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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
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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
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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
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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.
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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
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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)
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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
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 ==
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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)
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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)
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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 += ][][,][
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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+=
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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.
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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
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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.
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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)
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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)
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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)
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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.
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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).
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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)
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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
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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
.
==
+=
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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
++
+=
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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.
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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.
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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.
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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)
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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)
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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.
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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.
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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)
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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
=
=
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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
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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
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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).
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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.
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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,,;;)(
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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
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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 .
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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
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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).
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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)
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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
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...
...
;
...
...;
...
...
;
...
...;
...
...
...
;...
...
...
+++=
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):
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+=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
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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)
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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
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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.
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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.
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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
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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 ++= )(;)()()()( ;;
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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
)()()(
)()()()(
;;
)()(
;;
)(
;;
++
+=
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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