incomparison with these elementary particles, but fail to describe their be-havior. This means that the motion of individual atoms and moleculesremains unexplained by physical theories other than quantum theory inwhich discrete representations and a full concept of the so-called mate-rial wave are taken into account. I am convinced, indeed, that if we had


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12 I. Suhendro Spin-Curvature and the Unification of Fields

= z1z2 z1z2 ,tr = = z1z2 z1z2 = 0 ,

where = is the generalized Kronecker delta and =

. (The minus sign holds if the manifold is Lorentzian and

vice versa.) The particular equation tr = 0 is of course valid in Mdimensions as well. In M dimensions, the fundamental spin tensor ofour theory is defined as a bivector satisfying

g[]g

[]

=

+

1

M 1 . (1.3a)Hence the above relation leads to the identity

g[]g[] = . (1.3b)

In the particular case of M=2 and n =1, the vanishes and thefundamental spin tensor is none other than the two-dimensional Levi-Civita permutation tensor:

g[AB] =AB=g

0 11 0

,

g[AB]g[CD ] = CA

DB CB DA ,

g[AC]g[BC] = BA ,

where A, B = 1, 2. Lets now return to our four-dimensional manifoldR4. We now have

g[ij]g[kl] = klij +

1

3

ki

lj kj li

,

g[ij]g[kj] = ki .

Hence the eigenvalue equation is arrived at:

g[ij] = klij g[kl] . (1.3c)

We can now construct the symmetric traceless matrix Qki through

Qki klij uluj = ijpqklrs zp1zq2zr1zs2uluj ,

klij Qki uj ul ,

Qik = Qki ,


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We can derive all of (1.2) and (1.6) by means of (1.7). In a pseudo-five-dimensional space n =R4 n (a natural extension of R4 whichincludes a microscopic fifth coordinate axis normal to all the coordinatepatches ofR4), the algebra is extended as follows:

[gi, gj] = Ckij gk + g[ij]n (a)

[n, gi] = g[ij]gj (b)

i = [n, gi]

= g[ij]gj

(c)

, (1.8)

where the square brackets [ ] are the commutation operator, Ckij standsfor the commutation functions and n is the unit normal vector to the

manifold R4 satisfying (n n) = 1. Here we shall always assume that(n n) = + 1 anyway. In summary, the symmetric and skew-symmetricmetric tensors can be written in n =R4 n as

g(ik) = (gi gk) (d)g[ik] = ([gi, gk] n) (e)

. (1.8)

The (intrinsic) curvature tensor of the space R4 is given through therelations

Rijkl () = ijl,k ijk,l + mjl imk mjk iml ,

ai;j;k ai;k;j = Rlijk al 2l[jk]ai;l ,for an arbitrary vector ai. The torsion tensor

i[jk] is introduced through

the relation

;i;k ;k;i = 2r[ik],r ,which holds for an arbitrary scalar field . The connection of course canbe written as ijk =

i(jk) +

i[jk]. The torsion tensor

i[jk] together with

the spin tensor g[ij] shall play the role associated with the internal spinof an object moving in space-time. On the manifold R4, lets now turnour attention to the spin space Sp and evaluate the tangent componentof the derivative of the spin basis {l} with the help of (1.7):

(j i)T = g[ik],j g[lk] g[ik]g[lm]kmjl (1.9)since (j gi)T =

kij gk. Now with the help of (1.3), we have

(j i)T =

g[ik],j g

[lk] 13

j

li lij

l ,


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Letting Wjik = g[jm]mik, we have

g[ij],k = Wijk Wjik . (1.14)Solving for Wi[jk] by making cyclic permutations ofi, j and k, we get

Wi[jk] =1

2

g[ij],k g[jk],i + g[ki],j

+ Wj(ik) Wk(ij) .

Therefore

Wijk = Wi(jk) + Wi[jk] =

=1

2

g[ij],k g[jk],i + g[ki],j

+ Wi(jk) + Wj(ik) Wk(ij) .

Now recall that Wijk = g[im]mjk . Multiplying through by g

[il], we get

ijk =1

2g[li]

g[lj],k g[jk],l + g[kl],j

+

+ g[li]g[jm]m(lk) g[li]g[km]m(lj) + i(jk) .

(1.15)

On the other hand,

ijk =1

2g(li)

g(lj),kg(jk),l +g(kl),j

g(li)

g(jm)m[lk] g

(li)

g(km)m[lj] +

i[jk] .

(1.16)

From (1.15) the torsion tensor is readily read off as

i[jk] =1

2g[li]


+

+ g[li]g[jm]m(lk) g[li]g[km]m(lj) .

(1.17)

We denote the familiar symmetric Levi-Civita connection byijk

=

1

2g(li)

g(lj),k g(jk),l + g(kl),j

.

If we combine (1.13) and (1.17) with the help of (1.2), (1.3) and (1.4),

after a rather lengthy but straightforward calculation we may obtain asolution:

i[jk] =1

2g[li]


+

+ g[li]g[jm]

mlk

g[li]g[km]

mlj

+

1

3

m[mk]

ij m[mj]ik

.

(1.18)


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Chapter 1 A Four-Manifold Possessing an Internal Spin 19

Having developed the basic structural equations here, we shall see inthe following chapters that the gravitational and electromagnetic ten-sors are formed by means of the fundamental tensors g(ij)

2 (ij)

and g[ij]

2 [ij] alone (see Section 4.2). In other words, gravity andelectromagnetism together arise from this single tensor. We shall also in-vestigate their fundamental relations and ultimately unveil their union.


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Chapter 2

THE UNIFIED FIELD THEORY

2.1 Generalization of Kaluzas projective theory

We now assume that the space-time R4 is embedded in a generalfive-dimensional Riemann space IR

5. This is referred to as embedding

of class 1. We shall later define the space n =R4 n to be a specialcoordinate system in IR5. The five-dimensional metric tensor g(AB) ofIR5 of course satisfies the usual projective relations

g(AB) = eiAe

jBg(ij) + nAnB ,

eAi nA = 0 ,

where eAi = ixA is the tetrad. If now {gl} denotes the basis ofR4 and

{eA} of IR5:eA = e

iAgi + nAn ,

gi = eAi eA , g(ij) = e

Ai e

Bj g(AB) ,

eAi eiB = AB nAnB , eAi ejA = ji .The derivative of gi R4 IR5 is then

j gi = kij gk + ij n .

We also have the following relations:

gi;j = ij n , in = jigj , eA;B = 0 , (BeA = CABeC) ,

eAi;j = ij nA, nA;i = jieAj , eiA;B = nAij ejB, g(ij);k = g(AB);C = 0 .

In our work we shall, however, emphasize that the exterior curvaturetensor ij is in general asymmetric: ij

= ji just as the connection

ijk

is. This is so since in general j eAi = ieAj . Within a boundary , themetric tensor g(ij) may possess discontinuities in its second derivatives.Now the connection and exterior curvature tensor satisfy

kij = ekAj e

Ai + e

kA

ABCe

Bi e

cj (a)

ij = nAj eAi + nA

ABCe

Bi e

cj (b)

, (2.1)


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Chapter 2 The Unified Field Theory 21

wherej e

Ai = e

Ak

kij ABC eBi eCj + ij nA. (2.2)

We can also solve for ABC in (2.1) with the help of the projectiverelation eA = e

iAgi+nAn. The result is, after a quite lengthy calculation,

ABC = eAi Ce

iB + e

Ak

kij e

iBe

jC + ij e

iBe

jCn

A+

+ (CnB)nA ij eAi ejCnB .

(2.3)

If we now perform the calculation (kj

j k) gi with the help of

some of the above relations, we have in general

(k j j k) gi = RABC DeBi eCj eDk eA ++

keCj j eCk

ABCe

Bi eA +

+ (kj j k) eAi eA .(2.4)

Here we have also used the fact that

(C B BC) eA = RDABCeD .On the other hand, in = jigj ,and

kj gi = lij gl + ij n ,k ==

lij,k +

mij

lmk ij lk

gl +

ij,k +

lij lk

n .

Therefore we obtain another expression for (k j j k) gi:(kj j k) gi =

Rlijk + iklj ij lk gl ++

ij;k ik;j + 2l[jk]il

n ,(2.5a)

(k j j k) eAi +

keCj j eCk

ABCe

Bi SAijk . (2.5b)

Combining (2.4) and (2.5), we get, after some algebraic mani-pulations,

Rijkl = ikjl iljk + RABCD eAi eBj eCk eDl SAjkleAi (a)

ij;k ik;j = RABCDnAeBi eCj eDk 2l[jk]il + SAijknA (b)

. (2.6)We have thus established the straightforward generalizations of the

equations of Gauss and Codazzi.


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Now the electromagnetic content of (2.6) can be seen as follows:first we split the exterior curvature tensor ij into its symmetric andskew-symmetric parts:

ij = (ij) + [ij]

(ij) kij , [ij] fij

. (2.7)

Here the symmetric exterior curvature tensor kij has the explicitexpression

kij =1

2

nA j eAi + ieAj + nAA(BC)eBi eCj == 1

2eAi e

Bj (nA;B + nB;A) .

(2.8)

Furthermore, in our formalism, the skew-symmetric exterior curva-ture tensor fij is naturally equivalent to the electromagnetic field tensorFij . It is convenient to set fij =

12Fij . Hence the electromagnetic field

tensor can be written as

Fij = nA

j eAi ieAj

+ 2 nA

A[BC]e

Bi e

Cj =

= eAi eBj (nA;B nB;A)(2.9)

The five-dimensional electromagnetic field tensor is therefore

FAB = AnB BnA .

2.2 Fundamental field equations of our unified field theory.Geometrization of matter

We are now in a position to simplify (2.6) by invoking two conditions.The first of these, following Kaluza, is the cylinder condition: the lawsof physics in their four-dimensional form shall not depend on the fifthcoordinate x5 n. We also assume that x5 y is a microscopic co-ordinate in n. In short, the cylinder condition is written as (by firstputting nA = eA5 )

g(ij),5 = g(ij),AnA = eBi e

Cj (nB;C + nC;B) = 0 ,

where we have now assumed that in IR5 the differential expression eiA,B

eiB,A vanishes. However, from (2.2), we have the relation

eAi,j eAj,i = 2eAk k[ij] + Fij nA .


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We may represent a negative charge by a negative spin producedby a left-handed twist (torsion) and a positive one by a positive spinproduced by a right-handed twist. (For the conservation of charges(currents) see Sections 4.3 and 4.4.) Now (2.11c) tells us that when thespatial curvature, represented by the Ricci scalar, vanishes, we have anull electromagnetic field, also it is seen that the strength of the electro-magnetic field is equivalent to the spatial curvature. Therefore gravityand electromagnetism are inseparable. The electromagnetic source, thecharge, looks like a microscopic spinning hole in the structure of thespace-time R4, however, the Schwarzschild singularity is non-existent in

general. Consequently, outside charges our field equations read

Rijkl =1

4(FikFjl FilFjk ) , (2.12a)

Fij;j = 0 , (2.12b)

which, again, give us a picture of how a gravitational field emerges(outside charges).

In this way, the standard action integral of our theory may take theform

I =

R

RikR

ik

12

g d4x = (2.13)

=

R

1

16

FikF

ik2 RijklRijkl 1

4RijklF

ilFjk12g d4x .

Here R denotes the Ricci scalar built from the symmetric Christoffelconnection alone.

From the variation of which, we would arrive at the standardEinstein-Maxwell equations. However, we do not wish to stress heavyemphasis upon such an action-method (which seems like a forced shortcut) in order to arrive at the field equations of our unified field the-ory. We must emphasize that the equations (2.10)-(2.13) tell us howthe electromagnetic field is incorporated into the gravitational field in avery natural manner, in other words theres no need here to constructany Lagrangian density of such. We have been led into thinking of howto couple both fields using different procedures without realizing thatthese fields already encapsulate each other in Nature. But here ourspace-time is already a polarized continuum in the sense that there ex-ists an electromagnetic field at every point of it which in turn generatesa gravitational field.


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The five-dimensional index of the tetrad Ai is raised and loweredusing the metric tensor g(AB). The inverse of the tetrad is achievedwith the help of the metric tensor

GAB = eiAe

kBg(ik) + nAnB 2y(ik)eiAekB ,

which reduces to g(AB) due to the cylinder condition. Thefour-dimensional metric tensor g(ik) is used to raise and lower the four-dimensional index. Again, we bring in the electromagnetic field tensorFij via the cylinder condition, which yields ik =

12

Fik. The connectionof the space S5 is then

xi, y

= 1

2Fy,

1

2F y,

1

2yF,

1

2yF

,

where is the connection of the space n =R4 n with the electro-magnetic field tensor derived from it: F = 2

5 , F

= 25 , F5 = 0.

Therefore its only non-zero components are Fik. At the base of the spaceS5, the connection is

=

xi, 0

=

1

2Fy,

1

2F y, .

From the above expression, we see that

k = k 1

2 Fky, ,

5 = 1

2F,

5 = F .

As can be worked out, the five-dimensional connection of the back-ground space IR5 is related to that of S5 through

ABC

xi, y

= A B,C +

A

B

C

1

2F

A

BnC

12

yA k

BF

kie

iC .

Now the five-dimensional curvature tensor RABCD = RABCD xi, 0is to be related once again to the four-dimensional curvature tensorofR4, which can be directly derived from the curvature tensor of thespace S5 as Rijkl =

S5Rijkl

xi, 0

. With the help of the above geometricobjects, and after some laborious work-out, we arrive at the relation

RABCD = eAe

Be

Ce

DR

1

2FABFCD + ABCD ,


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R5ijk = R5ijk = 1

2

Fij;k Fik;j + 2r[jk]Fir

,

Rij5k = 0 .

Furthermore, we obtain the following equivalent expressions:

RABCD = RijkleiAe

jBe

kCe

lD +

1

2

Fik;l Fil;k + 2m[kl]Fim

eiAnBe

kCe

lD +

+1

2

Fjl;kFjk;l +2m[lk]Fjm

nAe

jBe

kCe

lD

1

2FABFCD + ABCD ,

RABCD = 12

(FAC;D FAD;C) nB + 12

(FBD;C FBC;D) nA

12

FABFCD ABCD + RijkleiAejBekCelD +

+ 2m[kl]FimeiAnB e

kCe

lD + 2

m[lk]FjmnAe

jBe

kCe

lD ,

where

ABCD =1

2

FMA

FMCnD FMD nC

nB

FMB

FMCnD FMD nC

nA

ABCD .

When the torsion tensor of the space R4 vanishes, we have the rela-

tion

RABCD =1

2(FAC;D FAD;C) nB + 1

2(FBD;C FBC;D) nA

12

FABFCD ABCD + RijkleiAejBekCelDwhich, again, relates the curvature tensors to the electromagnetic fieldtensor.

Finally, if we define yet another five-dimensional curvature tensor:

RABCD RABCD + 12FAB FCD ABCD ,

where RABCD, FAB and ABCD are the extensions of RABCD, FABand ABCD which are dependent on y, we may obtain the relationRABCDe

Ai e

Bj e

Ck e

Dl =

= Rijkl +1

2yFriRrjkl +

1

2yFrjRirkl +

1

2yFrkRijrl +

1

2yFrlRijkr .

End of Sub-remark


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the Weyl tensor, rather than the Riemann tensor, which is compatiblewith the electromagnetic field tensor. From the structure of the Weyltensor as revealed by (2.r), it is understood that the Weyl tensor ac-tually plays the role of an electromagnetic polarization tensor in thespace-time R4. In an empty region of the space-time R4 with a vanish-ing torsion tensor, when the Weyl tensor vanishes, that region possessesa constant sectional curvature which conventionally corresponds to aconstant energy density.

Lets for a moment turn back to (2.6). We shall show how to get thesource-torsion relation, i.e., (2.11d) in a different way. For this purposewe also set a constraint SAijk = 0 and assume that the background five-dimensional space is an Einstein space:

RAB = g(AB) ,

where is a cosmological constant. Whenever = 0 we say that thespace is Ricci-flat or energy-free, devoid of matter. Taking into ac-count the cosmological constant, this consideration therefore takes on aslightly different path than our previous one. We only wish to see whatsort of field equations it will produce.

We first write

Rijkl =1

4 (FikFjl FilFjk ) + RABCD eAi eBj eCk eDl ,Fij;k Fik;j = 2RABCD nAeBi eCj eDk 2l[jk]Fil .

Define a symmetric tensor:

Bik RABCD eAi nBeCk nD = Bki . (2.14)It is immediately seen that

RAB eAi e

Bk = g(ik) , (2.15a)

RAB nAnB = , (2.15b)

RAB eAi nB = 0 . (2.15c)Therefore

Rik = 14

FilFlk + RAB e

Ai e

Bk RABCDeAi nBeCk nD =

= 14

FilFlk + g(ik) + Bik .

(2.16)


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From (2.14) we also have, with the help of (2.15b), the following:

B = g(ik)Bik = RABCD nBnD

g(AC) nA nC

=

= RAB nAnB + RABCD nAnBnCnD == .

Hence we have

R =1

4FikF

ik + 3 . (2.17)

The Einstein tensor (or rather, the generalized Einstein tensor en-dowed with torsion)

Gik Rik 12

g(ik)R

up to this point is therefore

Gik = Bik 18

g(ik)FrsFrs 1

2g(ik)

1

4FilF

lk . (2.18)

From the relation

Fij;k Fik;j = 2RABCD nAeBi eCj eDk 2l[jk]Filwe see that

Fkj;k = 2R BCAD nAekBejCeDk 2jl[.k]Fkl =

= 2R CA ejCnA 2R BCAD nAnBejCnD 2 jl[k]Fkl =

= 2 jl[k]Fkl = Jj .

In other words,Ji = 2

l[ik]F

kl ,

which is just (2.11d). We will leave this consideration here and commitourselves to the field equations given by (2.10) and (2.11) for the restof our work.

Lets obtain the (generalized) Bianchi identity with the help of (2.10)and (2.11). Recall once again that

Rijkl =1

4(FikFjl FilFjk ) ,

Fij;k Fik;j = 2l[jk]Fil .


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We can also writeRik 1

2g(ik)R

;k

= 14

JkFik + i. (2.22)

On the other hand, repeating the same contraction steps on(2.20) gives

R;m2Rim;i =

imR 2Rim;i

gm =

=1

2n[kl]F

knF

lm +

1

2n[km]F

lnF

kl +

1

2n[lm]F

knF

lk+

+12

n[lk]FlnF

km =

= n[kl]Fk nF

lm +

n[km]F

lnF

kl =

= n[kl]FknF

lm 2m .

Therefore we haveRik 1

2g(ik)R

;k

= 12

gi, (2.23a)

where gi, a non-linear quantity, can be seen as a complementary tor-sional current:

gi = 12

JkFik + i = n[kl]FknF

li 2i. (2.23b)

In the most general case, by the way, the Ricci tensor is asymmetric.If we proceed further, the generalized Bianchi identity and its contractedform will be given by

Rijkl;m + Rijlm;k + Rijmk;l =

= 2

r[kl]Rijrm + r[lm]Rijrk +

r[mk] Rijrl

,

(2.24a)

Rik 1

2g(ik)R

;i

= 2g(ik)r[ji]Rjr +

r[ij]R

ijk...r . (2.24b)

Remark 2

Consider a uniform charge density. Again, our resulting field equation(2.21) reads

Rik 12

g(ik)R

;k

= 14

FikJk +

1

4g(ik)

Frs

Fsr;k Fsk;r

,


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Setting = 2 em0c2 where, again, e is the electric charge, m0 is thepoint-mass and c is the speed of light in vacuum, we have the Lorentzequation of motion:

Dui

Ds=

e

m0c2Fij u

j ,Dui

Ds ui;j uj .

The fifth component of the momentum is consequently given by

p5 = 2

e

c2. (2.26)

We will now show that the quantity is indeed constant along theworld-line. First we write u = uigi+ n , then, as before, we have

j u =

ui;j 1

2 Fij

gi +

;j + 1

2Fij u

i

n .

Applying the law of parallel transport in n =R4 n, i.e., 4uu = 0(in the direction of 4u) where 4u represents the ordinary tangent four-velocity field, we get two equations of motion:

ui;j 1

2 Fij

uj = 0 , (2.27a)

,j + 12

Fij uiuj = 0 . (2.27b)

The first of these is just the usual Lorentz equation in a general co-ordinate system, the one weve just obtained before using the straight-forward notion of vertical-horizontal n-covariant derivative. Mean-while, the second reads, due to the vanishing of its second term: dds = 0,which establishes the constancy of with respect to the world-line.Therefore it is justified that forms a fundamental constant of Naturein the sense of a correct parameterization. We also have d eds =

dm0ds = 0.

Again, theres a certain possibility for the electric charge and the mass,to vary with time, perhaps slowly in reality. We shall now consider the

unit spin vector field in the spin space Sp:4v uii = g[ik]uigk , (2.28)

which has been defined in Section 1.2. This spin (rotation) vector isanalogous to the ordinary velocity vector in the spin space representa-tion. For the moment, let v =

vi,

; vi = g[ik]uk where v = v

ii + n.


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55i = n (in) = 1

2n Fki gk = 0 = 5i5 = k55 ,

k5k = gk (kn) = 1

2gk Fjk gj =

1

2Fjj = 0 =

kk5 ,

R5i = k5i,k k5k,i + k5ilkl l5kkli =

= 12

Fki,k +

klkF

li kliFlk

=

= 1

2 Fki;k + 2l[ik]Fkl == 1

2(Ji + Ji) = 0

with the help of (2.11d). Hence Gik|k = Gik;k . (2.32b) can also be easily

proven this way.As a brief digression, consider a six-dimensional manifold V6=nm

where m is the second normal coordinate with respect to R4. Let

i g[5i], a 5, b 6, ik

i5k

, ik

i6k

, = g[56], and

i g[6i]. Casting (1.12) and (1.22) into six dimensions, the electro-magnetic field tensor can be written in terms of the fundamental spintensor as the following equivalent expressions:

Fik =5

4

r

g[ir],k g[kr],i

(i,k k,i) ==

5

4

rg[ik],r (i,k k,i)

,

(2.33a)

Fik =l

g[li],kg[ik],l +g[kl],ia i,k k,i b (i,k k,i)

2l

g[im]

mlk

g[km]

mli

2a

g[im]

mk g[km]mi

2b

g[im]

mk g[km]mi

.

(2.33b)

Since the basis in this space is given by g = gi, n , m, the funda-mental tensors are

g() =

g(ik)4x4

0 0

0 1 0

0 0 1

, (2.34a)


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g[] =

g[ik]4x 4

i i 0 i i 0

. (2.34b)

Remark 3 (on the modified Maxwells equations)

Using (2.25) we can now generalize Maxwells field equations throughthe new extended electromagnetic tensor Fik = i|k k|i where

Fik(old) = i;k k;i = i,k k,i + 2l[ik]l = Fik + 2

l[ik]l ,

where i|k = i;k 12 Fik. Now 5 = is taken to be an extra scalarpotential. Therefore Fik = i,k k,i Fik or

Fik =1

1 + (i,k k,i) (i,k k,i) . (2.a)

For instance, the first pair of Maxwells equations can therefore begeneralized into

E =1

1 +

1

c

A

t

, (2.b1)

B =

1

1 + x

A , (2.b2)

div B = 1(1 + )2

x A

+

1

1 + x A , (2.b3)

curl

1

1 + E

= 1

c

t

1

1 + B 1

c

B

t

1

1 + , (2.b4)

where A is the three-dimensional electromagnetic vector potential:A = (Aa), is the electromagnetic scalar potential, E is the electric

field, B is the magnetic field and is the three-dimensional (curvilin-ear) gradient operator and 2 = 4. Here we define the electricand magnetic fields in such a way that

F4a = (1 + )1 Ea, F12 = (1 + )

1 B3 ,

F31 = (1 + )1

B2 , F23 = (1 + )

1B1 .

We also note that in (2.b3) the divergence x A is in generalnon-vanishing when torsion is present in the three-dimensional curved


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sub-space. Direct calculation gives

div curl A = 12

abc

RdcabAd 2d[ab]Ac;d

,

(curl grad )c = ab.c d[ab],d .So the magnetic charge with density m in the infinitesimal volume

d is given by

=

1

2 abc

RdcabAd

2d[ab]Ac;d d .

End of Remark 3


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Chapter 3

SPIN-CURVATURE

3.1 Dynamics in the microscopic limit

We now investigate the microscopic dynamics of our theory. Lets in-troduce an infinitesimal coordinate transformation into n through thediffeomorphism

xi = xi + i

with an external Killing-like vector =

i,

: i = i(x), = (x)(not to be confused with the internal Killing vector which describesthe internal symmetry of a particular configuration of space-time orwhich maps a particular space-time onto itself). The function hereshall play the role of the amplitude of the quantum mechanical statevector |. Recall that R4 represents the four-dimensional physicalworld and n is a microscopic dimension. In its most standard form (x) Ce(2i/h) (Etpr(x,y,z)) is the quantum mechanical scalar wavefunction; h is the Planck constant, E is energy and p is the three-momentum. Define the extension of the space-time n =R4

n by

ij =1

2D g(ij) . (3.1)

We would like to express the most general symmetry, first, of thestructure of n and then find out what sort of symmetry (expressedin terms of the Killing-like vector) is required to describe the non-localstatics or non-deformability of the structure of the metric tensorg (the lattice arrangement). Our exterior derivative is defined as thevariation of an arbitrary quantity with respect to the external field .Unlike the ordinary Killing vector which maps a space-time onto itself,the external field and hence also the derivative D g(ij) map R4 onto,say, R4 which possesses a deformed metrical structure of g R4, g.

We calculate the change in the metric tensor with respect to the ex-ternal field, according to the scheme g(ij) = (gi gj ) g(ij) = gi gj,as follows:

D g(ij) = (gi D gj ) + (D gi gj) ; D gi = gi gi = i .The Lie derivative D denotes the exterior change with respect to the

infinitesimal exterior field, i.e., it dynamically measures the deformation


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Chapter 3 Spin-Curvature 45

of the geometry of the space-time R4.Thus

D g(ij) = (gi j ) + (i gj ) ,where

i =

k;i 1

2Fki

gk +

,i +

1

2Fki

k

n .

By direct calculation, we thus obtain

D g(ij) = i;j + j;i . (3.2)

3.2 Spin-curvature tensor of Sp

As an interesting feature, we point out that the change D g(ij) in thestructure of the four-dimensional metric tensor does not involve thewave function . The space-time R4 will be called static if g doesnot change with respect to . The four-dimensional (but not the five-dimensional) metric is therefore static whenever i;j + j;i = 0.

Now let R4 be an infinitesimal copy ofR4. To arrive at the latticepicture, let also R4 , R

4 , R

4 , . . . be n such copies ofR4. Imagine the

space Sn consisting of these copies. This space is therefore populatedby R4 and its copies. If we assume that each of the copies ofR4 has thesame metric tensor as R4, then we may have

= (0, ) .

We shall call this particular fundamental symmetry normal sym-metry or spherical world-symmetry. Then the n copies ofR4 existsimultaneously and each history is independent of the four-dimensionalexternal field 4 and is dependent on the wave function only. In otherwords, the special lattice arrangement = (0, ) gives us a conditionfor R4 and its copies to co-exist simultaneously independently of howthe four-dimensional external field deforms their interior metrical struc-ture. Thus the many sub-manifolds nR4 represent many simultaneousrealities which we call world-pictures. More specifically, at one point inR4, there may at least exist two world-pictures. In other words, a pointseen by an observer confined to lie in R4 may actually be a line or curvewhose two end points represent the two solutions to the wave function. The splitting ofR4 into its copies occurs and can only be perceivedon the microscopic scales with the wave function describing the en-tire process. Conversely, on the macroscopic scales the inhabitants ofn =R4 n may perceive the collection of the nR4 (sub-spaces ofR4)


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is the spin vector (3.6a). The meaning of the tensor (3.16) will becomeclear soon. It has no classical analogue. We are now in a positionto decompose (3.16) into its symmetric and alternating parts. Thesymmetric part of (3.16):

A(ij) =1

2

Si;j + Sj;i 1

2Fkik|j

1

2Fkj k|i

(3.17a)

may be interpreted as the tension of the spin field.Now its alternating part:

A[ij] =12Si;j Sj;i + 1

2Fkik|j 12 F

kj k|i

(3.17b)represents a non-linear spin field. (However, this becomes linear whenwe invoke the fundamental world-symmetry = (0, ).) If we employthis fundamental world- symmetry, (3.17a) and (3.17b) become

A(ij) =1

2(Si;j + Sj;i) + Rij , (3.18a)

A[ij] =1

2(Si;j Sj;i) . (3.18b)

With the help of (3.6a) and the relation

;i;k ;k;i = 2r[ik],r ,

(3.18b) can also be written

A[ij] = k[ij],k . (3.19)

Now

Aij = ;i;j 12

Fki k

;j

12

Fkj k;i +1

4Fkj Fki =

=

;i;j Rij + 12

Fkik

;j

+1

2Fkj k;i

.

Therefore the Ricci tensor can be expressed as

Rik = 12

1Frk r|i

1

;i;k + Si;k +1

2(Fri r);k

1

2Frk r;i

.

(3.20)


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Chapter 3 Spin-Curvature 51

We can still obtain another form of the wave equation of our quan-tum gravity theory. Taking the world-symmetry = (0, ), we have,from r = r n,

gi = hi ,i n + 12

Fri gr , (3.21)

where hi r,i is the basis of the space-time R4. Now the metric tensorof the space-time R4 is

g(ik) = (gi.gk) =

= hik ,ki +1

2 Frk ir ,ik + ,i,k +

+1

2Fri kr +

1

42g(rs)F

riF

sk ,

where hik (hi hk) is the metric tensor of R4, i (hi n) andik (hi gk). Direct calculation shows that

i = ,i ,

ik = g(ik) +1

2Fik .

Then we arrive at the relation

g(ik) = hik ,i,k 14 2g(rs)F

riF

sk . (3.22)

Now from (3.2) we find that this is subject to the condition

D g(ij) = 0 . (3.23)

Hence we obtain the wave equation

,i,k = 14

2g(rs)FriF

sk . (3.24a)

Expressed in terms of the Ricci tensor, the equivalent form of(3.24a) is

,i,k = 2Rik . (3.24b)Expressed in terms of the Einstein tensor Gik = Rik 12 g(ik)R,

(3.24b) becomesri

sk

1

2g(ik)g

(rs)

,r,s = 2Gik . (3.25)


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If in particular the space-time R4 has a constant sectional curvature,then Rijkl =

112

g(ik)g(jl) g(il)g(jk)

and Rik = g(ik), where 14 R

is constant, so (3.24b) reduces to

,i,k = 2g(ik) . (3.26)

Any axisymmetric solution of (3.26) would then yield equations thatcould readily be integrated, giving the wave function in a relativelysimple form. Multiplying now (3.24b) by the contravariant metric tensorg(ik), we have the wave equation in terms of the curvature scalar as

follows: g(ik),i,k = 2R . (3.27)Finally, lets consider a special case. In the absence of the scalar

source, i.e., in void, the wave equation becomes

g(ik),i,k = 0 . (3.28)

This wave equation therefore describes a massless, null electromag-netic field where

FikFik = 0 .

In this case the electromagnetic field tensor is a null bivector. There-fore, according to our theory, there are indeed seemingly void regions

in the Universe that are governed by null electromagnetic fields only.


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Chapter 4 Additional Considerations 55

Meanwhile,

n()A;i = n

()A;Be

Bi ,

n()A;i;k =

n()A;Be

Bi

;C

eCk =

= n();B;Ce

Bi e

Ck + n

()A;Be

Ai;k =

= n()A;B;Ce

Bi e

Ck +

n()A;B

()()ik n

()B ,

butn()A;B = n

()A;ie

iB = ()ki ekAeiB .

Hencen()A;Bn

()B = 0 ,

n()A;Bn

()A = 0 .

We also see that

n()A;i;kn

()A = n()A;B;Cn

()AeBi eCk .

Consequently, we have

n()A;i;k n()A;k;in()A = n()A;B;C n()A;C;Bn()AeBi eCk == RABCD n

()An()BeCi eDk .

On the other hand, we see thatn()A;i;kn()A;k;i

n()A = i;k k;i + grs

()rk

()si ()ri ()sk

+

+

() (i k k i ) +

+ RABCD n()An()BeCi e

Dk .

Combining the last two equations we get the Ricci equations:

i;k k;i = grs

()ri

()sk ()rk ()si

() (i k k i ) .

Now from the relation

gi,j = kij gk +

()()ij n

() ,


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we obtain the expression

gi,jk =

rij,k +

sij

rsk

grs ()()ij

()sk

gr +

+

()

()rk

rij +

()()ij,k +

()()ij

k

n() .

Hence consequently,

gi,jk

gi,kj = Rrijk + grs() ()sj ()ik ()sk ()ij gr ++

()

()ij;k ()ik;j + 2r[jk]()ir

n() +

+,

()

()ij

k ()ik j

n() .

On the other hand, we have

gi,jk =

ABC,D + EBC

AED

eBi e

Cj e

Dk eA+

+

eAi,jk + ABC e

Bi,j e

Ck +

ABC e

Bi,ke

Cj +

ABC e

Bi e

Cj,k

eA ,

gi,jk gi,kj = RABC DeBi eCj eDk + SAijk eA ,where, just as in Chapter 2,

SAijk = eAi,jk eAi,kj +

eCj,k eCk,j

ABCe

Bi .

Combining the above, we generalize the Gauss-Codazzi equa-tions into

Rijkl =

()

()ik

()jl ()il ()jk

+ RABCDe

Ai e

Bj e

Ck e

Dl SAjkleAi ,

()ij;k ()ik;j = RABCDn()eBi eCj eDk + SAijkn()A 2r[jk]()ir +

+

() ()ij k ()ik j .Finally, when Rn is embedded isometrically in RN, i.e., when

the embedding manifold RN is an Euclidean or pseudo-EuclideanN-dimensional space(-time) or if we impose a particular integrabilitycondition on the n-vectors in Rn the way we derived (2.10) in Chap-


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magnitude is independent of the four coordinates ofR4. In other words,

ui;k =1

2 Fik ,

,i= 12

Fkiuk .

From these we haveDui

Ds=

e

m0c2Fiku

k,

ui;i = 0 ,

d ds

= 0 .

Were now in a position to derive the field equations of gravoelec-trodynamics with the help of these relations. Afterwards, we shall showthat these relations, indeed, lead to acceptable equations of motion inboth four and five dimensions. We may also emphasize that the five-dimensional space IR5 is a Riemann space. First we note that

ui;k;l =1

2 Fik;l + 1

2,l Fik =

=1

2 Fik;l 1

4FrlFikur .

Nowui;k;l ui;l;k = Rriklur 2ui;rr[kl] =

=1

2(Fik;l Fil;k) + 1

4(FrkFil FrlFik) ur .

Therefore we have

Rijkl ui =1

4

Fik Fjl FilFjk

ui +

1

2

Fjk;l Fjl;k + 2r[kl]Fjr

.

But ui = Ai uA and = uAnA, so by means of symmetry, we can

lop off the uA:

Rijkl Ai =

1

4 FikFjl

FilFjk

Ai +

1

2 Fjk;l Fjl;k + 2r[kl]Fjrn

A .

From this relation, we derive the unified field equations

Rijkl =1

4(FikFjl FilFjk ) ,

Fij;k Fik;j = 2r[jk ,]Fir


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as expected. We have thus no need to assume that the curvature ofthe background five-dimensional space IR5 vanishes. To strengthen ourproof, recall that gi;j = ij n. Hence

gi;j;k = ij;kn + ij n;k =

= ij;kn ij rk gr ,gi;j;k gi;k;j = (ij;k ik;j) n +

ik

rj ij rk

gr .

However,gi;j;k

gi;k;j = R

rijk gr

2r[jk]

nir .

Combining the above, we have

Rijkl = ikjl iljk .ij;k ik;j = 2r[jk]ir .

Invoking the cylinder condition, again we get

Rijkl =1

4(FikFjl FilFjk ) ,

Fij;k Fik;j = 2r[jk]Fir .We now assume that the five-dimensional equation of motion in IR5

is in general not a geodesic equation of motion. Instead, we expect an

equation of motion of the formd2xA

ds2+ ABC

dxB

ds

dxC

ds= FAB uB,

where uA = eAi ui, and

ds2 = 5ds2 = g(AB)dxAdxB =

eiAe

kBg(ik) + nAnB

dxAdxB =

= g(ik)dxidxk = 4ds2,

ABC =1

2g(DA)

g(DB),C g(BC),D + g(CD),B

,

FAB FAB + AB ,

FAB = (nA;B nB;A) ,Fik = e

Ai e

Bk FAB ,

FABnB = 0 ,

ABeAi e

Bk = 0 .


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Note also that the connection transforms as

ABC = eAi e

iB,C + e

Ai

ijk e

jBe

kC +

+1

2Fike

iBe

kCn

A + nAnB,C 12

FikeAi e

kCnB .

Now with the help of the relation

eAi,k = eAr

rik ABCeBi eCk +

1

2Fikn

A,

we have

d2xA

ds2=

d

ds

eAi

dxi

ds

= eAi,k

dx

ds

i dx

ds

k

+ eAid2xi

ds2=

= eAid2xi

ds2+ eAi

ijk

dxj

ds

dxk

ds ABC

dxB

ds

dxC

ds.

Hence

d2xA

ds2+ ABC

dxB

ds

dxC

ds= eAi

d2xi

ds2+ ijk

dxj

ds

dxk

ds

= FABuB.

Setting AB = 0 and =e

m0c2, since FABe

iAu

B =FABeiAe

Bk u

k = Fikuk,

we then obtain the equation of motion:

Dui

Ds=

e

m0c2Fiku

k.

A straightforward way to obtain the five-dimensional equation ofmotion is as follows: from our assumptions we have

eAi uA;k

=1

2 Fik ,

FikuAnA + uA;Be

Ai e

Bk =

1

2 Fik .

Therefore recalling that uAnA = 0 and multiplying through by uk,

we get

uA;BeiAu

B = 12

Fikuk.

Again, noting that uA;BuB = uA;ku

k = DuA

Ds and multiplying through

by eCi , we getDuC

Ds nA Du

A

DsnC =

e

m0c2FCAu

A.


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4.3 On the conservation of currents

We need to recall the basic field equations:

Rijkl =1

4(FikFjl FilFjk ) ,

Fij;k Fik;j = 2r[jk]Fir ,

Rik = 14

FirFrk ,

Rik;k =

1

4 Fir;kF

rk

1

4 FirF

rk;k =

= 14

FikJk +

1

4Fik

k[rs]F

rs,

Ji = 2 g(ik)r[ks]Fsr .

To guarantee conservation of currents, we now introduce the one-form i n:

i = Rikgk + in ,

where i represents another current. Requiring, in the sense of thetheory of distributions, that the covariant derivative of i vanishes atall points ofR4 also means that its covariant divergence also vanishes:

i;i = 0 .

HenceRik;i gk + R

ikgi;k + i;i n +

i n;i = 0 ,

Rik;i gk 1

2RikFikn +

i;in

1

2iFki gk = 0 .

Then we have

Rik;k =1

2Fik

k,

i;i = 0 .

Comparing the last two equations above with the fourth equation,one must find

i = 1

2 Ji i[kl]Fkl .Meanwhile, in the presence of torsion the Bianchi identity for the

electromagnetic field tensor and the covariant divergence of the four-current J are

Fij;k + Fjk;i + Fki;j = 2

r[ij]Fkr + r[jk]Fir +

r[ki]Fjr

,


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Ji;i =

i[kl]Fkl;i

.

Now since i;i = 0, we see thati[kl]F

kl;i

= 0

and (in a general setting)

Ji = i[kl]Fkl = 2 i

for some constant , are necessary and sufficient conditions for the cur-

rent density vector J to be conserved. Otherwise both i

and Ji

aredirectly equivalent to each other. Of course one may also define anotherconserved current through

ji = Fik||k ji||i = 0 ,where the double stroke represents a covariant derivative with respectto the symmetric Levi-Civita connection.

In general the current J will automatically be conserved if the or-thogonality condition imposed on the twist vector, derived from thetorsion tensor, and the velocity vector:

i ui = 0 ,

where i = k[ki]

holds. For more details of the conservation law for charges, see Sec-tion 4.4 below.

4.4 On the wave equations of our unified field theory

We start again with the basic field equations of our unified field theory:

Rijkl =1

4(FikFjl FilFjk ) ,

Fij;k Fik;j = 2r[jk]Fir ,

Ji = 2 g(ik)s[kr]Frs .

We remind ourselves that these field equations give us a set of com-plete relations between the curvature tensor, the torsion tensor, the elec-tromagnetic field tensor and the current density vector. From these fieldequations, we are then able to derive the following insightful algebraic


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We can also express this as

Rik = g(AB)eAi 2eBk ,which can be written equivalently as

Rik = 12

g(AB)

eAi rreBk + eAk rreBi

.

Consider now a source-free region in the space-time R4. As weveseen, the absence of source is characterized by the vanishing of thetorsion tensor. In such a case, if the space-time has a constant sectional

curvature K, we obtain the tetrad wave equation for the empty region:

(2 +1

4R) eAi = 0 .

In other words,(2 + K) eAi = 0 .

Combining our tetrad wave equation 2eAi =12Jin

A R ki eAk with theequation for the Ricci tensor we have derived in the quantum limit (inSection 4.2), which is

Rik = 4

e

m0c2

22g(ik) ,

we obtain the following wave equation in the presence of the electro-

magnetic current density:

(2 +1

4R) eAi =

1

2Ji n

A,

R = C+ 4

r[ik]R

ikrs 2i[rs]Rri

dxs,

where C= 4K is constant.Finally, lets have a look back at the wave equation given by (3.15b):

(2 R) | = 0 .If we fully assume that the space-time R4 is embedded isometrically

in IR5 spanned by the time coordinate = ct and the four space co-ordinates u, v, w, y which together form the line-element ds2 = d2

du2 dv2 dw2 dy2 d2 d2 dy2, and if the wave functionrepresented by the state vector | does not depend on the microscopicfifth coordinate y, we can write the wave equation in the simple form:

2

2

2

2

| = R | .


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4.5 A more compact form of the generalized Gauss-Codazziequations in IR5

We write the generalized Gauss-Codazzi equations (see (2.6) for in-stance) in IR5 once again:

Rijkl = ikjl iljk + RABCD eAi eBj eCk eDl SAjkleAi ,

ij;k ik;j = RABCD nAeBi eCj eDk 2r[jk]ir + SAijk nA

in terms of the general asymmetric extrinsic curvature tensor. Here, as

usual,SAijk = e

Ai,jk eAi,kj + eCj,k eCk,jABC eBi .

From the fundamental relations

eAi;j = ij nA,

nA;i = ki eAk ,we have

eAi;j;k = ij;knA ij rkeAr .

Hence

eAi;j;k eAi;k;j = (ij;k ik;j ) nA + rjik rkij

eAr .

With the help of the generalized Gauss-Codazzi equations above andthe identity

eiAeBi =

BA nAnB,

we see that that the generalized Gauss-Codazzi equations in IR5 can bewritten somewhat more compactly as a single equation:

eAi;j;k eAi;k;j = Rrijk eAr RABC DeBi eCj eDk 2r[jk]irnA + SAijk .


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Appendix: The Fundamental Geometric Properties

of a Curved Manifold

Let us present the fundamental geometric objects of an n-dimensional

curved manifold. Let a =X i

xa Ei = aXiEi (the Einstein summation

convention is assumed throughout this work) be the covariant (frame)basis spanning the n

dimensional base manifold C with local coordi-

nates xa = xa Xk. The contravariant (coframe) basis b is then givenvia the orthogonal projection

b, a

= ba, where

ba are the compo-

nents of the Kronecker delta (whose value is unity if the indices coin-cide or null otherwise). The set of linearly independent local directionalderivatives Ei =

X i = i gives the coordinate basis of the locally flat

tangent space Tx (M) at a point x C. Here M denotes the topo-logical space of the so-called n-tuples h (x) = h

x1, . . . , xn

such that

relative to a given chart (U, h (x)) on a neighborhood U of a local co-ordinate point x, our C-differentiable manifold itself is a topologicalspace. The dual basis to Ei spanning the locally flat cotangent spaceTx (M) will then be given by the differential elements dX

k via the rela-tion dX

k, i = ki . In fact and in general, the one-forms dX

k indeed

act as a linear map Tx(M) IR when applied to an arbitrary vectorfield F Tx(M) of the explicit form F = Fi X i = fa xa . Then it iseasy to see that Fi = F Xi and fa = F xa, from which we obtain theusual transformation laws for the contravariant components of a vectorfield, i.e., Fi = aX

ifa and fi = ixaFi, relating the localized compo-

nents of F to the general ones and vice versa. In addition, we also seethat

dXk, F

= F Xk = Fk.

The components of the symmetric metric tensor g = gaba b of the

base manifold C are readily given by

gab = a, bsatisfying

gacg

bc

=

b

a ,where gab =

a, b

. It is to be understood that the covariant and

contravariant components of the metric tensor will be used to raise andthe (component) indices of vectors and tensors.

The components of the metric tensor g (xN) = ikdXi dXk de-

scribing the locally flat tangent space Tx(M) of rigid frames at a point


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Appendix: The Fundamental Properties of a Curved Manifold 71

xN = xN (xa) are given by

ik = Ei, Ek = diag (1, 1, . . . ,1) .In four dimensions, the above may be taken to be the components of

the Minkowski metric tensor, i.e., ik = Ei, Ek = diag (1, 1, 1, 1).Then we have the expression

gab = ikaXibX

k.

The line-element ofC is then given by

ds2 = g = gab ixakxb dXi dXk,where a = ix

adXi.Given the existence of a local coordinate transformation via

xi = xi (x) in C, the components of an arbitrary tensor field T Cof rank (p, q) transform according to

Tab...gcd...h = T...

... xa x

b . . . xgcx

dx . . . hx

.

Let i1i2...ipj1j2...jp

be the components of the generalized Kronecker delta.They are given by

i1i2...ipj1j2...jp

=j1j2...jpi1...ip= deti1j1

i2j1

. . . ipj1

i1j2 i2j2 . . . ipj2. . . . . . . . . . . .

i1jp i2jp

. . . ipjp

where j1j2...jp =

det(g) j1j2...jp and i1i2...ip = 1det(g)

i1i2...ip are

the covariant and contravariant components of the completely anti-symmetric Levi-Civita permutation tensor, respectively, with the ordi-nary permutation symbols being given as usual by j1j2...jq and

i1i2...ip .Again, if is an arbitrary tensor, then the object represented by

j1j2...jp =1

p!

i1i2...ipj1j2...jp

i1i2...ip

is completely anti-symmetric.Introducing a generally asymmetric connection via the covariant

derivativeba =

cab c ,

i.e.,cab = c, ba = c(ab) + c[ab] ,


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where the round index brackets indicate symmetrization and the squareones indicate anti-symmetrization, we have, by means of the local coor-dinate transformation given by xa = xa (x) in C

bea =

cabe

c

ea e

b ,

where the tetrads of the moving frames are given by ea = ax and

ea = xa. They satisfy eae

b =

ab and e

a e

a =

. In addition, it can

also be verified that

ea =

ea abcebec ,

bea = e

a

eb acbec .

We know that is a non-tensorial object, since its components trans-form as

cab = ec be

a + e

c

ea e

b .

However, it can be described as a kind of displacement field sinceit is what makes possible a comparison of vectors from point to pointin C. In fact the relation ba =

cabc defines the so-called metricity

condition, i.e., the change (during a displacement) in the basis can bemeasured by the basis itself. This immediately translates into

c gab = 0 ,

where we have just applied the notion of a covariant derivative to anarbitrary tensor field T:

mTab...gcd...h = mTab...gcd...h + apmTpb...gcd...h + bpmTap...gcd...h + + gpmTab...pcd...h pcmTab...gpd...h pdmTab...gcp...h phmTab...gcd...p

such that (mT)ab...gcd...h = mTab...gcd...h .

The condition c gab = 0 can be solved to give

cab =1

2gcd (bgda dgab + agbd) + c[ab] gcd

gae

e[db] + gbe

e[da]

from which it is customary to define

cab =1

2gcd (bgda dgab + agbd)

as the Christoffel symbols (symmetric in their two lower indices) and

Kcab = c[ab] gcd

gae

e[db] + gbe

e[da]


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Appendix: The Fundamental Properties of a Curved Manifold 73

as the components of the so-called contorsion tensor (anti-symmetric inthe first two mixed indices).

Note that the components of the torsion tensor are given by

a[bc] =1

2ea

ce

b bec + eb

c ec

b

,

where we have set

c =

ec , such that for an arbitrary scalar field

we have(ab ba) = 2c[ab]c .

The components of the curvature tensor R ofC

are then given viathe relation

(qp pq) Tab...scd...r = Tab...swd...r Rwcpq +Tab...scw...r Rwdpq + + Tab...scd...wRwrpqTwb...scd...r Rawpq Taw...scd...r Rbwpq . . .Tab...wcd...r Rswpq 2w[pq]wTab...scd...r ,

where

Rdabc = bdac cdab + eacdeb eabdec =

= Bdabc () +bKdac cKdab + KeacKdeb KeabKdec ,

where denotes covariant differentiation with respect to the Christoffelsymbols alone, and whereBdabc () = b

dac cdab + eacdeb eabdec

are the components of the Riemann-Christoffel curvature tensor ofC.From the components of the curvature tensor, namely, Rdabc, we

have (using the metric tensor to raise and lower indices)

Rab Rcacb = Bab () + cKcab KcadKdcb 2bc[ac] + 2Kcabd[cd] ,R Raa = B () 4gabac[bc] 2gacb[ab]d[cd] KabcKacb ,

where Bab () Bcacb () are the components of the symmetric Riccitensor and B () B

a

a () is the Ricci scalar. Note that Kabc gadKd

bcand Kacb gcdgbeKade.Now since

bba = bba =

bab = a

ln

det(g)

,

bab = a

ln

det(g)

+ 2b[ab] ,


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we see that for a continuous metric determinant, the so-called homoth-etic curvature vanishes:

Hab Rccab = accb bcca = 0 .

Introducing the traceless Weyl tensor W, we have the following de-composition theorem:

Rdabc = Wdabc +

1

n 2

db Rac + gacRdb dc Rab gabRdc

+

+

1

(n 1) (n 2) dc gab db gacR ,which is valid for n > 2. For n = 2, we have

Rdabc = KG

db gac dc gab

,

whereKG =

1

2R

is the Gaussian curvature of the surface. Note that (in this case) theWeyl tensor vanishes.

Any n-dimensional manifold (for which n > 1) with constant sec-tional curvature R and vanishing torsion is called an Einstein space. It

is described by the following simple relations:

Rdabc =1

n(n 1)

db gac dc gab

R ,

Rab =1

ngabR .

In the above, we note especially that

Rdabc = Bdabc () ,

Rab = Bab () ,

R = B () .

Furthermore, after some lengthy algebra, we obtain, in general, thefollowing generalized Bianchi identities:

Rabcd + Racdb + R

adbc =

= 2

da[bc] + b

a[cd] + c

a[db] +

aeb

e[cd] +

aec

e[db] +

aed

e[bc]

,


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Hence, noting that the components of the torsion tensor, namely,i[kl], indeed transform as components of a tensor field, it is seen that the

LUTij...skl...r do transform as components of a tensor field. Apparently, the

beautiful property of the Lie derivative (applied to an arbitrary tensorfield) is that it is connection-independent even in a curved manifold.


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Conclusion

We have shown that gravity and electromagnetism are intertwined in avery natural manner, both ensuing from the melting of the same under-lying space-time geometry. They obey the same set of field equations.However, there are actually no objectively existing elementary particlesin this theory. Based on the wave equation (73), we may suggest that

what we perceive as particles are only singularities which may be in-terpreted as wave centers. In the microcosmos everything is essentiallya wave function that also contains particle properties. Individual wavefunction is a fragment of the universal wave function represented by thewave function of the Universe in (73). Therefore all objects are essen-tially interconnected. We have seen that the electric(-magnetic) chargeis none other than the torsion of space-time. This charge can also bedescribed by the wave function alone. This doesnt seem to be contra-dictory evidence if we realize that nothing exists in the quantum realmsave the quantum mechanical wave function (unfortunately, we have notmade it possible here to carry a detailed elaboration on this statement).Although we have not approached and constructed a quantum theory ofgravity in the strictly formal way (through the canonical quantizationprocedure), internal consistency of our theory awaits further justifica-tion. For a few more details of the underlying unifying features of ourtheory, see Chapter Additional Considerations.


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Spin-Curvature and the Unification of Fields in a Twisted Spaceby Indranu SuhendroISBN 978-91-85917-01-3. Svenska fysikarkivet, 2008, 78 pages

The book draws theoretical findings for spin-curvature and the unification offields in a twisted space. A space twist, represented through the appropriate

formalism, is related to the anti-symmetric metric tensor. Kaluzas theory isextended and given an appropriate integrability condition. Both matter andthe isotropic electromagnetic field are geometrized through common fieldequations: trace-free field equations giving the energy-momentum tensorfor such an electromagnetic field solely via the (generalized) Ricci curvaturetensor and scalar are obtained. In the absence of electromagnetic fieldsthe theory goes to Einsteins 1928 theory of distant parallelism where only

matter field is geometrized (through the twist of space-time). The aboveresults in common with respective wave equations are joined into a unifiedfield theory of semi-classical gravoelectrodynamics.

Spinn-krokning och foreningen av falt i ett tvistat rumav Indranu SuhendroISBN 978-91-85917-01-3. Svenska fysikarkivet, 2008, 78 sidor

Boken skildrar teoretiska forskningsresultat for spinn-krokning och foreningav falt i ett tvistat rum. En rumsartad tvist, representerat genom lampligformalism, relateras till den antisymmetriska metriska tensorn. Kaluzasteori utokas och ges ett lampligt villkor for integrabilitet. Bade materiefaltoch isotropiska elektromagnetiska falt geometriseras genom gemensammafaltekvationer som ar sparfria faltekvationer som ger energi-impulstensornfor ett elektromagnetiskt falt enbart via den (generaliserade) Ricci krokning-stensorn och skalaren. I avsaknad av elektromagnetiska falt leder teorintill Einsteins teori fran 1928 om avlagsen parallellism dar bara materiefalt

geometriseras (genom en tvistning av rumstiden). Ovanstaende resultat till-sammans med respektive vagekvationer forenas till en forenad faltteori avsemiklassisk gravitationselektrodynamik.

Spin-Curvature and the Unification of Fields in a Twisted Space

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