a spin-curvature theory of quantum gravoelectrodynamics

7/31/2019 A Spin-Curvature Theory of Quantum Gravoelectrodynamics

1/85

1

ON A COMPREHENSIVE THEORY OFSEMI-CLASSICAL QUANTUM GRAVOELECTRODYNAMICS

Indranu Suhendro

Department of PhysicsKarlstad University

Karlstad 651 88Sweden

Abstract

We consider a unification of gravity and electromagnetism in which electromagneticinteraction is seen to produce a gravitational field. The field equations of gravity andelectromagnetism are therefore completely determined by the fundamentalelectromagnetic laws. Insight into this unification is that although gravity andelectromagnetism have different physical characteristics (e.g., they differ in strength), itcan be shown through the algebraic properties of the curvature and theelectromagnetic field tensors that they are just different aspects of the geometry ofspace-time. Another hint comes from the known speed of interaction of gravity andelectromagnetism: electromagnetic and gravitational waves both travel at the speed oflight. This means that they must somehow obey the same wave equation. This indeedis unity. Consequently, many different gravitational and electromagnetic phenomena

may be described by a single wave equation reminiscent of the scalar Klein-Gordonequation in quantum mechanics. Light is understood to be a gravoelectromagneticwave generated by a current-producing oscillating charge. The charge itself is generatedby the torsion of space-time. This electric (or more generally, electric-magnetic) chargein turn is responsible for the creation of matter, hence also the transformation ofmatter into energy and vice versa. Externally, the gravitational field manifests itself asthe final outcome of the entire process. Hence gravity and electromagnetism obey thesame set of field equations, i.e., they derive from a common origin. As a result, thecharge produces the so-called gravitational mass. Albeit the geometric non-linearity ofgravity, the linearity of electromagnetism is undisturbed: an idea which is central also inquantum mechanics. Therefore we preserve the most basic properties of matter suchas energy, momentum, mass, charge and spin through this linearity. It is our modestattempt to once again achieve a comprehensive unification which explains that gravity,

electromagnetism, matter and light are only different aspects of a single theory.

1. Introduction

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

and electromagnetism and perhaps also chromodynamics have been made by many great

past authors since the field concept itself was introduced by the highly original physicist,

M. Faraday in the 19th century. These attempts temporarily ended in the 1950s: in fact


2/85

2

Einsteins definitive version of his unified field theory as well as other parallel

constructions never comprehensively and compellingly shed new light on the relation

between gravity and electromagnetism. Indeed, they were biased by various possible

ways of constructing a unified field theory via different geometric approaches and

interpretations of the basic geometric quantities to represent the field tensors, e.g., the

electromagnetic field tensor in addition to the gravitational field tensor (for further

modern reference of such attempts, especially the last version of Einsteins

gravoelectrodynamics see, e.g., various works of S. Antoci). This is put nicely in the

words of Infeld: the problem of generalizing the theory of relativity cannot be solved

along a purely formal way. At first, one does not see how a choice can be made among

the various non-Riemannian geometries providing us with the gravitational and

Maxwells equations. The proper world-geometry which should lead to a unified theory

of gravitation and electricity can only be found by an investigation of its physical

content. In my view, one way to justify whether a unified field theory of gravitation and

electromagnetism is really true (comprehensive) or really refers to physical reality is to

see if one can derive the equation of motion of a charged particle, i.e., the (generalized)

Lorentz equation, if necessary, effortlessly or directly from the basic assumptions of the

theory. It is also important to be able to show that while gravity is in general non-linear,

electromagnetism is linear. At last, it is always our modest aim to prove that gravity and

electromagnetism derive from a common source. In view of this one must be able to

show that the electromagnetic field is the sole ingredient responsible for the creation of

matter which in turn generates a gravitational field. Hence the two fields are inseparable.

Furthermore, I find that most of the past theories were based on the Lagrangian

formulation which despite its versatility and flexibility may also cause some uneasiness

due to the often excessive freedom of choosing the field Lagrangian. This strictly formal

action-method looks like a short cut which does not lead along the direct route of true

physical progress. In the present work we shall follow a more fundamental (natural)

method and at the same time bring up again many useful classical ideas such as the

notion of a mixed geometry and Kaluza's cylinder condition and five-dimensional


3/85

3

formulation. Concerning higher-dimensional formulations of unified field theories, we

must remember that there is always a stage in physics at which direct but narrow physical

arguments can hardly impinge upon many hidden properties of Nature. In fact the use

of projective geometries also has deep physical reason and displays a certain degree of

freedom of creativity: in this sense science is an art, a creative art. But this should never

exclude the elements of mathematical simplicity so as to provide us with the very

conditions that Nature's manifest four-dimensional laws of physics seemingly take.

For instance, Kaluza's cylinder condition certainly meets such a requirement and

as far as we speak of the physical evidence (i.e., there should possibly be no intrusion of

a particular dependence upon the higher dimension(s)), such a notion must be regarded

as important if not necessary. An arbitrary affine (n + 1)-space can be represented by a

projective n-space. Such a pure higher-dimensional mathematical space 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

represent the events of the ordinary four-dimensional space-time by a collection of

congruence curves. It in no way points to the factual, exact number of dimensions of the

Universe with respect to which the physical four-dimensional world is only a sub-space.

In this work we shall employ a five-dimensional background space simply for the sake of

convenience and simplicity.

On the microscopic scales, as we know, matter and space-time itself appear to be

discontinuous. Furthermore, matter arguably consists of molecules, atoms and smaller

elements. A physical theory based on a continuous field may well describe pieces of

matter which are so large in comparison with these elementary particles, but fail to

describe their behavior. This means that the motion of individual atoms and molecules

remains unexplained by physical theories other than quantum theory in which discrete

representations and a full concept of the so-called material wave are taken into account. I

am convinced, indeed, that if we had a sufficient knowledge of the behavior of matter in

the microcosmos, it would, and it should, be possible to calculate the way in which

matter behaves in the macrocosmos by utilizing certain appropriate statistical techniques


4/85

4

as in quantum mechanics. Unfortunately, such calculations prove to be extremely

difficult in practice and only the simplest systems can be studied this way. Whats more,

we still have to make a number of approximations to obtain some real results. Our

classical field theories alone can only deal with the behavior of elementary particles in

some average sense. Perhaps we must humbly admit that our understanding and

knowledge of the behavior of matter, as well as space-time which occupies it, is still in a

way almost entirely based on observations and experimental tests of their behavior on

the large scales. This is a matter for experimental determination but a theoretical

framework is always worth constructing. As generally accepted, at this point one must

abandon the concept of the continuous representation of physical fields which ignores

the discrete nature of both space-time and matter although it doesnt always treat matter

as uniformly distributed throughout the regions of space. Current research has centered

on quantum gravity since the departure of the 1950's but we must also acknowledge the

fact that a logically consistent unification of classical fields is still important. In fact we

do not touch upon the formal, i.e., standard construction of a quantum gravity theory

here. We derive a wave equation carrying the information of the quantum geometry of

the curved four-dimensional space-time in Section 4 by first assuming the discreteness of

the space-time manifold on the microscopic scales in order to represent the possible

inter-atomic spacings down to the order of Plancks characteristic length.

Einstein-Riemann space(-time) 4R (a mixed, four-dimensional one) endowed with

an internal spin space pS is first considered. We stick to the concept of metricity and do

not depart considerably from affine-metric geometry. Later on, a five-dimensional

general background space IR5 is introduced along with the five-dimensional and (as a

brief digression) six-dimensional sub-spaces n and 6V as special coordinate systems.

Conventions: Small Latin indices run from 1 to 4. Capital Latin indices run from 1 to 5.Round and square brackets on particular tensor indices indicate symmetric and skew-symmetric characters, respectively. The covariant derivative is indicated either by a semi-

colon or the symbol . The ordinary partial derivative is indicated either by a comma orthe symbol . Einstein summation convention is, as usual, employed throughout thiswork. Finally, by the word spacewe may also mean space-time.


5/85

5

2.Geometric construction of a mixed, metric-compatible four-manifold

4R possessing an internal spin space

pS

Our four-dimensional manifold 4R is endowed with a general asymmetric

connectioni

jk and possesses a fundamental asymmetric tensor defined herein by

( )][)(][)(2

1ijijijijij

gg ++= (1)

where )()( 2 ijijg will play the role of the usual geometric metric tensor with

which we raise and lower indices of tensors while ][][ 2 ijijg will play the role of a

fundamental spin tensor (or of a skew- or anti-symmetric metric tensor). We shall also

refer both to as the fundamental tensors. They satisfy the relations

[ ]

==

=

)(

)(

)(

)(

][

][

)(

][

)(

)(

cgggg

bgg

agg

jk

ij

jk

ij

k

i

kj

ij

k

i

kj

ij

(2)

We may construct the fundamental spin tensor as a generalization of the skew-symmetric symplectic metric tensor in anyM-dimensional space(-time), whereM = 2,4,6,,M = 2 + p, embedded in (M+n)-dimensional enveloping space(-time), where n= 0,1,2, . In Mdimensions, we can construct p = M - 2null (possibly complex) normal

vectors (null n-legs) pzzz ,...,, 21 with 0=

nm zz ( pnm ,...,1,..., = and

M...,,1,..., = ). If we define the quantity

= where the skew-

symmetric, self-dual null bivector defines a null rotation, then these null n-legs are

normal to the (hyper)planeMnMR + 2 (containedin MR )defined in such a way that

( ) [ ] [ ]( )

gdet,1

,

0

...!...

......

......

|1|2121...

=

==

=

++==

gg

g

zzzzpMzzz pp


6/85

6

Here ... is the Levi-Civita permutation symbol. Hence in four dimensions we have

0

2

1

2121

2121

212121

===

=

===

zzzztr

zzzz

zzzzzz

where

= is the generalized Kronecker delta and

= .

(The minus sign holds if the manifold is Lorentzian and vice versa.) The particular

equation 0=tr is of course valid in M dimensions as well. In M dimensions, thefundamental spin tensor of our theory is defined as a bivector satisfying

[ ][ ] ( )

+=

1

1

Mgg (3a)

Hence the above relation leads to the identity

[ ][ ]

=gg (3b)

In the particular case ofM= 2 and n = 1, the

vanishes and the fundamental spin

tensor is none other than the two-dimensional Levi-Civita permutation tensor:

[ ]

[ ][ ]

[ ][ ] B

A

BC

AC

D

A

C

B

D

B

C

A

CD

AB

ABAB

gg

gg

gg

=

=

==01

10

where A,B = 1,2. Lets now return to our four-dimensional manifold 4R . We now have

[ ][ ] ( )

[ ][ ] k

i

kj

ij

l

i

k

j

l

j

k

i

kl

ij

kl

ij

gg

gg

=

+=3

1

Hence the eigenvalue equation is arrived at:

[ ] [ ]klkl

ijijgg = (3c)


7/85

7

We can now construct the symmetric traceless matrixk

iQ through

0

0

2121

=

=

=

=

k

ik

kiik

l

j

k

i

kl

ij

j

lsr

qpklrs

ijpq

j

l

kl

ij

k

i

uQ

tr

QQ

uuQ

uuzzzzuuQ

Q

whereiu is the unit velocity vector, 1=ii uu . Lets introduce the unit spin vector:

[ ]

[ ] 1

0,1

=

==

=

ki

ik

i

i

i

i

k

iki

uvg

vuvv

ugv

Multiplying both sides of(3c) by the unit spin tensor, we get

[ ] [ ]

r

kr

k

i

j

ij ugQug =

In other words,

ki

k

i vQv = (3d)

Now we can also verify that

k

i

kj

ij = (4)

Note that since the fundamental tensor is asymmetric it follows that

[ ] ( )ki

kj

ij

jk

ij

jk

ijgg =

)( (5)

The line-element of 4R can then be given through the asymmetric fundamental tensor:

ki

ik

ki

ikdxdxgdxdxds

)(

2 2 ==


8/85

8

There exists in general no relation such ask

iji

kjgg =)(

][. However, we have the

relations

=

=

)(

)(

][)()(

][

)(][][

)(

bgggg

agggg

ijjsir

rs

ijjsir

rs

(6)

We now introduce the basis lg which spans the metric space of 4R and its associate

{ }l which spans the spin space pS 4R (we identify the manifold 4R as having the

Lorentzian signature -2, i.e., it is a space-time). These bases satisfy the algebra

( ) ( )( ) ( )( )

=

==

==

=

=

)(

)(

)(

)(

)(

][

)(

][

][

eg

d

cg

bg

ag

ijji

i

jj

i

j

i

ijjiji

k

kii

k

iki

g

gg

gg

g

g

(7)

We can derive all of (2) and (6) by means of (7). In a pseudo-five-dimensional spacenRn = 4 (a natural extension of 4R which includes a microscopic fifth coordinate

axis normal to all the coordinate patches of 4R ), the algebra is extended as follows:

[ ][ ]

[ ] ( )

==

=

+=

)(,

)(,

)(,

][

][

][

cg

bg

agC

j

ijii

j

iji

ijk

k

ijji

ggn

ggn

nggg

(8)

where the square brackets [ ] are the commutation operator,k

ijC stands for thecommutation functions and n is the unit normal vector to the manifold 4R satisfying

1)( = nn . Here we shall always assume that 1)( += nn anyway. In summary,the symmetric and skew-symmetric metric tensors can be written in nRn = 4 as

[ ] [ ] ),(

)()(

ngg

gg

=

=

kiik

kiik

g

g (8d),(8e)


9/85

9

The (intrinsic) curvature tensor of the space 4R is given through the relations

( )

[ ] lil

jkl

l

ijkjkikji

i

ml

m

jk

i

mk

m

jl

i

ljk

i

kjl

i

jkl

aaRaa

R

;.;;;;

,,.

2 =

+=

for an arbitrary vector ia . The torsion tensor [ ]i

jk is introduced through the relation

[ ] rr

ikikki ,;;;; 2 =

which holds for an arbitrary scalar field . The connection of course can be written as

[ ]i

jk

i

jk

i

jk+=

)( . The torsion tensor [ ]i

jk together with the spin tensor [ ]ijg shall play

the role associated with the internal spin of an object moving in space-time. On the

manifold 4R , lets now turn our attention to the spin space pS and evaluate the tangent

component of the derivative of the spin basis { }l with the help of(7):

( ) ( ) lk

mj

lm

ik

lk

jikTij gggg =][

][

][

,][ (9)

since ( )k

k

ijTijgg = . Now with the help of(3), we have

( ) ( ) llijlijlkjikTij gg

= 3

1][,][

where we have puti

ijj = . On the other hand one can easily show that by imposingmetricity upon the two fundamental tensors (use (7) to prove this), the following holds:

( ) ( )kijkijijk == gg (10)

Thus, solving for a tetrad-independent connection, we have

[ ][ ] k

i

jkjr

iri

jkgg =

2

1

2

3, (11)


10/85

10

The torsion tensor is therefore

( ) ( )

( ) [ ] [ ] ljkili

jk

i

kj

jklkjl

ili

jk

i

kj

i

jk

gg

ggg

,

,][,][

][

][

4

3

4

1

4

3

4

1

+=

+=

(12)

where we have assumed that the fundamental spin tensor is a pure curl:

[ ] ijjiijg ,, = , [ ] [ ] [ ] 0,,, =++ jkiijkkij ggg

This expression and the symmetric part of the connection:

( ) mljkm

lim

lkjm

li

jklljkklj

lii

jkgggggggg

][)(

)(

][)(

)(

,)(,)(,)(

)(

)(2

1+=

therefore determine the connection uniquely in terms of the fundamental tensors alone:

( ) ( ) [ ] [ ] [ ]( )

[ ][ ] [ ]( )

[ ][ ] [ ]( )ljsjls

rs

kr

li

lkskls

rs

jr

li

jkrkjr

ir

rjk

ir

j

i

kjklljkklj

lii

jk

gggggggggg

ggggggggg

,,)(

)(

,,)(

)(

,,)(

)(

,)(,)(,)(

)(

4

3

4

3

4

3

2

1

2

1

+++=

(13)

There is, however, an alternative way of expressing the torsion tensor. The metric and

the fundamental spin tensors are treated as equally fundamental and satisfy the ansatz

0;][;)( == kijkij gg and 0; =kij .

Therefore, from 0;][ =kijg ,we have the following:

m

jkim

m

ikmjkij ggg += ][][,][


11/85

11

Lettingm

ikjmjik gW = ][ , we have

jikijkkijWWg =

,][ (14)

Solving for ][ jkiW by making cyclic permutations ofi,jand k, we get

( ))()(,][,][,][][

2

1ijkikjjkiijkkijjki

WWgggW ++=

Therefore

( ))()()(,][,][,][

][)(

2

1ijkikjjkijkiijkkij

jkijkiijk

WWWggg

WWW

+++=

+=

Now recall thatm

jkimijkgW = ][ . Multiplying through by

[ ]ilg , we get

( ) ijk

m

ljkm

lim

lkjm

li

jklljkklj

lii

jkgggggggg

)()(][

][

)(][

][

,][,][,][

][

2

1+++=

(15)

On the other hand,

( ) [ ]i

jk

m

ljkm

lim

lkjm

li

jklljkklj

lii

jkgggggggg

][][)(

)(

)(

)(

,)(,)(,)(

)(

2

1++=

(16)

From (15) the torsion tensor is readily read off as

( )m

ljkm

lim

lkjm

li

jklljkklj

lii

jk gggggggg )(][][

)(][

][

,][,][,][

][

][2

1

++=

(17)

We denote the familiar symmetric Levi-Civita connection by

( )( ) ( )( )jklljkklj

ligggg

jk

i,)(,,

2

1+=


12/85

12

If we combine (13) and (17) with the help of (2), (3) and (4), after a rather lengthy butstraightforward calculation we may obtain a solution:

( )

( )ik

m

mj

i

j

m

mkkm

li

jm

li

jklljkklj

lii

jk

lj

mgg

lk

mgg

gggg

][][][

][

][

][

,][,][,][

][

][

3

1

2

1

+

+

+=

(18)

Now the spin vector iik][ is to be determined from (12). If we contract (12) on theindices iandj, we have

( ) kikjkijijiik ggg =4

3

4

3,][,][

][

][ (19)

But from (14):

kij

ij

kij

iji

ikkgggg

,][

][

,)(

)(

2

1

2

1=== (20)

Hence (19) becomes

[ ] ikjij

kij

ij

k

i

ikggggS

,][

][

,

][

][4

3

8

3= (21)

Then with the help of(21), (18) reads

[ ] ( )

[ ] [ ]( ) [ ] [ ] [ ]( )i

knjm

i

jnkm

mni

kjmn

i

jkmn

mn

km

li

jm

li

jklljkklj

lii

jk

gggggg

lj

mgg

lk

mgg

gggg

,,,,

][

][

][

][

][

,][,][,][

][

4

1

8

1

2

1

++

+

+=

(22)

So far, we have been able to express the torsion tensor, which shall generate physical

fields in our theory, in terms of the components of the fundamental tensor alone.


13/85

13

In a holonomic frame, 0][ =i

jk and, of course, we have from (16) the usual Levi-Civita

(or Christoffel) connection:

( )jklljkklj

lii

jkgggg

jk

i,)(,)(,)(

)(

2

1+=

=

If therefore a theory of gravity adopts this connection, one may argue that in a strict

sense, it does not admit an integral concept of internal spin in its description. Such is the

classical theory of general relativity.

In a rigid frame (constant metric) and in a pure electromagnetic gauge condition, one

may have 0)(

=ijk

and in this special case we have from (15)

( )jklljkklj

lii

jkgggg

,][,][,][

][

2

1+=

which is exactly the same in structure as

jk

i with the fundamental spin tensor

replacing the metric tensor. We shall call this connection the pure spin connection,denoted by

( )jklljkklj

lii

jkggggL

,][,][,][

][

2

1+= (23)

Lets give an additional note to (20). Let's find the expression fori

ki , provided weknow that

( )

==

=

==

ik

ig

gg

gg

k

kij

ij

kij

iji

ikk

,

,][

][

,)(

)(

ln

2

1

2

1


14/85

14

Meanwhile, we express the following relations:

[ ]

[ ][ ]( )ikjijk

ikj

ij

kij

ij

i

ki

i

ikk

gg

gggg

S

,

,][

][

,][

][

][

4

3

4

3

8

3

=

=

==

[ ]

[ ][ ] ikj

ij

k

i

ikk

m

likm

lim

lkim

lii

ik

gg

ggggik

i

,

][)(

)(

][)(

)(

)(

4

3

4

1+=

=

=

Therefore

i

ik

i

ki

i

ki

i

ki

i

ki ][)(][)( =+=

kikj

ij gg =

2

1

2

3,][

][

which can also be derived directly from (11). From (11) we also see that

[ ] [ ] [ ]

[ ] [ ]r

jkri

r

ikrj

kij

r

ikrjkij

gg

ggg

=

+=3

1

3

2,

(24a),(24b)

Also, for later purposes, we derive the condition for the conservation of charges:

[ ] [ ]k

ikik

kgg =

3

1, (24c)

Having developed the basic structural equations here, we shall see in the followingchapters that the gravitational and electromagnetic tensors are formed by means of the

fundamental tensors )()( 2 ijijg and ][][ 2 ijijg alone (see Appendix B). Inother words, gravity and electromagnetism together arise from this single tensor. Weshall also investigate their fundamental relations and ultimately unveil their union.


15/85

15

3.Generalization of Kaluzas projective theory. Fundamental fieldequations of our unified field theory

We now assume that the space-time 4R is embedded in a general five-dimensional

Riemann space IR5. This is referred to as embedding of class 1. We shall later define the

space nRn = 4 to be a special coordinate system in IR5. The five-dimensional

metric tensor )(ABg ofIR5 of course satisfies the usual projective relations

=

+=

0

)()(

AAi

BAij

j

B

i

AAB

ne

nngeeg

whereA

i

A

ixe = is the tetrad. If now lg denotes the basis of 4R and { }Ae ofIR5:

==

==

+=

j

i

j

A

A

iB

AA

B

i

B

A

i

AB

B

j

A

iijA

A

ii

Ai

i

AA

eennee

geege

ne

,

,)()(eg

nge

The derivative of ig 4R IR5 is then

ngg ijkk

ijij +=

We also have the following relations:

0,,,

)(0,,

;)(;)(.;.;;

;.;

=====

====

CABkij

j

B

i

jA

i

BA

A

j

j

i

A

i

A

ij

A

ji

C

C

ABABBAj

j

iiijji

ggeneenne

eeegnng

In our work we shall, however, emphasize that the exterior curvature tensorij

is in

general asymmetric: jiij just as the connection ijk is. This is so since in generalA

ji

A

ij ee . Within a boundary , the metric tensor )(ijg may possessdiscontinuities in its second derivatives. Now the connection and exterior curvaturetensor satisfy

+=

+=c

j

B

i

A

BCA

A

ijAij

c

j

B

i

A

BC

k

A

A

ij

k

A

k

ij

eenen

eeeee

(25a),(25b)


16/85

16

where

A

ij

C

j

B

i

A

BC

k

ij

A

k

A

ijneeee += (26)

We can also solve forA

BC in (25) with the help of the projective relation

ngeAi

i

AA ne += . The result is, after a quite lengthy calculation,

B

j

C

A

i

i

j

A

BC

Aj

C

i

Bij

j

C

i

B

k

ij

A

k

i

BC

A

i

A

BC

neenn

neeeeeee

.)(

+

++=(27)

If we now perform the calculation ) ikjjk g with the help of some of theabove relations, we have in general

( ) ( )( )

A

A

ikjjk

A

B

i

A

BC

C

kj

C

jkA

D

k

C

j

B

i

A

BCDikjjk

e

eeeeeeR

e

eeg

+

+= .(28)

Here we have also used the fact that

( )D

D

ABCACBBC R ee .=

On the other hand, jj

ii gn .= , and

( )( ) ( )ng

ngg

lk

l

ijkijl

l

kij

l

mk

m

ij

l

kij

kijl

l

ijijk

+++=

+=

,.,

,

Therefore we obtain another expression for ) ikjjk g :

( ) ngg illjkjikkijllkijljiklijkikjjk R ][;;... 2 +++=

) ) AijkBiABCCkjCjkAikjjk Seeee .+ (29a),(29b)


17/85

17

Combining(28) and (29), we get, after some algebraic manipulations,

A

iAjkl

D

l

C

k

B

j

A

iABCDjkiljlikijkl eSeeeeRR +=

(30a),(30b)

[ ]A

Aijkil

l

jk

D

k

C

j

B

i

A

ABCDjikkijnSeeenR += 2

;;

We have thus established the straightforward generalizations of the equations of Gaussand Codazzi.

Now the electromagnetic content of(30) can be seen as follows: first we split the

exterior curvature tensor ij into its symmetric and skew-symmetric parts:

ijijijij

ijijij

fk

+=

][)(

][)(

,

(31)

Here the symmetric exterior curvature tensorij

k has the explicit expression

( )

( )ABBA

B

j

A

i

C

j

B

i

A

BCA

A

ji

A

ijAij

nnee

eeneenk

;;

)(

2

1

2

1

+=

++=

(32)

Furthermore, in our formalism, the skew-symmetric exterior curvature tensor ijf is

naturally equivalent to the electromagnetic field tensorij

F . It is convenient to set

ijijFf

2

1= . Hence the electromagnetic field tensor can be written as

( ) [ ]( )

ABBA

B

j

A

i

C

j

B

i

A

BCA

A

ji

A

ijAij

nnee

eeneenF

;;

2

=

+=(33)


18/85

18

The five-dimensional electromagnetic field tensor is therefore

ABBAABnnF =

We are now in a position to simplify(30) by invoking two conditions. The first ofthese, following Kaluza, is the cylinder condition: the laws of physics in their four-

dimensional form shall not depend on the fifth coordinate nx 5 . We also assume that

yx 5 is a microscopic coordinate in n . In short, the cylinder condition is written as(by first putting

AAen 5= )

( ) ( ) 0;;,5,)( =+== BCCBCjBiAAijij nneengg

where we have now assumed that in IR5 the differential expressioni

AB

i

BA ee ,,

vanishes. However, from (26), we have the relation

[ ]A

ij

k

ij

A

k

A

ij

A

jinFeee += 2

,,

Furthermore, the cylinder condition implies that 0;; =+ABBA nn and therefore we

can nullify(32). This is often called the assumption of weakness. The second condition

is the condition of integrability imposed on arbitrary vector fields, e.g., on i (say) in

4R . The necessary and sufficient condition for a vector field ii , (a one-form) to be

integrable is ijji ,, = . If this is applied to (28), we will then have

A

ijk

D

k

C

j

B

i

A

BCDSeeeR

..= . Therefore (30) will now go into

( )

il

l

jkjikkij

jkiljlikijkl

FFF

FFFFR

][;;2

4

1

=

=(34a),(34b)

These are the sought unified field equations of gravity and electromagnetism. They form

the basic field equations of our unified field theory.


19/85

19

Altogether they imply that

)(.4

1ik

j

kijikRFFR == (35a)

[ ] 0=ikR (35b)

ik

ikFFR

4

1= (35c)

i

k

l

l

ik

j

ijJFF =

.][

; 2 (35d)

These re lations are necessary and sufficient following the two conditions we havedealt with. These field equations seem to satisfy a definite need. They tell us a beautifuland simple relation between gravity and electromagnetism: (34a) tells us that both insideand outside charges, a gravitational field originates in a non-null electromagnetic field (asin Rainich's geometry), since according to (34b), the electromagnetic current is producedby the torsion of space-time: the torsion produces an electromagnetic source. Theelectromagnetic current is generated by dynamic electric-magnetic charges. In a strictsense, the gravitational field cannot exist without the electromagnetic field. Hence allmatter in the Universe may have an electromagnetic origin. Denoting by d a three-

dimensional infinitesimal boundary enclosing several charges, we have, from (35d)

[ ] = dFuek

r

ir

ik .2

We may represent a negative charge by a negative spin produced by a left-handedtwist (torsion) and a positive one by a positive spin produced by a right-handed twist.(For the conservation of charges (currents) see Appendices C and D.) Now(35c) tells usthat when the spatial curvature, represented by the Ricci scalar, vanishes, we have a nullelectromagnetic field, also it is seen that the strength of the electromagnetic field isequivalent to the spatial curvature. Therefore gravity and electromagnetism areinseparable. The electromagnetic source, the charge, looks like a microscopic spinning

hole in the structure of the space-time 4R , however, the Schwarzschild singularity isnon-existent in general. Consequently, outside charges our field equations read

( )

0

4

1

;=

=

ij

j

jkiljlikijkl

F

FFFFR(36a),(36b)

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


20/85

20

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

( )( )

( ) xdgFFRRRFFR

xdgRRRI

jkil

ijkl

ijkl

ijkl

ik

ik

ik

ik

4

2/1

2*

42/1*

4

1

16

1

=

=

(37)

Here R* denotes the Ricci scalar built from the symmetric Christoffel connection alone.From the variation of which, we would arrive at the standard Einstein-Maxwellequations. However, we do not wish to stress heavy emphasis upon such an action-

method (which seems like a forced short cut) in order to arrive at the field equations ofour unified field theory. We must emphasize that the equations (34)-(37) tell us how theelectromagnetic field is incorporated into the gravitational field in a very natural manner,in other words theres no need here to construct any Lagrangian density of such. Wehave been led into thinking of how to couple both fields using different procedureswithout realizing that these fields already encapsulate each other in Nature. But here ourspace-time is already a polarized continuum in the sense that there exists anelectromagnetic field at every point of it which in turn generates a gravitational field.

__________________________________________________________________________________

Remark 1

Without the integrability condition we have, in fairly general conditions, the relation

( ) ( )( ) [ ] [ ]( ) BiCjkClljkABCAllkijljik

A

il

l

jkjikkij

D

k

C

j

B

i

A

BCD

A

l

l

ijk

A

ikjjk

enee

neeeReRe

++

+++=

2

2

..

][;;..

(a)

( )

( ) ( )B

i

C

kj

C

jk

A

BC

A

ikjjk

B

i

C

jk

C

l

l

jk

A

BC

A

ikjjk

A

ijk

eeee

eneeS

+=

++][][. 2

Hence we have

A

iAjkl

D

l

C

k

B

j

A

iABCDjkiljlikijkleSeeeeRR += (b1)

[ ] ill

jk

A

Aijk

D

k

C

j

B

i

A

ABCDjikkij nSeeenR += 2;; (b2)


21/85

21

These are just the equations in (30). Upon employing a suitable cylinder condition and

putting [ ] ijij F2

1= (within suitable units), we have the complete set of field equations

of gravoelectrodynamics:

( ) AiAjkl

D

l

C

k

B

j

A

iABCDjkiljlikijkleSeeeeRFFFFR +=

4

1(c1)

( ) [ ] illjk

AAijk

Dk

Cj

Bi

AABCDjikkij FnSeeenRFF += ;;2

1(c2)

--------------------------------------------------------------------------------------------------------------

Sub-remark: Lets consider the space YRS =45

which describes a five-

dimensional thin shell where Y is the microscopic coordinate representation spanned

by the unit normal vector to the four-manifold 4R . The coordinates of this space are

characterized by ( )yxy i ,= where the Greek indices run from the 1 to 5 and wherethe extra coordinate y is taken to be the Planck length:

3c

Gy

h=

which gives the thickness of thin shell. Here G is the gravitational constant of

Newton, h is the Planck constant divided by 2 and c is the speed of light in vacuum.(From now on, since the Planck length is extremely tiny, we may drop any higher-order

terms in .) Then the basis

of the space 5S can in general be split into

)n

g

=

=

5

.

k

k

i

k

iiy


22/85

22

It is seen that the metric tensor of the space5

S , i.e.,

, has the following non-zero

components:

1

2

55

)()(

=

=

ikikik

yg

The simplest sub-space of the space5

S is given by the basis

( )ng

g

=

=

5

0,i

ii x

where lg is of course the tangent basis of the manifold 4R . We shall denote this

pseudo-five-dimensional space as the special coordinate system nRn = 4 whose

metric tensor

g can bearrayed as

=

10

044)( xik

gg

Now the tetrad of the space5

S is then given by ( )

= AA e , which can be split

into

AA

A

k

k

i

A

i

A

i

n

eye

=

=

5

.

Then we may find the inverse to the tetradA

i as follows:

AAA

k

A

i

k

i

A

i

A

yn

eye

,

5

.

==

+=


23/85


24/85

24

From the above expression, we see that

.5.5

,.

,2

1

2

1

FF

yFkkk

==

=

As can be worked out, the five-dimensional connection of the background space IR5 is

related to that of 5S through

( )

i

C

k

iBk

A

CB

A

CB

A

CB

AiA

BC

eFy

nFyx

.

.,

2

1

2

1,

+=

Now the five-dimensional curvature tensor ( )0,iABCDABCD

xRR = is to be related

once again to the four-dimensional curvature tensor of4

R , which can be directly derived

from the curvature tensor of the space 5S as ( )0,5 i

ijklS

ijkl xRR = . With the help of

the above geometric objects, and after some laborious work-out, we arrive at the relation

ABCDCDABDCBAABCDFFReeeeR

+=2

1

where we have a new geometric object constructed from the electromagnetic field

tensor:

( ) ( )

( ) ( )( ) ( )

A

BCD

A

BCD

ABCCBCBABCA

ABCCBABCCBABC

e

eenenFnxnxF

eenenFeneneF

..

.,,.

,...

2

10,0,

2

1

2

1

2

1

++

+=


25/85

25

Define another curvature tensor:

( ) Dl

C

k

B

j

A

iABCDABCDijkleeeeR

Then we have the relation

klijijklijkl

FFR2

1+=

By the way, the curvature tensor of the space nRn = 4 is here given by

( ) ( )

+= 0,0,

,,.xxR

Expanding the connections in the above relation, we obtain

( ) ( )

( ) ( )

( ) ( )

( )

,,....

,,.,..,,.,..

,..,..

,,.,,.,.

,,.

41

4

1

4

1

2

1

2

1

2

1

2

1

yyFFFF

yyFyFFyyFyFF

yFFyFF

yyFyFF

R

+

++

++

++

+=

We therefore see that the electromagnetic field tensor is also present in the curvature

tensor of the space nRn = 4 . In other words, electromagnetic and gravitational

interactions are described together on an equal footing by this single curvature tensor.


26/85

26

Direct calculation shows that some of its four-dimensional and mixed components are

[ ]( )

0

22

1

5

;;5

5

.

,,.

=

+==

+=

kij

ir

r

jkjikkijijkijk

i

rl

r

jk

i

rk

r

jl

i

ljk

i

kjl

i

jkl

R

FFFRR

R

Furthermore, we obtain the following equivalent expressions:

( )

( )ABCDCDAB

l

D

k

C

j

BAjm

m

lkljkkjl

l

D

k

CB

i

Aim

m

klkillik

l

D

k

C

j

B

i

AijklABCD

FFeeenFFF

eeneFFFeeeeRR

+++

++=

2

12

2

1

22

1

][;;

][;;

( ) ( )l

D

k

C

j

BAjm

m

lk

l

D

k

CB

i

Aim

m

kl

l

D

k

C

j

B

i

Aijkl

ABCDCDABADBCCBDBCADDACABCD

eeenFeeneFeeeeR

FFnFFnFFR

][][

;;;;

22

2

1

2

1

2

1

+++

+=

where

( ) ( )( ) ABCDACMDDMCMBBCMDDMCMAABCD nnFnFFnnFnFF = ....2

1

When the torsion tensor of the space 4R vanishes, we have the relation

( ) ( )l

D

k

C

j

B

i

Aijkl

ABCDCDABADBCCBDBCADDACABCD

eeeeR

FFnFFnFFR

+

+= 2

1

2

1

2

1;;;;

which, again, relates the curvature tensors to the electromagnetic field tensor.


27/85

27

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

ABCDCDABABCDABCDFFR ~~~

2

1~+

where ABCDR~

, ABF~

and ABCD~

are the extensions of ABCDR , ABF and ABCD which

are dependent on , we may obtain the relation

ijkr

r

l

ijrlrkirkl

rjrjkl

riijkl

Dl

Ck

Bj

AiABCD

RFy

RFyRFyRFyReeee

.

...

2

1

2

1

2

1

2

1

+

+++=

--------------------------------------------------------------------------------------------------------------

Lets now write the field equations of our unified field theory as

( )

[ ] ijkill

jkjikkij

ijkljkiljlikijkl

FFF

FFFFR

+=

+=

2

4

1

;;

(d)1,2

( ) AAijk

D

k

C

j

B

iABCDijk

A

iAjkl

D

l

C

k

B

j

A

iABCDijkl

nSeeeR

eSeeeeR

=

=

2

(e)1,2

Consider the invariance of the curvature tensor under the gauge transformation

k

i

j

i

jk

i

jk ,' += (f)

for some function )(x = . This is analogous to the gauge transformation of the

electromagnetic potential, i.e.,iii ,

' += , with a scaling constant ,

which leaves the electromagnetic field tensor invariant.. We define the electromagnetic

potential vectori and pseudo-vector i via ii

k

ki += where is a constant.


28/85

28

Then we see that the electromagnetic field tensor can be expressed as

( )ikkiikkiik

F,,,,

1

== (g)

More specifically, the two possible electromagnetic potentialsi

and i transform

homogeneously and inhomogeneously, respectively, according to

C

i

BA

BCA

A

ik

k

AA

A

ii

A

A

ii

enneee

e

+=

=

,

The two potentials become equivalent in a coordinate system where g equals a

constant. Following (g), we can express the curvature tensor as

)()()()(21 jkiljlikjkiljlikijklggggFFFFR += (h)

where 1 and 2 are invariants. (The term klij FF0 would contribute nothing.)

Hence

)(2.13

ik

l

kilikgFFR += (i)

Putting4

11 = in accordance with (d)1 and contracting (i) on the indices iand k we see

thatik

ik FFR 48

1

12

12 = . Consequently, we have the important relations

( ) ( )

( ) rsrsjkiljlik

jkiljlikjkiljlikijkl

FFgggg

RggggFFFFR

)()()()(

)()()()(

48

1

12

1

4

1

+=(j)


29/85

29

rs

rsikik

l

kilikFFgRgFFR

)()(.16

1

4

1

4

1+= (k)

Comparing (j) and (e)1 we find

( ) ( ) rsrsjkiljlikjkiljlik

A

iAjkl

D

l

C

k

B

j

A

iABCDijkl

FFggggRgggg

eSeeeeR

)()()()()()()()(

48

1

12

1=

=

(l)

Hence also

rs

rs

rs

rsikikik

FFR

FFgRg

4

1

16

1

4

1)()(

=

=

(m)1,2

Note that our above consideration produces the following traceless field equation:

= rs

rsik

l

kilikikFFgFFRgR

)(.)(4

1

4

1

4

1 (n)

In a somewhat particular case, we may set ( )ikkiik tuucRg +=

2

)(4

1 where

is a coupling constant and ikt is the generalized stress-metric tensor, such that

eR = ,where now tce +=2 is the effective material density. We also have

( ) ( )

++= rs

rsik

l

kilikkiikikFFgFFtuucRgR

4

1

4

1

2

1.

2

)(

(o)


30/85

30

The above looks slightly different from the standard field equation of general relativity:

( )

+= rsrsik

l

kilikkiikikFFgFFtuuckRgR

4

1

2

1.

2*

)(

* (p)

which is usually obtained by summing altogether the matter and electromagnetic terms.

Hereik

R* and R*

are the Ricci tensor and scalar built out of the Christoffel connection

and k is the usual coupling constant of general relativity. Lets denote by

,0m and c , the point-mass, material density and speed of light in vacuum. Then the

vanishing of the divergence of (p) leads to the equation of motion for a charged particle:

ki

k

ki

k

i

uFcm

euu

Ds

Du.2

0

;=

However, this does not provide a real hint to the supposedly missing link between

matter and electromagnetism. We hope that theres no need to add an external matter

term to the stress-energy tensor. We may interpret (n), (o) and (p) as telling us that

matter and electromagnetism are already incorporated, in other words, the

electromagnetic field produces material density out of the electromagnetic current J . In

fact these are all acceptable field equations. Now, for instance, we have

( )tuJcRi

i

e+== 2 . From (m)2, the classical variation follows:

04

1 4

4

=

=

=

xdgFFR

xdgI

ik

ik

(q)

which yields the gravitational and electromagnetic equations of Einstein and Maxwell

endowed with source since the curvature scalar here contains torsion as well.


31/85

31

Finally, lets investigate the explicit relation between the Weyl tensor and theelectromagnetic field tensor in this theory. In four dimensions the Weyl tensor is

( )

( ) Rgggg

RgRgRgRgRC

jkiljlik

iljkjkilikjljlikijklijkl

)()()()(

)()()()(

6

1

2

1

+

+=

Comparing the above equation(s) with (j) and (k), we have

( ) ( )

( )rlirjk

r

kjril

r

kirjl

r

ljrik

rs

rsjkiljlikjkiljlikijkl

FFgFFgFFgFFg

FFggggFFFFC

.)(.)(.)(.)(

)()()()(

8

1

24

1

4

1

+

+=

(r)

We see that the Weyl tensor is composed solely of the electromagnetic field tensor in

addition to the metric tensor. Hence we come to the conclusion that the space-time 4R

is conformally flat if and only if the electromagnetic field tensor vanishes. This agreeswith the fact that, when treating gravitation and electromagnetism separately, it is the

Weyl tensor, rather than the Riemann tensor, which is compatible with theelectromagnetic field tensor. From the structure of the Weyl tensor as revealed by (r), itis understood that the Weyl tensor actually plays the role of an electromagnetic

polarization tensor in the space-time 4R . In an empty region of the space-time 4R with

a vanishing torsion tensor, when the Weyl tensor vanishes, that region possesses aconstant sectional curvature which conventionally corresponds to a constant energydensity.

__________________________________________________________________________________

Let's for a moment turn back to (30). We shall show how to get the source-torsionrelation, i.e., (35d) in a different way. For this purpose we also set a constraint

0. =AijkS and assume that the background five-dimensional space is an Einstein space:

)(ABABgR =

where is a cosmological constant. Whenever 0= we say that the space is Ricci-flat or energy-free, devoid of matter. Taking into account the cosmological constant, thisconsideration therefore takes on a slightly different path than our previous one. We onlywish to see what sort of field equations it will produce.


32/85

32

We first write

( )

il

l

jk

D

k

C

j

B

i

A

ABCDjikkij

D

l

C

k

B

j

A

iABCDjkiljlikijkl

FeeenRFF

eeeeRFFFFR

][;;22

4

1

=

+=

Define a symmetric tensor:

ki

DC

k

BA

iABCDik

BneneRB = (38)

It is immediately seen that

0

)(

=

=

=

BA

iAB

BA

AB

ik

B

k

A

iAB

neR

nnR

geeR

(39a),(39b),(39c)

Therefore

ikik

l

kil

DC

k

BA

iABCD

B

k

A

iAB

l

kilik

BgFF

neneReeRFFR

++=

+=

)(.

.

4

1

4

1

(40)

From (38) we also have, with the help of(39b), the following:

( )

=

+=

==DCBA

ABCD

BA

AB

CAACDB

ABCDik

ik

nnnnRnnR

nngnnRBgB )()(

Hence we have

+= 34

1 ikikFFR (41)


33/85

33

The Einstein tensor (or rather, the generalized Einstein tensor endowed with torsion)

RgRGikikik )(

2

1

up to this point is therefore

l

kilik

rs

rsikikikFFgFFgBG

.)()(4

1

2

1

8

1= (42)

From the relation

il

l

jk

D

k

C

j

B

i

A

ABCDjikkijFeeenRFF

][;;22 =

we see that

jklj

kl

klj

kl

Dj

CB

ABC

DA

Aj

C

C

A

klj

kl

D

k

j

C

k

B

ABC

DA

kj

k

JF

FnennRneR

FeeenRF

==

=

=

][.

][....

][...;

2

222

22

In other words,

k

l

l

ikiFJ

.][2 =

which is just (35d). We will leave this consideration here and commit ourselves to the

field equations given by(34) and (35) for the rest of our work.

Lets obtain the (generalized) Bianchi identity with the help of (34) and (35).

Recall once again that

( )

il

l

jkjikkij

jkiljlikijkl

FFF

FFFFR

][;;2

4

1

=

=


34/85


35/85


36/85

36

In the most general case, by the way, the Ricci tensor is asymmetric. If we proceed

further, the generalized Bianchi identity and its contracted form will be given by

[ ] [ ] [ ]( )ijrlrmkijrkrlmijrmrkllijmkkijlmmijkl RRRRRR ++=++ 2;;; (48a)

[ ] [ ]ijk

r

r

ij

j

r

r

ji

ik

i

ikik RRgRgR....

)(

;

)( 22

1+=

(48b)

Remark 2

Consider a uniform charge density. Again, our resulting field equation (45) reads

( )( )srk

s

kr

r

s

ikki

k

k

ikik FFFgJFRgR;.;..

)(

.

;

)(

4

1

4

1

2

1+=

where we have set ( )( )s rks

kr

r

s

iki FFFg;.;..

)(

4

1= . Recall the Lorentz equation of

motion:ki

k

i

uFeDs

Ducm

.

2

0= . Setting

=e

4

1, we obtain

= ik

ikik

i

RgRDs

Ducm

;

)(2

02

11

or

[ ]

=

rs

srk

ik

k

ikik

i

RgRgRDsDucm .)(

;

)(20 2

211

In the absence of charge density (when the torsion tensor is zero), i.e., in the limit

, we get the usual geodesic equation of motion of general relativity:

02

2

=

+ds

dx

ds

dx

jk

i

ds

xd kji


37/85

37

The field equation can always be brought into the form

= TgTkR ikikik )(2

1

Here the energy-momentum tensor need not always vanish outside the world-tube and

in general we can write iikk

gT2

1;

= . Now the Einstein tensor is

ikrs

rsiklkilikikik TFFgFFRgRG === )(.)(

81

41

21

The right-hand side stands more appropriately as the field strength rather than the

classical conservative source term asik

ik FFT2

1= (again, is a coupling

constant). The equation of gravoelectrodynamics can immediately be written in the form

( ) ( )

( )rs

rsik

l

kilik

ik

rs

rsik

l

kilikik

FFgFFT

TFFgFFRgR

4

1

4

1

4

1

4

1

.

.

=

=

=

as expected. In this field equation, as can be seen, the material density arises directly

from electromagnetic interaction.

__________________________________________________________________________________

The (sub-)spaces nRn = 4 and mV n = 6 . The n covariant

derivative

We now consider the space nRn = 4 IR5, a sub-space of IR5with basis

{ }A satisfying ( )ng ,iA = . Therefore this basis spans a special coordinate system inIR5. We define the n covariant derivative to be a projective derivative which acts

upon an arbitrary vector field of the form ( ) ,i= or, more generally, upon an


38/85


39/85


40/85

40

line. Therefore it is justified that forms a fundamental constant of Nature in the sense

of a correct parameterization. We also have 00 ==

ds

md

ds

ed. 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 pS :

ki

iki

i ugu gv][

4 = (52)

which has been defined in Section 2. This spin (rotation) vector is analogous to the

ordinary velocity vector in the spin space representation. For the moment, let

( )k

ikiiugvv

][;, == v where nv += i

iv . Therefore we see that

n

nn

nnggv

++

=

+++=

+++=

kl

jkljikj

iki

j

jjji

i

i

i

j

jjji

i

i

i

jj

vFgFgv

vv

uu

.][,

][

;

;;;;

;;;;

2

1

2

1

(53)

with the help of (7). If the law of parallel transport 04

= uu

applies for the velocity

field u , it is intuitive that in the same manner it must also apply to the spin field v :

04

= vu

(54)

This states that spin is geometrically conserved. We then get

02

1

021

.][,

][

;

=

+

=

jkl

jklj

j

kj

iki

j

uvFg

uFgv

(55a),(55b)

which are completely equivalent to the equations of motion in (51a) and (51b).


41/85

41

Lets also observe that

ik

k

ik

kGG

;|= (56a)

jkiijkkijjkiijkkij FFFFFF ;;;||| ++=++ (56b)

i.e., the vertical-horizontal n covariant derivative operator when applied to theEinstein tensor and the electromagnetic field tensor equals the ordinary covariantderivative operator. We shall be able to prove this statement. First

i

j

kk

j

iik

j

ik

j

ik

j FYFXGGG ..;|; 2

1

2

1

= whereiii

YGX ==5

(due to the

symmetry of the tensorikG , 55 ii RG = ), so that ik

kik

k

ik

kFYGG

.;|2

1= . Now the

five-dimensional curvature tensors (the Riemann and Ricci tensors) in n are

C

EB

E

AC

C

EC

E

AB

C

BAC

C

CABAB

A

ED

E

BC

A

EC

E

BD

A

DBC

A

CBD

A

BCD

R

R

+=

+=

,,

,,.

In this special coordinate system we have

( )

( )

( ) kk

j

jj

j

k

k

k

kk

k

k

ik

k

iii

k

ij

j

i

k

i

kk

i

FF

F

FF

5..5

55

5

5.

5

5

..5

02

1

2

1

02

1

2

1

2

1

=====

=====

===

ggng

gnnn

ggng

( )

( )

( ) 02

1

22

1

21

.][;.

..,.

55,5,55

=+=

+=

+=

+=

ii

k

l

l

ik

k

ki

lk

kli

li

klk

kki

k

li

l

k

l

kl

k

i

k

ik

k

kii

JJ

FF

FFF

R

with the help of(35d). Henceik

k

ik

kGG

;|= . (56b) can also be easily proven this way.


42/85

42

As a br ief digression, consider a six-dimensional manifold mVn = 6 where m is the

second normal coordinate with respect to 4R . Let[ ]ii g 5 , 5a , 6b ,

k

ii

k5

.,

k

ii

k6

., [ ]56g and [ ]ii g6 . Casting (12) and (22) into six

dimensions, the electromagnetic field tensor can be written in terms of the fundamental

spin tensor as the following equivalent expressions:

[ ] [ ]( ) ( )( )

[ ] ( )( )ikkirikr

ikkiikrkir

r

ik

g

ggF

,,,

,,,,

4

5

4

5

=

=

(57a)

) )

[ ] [ ]( )

[ ] [ ]( )m

ikm

m

kim

m

ikm

m

kimkmim

l

ikkiikkiikllikkli

l

ik

ggb

ggali

mg

lk

mg

bagggF

..

..][][

,,,,,][,][,][

2

22

+=

(57b)

Since the basis in this space is given by ( )m,ngg ,i= , the fundamental tensors are

[ ] [ ][ ]

=

=

0

0,

100

010

0044

][

44)(

)(

i

ii

ixikxikg

g

g

g

(58a),(58b)

__________________________________________________________________________________

Remark 3 (on the modified Maxwell's equations)

Using (49) we can now generalize Maxwell's field equations through the new extended

electromagnetic tensor ikkiikF || = where

[ ] [ ] ll

ikikl

l

ikikkiikkioldik

FF +=+== 22,,;;)(


43/85

43

where ikkiki F2

1;|

= . Now =5 is taken to be an extra scalar potential.

Therefore ikikkiik FF = ,, or

( ) ( )ikkiik

F,,

11 +=

ikki ,, (a)

For instance, the first pair of Maxwell's equations can therefore be generalized into

( )

+=

rr

r

t

A

c

E1

11

(b1)

( ) AxBrrr

+= 11 (b2)

( ) ( ) ( ) AxAxBdivrrrrrrr

+++= 12 11 (b3)

( )( ) ( ) ( ) 111 11111 +

+

=+ t

B

cB

tcEcurl

rrr

(b4)

where Ar

is the three-dimensional electromagnetic vector potential: ( )aAA =r

, is the

electromagnetic scalar potential, E

r

is the electric field,

r

is the magnetic field and

r

is

the three-dimensional (curvilinear) gradient operator and 42 = . Here we

define the electric and magnetic fields in such a way that

( ) ( )3

112141,1 BFEF

aa +=+=

( ) ( ) 1123

2

1311,1 BFBF

+=+=

We also note that in (b3) the divergence Axrrr

is in general non-vanishing whentorsion is present in the three-dimensional curved sub-space. Direct calculation gives

[ ]( )

( ) [ ] dd

ab

ab

cc

dc

d

abd

d

cab

abc

gradcurl

AARAcurldiv

,..

;. 22

1

=

=r

So the magnetic charge with density m in the infinitesimal volume d is given by

[ ]( ) = dAAR dcd

abd

d

cab

abc

;.2

2

1


44/85

44

4.Dynamics in the microscopic limit. Spin-curvature tensor of pS .Wave equation describing the geometry of

4R

We now investigate the microscopic dynamics of our theory. Let's introduce an

infinitesimal coordinate transformation into n through the diffeomorphism

iii xx +='

with an external Killing-like vector ( ) )(),(:, xxiii === (not to beconfused with the internal Killing vector which describes the internal symmetry of aparticular configuration of space-time or which maps a particular space-time onto itself).The function here shall play the role of the amplitude of the quantum mechanical

state vector . Recall that 4R represents the four-dimensional physical world and nis

a microscopic dimension. In its standard form ( ) )),,(()/2( zyxtEhieCx rp is thequantum mechanical scalar wave function; his the Planck constant,E is energy andp is

the three-momentum.Define the extension of the space-time nRn = 4 by

)(

2

1ijij

gD= (59)

We would like to express the most general symmetry, first, of the structure of n and

then find out what sort of symmetry (expressed in terms of the Killing-like vector) isrequired to describe the non-local statics or non-deformability of the structure ofthe metric tensor g (the lattice arrangement). Our exterior derivative is defined as the

variation of an arbitrary quantity with respect to the external field . Unlike the ordinaryKilling vector which maps a space-time onto itself, the external field and hence also the

derivative )( ijgD map 4R onto, say, 4'R which possesses a deformed metrical

structure ofg 4R , g' . We calculate the change in the metric tensor with respect tothe

external field, according to the scheme ( ) )( ''' )()( jiijjiij gg gggg == , as follows:

iiiijijiij DDDgD ==

+

= ggggggg '

)(;

The Lie derivative D denotes the exterior change with respect to the infinitesimal

exterior field, i.e., it dynamically measures the deformation of the geometry of the space-

time 4R .


45/85

45

Thus

jijiijgD gg += )(

where

ng

++

= k

kiik

k

i

k

iiFF

2

1

2

1,.;

By direct calculation, we thus obtain

ijjiijgD

;;)( += (60)

As an interesting feature, we point out that the change )(ijgD in the structure of the

four-dimensional metric tensor does not involve the wave function . The space-time

4R will be called static ifg does not change with respect to . The four-dimensional

(but not the five-dimensional) metric is therefore static whenever 0;; =+ ijji .

Now let 4R be an infinitesimal copy of 4R . To arrive at the lattice picture, let also

...,,, ""4

'"

4

"

4RRR be n such copies of

4R . Imagine the space n consisting of these

copies. This space is therefore populated by4R and its copies. If we assume that each

of the copies of4R has the same metric tensor as 4R , then we may have

( ),0=

We shall call this particular fundamental symmetry normal symmetry or spherical world-symmetry. Then the ncopies of4R exist simultaneously and each history is independent

of the four-dimensional external field 4 and is dependent on the wave function only.In other words, the special lattice arrangement ( ),0= gives us a condition for

4R and its copies to co-exist simultaneously independently of how the four-dimensional

external field deforms their interior metrical structure. Thus the many sub-manifolds

n4R represent many simultaneous realities which we call world-pictures. More

specifically, at one point in 4R , there may at least exist two world-pictures. In other


46/85


47/85


48/85

48

We can also calculate the exterior variation of the electromagnetic field tensorik

F :

( )

( ) ( )[ ]( ) ( )

( ) ( )

kj

j

ikj

j

iki

kiki

kiki

kiik

FF

D

DD

DFD

=

+=

+=

=

gggSS

gngnS

gngn

gn

..2

1,22

2,2

22

2

(65a)

Hence we can write

( ) [ ]( )( )

( ) ( )

++=

=

=

jk

j

ijk

j

iik

kj

j

i

l

klj

j

iki

kkj

j

ikiik

XFgDFH

FgDF

FFD

.)(.

.)(.

.

2

12

2

12

,2

XgggS

gSgS

(65b)

where we have just defined the spin-curvature tensor ikH :

( )kiik

H = S (66)

which measures the internal change of the spin in the direction of the spin basis. Wefurther posit that the spin-curvature tensor satisfies the supplementary identities (which

are deduced from the conditions 04 = uu and 04 = Su )

0=iik uH (67a)

0=Htr (67b)

The transverse condition (67a) reproduces the Lorentz equation of motion while (67b)

describes the internal properties of the structure of physical fields which corresponds to

the quantum limit on our manifold (for details see Appendix B).


49/85

49

The explicit expression ofik

H can be found to be

( )kiik

H = S

[ ] ikl

iklSFg

;.2

1= (68)

from (64a) and (64b). Furthermore, still with the help of(64a) and (64b), we obtain, after

some simplifications,

( )lkkl

l

iikikFH

;;.;;4

1 += ( )

il

l

klk

l

iFFF

;.. 2

1

4

1 + (69)

However,we recall that

( )jkiljlikijklFFFFR =

4

1,

l

kilikFFR

.4

1=

and therefore we obtain the spin-curvature relation in the form

( )lkkl

l

iikikikFRH

;;.;;4

1 += ( )

il

l

kF ;.2

1+ (70)

If we contract (70) with respect to the indices iand k, we have

H = ( R ) ii

ki

ik JF 2

1; ++ (71)

where is the covariant four-dimensional Laplacian, again, R is the curvature scalar

andiJ is the current density vector. However, using(67a) and (67b) and associating

with the space nRn = 4 the fundamental world-symmetry ( ),0= , then weobtain, from (70), the equation of motion

ikikik RH +=;; (72)


50/85

50

From (67a), (67b) and (72), we obtain the wave equation

( R ) 0= (73a)

This resembles the scalar Klein-Gordon wave equation except that we have the

curvature scalar R in place of ( )20

2 / hcmM = (we normally expect this in generalizingthe scalar Klein-Gordon equation). Note also that the ordinary Klein-Gordon and Diracequations do not explicitly contain any electromagnetic terms. This means that the

electromagnetic field must somehow already be incorporated into gravity in terms ofM . Since is just the amplitude of the state vector , we can also write

( R ) 0= (73b)

If the curvature scalar vanishes, there is no source (or actually, no electromagnetic

field strength) and we have 0= which is the wave equation of masslessparticles.

__________________________________________________________________________________

Remark 4

Recall (65):

++= jk

j

ijk

j

iikik XFgDFHFD .)(.2

12 (a)

where

( )kiik

ikikkiik

H

gD

=

=+=

S

2;;)(

Meanwhile, for an arbitrary tensor field T , we have in general


51/85

51

...

...

;

...

...;

...

...

;

...

...;

...

...

...

;...

...

...

+++=

j

m

im

kl

i

m

mj

kl

m

l

ij

km

m

k

ij

ml

ij

mkl

mij

kl

TT

TTTTD

Therefore

l

kil

l

ilklik

l

ikFFFFD

;;; ++= (b)

Comparing this with (a),we have, for the spin-curvature tensor ikH ,

( ) ( )lk

l

ilkkl

l

i

l

kil

l

ilklik

l

ikXFFFFFH

.;;.;;;2

1

4

1

2

1+++= (c)

in terms of the electromagnetic field tensor.

__________________________________________________________________________________

Finally, lets define the following tensor:

ik

k

jjiijFSA

|.;2

1 (74)

where, as before,

kiikikF

2

1;

=

is the n covariant derivative of k , the notion of which we have developed in Section3 of this work, and

k

k

iiiFS .,

2

1=

is the spin vector (64a).The meaning of the tensor (74) will become clear soon. It has noclassical analogue. We are now in a position to decompose (74) into its symmetric andalternating parts. The symmetric part of(74):


52/85

52

+=ik

k

jjk

k

iijjiijFFSSA

|.|.;;)(2

1

2

1

2

1 (75a)

may be interpreted as the tension of the spin field.

Now its alternating part:

[ ]

+=ik

k

jjk

k

iijjiijFFSSA

|.|.;;2

1

2

1

2

1 (75b)

represents a non-linear spin field. (However, this becomes linear when we invoke the

fundamental world-symmetry ( ),0= .) If we employ this fundamental world-

symmetry, (75a) and (75b) become

( )ijijjiij

RSSA ++= ;;)(2

1(76a)

[ ] ( )ijjiij SSA ;;21 = (76b)

With the help of(64a) and the relation

[ ] rr

ikikki ,;;;;2 =

(76b) can also be written

[ ] [ ] kk

ijijA ,= (77)

Now

( ) ki

k

jik

k

jjk

k

ijiijFFFFA

.;.;.;; 4

1

2

1

2

1+=

( )

++=ik

k

jjk

k

iijjiFFR

;.;.;; 2

1

2

1


53/85

53

Therefore the Ricci tensor can be expressed as

( )

++=

ir

r

kkr

r

ikikiir

r

kikFFSFR

;.;.;;;

1

|.

1

2

1

2

1

2

1 (78)

We can still obtain another form of the wave equation of our quantum gravity theory.

Taking the world-symmetry ( ),0= , we have, from nrr = ' ,

r

r

iiiiF gnhg

.,

2

1 += (79)

where ii ,'rh is the basis of the space-time 4'R . Now the metric tensor of the space-time 4R is

( )

s

k

r

irskr

r

i

kikiir

r

kikik

kiik

FFgF

Fh

g

..)(

2

.

,,,.,

)(

4

1

2

1

2

1

.

++

++=

= gg

where ( )kiik

h hh is the metric tensor of 4'R , )nh ii ( and ( )kiik gh .Direct calculation shows that

ikikik

ii

Fg

2

1)(

,

+=

=

Then we arrive at the relation

s

k

r

irskiikikFFghg

..)(

2

,,)(4

1 = (80)

Now from (60) we find that this is subject to the condition

0)(

=ij

gD (81)


54/85

54

Hence we obtain the wave equation

s

k

r

irski FFg ..)(2

,,4

1 = (82a)

Expressed in terms of the Ricci tensor, the equivalent form of(82a) is

ikkiR2

,, = (82b)

Expressed in terms of the Einstein tensor RgRG ikikik )(2

1

= , (82b) becomes

iksr

rs

ik

s

k

r

iGgg

2

,,

)(

)(2

1 =

(83)

If in particular the space-time 4R has a constant sectional curvature, then

( ))()()()(

12

1jkiljlikijkl

ggggR = and )(ikik gR = , where R4

1 is constant,

so (82b) reduces to

)(

2

,, ikki g = (84)

Any axisymmetric solution of (84) would then yield equations that could readily beintegrated, giving the wave function in a relatively simple form. Multiplying now(82b) by

the contravariant metric tensor)(ikg , we have the wave equation in terms of the

curvature scalar as follows:

Rg kiik 2

,,

)( = (85)

Finally, lets consider a special case. In the absence of the scalar source, i.e., in void,the wave equation becomes

0,,

)( =ki

ikg (86)

This wave equation therefore describes a massless, null electromagnetic field where

0=ikikFF


55/85

55

In this case the electromagnetic field tensor is a null bivector. Therefore, according toour theory, there are indeed seemingly void regions in the Universe that are governedby null electromagnetic fields only.


56/85

56

5. Conclusion

We have shown that gravity and electromagnetism are intertwined in a very naturalmanner, both ensuing from the melting of the same underlying space-time geometry.They obey the same set of field equations. However, there are actually no objectivelyexisting elementary particles in this theory. Based on the wave equation (73), we maysuggest that what we perceive as particles are only singularities which may be interpretedas wave centers. In the microcosmos everything is essentially a wave function that alsocontains particle properties. Individual wave function is a fragment of the universal wavefunction represented by the wave function of the Universe in (73). Therefore all objectsare essentially interconnected. We have seen that the electric(-magnetic) charge is none

other than the torsion of space-time. This charge can also be described by the wavefunction alone. This doesnt seem to be contradictory evidence if we realize that nothingexists in the quantum realm save the quantum mechanical wave function (unfortunately,we have not made it possible here to carry a detailed elaboration on this statement).Although we have not approached and constructed a quantum theory of gravity in thestrictly formal way (through the canonical quantization procedure), internal consistencyof our theory awaits further justification. For a few more details of the underlyingunifying features of our theory, see the Appendices.


57/85

57

Appendix A: Embedding of generalized Riemannian manifolds (withtorsion) in N= n+ pdimensions

In connection to Section 3 of this work where we considered an embedding of class 1,we outline the most general formulation of embedding theory of class p inN = n + pdimensions where n now is the number of dimensions of the embedded Riemannian

manifold. First, let the embedding space NR be anN-dimensional Riemannian manifold

spanned by the basis { }Ae . For the sake of generality we take NR to be an N-dimensional space-time. Let also nR be an n-dimensional Riemannian sub-manifold

(possessing torsion) in NR spanned by the basis lg where now the capital Latin

indices A,B, run from 1 to N and the ordinary ones i,j, from 1 to n. If now

( )BAAB

g ee = and jiijg gg = denote the metric tensors of NR and nR ,

respectively, and if we introduce the p-unit normal vectors (also called n-legs))(

n (where the Greek indices run from 1 to p and summation over any repeated Greekindices is explicitly indicated otherwise there is no summation ), then

+=

=

+=

+

++=

+=

=

=+=

===

+=

=

A

ij

C

j

B

i

A

BC

k

ij

A

k

A

ji

A

ij

A

ji

C

j

B

i

A

BCA

A

jiAij

BjC

Ai

kikjCB

A

Aj

C

i

Bij

j

C

i

B

k

ij

A

k

i

CB

A

i

A

BC

C

j

B

i

A

BC

k

A

A

ji

k

A

k

ij

C

C

ABBA

ijjiijk

k

ijji

AAi

BAij

j

B

i

AAB

AB

B

j

A

iij

neeee

ne

eenen

neegnn

neeeeeee

eeeee

ne

nngeeg

geeg

)()()(

,

)()()(

;

)(

,

)()(

)()()()(,

)(

)()()(

,

,

,

)()()(

;

)()()(

,

)()()()(

)(

)()(

,

1,)(

0

ee

ngngg

nn


58/85

58

Now since +=

)()()(nge Ai

i

AA ne and)(

;

)(

jA

A

iijne= for the

asymmetricp-extrinsiccurvatures,we see that

+=

=

)()()(

;

)()(

;

)(

;

)()()()(

B

A

jAjB

jAB

AA

Bij

i

B

nnnn

nnne

Lets define the p-torsion vectors by

ii

A

iAinn

=

= )()(;

Hence

( )

Ai

k

AkiiAnen += )()()( ;

or

+=

)()()()(

; ngn ik

kii

Now

+

+=

,

)()(

)()(

;

)()()()()()(

;;

Aki

r

ArkiAkiAskri

rsi

AikkiA

n

enngen

Hence we obtain the expression

kikiskri

rsA

kiAgnn ++= )(;)()()()( ;;


59/85

59

Meanwhile,

( )

B

ikBA

C

k

B

iCBA

A

kiBA

C

k

B

iCBA

C

kC

B

iBAkiA

B

iBAiA

nneen

eneen

eenn

enn

)()()()(

;

)(

;;

;

)(

;

)(

;;

;

)(

;

)(

;;

)(

;

)(

;

+=+=

=

=

but

i

B

k

Aki

i

BiABAeeenn

)()(

;

)(

;

==

Hence

0

0

)()(

;

)()(

;

=

=A

BA

B

BA

nn

nn

We also see that

C

k

B

i

A

CBA

A

kiAeennnn )()(

;;

)()(

;;

=

Consequently, we have

( ) ( )D

k

C

i

BA

ABCD

C

k

B

i

A

BCACBA

A

ikAkiA

eennR

eennnnnn

)()(

)()(

;;

)(

;;

)()(

;;

)(

;;

=

=

On the other hand, we see that

( )

( ) Dk

C

i

BA

ABCDikki

skrisirk

rs

ikki

A

ikAkiA

eennR

gnnn

)()()(

)()()()(

;;

)()(

;;

)(

;;

++

+=


60/85

60

Combining the last two equations we get the Ricci equations:

( ) ( ) =

ikkisirkskri

rs

ikkig

)()()()()(

;;

Now from the relation

+=

)()()(,

nggijk

k

ijji

we obtain the expression

)()()()(

,

)()()(

)()()(

,,

n

gg

+++

+=

kijkij

r

ijrk

rskij

rsr

sk

s

ij

r

kijjkig

Hence consequently,

( )

[ ]( )

( )

+

++

+=

,

)()()()(

)()()(

;

)(

;

)(

)()()()()(

.,,

2

n

n

ggg

jikkij

ir

r

jkjikkij

rijskiksj

rsr

ijkkjijki gR

On the other hand, we have

( )

( ) ACkjBiABCCjBkiABCCkBjiABCAjkiA

D

k

C

j

B

i

A

ED

E

BC

A

DBCjki

eeeeeee

eee

e

eg

,,,,

,,

++++

+=

( )A

A

ijk

D

k

C

j

B

i

A

BCDkjijkiSeeeR egg

..,,+=

where, just as in Section 3,

( ) Bi

A

BC

C

jk

C

kj

A

kji

A

jki

A

a spin-curvature theory of quantum gravoelectrodynamics

Documents