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 Contributions to the EinsteinKursunoglu field equations.
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 Pizzo, Joseph Francis, 1939
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 1964
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 English
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Curvature ( jstor ) Differential equations ( jstor ) Electric fields ( jstor ) Electrodynamics ( jstor ) Electromagnetic fields ( jstor ) Gravitational fields ( jstor ) Magnetic fields ( jstor ) Mathematical vectors ( jstor ) Tensors ( jstor )
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 University of Florida
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Full Text 
CONTRIBUTIONS TO THE
EINSTEINKURSUNOGLU FIELD EQUATIONS
By
JOSEPH FRANCIS PIZZO JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1964
To Paula
ACKNOWLEDGMENTS
The author extends his gratitude to the members of his committee. In particular, he is deeply grateful to Dr. J. Kronsbein, who suggested this problem and has given much of his time in helping to bring about the solution.
The author would also like to thank both his wife, Paula for her help and suggestions in the preparation of the first draft, and Miss Nana Royer for the typing of the final copy.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . .. .. l l
CHAPTER
1. INTRODUCTION . . . . . . . . . .
2. CURVATURE, DISPLACEMENT AND FIELD EQUATIONS .... 4 3. REDUCTION AND SOLUTION . . .... . .. . 20
4. SPECIAL SOLUTIONS ......... .. .. . 30
Minkowski Space ....... .......... 30
A More General Case .. ... ... .. .. 33
5. SUMMARY ... .... ..... ..... ... 51
BIBLIOGRAPHY . . . . . . . . . . . 54
BIOGRAPHICAL SKETCH .................. 56
Iv
CHAPTER I
INTRODUCTION
In 1916, Einstein's classic paper, in which he formulated the general theory of relativity, appeared in Annalen der Physik. [1]1 In this paper he was able to describe the phenomena of gravitation in terms of geometrical concepts. The field equations of gravitation were shown to be derivable from a variational principle, using a symmetric, second rank, covariant tensor. This tensor (which we will call the fundamental tensor, and will denote by gik) represents the gravitational potentials.
The theory was beautiful, and moreover it worked! Einstein was
still not satisfied. He reasoned that there are other fields in Nature besides gravitation. How do they fit into this picture? An example is the electromagnetic field and equations. There is no natural way for them to be included. To express the covariant electrodynamic equations, Maxwell's equations are written, then covariant derivatives are taken in place of ordinary partial derivatives. In other words, electrodynamics must be introduced separately. This type of inclusion has been considered arbitrary and unsatisfactory by many theoreticians. Indeed, Einstein himself has stated, "A theory in which the gravitational field and the electromagnetic field do not enter as logically distinct
i. The numbers in square brackets refer to the bibliography at the end of the dissertation.
I
2
structures would be much preferable." (See [7], page 115) This then is the aim of Einstein's unified field theory: to derive all fields from one, single nonsymmetric tensor.
Einstein formulated the unified field theory according to the
same pattern he had used for his theory of gravitation, with the exception that now he did not require the fundamental tensor to be symmetric. One of the resultant field equations turns out to be a set of sixtyfour algebraic equations in sixtyfour unknown functions of the gik' These equations have to be solved before the other three field laws (second order partial differential equations) can be set up. This problem occupied much of Einstein's later years, and although he found a solution to the sixtyfour equations, it was in his own expressed opinion, too complicated to be of any further use.
This is essentially where the problem has stood until this time. There has been no verification of Einstein's hypothesis, that other fields are included in the fundamental tensor, because the differential equations have never been solved. In this dissertation, we are able to show that, by a special restriction on the fundamental tensor, the unified field equations become those of gravitation and electrodynamics, while the components of gik represent the gravitational and electromagnetic fields. That which was introduced artificially before, now comes about naturally from one single tensor, in this special case.
The geometrical concepts upon which the unified field theory is
based, are developed in the second chapter. Also the different versions of the theory (Einstein's, Schroedinger's, and Kursunoglu's) are discussed here. The correspondence of all of the theories to the gravitational equations of general relativity is shown for limiting cases.
3
Chapter three embodies our version of the theory. The sixtyfour equations are shown to be reducible to twentyfour, which still cannot be solved in a useful form. Nevertheless, they suggest a modification in the fundamental tensor which allows a tractable form for the solution of the equations to be obtained. The three sets of differential equations, referred to above, are constructed. One set is the same as the gravitational equations. The next is seen to be the first set of covariant electrodynamic equations in the absence of charges and currents. The last is actually a set of third order partial differential equations and needs further investigation. It turns out we are able to solve this problem in our version of the theory.
In chapter four two cases are considered. First, the equations are examined when the radius of the Einstein universe is taken infinite, in which case the gravitational equations are satisfied identically. The other two sets of equations are shown to be Maxwell's equations in the absence of matter. Next we take the more general case of a finite radius. The first two sets of differential equations were investigated before. Now, a term by term examination reveals that the last field law implies the second set of covariant electrodynamic equations. The last part of this chapter is devoted to an exact solution of these equations in a special coordinate system.
Throughout this dissertation an elementary understanding of tensor algebra and a basic knowledge of general relativity are assumed.
CHAPTER 2
CURVATURE, DISPLACEMENT, AND FIELD EQUATIONS
There are three varieties of unified field theory: Einstein's,
Schroedinger's, and Kursunoglu's. Throughout each of these, two fundamental entities are dominant; the displacement field and the curvature tensor. These concepts require some elaboration before they are used to derive the field equations.
In relativity, we deal with quantities known as tensors, which have an appealing property; a tensor equation remains unchanged, regardless of the coordinate system in which it is expressed. That is, tensor equations are covariant. When the laws of physics are written as tensor equations, all reference frames are treated equally. The idea of a preferred system no longer exists. This formulation breaks down when we try to compare vectors at infinitesimally separated points.
ii
Consider, for example, the vector A at the point x and the vector A + dA at the point x + dx The difference between the two is
(A + dA ) A' = dA (2.1) To illustrate the problem that has now arisen, suppose the vectors are equal in a particular coordinate system.2
dA = 0 = A,j dxj. (2.2)
2. The comma indicates ordinary partial derivatives.
4
5
Requirements of covariance would demand that this relation be true in any coordinate system. 'Upon transformation dA () dx (2.3)
Since A is a vector, use can be made of its transformation law.
b
i 8 & k bdx
dA = ( A) (2.4) bx 2x k
i k x A b dx b (2.5) k 3xbixk
The vanishing of the second term would insure the equality of the two vectors to be a covariant equation. Yet, in general, this is not so. An alternate way to require covariance of equation (2.2) is just to say that the difference between two vectors should transform like a vector.
The difficulty in (2.5) is due to the fact that the vectors were compared at different points. It becomes necessary to find some prescription for translating vectors so they may be compared at the same point. This method is called parallel translation, and is accomplished in such a manner as to make the equality of vectors a covariant relation. That is, the difference between the vectors, when compared at the same point, will be a vector.
The vector A translated parallel from x to x + dx will be denoted by A + B'A Then at x + dx the difference between the two vectors will be
DA = (Ai + dAi) (A + Ai) = dAi A .Ai (2.6)
6
The expression BAi will depend on the vector Ai and dxi and can be represented by
8A = st As dxt (2.7)
where theV st is called the Displacement Field. Its components,
i
which are to be determined, are functions of the x. In this form,
DAi = [A 't + st As] dxt (2.8)
The term in brackets is denoted by a special symbol,
i i i
A jt = A ,t + st (2.9)
and is called a covariant derivative.
The difference, DA between the two vectors at the same point
is to be a vector itself. This requires the covariant derivative to be a tensor. Therefore the transformation law for the displacement field must be
 a
i I b c a I 2xa
(2.10) st a, s $t bc a sa S t
It is clear that ordinary derivatives do not form a tensor and covariant derivatives must be taken instead.
Before the displacement field can be determined beyond its transformation law, (2.10), we must know more about the distribution of matter and charges which dictate the structure of the space.
It is clear that the displacement field is somehow related to the curvature of the space. So far, in this presentation, the notion of curvature has been a vague one, at best. Using the idea of parallel displacement, a mathematical picture of curvature may be displayed.
7
Let a vector, A be transferred parallel to itself along the boundary of an infinitesimal surface element and brought back to its starting point, x(s). We propose to compute the change, 4A, in the vector after one complete circuit. This change is given by
AA= (2.11)
Expression (2.7) is used for BA The quantities r st and As are expanded about their values at the initial point, and terms kept to first order, since the curve is infinitely small. The expansions are
AS(xJ) = As J(s) pq (s) xq(s)) (2.12)
and
st(xl) st x(S) s x(s) (x (s)) s (2.13)
Let
t t t (2.14)
(s) (2.14)
The change in the vector is now written
dA = i st,q As qd t + r stp pq Apjdt (2.15) With an appropriate renaming of dummy indices this becomes
i i s
Ai = [ pt,q r st p APlqd (2.16) It is understood that the terms preceding the integral are the initial values. As usual, we are interested in the displacement field and want to investigate the term in brackets to determine tensor character. It
8
can be shown that the integral transforms as a tensor. Moreover, it is an antisymmetric tensor since
f llqdt = d(At) ItdT (2.17) and the first term on the right hand side vanishes, because it is an exact differential. The integral will be denoted by3
ft = 9Tlqdt (2.18) In equation (2.16), the quantity in parentheses can be expressed as a sum of parts which are symmetric and antisymmetric on the indices t and q. The contraction of the symmetric part with the integral vanishes leaving
i i i s i s
AA' pt,q pq,t sti pq sqF pt) Apd. (2.19) It is known that AA' is a vector since it is obtained by parallel displacement. The term APftq is a third rank tensor, therefore the tensor character of
ptq = i pt,q + pq,t + st pq sq pt (2.20) is established. This is the wellknown RiemannChristoffel tensor. R ptq is also referred to as the curvature tensor, because whenever it vanishes, there is no change in a vector after it is displaced parallel to itself around a complete circuit. The space is then said to have no curvature, or to be flat.
Contraction on the indices i and q gives the second rank tensor,
3. The hook under the indices is used to indicate antisymmetry.
9
Rik = Pik,p ip,k iq pk Pqpl ik (2.21) which is named the Ricci tensor, or the contracted curvature tensor. It is to play a central role in the variational principle from which the field equations are to be derived.
For a variational method, we postulate an invariant space integral which involves the displacement field. Invariance of this integral presupposes the existence of a scalar density, L, which can be formed by the contraction of the Ricci tensor with a contravariant tensor density,j ik. The field laws are to be derived from
6fLdv = 0 (2.22) where d' is an element of fourvolume and
L = ik Rik (2.23) is considered as a function of the ik and bik, which are to be varied independently, their variations vanishing on the boundary of integration.
This postulate would lead to the equations of the purely gravitational field if the condition of symmetry were imposed on j ik and
b
ik. (This will be demonstrated in the next chapter However, the present theory is an attempt to generalize these equations, and the constructions which were essential for the setting up of covariant
4. A scalar density transforms like a scalar but with the inverse of the Jacobian determinant included. This is necessary to cancel the Jacobian which results from the transformation of the element of fourvolume. The result is that the integral transforms like an invariant.
10
equations are independent of the assumption of symmetry! Instead of symmetry, an analogous condition is posited. It is referred to as Transposition Invariance. Even though I ik and b ik are nonsymmetric, their transformation laws would be invariant if the indices of these variables were to be transposed and then the free indices interchanged. This is what is meant by transposition invariance. In addition to the variational principle, it will be postulated that all field laws shall be transposition invariant.5
If we proceed to substitute (2.21) for Rik into equation (2.23)
and carry out the variation indicated in (2.22), we would find that the resulting equations are not transposition invariant. To circumvent such a distasteful result, four new arbitrary variables, Xk, are introduced by making a formal change in the description of the field. In
b
place of P ik' the following substitution is made b *b
ik = ik + 6bi k (2.24)
*b b
The r ik is a displacement field just as the F ik* The &k are treated in the variational procedure as independent variables. After the
5. An example of an equation which is transposition invariant is
S s
9ik,b skr ib gisf bk= 0
Transposing the indices on the variables yields
s s
9ki,b gks bi si r kb 0
Interchanging the free indices (i and k) gives back the original equation
gik,b gis rbk gsk ib = 0
11
variation is carried out, the Xk will be chosen to make the field equations transposition invariant, and then they will be eliminated from the system.
b
The contracted curvature tensor, with r ik replaced by (2.24) becomes
Rik(r ) = Rik(r ) + (i,k Ik,) (2.21a) where Rik([F)is the same as expression (2.21), except that a star is
b
put over all the F ik The variational integral is now
6(ik[Rk( + (i,k k,i) dT = 0 (2.25)
b
After the variations in 1ik ik' and Xk are made, the integrand, denoted by I, takes on the following appearance
S= [Rik( + (i,k k,) ik 1jk 'k )61
+ iks it, k + t )it ts itk
Smt r mt6ks k k st6 p ik
ks k
+ is 61 si611 i i k + s6 iks (2.26)
The last term is a generalized divergence and since it appears in a fourvolume integral, it can be converted to an integral over the threemanifold enclosing this volume. The variations vanish on the boundary, hence the last term in equation (2.26) can be ignored. The variations in the remaining terms are all independent of each other so each coefficient must vanish separately, giving the three equations
Rik(F ) + (Lik Lk, ) = 0 (2.27)
12
i kk = 0 (2.28) ki .t k At ik + tk ts + st )i st 6ks(l i t
+ 1)mt mt) = (2.29)
The four extra variables, k, may be given any value. The three equations will be transposition invariant if Xk is chosen such that
ik = 0 (2.30)
V
Now Xk can be eliminated entirely from the equations by writing (2.27)
as two equations.6
R.ik =0 (2.31) Rikb + Rkb,i + Rbi,k = 0 (2.32) y V v V
b
The star may just as well be excluded from above the ik' renaming them.
By contraction on the indices k and s in (2.29), we can verify that the expression in parentheses vanishes and the equation can be written
ik tk ts + iikit = 0 (2.33) Before further simplification is made, it should be noted that the field
law, (2.28) is already implied in this expression. This can be seen from the following consideration. The equation formed by exchanging
free indices,
ki, + ti + .t ki st 0 (2.33a)
6. The dash under the indices indicates symmetry.
13
is just as valid. If both equations are contracted on the indices k and s and (2.33a) subtracted from (2.33), then clearly
(1)is, si,) = 0 = s (2.34) The covariant form of a tensor density is defined so that
=ipi 6 = pi si (2.35) Equation (2.33) is multiplied through by 'jik and the summation carried out to give
t ) iki,s (2.36) which is replaced in (2.33)
)ik I ik, ab tk it k (237)
,s 2 1) ab ) ,s +) ts +) st = 0 (2.37) The definition of a density,
i ,k l1/2 ik 1/2 ik
= g g = (det. gik) g (2.38) and the rule for the derivative of a determinant, ab ab
9,s = 9ab,s ggabg (239) are used to bring (2.36) into the final form, s s
gik,b gskp ib gisr bk = 0 (2.40)
This set of sixtyfour equations gives the relations for the sixtyfour
b b
ik in terms of the gik' and their partial derivatives. Once the P ik
14
are known, they are substituted in the other three field equations,7
k
ik = 0 (2.30) Rik = 0 (2.31)
R ik,b + Rkb,i + Rbi,k = 0 = R[ik,b] (2.32)
for a solution of gik. This is the formalism of Einstein's unified field theory [5].
Schroedinger's unified field theory is quite similar to that of Einstein's. Parallel transfer, displacement fields, and the curvature tensor all form the basis for the theory. The difference appears in the integrand of the variational principle. Whereas Einstein took )kRik as the.scalar density, Schroedinger chooses [181
2 1/2
L = k (det. Rik) (2.41) which is the simplest scalar density that can be built out of the curvature tensor. The constants 2 and X have no influence on the result. A contravariant tensor density'is introduced here also:
ik = k (2.42) cik
but it can be eliminated in the end, leaving equations which contain
b
only the P ik'
When the variation,
6fLdT = I 6RikdT = 0 (2.43) ik
7. The square brackets in (2.32) indicate summation over the cyclic permutations of the enclosed indices. This convention will be used hereafter.
15
is carried out with (2.41) as integrand, the result is the following set of equations,
s s
ik,b sk ib gisB bk =0 (2.44) Rik = Xgik (2.45)
These can be taken as the field laws as they stand, but Schroedinger takes the theory further. Equation (2.45) can be substituted in (2.44) to eliminate the gik. The field equations are now sixtyfour differential
b
equations involving nothing but the sixtyfour F ik s s
S S
R.ik,b R ib Ri bk = 0 (2.46) This is known as the "purely affine theory."
Kursunoglu's approach to the theory [16] retains the basic ideas of displacement field, curvature tensor, and the fundamental tensor gik (or gik) but the scalar density for the variational integral is quite different and entails some new concepts. These will be defined before the variational principle and the field equations are displayed.
The fundamental tensor, gik' is expressed in Kursunoglu's theory, in terms of its symmetric part, aik, and its antisymmetric part, lik'
gik = aik + (2) /2C2ro ik (2.47) where G is the gravitational constant and ro is a "fundamental length." The determinant, g, of gik is constructed in terms of what will be called the "Kursunoglu invariants," namely
1 = 1 kik (2.48)
2 ik
16
and
1 (a)1/2ij kb kb (2.49)
8 ij kb .
42 2842
g = a(l + 2Gc ro 02 4G c r o 2) (2.50) The action principle from which the equations are to be deduced is
6A[ ikRi 2r 2 b a)]d= 0 (2.51) where b is the determinant of the tensor, 1/2 1/2 4 2 s b. ik= a g (aik + 2Gc r is (2.52) The variations carried out in (2.51) produces the Kursunoglu field equations.
S s
gik,b gskF ib gis bk = 0 (2.53)
2
R. k =r 2(ai bk) (2.54)
2
R[ikb]= r 2(ikb + kbi + bik) (2.55)
ik,b o k,b kb,i bi,k
k =0 (2.56) The nomenclature, "fundamental length," is rationalized by examining the field equations. The vanishing of ro gives the general relativistic case in the absence of charges. This existence of free charges is now associated with a finite fundamental length.
All three field theories discussed here have at least one thing in common: they have been constructed so that there exists a correspondence principle which takes the unified field equations over the wellknown gravitational equations of general relativity when the antisymmetric field is absent. This is easy to see. First consider Einstein's
17
set of equations and require the gik to be composed of only a symmetric part, aik'
gik = aik = aki (2.57)
b
The solution of equation (2.40) for the V ik is readily found by permuting the indices to get three equations
S S
aik,b ask ib ais bk = 0 (2.40)
s s
akb. asb ki aks = 0 (2.5) abi,k asi bk abs i ki 0 (2.59) A combination of these three equations gives
s
akb,i + abk a ik,b = 2a sb ki (2.60)
Equation (2.35) applies here
b I abp
ki = 1a (akp i + apik a.ikp) (2.61)
b
The symmetry of the ki is a consequence of the symmetry of the aik
F ki = ik i bk (2.62) The last notation is a matter of convention. Equation (2.61) is the T'hristoffel symbol," where it is understood that only the symmetric part of the gik is used. The symmetric tensor, aik' is identified as the metric tensor of the space.
Since equation (2.30) is satisfied identically, the Xk may as well be chosen zero, in which case (2.31) and (2.32) could be recombined as
Rik k ,p + 1ip ,k P q q k (2.63)
18
This is precisely the gravitational field equation in empty space. (See, for example, [91, page 81) A particular solution to these equations corresponds to the field of an isolated particle continually at rest. The famous explanation of the discrepancy in the advance of Mercury's perihelion is a result of the solution to (2.63). There is no question about the physical significance of the gravitational field equations, so this gives a certain measure of confidence to the generalization, (2.30), (2.31), (2.32), and (2.40).
A similar situation exists with Schroedinger's theory in the limit of a symmetric gik The field equations, (2.45), become the same as Einstein's with the addition of a term involving X, which is now identified with the cosmological constant. Actually, the limiting case of Schroedinger's theory is the original form of the gravitational equations. The cosmological constant is so small that it need not be included on a scale such as our solar system. (See [91, p. 100)
In Kursunoglu's theory, the correspondence to general relativity
is achieved by the vanishing of the fundamental length. Again we revert to the equations of general relativity. This method is much like quantum mechanical correspondence to classical mechanics when Planck's constant vanishes.
Therefore all three theories have at least some basis, due to the
fact that they reduce in the lin ht to the wellknown and tested equations of gravitation derived by Einstein in his general theory of relativity. What has been presented so far is only the problem: a nonsymmetric, fundamental tensor, gik is chosen to represent the fields in Nature.
b
The displacement field, P ik' is introduced to insure that the field laws will not be dependent upon the choice of a coordinate system. From
19
a postulatory basis, a variational principle is used to derive a set of field equations. A solution to these equations will presumably give a description of nature in which all fields are united in the single tensor, gik' in the same manner as the electric and magnetic fields are unified in the electromagnetic tensor.
Now that the problem is laid before us, the next step is to provide a solution to these field equations.
CHAPTER 3
REDUCTION AND SOLUTION
It is evident that the three versions of unified field theory, which have been presented, are little more than postulates. Their tenuous claim to validity comes from the correspondence to the known equations of general relativity and the correct count of functions. So far there has been no indication that they give a field description of Nature in which all fields are united in a single tensor.
in keeping with this spirit, we choose to deal with yet another form of the equations, which can be considered as an adaptation of either Schroedinger's or Kursunoglu's equations to an Einstein model of the universe.
s s
ikb sk ib gisP bk = 0 (3.1)
2 i4 4 96 k
Rik = 6 (3.2)
0 otherwise
R= 9 (3.3) V V
kik = 0 (3.4)
V
In the limit of a vanishing antisymmetric field, the antisymmetric part
b b
of b ik vanishes and the symmetric component, [ ik' is the Christoffel symbol. Then (3.2) is seen to be exactly the set of differential equations which describe the Einstein Cylindrical Model of the Universe (See 20
21
[9], p. 159). Equations (3.3) and (3.4) are satisfied identically. Due to this correspondence, these equations should be considered as valid as the other three sets.
In all four cases, it has not been possible to ascertain if the theory contains any information other than the simple reduction to the gravitational equations. The reason for this is that a simple, general
b b expression for the F ik has not been found. True, the 1 ik have been expressed in special coordinates, but the differential equations have not been solved using these values. In this chapter and the subsequent
b
ones an appealingly simple form for the r ik in a special case, along with new information from the field equations, will be presented.
b
The relation for ik (3.1) is common to all four versions. It is a set of sixtyfour equations from which the sixtyfour f ik are to be determined as functions of the sixteen gik and their derivatives.
A solution to these equations has been given for a system of
spherically symmetric coordinates. [17, 21] Kursunoglu [16] sets up
b
the differential equations, (2.31) and (2.32), using the P ik expressed in terms of the gik' but can offer no explicit solution. In fact, no solution has yet been given to equation (2.31) in these special coordinates. It is for this reason that a completely general (coordinate independent) solution to (3.1) is desired.
A little manipulation of (3.1) is enough to show that a general
solution is far from trivial. Nevertheless, several formalistic solutions have been offered. [4, 10, 22] Formalistic, in this context, is
b
meant to imply that an explicit expression for r ik is never written
b
down. What is given is a prescription for determining r ik A brief demonstration of Einstein's formalistic solution [41 may illustrate this
22
S 5
better. In (3.1), gsk ib and g is bk are considered as individual terms and named
S
gsk ib Vikb (3.5) gisr bk = Wbki (3.6) The Vk and b are related to each other by
tbV = gbtW (kb
g ikb gb W/b (3.7) This allows the Viklb to be expressed as a Wikib in (3.1)
ik,b Wbkli g tkWibp = 0 (3.8)
b
Now, if the WikJb are found, then the ik are known as a consequence of (3.6)
p bk giPWbki (3.9) Equation (3.8) implies another expression for Wbkli
Wbkli = gik,b g akWibk, (3.10) which is reinserted in place of the last term in (3.8) gik,b Wbkli g k k'a gb'gpb Wkib
kak k'ab,i ak p k'ib' = 0. (3.11) If this procedure is repeated once more,
k'a k'a b'p
gik,b g gakk'b,i + g ak pbp'i,k
= Wbkui + gb'agabgk'p gi'c 9 (3.12)
=Vh14gl9'gpkg gi~bkIi' (3.12)
23
Einstein now defines
k gk'a i'a k'p
Aiklb 9bk,i 9akgk'i,b + g 9aig gpkgi'b,k' (3.13) and
i'k'b' = i' k' b' gI'a gk'pgpk b'c
U ikb i k b + gai pk cb (314) so that
U kb ikb Wi 'k'Ib' = Akb (3.15) The problem is to find U, the inverse of U, so that ilk'b'
W = Ui'k'b' A (316) Wikb U ikb Ai'k'lb (3.16) and the knowledge of W.ikb then gives the bik by (39).
Einstein's prescription for finding the inverse is presented in four pages and the answer becomes much too complicated to write down explicitly here. To quote Einstein in this paper,"Such a solution can indeed by arrived at. .but it is cumbersome, and not of any practical utility for solving the differential equations." This statement also applies to the solutions obtained by Mishra [22] and Hlavaty [101.
So, for all practical purposes, we are still left with the problem of sixtyfour equations in sixtyfour unknowns. One additional reduction is possible. The number of equations and unknowns can be reduced to
b
twentyfour if we treat symmetric and antisymmetric components of P ik as separate quantities. [21] (This is a natural thing to do since they transform separately. In fact, it can be seen from (2.10) that the antisymmetric components transform like a tensor, whereas the symmetric ones do not.) The fundamental tensor is also written as the sum of its parts. Consider the two equivalent equations
24
s s s s gik,b + ik,b sk i s ib+ b + g sk ib + skV ib v VI
gki,b+ ki,b= gsi kb+ gsi F kb + gsip kb+ gsi kb
V V
S S S S
+ 9ks bi + ks Fbi k bi gks bi (3.17)
The sum and difference are two new equations
S S S S gik,b 9sk ib + 9is r bk + sk b + 9isF bk (3.18) v v v
gik,b sk Flb + is F bk skF ib is bk (3.19)
This procedure is repeated twice by permuting the indices i, k, and b. Two more pairs of equations are obtained.
S S S S kb,I gsb P ki + iks P lb + 'sb[ ki k+ g Pib (3.20) V V v
s S S S
9kb,i = sb ks ib sb ki ks ib (3.21)
V . F V and
s s s s 9b1,k gsi Fbk+ 9bs1 ki + gSir bk+ bsF ki (3.22) S S S S bi,k 5si F bk + gbsp ki + 5i bk+ gbsV ki (3.23)
V V V
The following combinations are taken
S S S
gk,b + kb,i gb,k = sb ik si bk + ks ib (3.24) v v v
25
Using (2.62) and (2.61) as definition,
b b (is sk
f ik ib + is pk( Fip) (3.25) Substitution of (3.25) into (3.19) and use of definition (2.9) for a covariant derivative gives the set of twentyfour equations,
p p m
9ik;b ~ gk ib g bk g k(gmb is
m m m
+ gm rsb) + gip(gmkl bs + gbmF sk)1 (3.26)
V V 1. v/
b
to be solved for the twentyfour ik When they are found, substib vb tution in (3.25) yields ik Therefore the sixtyfour ik are known once (3.26) is solved. This still does not give a tractable form for solving (3.2). Einstein's hypothesis cannot be tested unless we find some way of solving the differential equations and these in
b
turn cannot be solved until a useful, general form for the F ik is obtained.
We are not completely stymied by the formidability of the equations.
A further advancement can be made. It can be shown that, within the framework of the theory, the gravitational and electromagnetic fields are contained in the single tensor, gik' and the field equations are those of electrodynamics as well as gravitation. To see how this comes about, an alteration in the fundamental tensor is made. It is chosen to be a CliffordHermitian tensor. This means the antisymmetric part is chosen as
gik Oik (3.27) where E is so small that its squares and higher powers will be neglected. Then the tensor,
26
gik = 9ik + eoik (3.28) is called a CliffordHermitian tensor after W. K. Clifford.
For the CliffordHermitian field, (3.26) becomes
ik;b gpk ib + i, bk (3.29) V v
If i, k, and b are permuted, two more equations are obtained.
P P
Elkb;i = gb ki + gr ib (3.30) P P
EObi;k= gL bk gbpF ki (3.31) The following combination of the three equations is taken
SE(ik;b kb;i Obi;k g2p ik (3.32) Which is immediately solved for
ik b=g ( ik;p kp;i pi;k) (3*33)
This result causes (3.25) to become ik b= (3.34)
and the displacement field is known
ik b + 2 ik;p kp;i pi;k)
This is a general form insofar as no particular coordinate system has been specified. It is also in a useful form for the field equations (3.2), (3.3), and (3.4).
27
First consider (3.2). If (3.4) is kept in mind, then Rk is given by
Rik = (Rik + Rki)
ipk),p + 2(jip pk + kpp ,i) + ip'Pq *p k qp i k
= 2 ik i# 4 k (3.36)
If i or k take on the value 1"4", the second part of (3.2) is identically satisfied. Equation (3.36) is a second order partial differential equation in the gik. It is easily verified that ip= logETg] (3.37) so that
i k ,p + p ,k + i q)(p k q pI k) G2 i k 38)
in analogy with general relativity, gik must be interpreted as the metric in the space. If we choose the physical space to possess spherical symmetry and the time dimension to be uncurved then the line element may be brought into the form8
ds2 = 2c2 + 6(2 sin2di + t 2 sin2sin2ind2 c2dt (323) in which case the metric is
8. "~" is interpreted as the radius of this world.
28
+ G 0 0 0
2 2
0 +i.sin2 0 0 ik 0 0 + in 2sin28 0 (3.40)
0 0 0 1
The left hand side of (3.38) is calculated using (3.40) and indeed the differential equations are satisfied by (3.40). The form (3.39) represents what is called an Einstein cylindrical model of the universe. The contravariant tensor to (3.40) is simply ik 1
g = (3.41)
gik
since gik is represented by a diagonal matrix, (3.40).
With the gik solved,9 we can move on to (3.4), using (3.33)
k ik =0 = E g (;p kp;i pi; (3.42)
sk 2 ik;p kp;s pi;k
Since gk is symmetric and 4kp is antisymmetric, the second term vanishes. Furthermore, p and k are dummy indices so that the first and last terms add.
2 k;p = 0 (3.43) In this form the field equation is significant since it is recognized as one of the covariant electrodynamic field equations in the absence of charges and currents [19], with the *ik interpreted as components of the electromagnetic tensor.
g ik;p = 0 (3.44)
9. The solution will be examined in more detail in the next chapter.
29
The remaining set of field equations (3.3) may be written
Ri + R ( + + .) (3.45)
ik;b Rkb;i Rbi;k 2 k;b + kb;i bi;k
V V v
From definition (2.21), and (3.4) and the results of the CliffordHermitian field, it follows
Rik [ +pp p q p q Rik ik,p Ik q p iq p qk ipj
+ (ip%,k kppi) (3.46) The term in parentheses vanishes due to (3.38) and the remaining term in brackets will be recognized as the definition of the covariant derivative of the tensor, ik
P
R (3.47) Then by (3.45)
(p P P P 2
k;pb + kb;pi + b;pk) 2 ik;b + kb;i +bi;k)
(3.48)
It remains to be shown in the next chapter that a solution of (3.47) is the second set of covariant electrodynamic equations.
So far, by use of a CliffordHermitian tensor, we have been able
b
to present a reasonable form for the [ ik and a solution to field equation (3.2). Also (3.3) and (3.4) have been put into a more familiar form and are ready to be solved in the next chapter by use of a special coordinate system which will make the equations especially
transparent.
CHAPTER 4
SPECIAL SOLUTIONS
In the preceding chapter the field equations were fashioned into a suitable form for solution. The three equations which must now be solved are
S p kp 1ip, k ip k qPik q, (3.38)
g (ik;p 0 (3.44) rp + + p 2e (0 + k + k) (3.48)
Ik;pb + kb;pi + bl;pk 2 Ik;b kb;l bl;k Minkowski Space
First, a very special situation will be considered. It is the opposite of the limiting case where the antisymmetric field vanishes. Now, the field equations will be examined when "61" is chosen infinitely large and the CliffordHermitian tensor Is assumed to have Minkowski form,
91k 51k + EIk (4.1) In light of this, all Christoffel symbols vanish and (3.38) is satisfied identically. There Is no gravitational field.
It is to be expected, since there is no gravitational field, that what is left of the tensor gk should represent the electromagnetic 30
31
field, and the field equations should be Maxwell's equations in a Mlinkowski space. To see that this is so, let us investigate (3.44) and (3.48) for10
g1k I6k (4.2) First, it is seen that there Is no distinction between contravariant and covarlant tensors. Furthermore, any covariant derivative may be replaced by an ordinary derivative since all Christoffel symbols are zero. Equation (3.44) can be expressed S = 0 (4.3)
'pp
Indeed, if the lIk represents the electromagnetic field, O h3 h2 lel Sk h3 0 hl e2
h2 h 1 0 ie
leI le2 ie3 0 then the second set of field equations (3.44) becomes one set of the Maxwell's equations in an empty space, as expected for I = 1, 2, 3 (4.5) 7 e 0 for i = 4 (4.6)
The third field equation, (3.48) can be written
10. We will choose x4 = ict
32
( ib + Pk bi+ ) ( + + *ik) (4.7)
Ik,b + kbI + bv, k) p +2 Ik,b + kb,i +bik, k
/ V
The solutions, (3.33) for F ik are substituted, using 6ik for glk In accord with (4.2).
[( k,s ks,i si,k ,b + (kb,s 'bs,k 'sk,b),i
bi,s is,b sb,I,k],s + 2 ik,b kb,i bl,k or
I(4 + + 4 ) + 2 (p + + 0 (4.9) 2 k,b + kb,i bi,k ,ss (2 ik,b + 0kb,i + 0bi,k = 0 (4.9) A solution to this equation Is
ik,b + kb,l + bi,k = 0 (4.10) which, in view of (4.4), represents the other set of Maxwell equations in a Minkowski space.
V b= 0 ; I, j, k # 4 (4.11) I )h
7x e = T ; i, j, or k = 4 (4.12)
Therefore, as expected, the field equations In Minkowski space become the Maxwell equations, while the fundamental tensor represents both the metric and the components of the electromagnetic field. These equations, (4.5), (4.6), (4.11) and (4.12) are well knowntheir solution and validity need no elaboration.
33
A More General Case
We now move to a more general case, where "t" is taken as finite. A solution to (3.38) for the gik has been given in chapter three, but a slightly different version of the solution will be given here. First, note must be taken that (3.38) involves only the symmetric part of the fundamental tensor. By an extension of the example in Minkowski space, we should anticipate that the glk will depend upon the distribution of matter in the universe. At this point the following model is adopteda static homogeneous universe. This means that all parts are considered extrinsically and permanently alike. In this case, the line element can be put in the general spherically symmetric form.
2 2 2 2 2 2 2 v2 (4.13)
ds = e dr + r d2 + r sin d ec dt (4.13) where
v = v(r) and X = X(r) (4.14) Most of the glk are already known:
g1k 0 i k (4.15)
922 = r2 (4.16)
22
g33 = sin 2 (4.17) From the differential equations, (3.39), we need only determine
gli = e (4.18) and g4 = e (4.19)
34
The Christoffel symbols are computed according to (2.61) and substituted In (3.38). It is seen that the solution is
2
e (1 r 1 (4.20) e = 1 (4.21) Thus in matrix form, the gik are
2
(1 r 0 0 0 0 r2 0 0 g = (4.22)
2 2
0 0 r sin2 0 (4.22)
0 0 0 1
This model is known as the Einstein cylindrical universe. (See [24], page 359) A change of variables will serve to show more distinctly the character of this universe. Let r 1/2
x2 = r cos 0
x3 = r sine cos@
x4 = r sinOesin (4.23) In which case,
ds2 = (dx12 + dx2 + d d ) c2dt2 (4.24)
35
and
2 2 2 2 2
x2 + x2 + + x4 = 2 (4.25) This illustrates that the physical space of the Einstein universe may be Interpreted as the threedimensional bounding manifold of a sphere of radius aj In a fourdimensional Euclidean space with the cartesian coordinates given above. The time dimension is uncurved. Hence the name cylindrical universecurved space and straight time.
Even though the line element displays spherical symmetry, there is no symmetry of form among the i kand the Rik. To make the field equations more transparent, we adopt a method used by Kronsbein (12]. The sphere represented by (4.25) is radially projected from its centerl2 In the fourdimensional space, (xi), into the threedimensional space, e, with coordinates X (Greek letters take on values 1, 2, 3) by the projection
11
The solution shown in the previous chapter, (3.41), may be obtained directly from (4.24) by the transformation
x = (cos Y
x2 = C sinY cos9
x3 = r.sin Tsine cos Y
x4 = 6Lsin Ysne snY
which expresses the spatial part in fourdimensional spherical coordinates.
12Some people prefer to call this a gnomonic projection.
36
x, 1/2
= 1/2 (4.26) where
A = 6(2 (X2 + (X3)2 (4.27) In this space, XA and X4 ct we will denote the symmetric part of the fundamental tensor by aik. It is computed by the standard method to give
[ (Xl)2] I x12 IX13 0 aIk X.1X2 [A (X2)2 X2X3 0
2
xx3 X2X3 [A (X3)2] 0 A2
0 0 0 .2 (4.28) The determinant is
Iaikl = (4.29) The contravarlant tensor to (4.28) is found to be
2 12 12 13
I[2 + (XI)2] X X XIX 0 Ik A XIX2 2 + (X22] X2X3 0
xx3 x x 20 SX [ + (X (4.30)
o o o (4.30)
37
With (4.28) and (4.30), the Christoffel symbols can be constructed. They are very Important because they are the symmetric part of the displacement field.13
al a
There is one more quantity which we need to calculate n this space(431) There Is one more quantity which we need to calculate in this space for future use. It is the analog of the RiemannChristoffel tensor, (2.20). It will be denoted by
GIkb 1 Vk ,b I ib) ,k jii k b) ibqpk (4.32) Notice the antisymmetry.
GpIkb = Gpibk (4.33) The components will be listed here for future reference:
= XaXB
G ..
r = E (Xa)2] (4.34) aar A2
All others are zero
For the antisymmetric part of the fundamental tensor, the following symbols will be used for the elk'
In (4.31) and (4.34), the repeated Index does not imply summation.
38
0 h3 h2 el
h3 0 h1 e h2 h1 0 e3
e e2 e3 0 (4.35) where the "ea" and "ha" are functions of (X1, X2, X3, t).
We now have the necessary material to calculate the antisymmetric
b
components of F k in our special coordinates. The actual calculation
b
of all twentyfour k is a long, tedious process. Only a sample v 3
calculation (the component, 12 will be used as an example) and the final results are given here. By (3.33),
3 E[ai3 2 + a32 + a33(  )] (4.36)
12 12;1 12;2 2 12;3 23;1 31;2 ,3 ( a312 + a32 + a33[(  ) 12 12;1 12;2 2 12,3 23,1 31,2
S1I2 113 li (437) Upon substitution of (4.30), (4.31), and (4.35), and consequent cancellation and simplification, it is found that
V3 A 4 (3X3h3 + X3Xlh3, + X3X2h3,2 + [62 + (X3)2]h33
[ 2 + (X3)2][(h h2,2 + h h) + (Xlhl + X2h2
+ X3h3)]) (4.38)
The other b are computed in the same manner and soon the following
39
pattern is recognizedl4
r~. E (3XahX + aa hX,) IEYaaX ( h ,, + X h)] (4.39)
and
a + a X a e +
4 +xe + a e + 2 (e + e B)
V
S ah,4 1 (4.40)
also
4 (ea e e (4.41)
aB a B X, 4)
V
and
L0 ea4 (4.42)
b
These are the twentyfour in the special coordinates. Combined with the [ ik given by (4.31), the complete set of the sixtyfour
ik is known in this space. This completes the information needed to set up the field equations, (3.44) and (3.48).
It has been pointed out that (3.4), which reduces to (3.45) is identically a set of the covariant electrodynamic equations in the presence of a gravitational field. To see what they look like in this
b
space, it will be easier to go back to (3.4), since the b ik have already been computed. The equations represented are:
2 3 4
12 + 13 + 14 = 0 (4.43)
V V V
148 and E aX are the usual permutation symbols whose value is zero If any two indices are alike, +1 if (aBX) is an even permutation of (123), 1 if (aBX) is an odd permutation of (123).
40
21 + 23 24 =0 (4.44)
F131 + 232 +P434 = 0 (4.45)
1 2 3
141 + 42 r 43 0 (4.46) The b are substituted from equations (4.39) to (4.42). After simplification, the four equations are:
3(X2h3 X h2) + [2 + (X2 2]h3,2 [2 + (X)2]h2,3 + X2X(h3,3 h2,2) + XI(X2h3,l X3h2,1) el,4 = 0 (4.47) 3(X 3h Xh3) + + 2 + (X3)2h1,3 [2 + (XI ) 2]h3,1 + X l(hll h,) + X2(X3h,2 Xlh.2 e2,4 = 0 (4.48) 3(Xlh2 X2hl) + [2+ (x)2 h2,1 [2 + (X2 )2]hl,2 + X X2(h, hl,) + X3(Xh X2h1) 6 e = 0 (4.49)
2,2 1 l 2,3 1,3 X" 3 ,4
2(XI e + X2e2 + X3e3) + [dL2 + (xI)2 el,l + XIX2el,2 + xX3el,3
+ X2X e2,1 + [2 + (X2 )2e2,2+ X2X3e2,3
+ X3,1 e32 + X e32 2 + (X3)2]e3, = 0 (4.50) These equations are exceedingly complicated as they stand. They will be left this way for the time being. After the other set of four field equations (3.48), has been written in this space, a simplification and solution will be presented for all eight equations.
A tensor identity which will be of great utility in the investigation of (3.48) is
41
Ak Ak=A ik G... + ik GS + .
bm...;pq bm... ;qp sm bpq + A bs'** mpq
sk. I is... k
A sk... G Ais Gk (4.51) bm* spq bm"* spq where A b is any tensor of arbitrary rank and Gs is defined by (4.32). This identity allows the left hand side of (3.48) to be written
lk;pb + kb;pi + bi;pk ik;b + kb;i + bi;k) ;p
V V V V V + P G t ipb p iGtkb t IG tbp p t t t +F tkG kp +P kt bpl + kb Gtip V V V + G p + kGt G (4.52) tltbpk kt Ipk bi tkp v V v If G Ikp Is written out as prescribed In (4.32), then it is seen to be
Gpkp = Rik (4.53)
It Is clear that if any of the i, k, b are equal, then (4.52) vanishes, as does the right hand side of (3.48). Equation (3.48) represents only four distinct equations. They are:
S p _ (4 + + ) (4.54) 13;p2 + 32;pl + 21;p3 = 2 13;2 32;1 21;3 P P P 2E
S + P +. ([ + + 4 )(4.55)
14;p2 + 42;p1 21;p4 ( 14,2 + 42;1 + 21;4) (455)
+ 43 + 34 + 2C) (4.56)
3;p2 + 32;p4 + 24;P3 = T (43;2 + 032;4 + 24;3) (456)
VV
13p ;p ;pl + ;P3 R2 13;4 34;1 + 41;3 The left hand side of (4.54) is expanded like (4.52).
42
P13;p2 1 3;pl P2;3 P 2 P 2 1 2;3);p
it3Gtll2 +1 t3G tl32 1ltGt312 + 31tGt332
1113R12 + 1R22 13R 2
2 t t p t tI'r
+ t2G 321 +P' t2G 331 3t 221 3tG231
+ 2 3
t t t t
+ 2 R 11 + + R 3
+ ti Gl 213 t 223 2 G t 113 2t 123
+ I21R + 21R +. 31R33 (4.58)
If the Index "4" appears in a Gtpq it vanishes. For this reason, the tpq+
following type of combinations must be observed.
1 G = (4.59) G 121 + G 123 2 R12
When the summation is carried out on the index "t" as indicated in (4.58), and combinations like (4.59) heeded, a fortunate cancellation occurs which leaves
Pp p P
S13;p2 + 32;pl +P 2l;p3 ('13;2 + 32;1 1 P 21;3);p (4.60) The field equation (4.54) is then
( ; + 2 1 + )  ( + + ) (4.61)
13;2 32;1  21;3 ;p 2 13;2 32;1 21;3
b
At this point, the set of b ik' as given in equations (4.39)
through (4.42), could be substituted, but the left hand side of (4.61)
43
can still be reduced. By (3.33),
P E SP+ 0 + 0
[ [13;21;p = 2 a 13;s2 32;sl 21;s3 3s;12 sl;32 2s;31 s3;21 ls;23 )s2;13);p (4.62) The summation Is carried out, as indicated, over s. Then (4.51) is used to make the following type of rearrangement
0 = + GP + Gp (4.63) ab;cd ab;dc pb acd ap bcd throughout each term in (4.62), giving
as [13;2;sp + [alP( GI + 4G G2 2
3;2p 2 sp 13 112 23 112 312
31G3 213 + 113 3 23G113 );p + 2p 12 1 321
1 2 2 3 + P31 221 + 32 221 + 21 G 223 + 31 223 + 23 G3123;p
+ a3P(423G2132 + 12G 2332 + 13 G3332 + 121 331
0 32G3331 + G1 23;p (4.64)
+ 32 331 311231);p
The covariant derivative of the G ikb are all zero, since, according to equations (4.34) and (4.28),15
 a (4.65) O is a constant and
aaB;p =0 (4.65a)
15No summation is implied by repetition of the index y in these equations.
44
therefore
GT y;p 0 (4.66) The following equations are now substituted in (4.64): (4.65), (4.28), and (4.30). Cancellation and simplification leave
P' Ip aSP+ 2 (4.67)
[13;2];p 2 [13;2];sp ( [13;2]
V
This result reinserted in the left hand side of (4.54) gives finally
aSP( 13;2 32;1 21;3 ;sp = 0 (4.68)
The same procedure and the same identities are used on equations (4.55), (4.56), and (4.57), giving
aSP 14;2 + 42; + 21;4);sp = 0 (4.69) asp (43;2 + )32;4 24;3 ;sp = 0 (4.70) asp 13;4 + 34; +41;3);sp = 0 (4.71) Solutions to (4.67) through (4.70) are, respectively,
13,2 32,1 21,3 = 0 (4.72)
024,1 + 041,2 + 12,4 = 0 (4.73) 034,2+ 042,3 + 23,4 = 0 (4.74) 14,3 + 43,1 + 31,4 = 0 (4.75) Using ea and ha from (4.35) we see that this Is the second set of covariant electrodynamic equations.
45
hl, + h2,2 + h3,3 = 0 (4.76) e2,1 el,2 + h3,4 = 0 (4.77) e3,2 e2,3 + h, = 0 (4.78) el,3 e3,1 + h2,4 = 0 (4.79) These four equations are simple compared with the other set, (4.47) through (4.50). In order to simplify the latter, we are willing to slightly complicate the former. The result will be a symmetry of form for both sets, and moreover, a solution will follow easily. This Is accomplished by replacing the e. and ha by the following quantities:
ha(X ,X2 X3 t) = 2 Ha + XBHBXa + (RL XHBX] (4.80) e (XI X2 X3,t) = + [LEa + EyEX'] (4.81) where
Ha = Ha(t) (4.82) and
Ea = E (t) (4.83) are yet to be determined.
These values are substituted in the field equations in place of
ea and ha. In the first set, (4.50) is satisfied identically by (4.81). Equation (4.47), after simplification, reduces to
2H1 3 2H 2 2H
1,4) ( E2,4) E3,4) (4.84) ) )+ dR. E2(?
46
or
2( b X 2 + 1 2 2 '
( H ) + H ( 2 (, (4.85) which implies
l H (4.86) This same result is implied by (4.48) and (4.49).
In the second set of equations, (4.76) is satisfied identically by (4.80). When (4.80) and (4.81) are substituted in (4.77), the result Is
[(62 + (X3)23(2E3 +(H3, ,4) + (X3 + 2)(2E l + (RHI,4)
+ (X23 6 XI)(2E2 + H2,4) = 0 (4.87) or
[L2 + (x3)2)(2 + ) + (X x +6R x2)(2 + 1
+ (x2x3 R xI)(2+ )2 = 0 (4.88)
which implies
1 2 ,
SF E (4.89)
This result Is also implied by (4.78) and (4.79).
These two sets of equations are written together to emphasize the symmetry of form which has been brought about. In place of the original sets, (4.47) through (4.50) and (4.76) through (4.79), we have
1 dE 2S= (4.86) c dt G.
Sd 2 (4.89) c 7E
47
Substitution of these equations into one another allows i and W to be eliminated from each equation respectively, giving two ordinary wave equations,
2 d2R 4 = 0 (4.90)
c2 dt2 R2
Sd2+ 4 0 (4.91)
c dt2 ~
These represent an electromagnetic wave of circular frequency
2c
=.c (4.92)
where
Ea(t) Ga sin wt (4.93) Ha(t) = Ga cos Wt (4.94) The G* is a constant.
a
So we finally have the solution to the two sets of equations,
(4.47) through (4.50) and (4.76) through (4.79). It is, in the special coordinates,
h a 16X2 G Xa + R Ga G X cos (Vt) (4.95) a 2 B a and
ea = [ Ga + 'aB Go X ] sin ( ) (4.96)
With the $1k given by (4.35), the above is the solution to the last two unified field equations, (3.44) and (3.48), which we had set out to solve at the beginning of this chapter.
In vector notation, the solutions are
48
hH V. H + ('H jjX+ (jX)a (4.97) e= Id[? + (( ] (4.98) The reason for using vector notation will soon be apparent. Now that the solutions are known, we want to visualize them in the spherical space.
First, consider the "straight lines" in this space. The geodesic equation,
d2Xi dXk dXb
+2 kb d = 0 (4.99)
becomes
d + 2 XI dX= 0 (4.100)
since
I k b
kb dX dX (4.101)
kbd  0 ( .0
and
kb kb35)
This shows that the geodesics of a space are not altered by the presence of electromagnetic fields in the space to the first order which we have investigated. (A question which Immediately comes to mind is, "Well, do the properties of the space affect the electromagnetic fields?" Of course they do. We have seen how the electrodynamic equations were complicated. Exactly how they are affected remains to be seen below.)
49
From [12}, the above equation (4.100) can be written
d2 a 2 dXa %d d d2 A d (. d
24
y=2 0 (4.102)
An explicit1 integral of (4.102) is
I sin R sin (; + 6)
cos (X + E)
2 sin pL sin (I + 6)
cos (X + E)
X = tan (X + E) (4.103) We have set R = 1 for simplicity. This is the mathematical form for the "straight lines."16 A geometrical picture may be found in [121.
Now when the sphere, (4.25) is rotated with angular velocity (G, in E4, it induces transformations on X in Points move with elliptic velocities
L(~ ~ + (+* iR' + ( x x (4.104) It can been seen that the geodesic equations fulfill (4.104) so that the points move along straight lines In the space.
Now, replace i 7In (4.104) by fiand the result is the same as (4.97). We see that the magnetic vector, R, has the form of the associated contravariant elliptic velocity, (4.104), and lies along the geodesics of the space. So not only is the magnetic vector altered by the space, it is altered in this very special way.
16The geodesics will sometimes be referred to as Clifford lines.
50
If we tensor multiply the contravariant velocity by gik, we get the covariant velocity vector, which turns out to be the same form as the electric vector, (4.98)
XiI [ i + (6x7 x i] (4.105) so that the electric vector lies along the covariant elliptic velocity vector. The solutions for the unified field equations are, therefore, standing waves lying along Clifford lines, and having fixed frequency, 2c
It may be pointed out, finally, that when the radius of the
Einstein universe becomes infinitely large, the field equations and their solutions naturally go over to Maxwell's equations and their solutions in a Minkowski space.
CHAPTER 5
SUMMARY
There is a possibility that the main features of the theory presented have been obscured by the preponderance of mathematics. For this reason, it may be well to summarize briefly the results of this dissertation.
The second chapter contains no new results. It is intended as a background to familiarize the reader with the different versions of the unified field theory. The similarities and differences are pointed out. All three have the same goalto derive all of the fields and field laws governing Nature from a single tensor, which need not have symmetry. The main difference is the choice of integrand for the variational principle. This, of course, leads to different forms of the field equations; however, the same set of sixtyfour algebraic equations appears in each version. The most important similarity In the theories is that they are based on the same geometrical concepts, and they all go over into the gravitational equations of general relativity in a limiting case. Notwithstanding all the beautiful mathematical formalism, this correspondence to the laws of gravitation is the only physical content which has been derived from any of the theories up to now.
The reason for this difficulty was pointed out in the next chapter. Before Einstein's hypothesis (which was discussed in the Introduction) can be tested, the differential field equations must be solved. Before
51
52
they can be solved, the sixtyfour algebraic equations must be solved in a general case (without reference to any coordinates) for the components of the displacement field. Since this general solution has not been accomplished, there is no way of telling whether all the field laws are included in the theories. The sixtyfour equations are shown to be reducible to twentyfour, but this has not enabled the solution to be found up to the present.
At this point, we present our version of the theory, in which the fundamental tensor is modified. It must be noted that this is not a linearization such as that used by Kursunoglu [14] or Einstein [3]. These linearizations exclude interaction between the symmetric and antisymmetric components of glk. The present work does not preclude this possibility. In fact, the antisymmetric components of the displacement field were seen to be a combination of the symmetric and antisymmetric parts of the fundamental tensor. So instead of a linearization, we have a perturbation type of technique. It is basically a first order antisymmetric perturbation of the gravitational field producing symmetric tensor.
With this method, the algebraic equations are readily solved for
b
rk without the specification of any coordinate system. It is found that the Rlk become the same expressions used in general relativity. With this in mind, we adapt the set of field equations involving the contracted curvature tensor to an Einstein model of the universe.
In the limit of an infinite radius of the Einstein universe, the remaining differential equations go directly over to the Maxwell equations of Minkowski space. For the general case (a finite radius), a transformation was made to a symmetrical arrangement of coordinates. Investigation of the remaining two differential field laws leads to the
53
important discovery that they are the covariant electrodynamic equations in the absence of charges and currents.
As a consequence of the symmetry of the coordinates, we are able to give an exact solution to these equations. This solution indicates that the electromagnetic is bent along the geodesic caused by the gravitational field, while an investigation of the geodesic equations shows that the gravitational field is unaffected by the electromagnetic field in this case,
Perturbation to higher order terms in E still makes it possible to obtain the displacement field explicitly but this does not, up to the present, imply success in solving the associated differential equations. In these cases, the presence of addition fields will distort the geodesics of the pure gravitational field, but it is not known whether the additional fields are only electromagnetic in Nature.
In summary, the most important result of this disseration is the
realization that, to the first order, the covariant electrodynamic field equations, as well as the gravitational equations, are included in the unified field theory. Heretofore this was conjectured but never shown. It is clear that what has been done is far from an ultimate goal of the theory. Nevertheless, we feel that our contributions should be an impetus to further work in the field. The most general case must be pursued. Along these lines, Kursunoglu has made the greatest innovations since the theory was formulated by Einstein. It would certainly be desirable to study his plan in which there is a possibility of deriving nuclear fields together with those previously discussed. [161
BIBLIOGRAPHY
1. Einstein, A. The Principle of Relativity. New York: Dover Publications, 1923, pp. 111164.
2. Einstein, A., Ann. Math. 46, 578 (1946).
3. Einstein, A. and Strauss, E., Ann. Math. 47, 731 (1946). 4. Einstein, A. and Kaufman, B., Ann. Math. 59, 230 (1954). 5. Einstein, A. and Kaufman, B., Ann. Math. 62, 128 (1955).
6. Einstein, A., Can. J. Math. 2, 120 (1950).
7. Einstein, A. The Meaning of Relativity. 5th ed. Princeton: Princeton University Press, 1955.
8. Einstein, A., Revs. Modern Phys. 20, 35 (1948).
9. Eddington, A. S. The Mathematical Theory of Relativity. Cambridge: Cambridge University Press, 1924.
10. Hlavat, V. Geometry of Einstein's Unified Field Theory. Groningen:
P. Noordhoff Ltd., 1957.
11. Hlavat9, V., J. of Math. and Mech. 7, 833 (1958). 12. Kronsbein, J., Phys. Rev. 109, 1815 (1958). 13. Kronsbeln, J. Phys. Rev. 112, 1384 (1958). 14. Kursunoglu, B., Phys. Rev. 88, 1369 (1952). 15. Kursunoglu, B., Revs. Modern Phys. 29, 412 (1957). 16. Kursunoglu, B., II Nuovo Cimento 15, 729 (1960). 17. Papapetrou, A., Proc. Roy. Irish Ac. 52, A, 69 (1948). 18. Schr6edinger, E. SpaceTime Structure. Cambridge: Cambridge University Press, 1960.
19. Synge, J. L. and Schild, A. Tensor Calculus. Toronto: University
of Toronto Press, 1949.
20. Tolman, R. C. Relativity, Thermodynamics, and Cosmology. Oxford:
Oxford University Press, 1934.
54
55
21. Tonnelat, M. A. La Theorie Du Champ Unifie' D'Einstein. Paris:
GauthierVillars, 1955.
22. Mishra, R. S., J. of Math. and Mech. 7, 877 (1958). 23. Landau, L. D. and Lifshitz, E. M. The Classical Theory of Fields,
2nd ed. Reading, Mass.: AddisonWesley Publishing Company, Inc.,
1962.
24. Miller, C. The Theory of Relativity. Oxford: Oxford University Press,
1952.
25. Kronsbein, J. Electromagnetic Fields in Einstein's Universe
[Unpubl ished].
BIOGRAPHICAL SKETCH
Joseph Francis Pizzo, Jr. was born on October 30, 1939, in Houston, Texas. There he attended St. Thomas High School and received his B. A. from the University of St. Thomas in 1961.
In September, 1961, Mr. Pizzo began graduate studies at the University of Florida. He received his Ph.D. in August, 1964.
Mr. Pizzo is married to the former Paula Awtry of Dallas, Texas. They have one child, a son.
56
This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy.
August 8, 1964 Dean, Collegg6f rt and Sciences
Dean, Graduate School Supervisory Committee:
Cha irman
Cocha i rma f
B ~l . .. ... .. .

Full Text 
19
a postulatory basis, a variational principle is used to derive a set
of field equations. A solution to these equations will presumably
give a description of nature in which all fields are united in the single
tensor, g.^ in the same manner as the electric and magnetic fields are
unified in the electromagnetic tensor.
Now that the problem is laid before us, the next step is to pro
vide a solution to these field equations.
7
Let a vector, A be transferred parallel to itself along the boundary
of an infinitesimal surface element and brought back to its starting
point, x* y We propose to compute the change, Aa', in the vector
after one complete circuit. This change is given by
aa' = pA' .
(2.11)
Expression (2.7) is used for &A1. The quantities P and AS are ex
panded about their values at the initial point, and terms kept to first
order, since the curve is infinitely small. The expansions are
s
AS(xJ) = A;
(s)
 r a
I pq
(s)
(x
xM, \)
(sr
(2.12)
and
r st r st xj(s) + r st.,
(s)
(xq xq(s)) (2.13)
Let
t t t
X x (5) T) .
The change in the vector is now written
^,r,.t.q*w + ristr>M*Wdi>t
(2.14)
(2.15)
With an appropriate renaming of dummy indices this becomes
 'r'pt.q+r'strspq] flP#nqd,it (z,6>
it is understood that the terms preceding the integral are the initial
values. As usual, we are interested in the displacement field and want
to investigate the term in brackets to determine tensor character. It
6
The expression &A will depend on the vector A and dx and can be
represented by
5A1 = P AS dx1
I st
(27)
where the) is called the Displacement Field. Its components,
which are to be determined, are functions of the x'. In this form
da! = ca*,t + r st aSj dxt
(2.8)
The term in brackets is denoted by a special symbol,
A
Jt
= A
t
+
(2.9)
and is called a covariant derivative.
The difference, DA between the two vectors at the same point
is to be a vector itself. This requires the covariant derivative to be
a tensor. Therefore the transformation law for the displacement field
must be
(2.10)
It is clear that ordinary derivatives do not form a tensor and
covariant derivatives must be taken instead.
Before the displacement field can be determined beyond its trans
formation law, (2.10), we must know more about the distribution of
matter and charges which dictate the structure of the space.
It is clear that the displacement field is somehow related to the
curvature of the space. So far, in this presentation, the notion of
curvature has been a vague one, at best. Using the idea of parallel
displacement, a mathematical picture of curvature may be displayed.
43
can still be reduced. By (333),
,P
rr = aS ^ f + 0 +0 0 0
[I3;2];p 2 13; s2 32;sl 21 ;s3 3s ; 12 si;
* jf
32
?2s;31 ls;23 4>s2;13) ;p (4,62)
The summation Is carried out, as indicated, over s. Then (4.51) is
used to make the following type of rearrangement
0 , = <*> + $ .Gp j ^ cPk a (4.63)
ab;cd ab;dc pb acd ap bed
throughout each term in (4.62), giving
.P
[
v
P [I3;2l;p = 2 a P<>[ 13;2];sp + ^ P(Cl3G 112 + *23 112 + *12G 312
+ $31G 213 + ^l5 113 + 113^ ;p + 3 P^12G 321
+ ^l13 221 + ^2G 221 + ^l6 223 + ^l0 223 + 023G 123^ ;p
+ a P^23G 132 + 0I2G 332 + 01 3q3332 + <>12G 331
+ 0 C5 + 0 G }
32^ 331 31 23 r ;p
(4.64)
The covariant derivative of the G ... are all zero, since, according to
equations (4.34) and (4.28),
15
j kb
.7
1
G aBY = ^2 aaB
(4.65)
0tis a constant and
aB;p 0
(4.65a)
15
tions.
No summation is implied by repetition of the index 7 in these equa
24
9j_k,b + 9k,b 9sk
r5ib + gski
s s
l ib + 9sk i ib + 9sk \
s
v
V V ~~ V
V
s
n
s
p
s s
n n
(3.U)
+ r bk+9
'is 1 bk + 9
v
s
is i bk 9is I bk
V V V
S S
s
9ki ,b + 9ki,b 9sÂ¡ P kb *" 9s? T kb + 9siP kb + 9 si P kb
s s s
s
+ 9ksP bl+ 9ksr bi + kjT bi+ 9
ksP bi
V V
(3.17)
The sum and difference are two new equations
s s
. s
ts
9?kb 9sk P j_b + 9is P bk + 9sk
ib + 9isP
V V
bk
V
(3.18)
s s
S
s
9Â¡k,b = 9sk T ib + 9is! bk + 9sk [
V v
"n> + 9Â¡vsl
(3.19)
This procedure is repeated twice by permuting the indices i,
k, and b
Two more pairs of equations are obtained.
s s
S
s
9kb, i 9sb T kj. + 9ks P ib + 9sb [
n ki + 9ks [
V V
"ib
V
(3.20)
s s
s
S
9kb, i 9sb.P k + 9ks P ib + 9sb
r kÂ¡ + gks
V
r ib
(3.21)
and
s s
S
s
9bi k ~ 9s i P bk ^ 9bs P kj_ r 9sj
bk + 9bs1
V V
ki
(3.22)
s s
n tj
s
s
9bi,k 9si  bk 9bs ki + 9sil
V V v v
bk + 9bsP
V
' J
(3.23)
The following combinations are taken
r*s
S
s
Pk.b + 9kb,Â¡ + 9bi,k 2[gsb Ik + 9sil
1 bk+ 9ksf
V V
ib* .
V
(3.24)
are considered as individual
22
n
better. In (3.1), gsk Â¡b and gjs.p
bk
terms and named
9sk r ib = V* kb *
9isT bk Wbkf I
The V.kb and W.kjb are related to each other by
tb bt
9 Vikb 9 Wikb
This allows the Vjkjb to be expressed as a in (3.1)
9 i k, b Wbki gP 9tkWib(p = 0 '
b
Now, if the W.^jk are found, then the J .^ are known as
of (3.6)
r
bk
 fP,
9 Wbki *
Equation (38) implies another expression for W,
bkf
Wbki 9ik,b 9 9akWibk
which is reinserted in place of the last term in (38)
k 'a
9ik,b Ubk I "Â¡UkVb.l t sk a9akgb P9pbWkib' 0
If this procedure is repeated once more,
k1 a k1 a b 1 p
9ik,b 9 9ak9k'b,i 9 9ak9 9pb9p'i,k
ii ^ b 'a k'P *'c w
Wbki + 9 9ab9 9pk9 gciWb'k1{i' '
(3.5)
(3.6)
(37)
(3.8)
consequence
(3.9)
(3.10)
(3.11)
(3.12)
25
Using (2.62) and (2.61) as definition,
r ik bkV ^ (9Â¡sr Pk+^rv 2s)
V v V n/
Substitution of (325) into (319) and use of definition (2.9) for a
covariant derivative gives the set of twentyfour equations,
P P
9ik;b = 9pk T ib + 9ip P bk + gE~ ^9pk^9mb T
m
m
V v
m
+ 9im P sb^ + 9ip^9mkP bs + 9bm P sk^
(326)
V
to tie solved for the twentyfour I .. When they are found, substi
b b
tution in (3.25) yields p .^ Therefore the sixtyfour j ^ are
known once (326) is solved. This still does not give a tractable
form for solving (32). Einstein's hypothesis cannot be tested unless
we find some way of solving the differential equations and these in
P*3
turn cannot be solved until a useful, general form for the J .^ is ob
tained.
We are not completely stymied by the formidabi1ity of the equations.
A further advancement can be made. It can be shown that, within the frame
work of the theory, the gravitational and electromagnetic fields are con
tained in the single tensor, g.^, and the field equations are those of
electrodynamics as well as gravitation. To see how this comes about,
an alteration in the fundamental tensor is made. It is chosen to be
a CliffordHermitian tensor. This means the antisymmetric part is
chosen as
9ik = ^ik
V
(3.27)
where e is so small that its squares and higher powers will be neglected.
Then the tensor,
53
important discovery that they are the covariant electrodynamic equations
in the absence of charges and currents.
As a consequence of the symmetry of the coordinates, we are able
to give an exact solution to these equations. This solution indicates
that the electromagnetic is bent along the geodesic caused by the gravi
tational field, while an investigation of the geodesic equations shows
that the gravitational field is unaffected by the electromagnetic field
in this case.
Perturbation to higher order terms in still makes it possible to
obtain the displacement field explicitly but this does not, up to the
present, Imply success in solving the associated differential equations.
In these cases, the presence of addition fields will distort the
geodesics of the pure gravitational field, but it is not known whether
the additional fields are only electromagnetic in Nature.
In summary, the most important result of this disseration is the
realization that, to the first order, the covariant electrodynamic field
equations, as well as the gravitational equations, are included in the
unified field theory. Heretofore this was conjectured but never shown.
It is clear that what has been done is far from an ultimate goal of the
theory. Nevertheless, we feel that our contributions should be an
impetus to further work in the field. The most general case must be pur
sued. Along these lines, Kursunoglu has made the greatest innovations
since the theory was formulated by Einstein. It would certainly be
desirable to study his plan in which there Is a possibility of deriving
nuclear fields together with those previously discussed. [16]
12
(2.28)
(2.29)
The four extra variables, may be given any value. The three equa
tions will be transposition invariant if ^ 's chosen such that
I Ik0 (2.30)
V
Now can be eliminated entirely from the equations by writing (2.27)
6
as two equations.
Rik=
R., + R, . + R, ,
ik,b kb,i bi,k
v v' v
= 0
(231)
(2.32)
b
The star may just as well be excluded from above the p1 ^, renaming them.
By contraction on the indices k and s in (2.29), we can verify that the
expression in parentheses vanishes and the equation can be written
DVlfPtsMf Tstifr0 (2.33)
Before further simplification is made, it should be noted that the field
law, (2.28) is already implied in this expression. This can be seen
from the following consideration. The equation formed by exchanging
free indices,
tk
it
i <
a Â¡,k n
(2.33a)
6. The dash under the indices indicates symmetry.
29
The remaining set of field equations (33) may be written
Rik;b + Rkb;i + Rbi;k = ^2 (Â£f>i k;b + ^kb; i + ^bi;^ 0 M)
v v v ^
From definition (2.21), and (3.4) and the results of the Clifford
Hermitian field, it follows
P n p
Ik [ r lk,p + r Ik \q p'j P Â¡q,qp T qk iPp)'
The term in parentheses vanishes due to (338) and the remaining term
in brackets will be recognized as the definition of the covariant
pP
derivative of the tensor, ( .^ .
V
? k
v
 T Ik;,
(347)
Then by (3.45)
r
p
ik;pb 1 kb;pi + ^ bi ;pk^
(3.48)
It remains to be shown in the next chapter that a solution of
(347) is the second set of covariant electrodynamic equations.
So far, by use of a CliffordHermitian tensor, we have been able
b
to present a reasonable form for the P and a solution to field
I 1 k
equation (32). Also (33) and (3.4) have been put into a more
familiar form and are ready to be solved in the next chapter by use
of a special coordinate system which will make the equations especially
transparent.
3
Chapter three embodies our version of the theory. The sixtyfour
equations are shown to be reducible to twentyfour, which still cannot
be solved in a useful form. Nevertheless, they suggest a modification
in the fundamental tensor which allows a tractable form for the solution
of the equations to be obtained. The three sets of differential equa
tions, referred to above, are constructed. One set is the same as the
gravitational equations. The next is seen to be the first set of co
variant electrodynamic equations in the absence of charges and currents.
The last is actually a set of third order partial differential equations
and needs further investigation. It turns out we are able to solve this
problem in our version of the theory.
In chapter four two cases are considered. First, the equations are
examined when the radius of the Einstein universe is taken infinite, in
which case the gravitational equations are satisfied identically. The
other two sets of equations are shown to be Maxwell's equations in the
absence of matter. Next we take the more general case of a finite radius.
The first two sets of differential equations were investigated before.
Now, a term by term examination reveals that the last field law implies
the second set of covariant electrodynamic equations. The last part of
this chapter is devoted to an exact solution of these equations in a special
coordinate system.
Throughout this dissertation an elementary understanding of tensor
algebra and a basic knowledge of general relativity are assumed.
This dissertation was prepared under the direction of the chairman
of the candidate's supervisory committee and has been approved by all
members of that committee. It was submitted to the Dean of the College
of Arts and Sciences and to the Graduate Council, and was approved as
partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
August 8, 1964
Dean, Graduate School
Supervisory Committee:
ChaKrman
/l'
Cocha i rmarf
/
f> .s n
\, x\
'kjvOWVV HV
[T
g /S
49
From [12], the above equation (4.100) can be written
d2x 2 dX^ /Â£* dxt
(X 7) = 0
dA
2 A dA
dA'
2 4
JL.= 0
dA
An explicit integral of (4.102) is
yI s in p. sin (A + )
cos (A + e)
y2 sin M sin (A + 5)
cos (A + e)
X"5 = tan (A + e)
(4.102)
(4.103)
We have set (^ = 1 for simplicity. This is the mathematical form for
16
the "straight lines." A geometrical picture may be found in El21.
 &
Now when the sphere, (4.25) is rotated with angular velocity u,
ct
in E^, it induces transformations on X in ^. Points move with
elliptic velocities
&?
It can been seen that the geodesic equations fulfill (4.104) so that
the points move along straight lines In the space.
Now, replace 5* In (4.104) by H* and the result Is the same as
(4.97). We see that the magnetic vector, h*, has the form of the
associated contravariant elliptic velocity, (4.104), and lies along
the geodesics of the space. So not only is the magnetic vector altered
by the space, It Is altered in this very special way.
The geodesics will sometimes be referred to as Clifford lines.
31
field, and the field equations should be Maxwell's equations in a
Minkowski space. To see that this is so, let us investigate (3.44)
and (3.48) for ^
9k6!k (1*2)
First, it is seen that there is no distinction between contravariant
and covariant tensors. Furthermore, any covariant derivative may be
replaced by an ordinary derivative since all Christoffel symbols are
zero. Equation (344) can be expressed
0. = 0 .
>P.P
(4.3)
Indeed, If the 0.^ represents the electromagnetic field,
0
Ik
0
h3
"h2
ie
h3
0
h.
ie,
ie.
i e
ie
(4.4)
then the second set of field equations (344) becomes one set of the
Maxwell's equations in an empty space, as expected
Vxii = j; for 1 = 1, 2, 3
(4.5)
V e = 0
for i = 4
(4.6)
The third field equation, (3.48) can be written
10, We w?11 choose x =
i ct .
13
is just as valid. If both equations are contracted on the indices k
and s and (2.33a) subtracted from (2.33), then clearly
(1)S.S '1)5i.s> = 0 = <234>
The covariant form of a tensor density is defined so that
i).Pr5vi)p ,r
(2.35)
Equation (2.33) is multiplied through by ji.^ and the summation
carried out to give
t
P st 2 ]) ikll ,s
(2.36)
which is replaced in (2.33)
DVh'W.sMiVts+uVst'0 <237>
The definition of a density,
c/iik 1/2 ik . x i/ IK
jj = g g = (det. g.k) g
1/2 ik
(2.38)
and the rule for the derivative of a determinant,
ab ab
9,s = 99 9ab,s = 99ab9 ,s
(2.39)
are used to bring (2.36) into the final form,
9Â¡k,b 9skP ib gisr bk *
(2.40)
This set of sixtyfour equations gives the relations for the sixtyfour
b b
P *k in terms of the g.^, and their partial derivatives. Once the p
ik
CHAPTER 5
SUMMARY
There is a possibility that the main features of the theory pre
sented have been obscured by the preponderance of mathematics. For
this reason, it may be well to summarize briefly the results of this
dissertation.
The second chapter contains no new results. It is intended as
a background to familiarize the reader with the different versions
of the unified field theory. The similarities and differences are
pointed out. All three have the same goalto derive all of the fields
and field laws governing Nature from a single tensor, which need not
have symmetry. The main difference is the choice of integrand for
the variational principle. This, of course, leads to different
forms of the field equations; however, the same set of sixtyfour alge
braic equations appears in each version. The most important similarity
in the theories is that they are based on the same geometrical con
cepts, and they all go over into the gravitational equations of
general relativity in a limiting case. Notwithstanding all the beauti
ful mathematical formalism, this correspondence to the laws of gravi
tation is the only physical content which has been derived from any of
the theories up to now.
The reason for this difficulty was pointed out in the next chapter.
Before Einstein's hypothesis (which was discussed in the introduction)
can be tested, the differential field equations must be solved. Before
51
5
Requirements of covariance would demand that this relation be true in
any coordinate system. Upon transformation
dA = (A'*) dxJ'
&cJ
(2.3)
Since A* is a vector, use can be made of its transformation law.
(2.4)
(2.5)
i k. xb dx*
dA = r (r A ) r
3x dx dx*
i dx lnk k d^x' b
dA = r dA + A rr dx
dxk dxbdxk
The vanishing of the second term would insure the equality of the two
vectors to be a covariant equation. Yet, in general, this is not so.
An alternate way to require covariance of equation (2.2) is just to
say that the difference between two vectors should transform like a
vector.
The difficulty in (2.5) is due to the fact that the vectors were
compared at different points. it becomes necessary to find some pre
scription for translating vectors so they may be compared at the same
point. This method is called parallel translation, and is accomplished
in such a manner as to make the equality of vectors a covariant relation.
That is, the difference between the vectors, when compared at the same
point, will be a vector.
The vector A* translated parallel from x' to x* + dx* will be denoted
Â¡I Â¡ J
by A + 5A Then at x + dx the difference between the two vectors
will be
DA1 = (A1 + dA*) (A1 + Sa') = dA1 5A* .
(2.6)
17
set of equations and require the g.^ to be composed of only a symmetric
part, a.k
'ik
= a
ik
= a
ki
(2.57)
b
The solution of equation (2.40) for the P ^ is readily found by per
muting the indices to get three equations
s s
i k,b
 a  1 ..
sk! ib
' Â¡s
1
(2.40)
s
s
kb, i
3sbP ki
 aksi
" ib 0
(2.58)
s
s
bi,k
w a { 1 ,
s i l bk
abs I
= 0 .
ki
(2.59)
A combination of these three equations gives
akb,i + abi,k aik,b 2aSb P ki
(2.60)
Equation (2.35) applies here
r\,
abp(a. + a . a )
2 kp,i pi,k ik,p'
(2.61)
The symmetry of the p is a consequence of the symmetry of the a.^.
nb = nb = ^ b
I ki I ik (i k)
(2.62)
The last notation is a matter of convention. Equation (2.61) is the
"Christoffel symbol," where it is understood that only the symmetric part
of the q., is used. The symmetric tensor, a , is identified as the
i k ik
metric tensor of the space.
Since equation (2.30) is satisfied identically, the may as well
be chosen zero, in which case (2.31) and (2.32) could be recombined as
(2.63)
41
Ik i k ik
A A = A
bm;pq bm;qp
_s ik
G + A Gs +
sm bpq bs mpq
. sk i .is* * k
 A G A G
bm spq bm spq
(4.51)
Ik*** s
where A is any tensor of arbitrary rank and G ^ is defined
by (4.32). This identity allows the left hand side of (3.48) to be
wr itten
r
ikjpb
v
pp ,+Pp (rp +Pp +pp )
kb;pi bi:Dk 'I ikh kb:i I bi:k .
v
bi ;pk
v
i k; b
v
kb; i
v
bi;k ;p
rpt/,pbnp,/kpb+rv
tkGtkpÂ¡+r kA,,*n
V
\k tbp
kt bp i
V
pK
kb tip
rv^rv.^r'
kt i pk
v
bi tkp
(4.52)
If G*3.^ is written out as prescribed in (4.32), then it is seen to be
Gk.. = R..
tkp i k
(453)
It is clear that if any of the i, k, b are equal, then (4.52)
vanishes, as does the right hand side of (348). Equation (3.48)
represents only four distinct equations. They are:
P U ;p2 ^ l1 32 ;pl +P y ;p3 ~ $2 (0 13 ;2 + ^32 ; 1 + *21 ;3}
P I4;p2  42;pl +n 21 ;p4 = ~^2 ^14,2 + ^42; 1 + *21 ;4^
V s/ V
P 43;p2+P 32;p4+P 24;p3 = ^2 (*43 ;2 + ^32 ;4 + *24;3*
P y3;p4+r 34;pl +P 41 ;P3 = ^2 (*13 ;4 + *34; 1 + *4l ;3}
(4.54)
(4.55)
(4.56)
(4.57)
The left hand side of (4.54) is expanded like (4.52).
CONTRIBUTIONS TO THE
EINSTEINKURSUNOGLU FIELD EQUATIONS
By
JOSEPH FRANCIS PIZZO JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1964
w
To Paula
ACKNOWLEDGMENTS
The author extends his gratitude to the members of his committee.
In particular, he is deeply grateful to Dr. J. Kronsbein, who suggested
this problem and has given much of his time in helping to bring about
the solution.
The author would also like to thank both his wife, Paula for her
help and suggestions in the preparation of the first draft, and Miss
Nana Royer for the typing of the final copy.
ii 1
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS Â¡II
CHAPTER
1. INTRODUCTION 1
2. CURVATURE, DISPLACEMENT AND FIELD EQUATIONS .... 4
3. REDUCTION AND SOLUTION 20
4. SPECIAL SOLUTIONS 30
Minkowski Space 30
A More General Case 33
5. SUMMARY 51
BIBLIOGRAPHY 54
BIOGRAPHICAL SKETCH 56
CHAPTER 1
INTRODUCTION
In 1916, Einstein's classic paper, in which he formulated the
general theory of relativity, appeared in Annalen der Physik. [l]^
In this paper he was able to describe the phenomena of gravitation in
terms of geometrical concepts. The field equations of gravitation were
shown to be derivable from a variational principle, using a symmetric,
second rank, covariant tensor. This tensor (which we will call the
fundamental tensor, and will denote by g.^) represents the gravitational
potentials.
The theory was beautiful, and moreover it worked.' Einstein was
still not satisfied. He reasoned that there are other fields in Nature
besides gravitation. How do they fit into this picture? An example is
the electromagnetic field and equations. There is no natural way for
them to be included. To express the covariant electrodynamic equations,
Maxwell's equations are written, then covariant derivatives are taken in
place of ordinary partial derivatives. In other words, electrodynamics
must be introduced separately. This type of inclusion has been con
sidered arbitrary and unsatisfactory by many theoreticians. Indeed,
Einstein himself has stated, "A theory in which the gravitational field
and the electromagnetic field do not enter as logically distinct
1. The numbers in square brackets refer to the bibliography at the
end of the dissertation.
1
2
structures would be much preferable." (See [73, page 115) This then is
the aim of Einstein's unified field theory: to derive all fields from
one, single nonsymmetric tensor.
Einstein formulated the unified field theory according to the
same pattern he had used for his theory of gravitation, with the excep
tion that now he did not require the fundamental tensor to be symmetric.
One of the resultant field equations turns out to be a set of sixty
four algebraic equations in sixtyfour unknown functions of the g.^.
These equations have to be solved before the other three field laws
(second order partial differential equations) can be set up. This pro
blem occupied much of Einstein's later years, and although he found a
solution to the sixtyfour equations, it was in his own expressed
opinion, too complicated to be of any further use.
This is essentially where the problem has stood until this time.
There has been no verification of Einstein's hypothesis, that other
fields are included in the fundamental tensor, because the differential
equations have never been solved. In this dissertation, we are able
to show that, by a special restriction on the fundamental tensor, the
unified field equations become those of gravitation and electrodynamics,
while the components of g.^ represent the gravitational and electro
magnetic fields. That which was introduced artificially before, now
comes about naturally from one single tensor, in this special case.
The geometrical concepts upon which the unified field theory is
based, are developed in the second chapter. Also the different versions
of the theory (Einstein's, Schroedinger's, and Kursunoglu's) are dis
cussed here. The correspondence of all of the theories to the gravita
tional equations of general relativity is shown for limiting cases.
3
Chapter three embodies our version of the theory. The sixtyfour
equations are shown to be reducible to twentyfour, which still cannot
be solved in a useful form. Nevertheless, they suggest a modification
in the fundamental tensor which allows a tractable form for the solution
of the equations to be obtained. The three sets of differential equa
tions, referred to above, are constructed. One set is the same as the
gravitational equations. The next is seen to be the first set of co
variant electrodynamic equations in the absence of charges and currents.
The last is actually a set of third order partial differential equations
and needs further investigation. It turns out we are able to solve this
problem in our version of the theory.
In chapter four two cases are considered. First, the equations are
examined when the radius of the Einstein universe is taken infinite, in
which case the gravitational equations are satisfied identically. The
other two sets of equations are shown to be Maxwell's equations in the
absence of matter. Next we take the more general case of a finite radius.
The first two sets of differential equations were investigated before.
Now, a term by term examination reveals that the last field law implies
the second set of covariant electrodynamic equations. The last part of
this chapter is devoted to an exact solution of these equations in a special
coordinate system.
Throughout this dissertation an elementary understanding of tensor
algebra and a basic knowledge of general relativity are assumed.
CHAPTER 2
CURVATURE, DISPLACEMENT, AND FIELD EQUATIONS
There are three varieties of unified field theory: Einstein's,
Schroedinger's, and Kursunoglu's. Throughout each of these, two funda
mental entities are dominant; the displacement field and the curvature
tensor. These concepts require some elaboration before they are used
to derive the field equations.
In relativity, we deal with quantities known as tensors, which
have an appealing property; a tensor equation remains unchanged, re
gardless of the coordinate system in which it is expressed. That is,
tensor equations are covariant. When the laws of physics are written
as tensor equations, all reference frames are treated equally. The
idea of a preferred system no longer exists. This formulation breaks
down when we try to compare vectors at infinitesimally separated points.
Consider, for example, the vector A* at the point x' and the vector
A* + dA* at the point x' + dx'. The difference between the two is
(A* + dA1) A1 = dA* .
(2.1)
To illustrate the problem that has now arisen, suppose the vectors are
2
equal in a particular coordinate system.
dA = 0
= A* ,j dxJ* .
(2.2)
2. The comma indicates ordinary partial derivatives.
4
5
Requirements of covariance would demand that this relation be true in
any coordinate system. Upon transformation
dA = (A'*) dxJ'
&cJ
(2.3)
Since A* is a vector, use can be made of its transformation law.
(2.4)
(2.5)
i k. xb dx*
dA = r (r A ) r
3x dx dx*
i dx lnk k d^x' b
dA = r dA + A rr dx
dxk dxbdxk
The vanishing of the second term would insure the equality of the two
vectors to be a covariant equation. Yet, in general, this is not so.
An alternate way to require covariance of equation (2.2) is just to
say that the difference between two vectors should transform like a
vector.
The difficulty in (2.5) is due to the fact that the vectors were
compared at different points. it becomes necessary to find some pre
scription for translating vectors so they may be compared at the same
point. This method is called parallel translation, and is accomplished
in such a manner as to make the equality of vectors a covariant relation.
That is, the difference between the vectors, when compared at the same
point, will be a vector.
The vector A* translated parallel from x' to x* + dx* will be denoted
Â¡I Â¡ J
by A + 5A Then at x + dx the difference between the two vectors
will be
DA1 = (A1 + dA*) (A1 + Sa') = dA1 5A* .
(2.6)
6
The expression &A will depend on the vector A and dx and can be
represented by
5A1 = P AS dx1
I st
(27)
where the) is called the Displacement Field. Its components,
which are to be determined, are functions of the x'. In this form
da! = ca*,t + r st aSj dxt
(2.8)
The term in brackets is denoted by a special symbol,
A
Jt
= A
t
+
(2.9)
and is called a covariant derivative.
The difference, DA between the two vectors at the same point
is to be a vector itself. This requires the covariant derivative to be
a tensor. Therefore the transformation law for the displacement field
must be
(2.10)
It is clear that ordinary derivatives do not form a tensor and
covariant derivatives must be taken instead.
Before the displacement field can be determined beyond its trans
formation law, (2.10), we must know more about the distribution of
matter and charges which dictate the structure of the space.
It is clear that the displacement field is somehow related to the
curvature of the space. So far, in this presentation, the notion of
curvature has been a vague one, at best. Using the idea of parallel
displacement, a mathematical picture of curvature may be displayed.
7
Let a vector, A be transferred parallel to itself along the boundary
of an infinitesimal surface element and brought back to its starting
point, x* y We propose to compute the change, Aa', in the vector
after one complete circuit. This change is given by
aa' = pA' .
(2.11)
Expression (2.7) is used for &A1. The quantities P and AS are ex
panded about their values at the initial point, and terms kept to first
order, since the curve is infinitely small. The expansions are
s
AS(xJ) = A;
(s)
 r a
I pq
(s)
(x
xM, \)
(sr
(2.12)
and
r st r st xj(s) + r st.,
(s)
(xq xq(s)) (2.13)
Let
t t t
X x (5) T) .
The change in the vector is now written
^,r,.t.q*w + ristr>M*Wdi>t
(2.14)
(2.15)
With an appropriate renaming of dummy indices this becomes
 'r'pt.q+r'strspq] flP#nqd,it (z,6>
it is understood that the terms preceding the integral are the initial
values. As usual, we are interested in the displacement field and want
to investigate the term in brackets to determine tensor character. It
8
can be shown that the integral transforms as a tensor. Moreover, it is
an antisymmetric tensor since
= MnV) (2.17)
and the first term on the right hand side vanishes, because it is an
exact differential. The integral will be denoted by^
f^ = ^TfdT]1 (2.18)
In equation (2.16), the quantity in parentheses can be expressed as a
sum of parts which are symmetric and antisymmetric on the indices t
and q. The contraction of the symmetric part with the integral
vanishes leaving
W 
M pt,q 1 pq,t 1 stl pq sql pt '
It is know that Aa* is a vector since it is obtained by parallel dis
placement. The term A^f^ is a third rank tensor, therefore the
tensor character of
ptq
= rPt,q+r pq.t+r str pqr
r
sql pt
(2.20)
is established. This is the wellknown RiemannChristoffel tensor.
R* is also referred to as the curvature tensor, because whenever it
ptq
vanishes, there is no change in a vector after it is displaced parallel
to itself around a complete circuit. The space is then said to have
no curvature, or to be flat.
Contraction on the indices i and q gives the second rank tensor,
3. The hook under the indices is used to indicate antisymmetry.
9
Rik
rVp*r
ip3k
q
Pk
qp
r
I ik
(2.21)
which is named the Ricci tensor, or the contracted curvature tensor.
It is to play a centra.! role in the variational principle from which
the field equations are to be derived.
For a variational method, we postulate an invariant space integral
which involves the displacement field. Invariance of this integral pre
4
supposes the existence of a scalar density, L, which can be formed
by the contraction of the Ricci tensor with a contravariant tensor
density, Â£]j 1 The field laws are to be derived from
6/ LdT = 0 ,
(2.22)
where dt is an element of fourvolume and
(2.23)
ik
is considered as a function of the jj ^ and P which are to be
varied independently, their variations vanishing on the boundary of
integration.
This postulate would lead to the equations of the purely gravi
tational field if the condition of symmetry were imposed on and
Pb
I (This will be demonstrated in the next chapter,) However, the
present theory is an attempt to generalize these equations, and the
constructions which were essential for the setting up of covariant
4. A scalar density transforms like a scalar but with the inverse
of the Jacobian determinant included. This is necessary to cancel the
Jacobian which results from the transformation of the element of four
volume. The result is that the integral transforms like an invariant.
10
equations are independent of the assumption of symmetry.' Instead of
symmetry, an analogous condition is posited. it is referred to as
.. b
i k r~i
Transposition Invariance. Even though and  are nonsymmetric,
their transformation laws would be invariant if the indices of these
variables were to be transposed and then the free indices interchanged.
This is what is meant by transposition invariance. In addition to the
variational principle, it will be postulated that all field laws shall
be transposition invariant."'
If we proceed to substitute (2.21) for R.^ into equation (2.23)
and carry out the variation indicated in (2.22), we would find that the
resulting equations are not transposition invariant. To circumvent
such a distasteful result, four new arbitrary variables, are intro
duced by making a formal change in the description of the field. In
b
place of p the following substitution is made
U b
(2.24)
r ik r.
sb ..
., 4 V
i k i k
*b b
The  ^ is a displacement field just as the j The are treated
in the variational procedure as independent variables. After the
5. An example of an equation which is transposition invariant is
s s
9ik,b 9sk p ib 9isp bk ^
Transposing the indices on the variables yields
s s
9k,b 9ks P bi 9si P kb ~ ^
Interchanging the free indices (i and k) gives back the original
equation
s s
9ik,b 9is P bk 9sk ib ~ ^
] 1
variation is carried out, the A. will be chosen to make the field
k
equations transposition invariant, and then they will be eliminated
from the system.
b
The contracted curvature tensor, with J""1 ^ replaced by (2.24)
becomes
*
Rik Rlk <2'2la>
where R.,(P)is the same as expression (2.21), except that a star is
b
put over all the p1 The variational integral is now
rk
^[Ulk[Rik+ <\k \)u dT (225)
Â¡k *b
After the variations in jj j1 .and A.^ are made, the integrand,
denoted by I, takes on the following appearance
I = [R
ik
*
+ (A..
i,k
+
k
s
(2.26)
The last term is a generalized divergence and since it appears in a four
volume integral, it can be converted to an integral over the three
manifold enclosing this volume. The variations vanish on the boundary,
hence the last term in equation (2.26) can be ignored. The variations
in the remaining terms are all independent of each other so each
coefficient must vanish separately, giving the three equations
R..
t k
*
(D
Vi} 
(2.27)
12
(2.28)
(2.29)
The four extra variables, may be given any value. The three equa
tions will be transposition invariant if ^ 's chosen such that
I Ik0 (2.30)
V
Now can be eliminated entirely from the equations by writing (2.27)
6
as two equations.
Rik=
R., + R, . + R, ,
ik,b kb,i bi,k
v v' v
= 0
(231)
(2.32)
b
The star may just as well be excluded from above the p1 ^, renaming them.
By contraction on the indices k and s in (2.29), we can verify that the
expression in parentheses vanishes and the equation can be written
DVlfPtsMf Tstifr0 (2.33)
Before further simplification is made, it should be noted that the field
law, (2.28) is already implied in this expression. This can be seen
from the following consideration. The equation formed by exchanging
free indices,
tk
it
i <
a Â¡,k n
(2.33a)
6. The dash under the indices indicates symmetry.
13
is just as valid. If both equations are contracted on the indices k
and s and (2.33a) subtracted from (2.33), then clearly
(1)S.S '1)5i.s> = 0 = <234>
The covariant form of a tensor density is defined so that
i).Pr5vi)p ,r
(2.35)
Equation (2.33) is multiplied through by ji.^ and the summation
carried out to give
t
P st 2 ]) ikll ,s
(2.36)
which is replaced in (2.33)
DVh'W.sMiVts+uVst'0 <237>
The definition of a density,
c/iik 1/2 ik . x i/ IK
jj = g g = (det. g.k) g
1/2 ik
(2.38)
and the rule for the derivative of a determinant,
ab ab
9,s = 99 9ab,s = 99ab9 ,s
(2.39)
are used to bring (2.36) into the final form,
9Â¡k,b 9skP ib gisr bk *
(2.40)
This set of sixtyfour equations gives the relations for the sixtyfour
b b
P *k in terms of the g.^, and their partial derivatives. Once the p
ik
14
are known, they are substituted in the other three field equations,^
k
r ,k
(2.30)
V
Â¡k=
(2.31)
Rik,b + Rkb,i + Rbi,k = = R[ik,b]
(2.32)
I IN W INU ,1 U I IN t IN
V V V V
for a solution of g.^ This is the formalism of Einstein's unified
field theory [5].
Schroedinger's unified field theory is quite similar to that of
Einstein's. Parallel transfer, displacement fields, and the curvature
tensor all form the basis for the theory. The difference appears in
the integrand of the variational principle. Whereas Einstein took
i k
j R.k as thescalar density, Schroedinger chooses [18]
L (det. R.k),/2 (2.41)
which is the simplest scalar density that can be built out of the
curvature tensor. The constants 2 and X have no influence on the result.
A contravariant tensor density is introduced here also:
r ik 3l
1)
(2.42)
but it can be eliminated in the end, leaving equations which contain
b
only the p .^.
When the variation,
6^LdT = 0RikdT = 0
i k
(2.43)
7. The square brackets in (2.32) indicate summation over the cyclic
permutations of the enclosed indices. This convention will be used
hereafter.
15
is carried out with (2.4l) as integrand, the result is the following set
of equations,
9lk,b 9skP ib 9isP bk 0 (2.Mt)
Rik=X9lk <245)
These can be taken as the field laws as they stand, but Schroedinger
takes the theory further. Equation (2.45) can be substituted in (2.44)
to eliminate the g.,. The field equations are now sixtyfour differential
'k b
equations involving nothing but the sixtyfour
s s
R., R T *k ~ R P . = 0 (2.46)
ik,b ski ib is I bk '
This is known as the "purely affine theory."
Kursunoglu's approach to the theory [16] retains the basic ideas
of displacement field, curvature tensor, and the fundamental tensor
I k
g.^ (or g ) but the scalar density for the variational integral is
quite different and entails some new concepts. These will be defined
before the variational principle and the field equations are displayed.
The fundamental tensor, g.^, is expressed in Kursunoglu's theory,
in terms of its symmetric part, a.^, and its antisymmetric part, 4>.^.
gik aik + (2G)'/2cVik
where G is the gravitational constant and rQ is a "fundamental length."
The determinant, g, of g.^ is constructed in terms of what will be
called the "Kursunoglu invariants," namely
=Uik*
2 ik
(2.48)
16
and
(a)
J/2eijkb *
j kb
(2.49)
,, or. 4 2n 2 8 4.2.
g = a(l t 2Gc rQ JL 4G c ro X. )
(2.50)
The action principle from which the equations are to be deduced is
6/[ Â£jj'kRJk. 2ro"2(^b ^a)]dT = 0 (2.51)
where b is the determinant of the tensor,
brk a'/23'l/2(aik **\\s*\) (252)
The variations carried out in (2.51) produces the Kursunoglu field
equations.
s
9Â¡k,b 9skT ib
s
(2.53)
(a
ik
(2.54)
R[Â¡k,b]
v
= r
2
(ik.!
kb, i
V J
b i k
= 0 .
(2.55)
(2.56)
The nomenclature, "fundamental length," is rationalized by examining
the field equations. The vanishing of r^ gives the general relativis
tic case in the absence of charges. This existence of free charges is
now associated with a finite fundamental length.
All three field theories discussed here have at least one thing in
common: they have been constructed so that there exists a correspond
ence principle which takes the unified field equations over the well
known gravitational equations of general relativity when the anti
symmetric field is absent. This is easy to see. First consider Einstein's
17
set of equations and require the g.^ to be composed of only a symmetric
part, a.k
'ik
= a
ik
= a
ki
(2.57)
b
The solution of equation (2.40) for the P ^ is readily found by per
muting the indices to get three equations
s s
i k,b
 a  1 ..
sk! ib
' Â¡s
1
(2.40)
s
s
kb, i
3sbP ki
 aksi
" ib 0
(2.58)
s
s
bi,k
w a { 1 ,
s i l bk
abs I
= 0 .
ki
(2.59)
A combination of these three equations gives
akb,i + abi,k aik,b 2aSb P ki
(2.60)
Equation (2.35) applies here
r\,
abp(a. + a . a )
2 kp,i pi,k ik,p'
(2.61)
The symmetry of the p is a consequence of the symmetry of the a.^.
nb = nb = ^ b
I ki I ik (i k)
(2.62)
The last notation is a matter of convention. Equation (2.61) is the
"Christoffel symbol," where it is understood that only the symmetric part
of the q., is used. The symmetric tensor, a , is identified as the
i k ik
metric tensor of the space.
Since equation (2.30) is satisfied identically, the may as well
be chosen zero, in which case (2.31) and (2.32) could be recombined as
(2.63)
18
This is precisely the gravitational field equation in empty space.
(See, for example, [9], page 81) A particular solution to these equa
tions corresponds to the field of an isolated particle continually at
rest. The famous explanation of the discrepancy in the advance of
Mercury's perihelion is a result of the solution to (2.63). There is
no question about the physical significance of the gravitational field
equations, so this gives a certain measure of confidence to the generali
zation, (2.30), (2.31), (2.32), and (2.40).
A similar situation exists with Schroedinger1s theory in the limit
of a symmetric g.,. The field equations, (2.45), become the same as
i K
Einstein's with the addition of a term involving X, which is now identi
fied with the cosmological constant. Actually, the limiting case of
Schroedinger's theory is the original form of the gravitational equa
tions. The cosmological constant is so small that it need not be in
cluded on a scale such as our solar system. (See [9], p. 100)
In Kursunoglu's theory, the correspondence to general relativity
is achieved by the vanishing of the fundamental length. Again we revert
to the equations of general relativity. This method is much like quantum
mechanical correspondence to classical mechanics when Planck's constant
vanishes.
Therefore all three theories have at least some basis, due to the
fact that they reduce in the limit to the wellknown and tested equations
of gravitation derived by Einstein in his general theory of relativity.
What has been presented so far is only the problem: a nonsymmetric,
fundamental tensor, g is chosen to represent the fields in Nature.
Ik b
The displacement field, p *s introduced to insure that the field
laws will not be dependent upon the choice of a coordinate system. From
19
a postulatory basis, a variational principle is used to derive a set
of field equations. A solution to these equations will presumably
give a description of nature in which all fields are united in the single
tensor, g.^ in the same manner as the electric and magnetic fields are
unified in the electromagnetic tensor.
Now that the problem is laid before us, the next step is to pro
vide a solution to these field equations.
CHAPTER 3
REDUCTION AND SOLUTION
It is evident that the three versions of unified field theory,
which have been presented, are little more than postulates. Their
tenuous claim to validity comes from the correspondence to the known
equations of general relativity and the correct count of functions.
So far there has been no indication that they give a field description
of Nature in which all fields are united in a single tensor.
In keeping with this spirit, we choose to deal with yet another
form of the equations, which can be considered as an adaptation of
either Schroedinger's or Kursunoglu's equations to an Einstein model
of the universe.
s s
1.. ~
i k,b
9sk
r Â¡b 9s r bk
(3.1)
2_
g j k i 4 k
(32)
0
, otherwise
[ik.b]
V
= 
Â¡2 3[lk,b]
(3.3)
rk
1 ik
= 0
(3.4)
v
In the limit of a vanishing antisymmetric field, the antisymmetrJc part
b b
of j vanishes and the symmetric component, p .^, is the Chrlstoffel
symbol. Then (3.2) is seen to be exactly the set of differential equa
tions which describe the Einstein Cylindrical Model of the universe (See
20
21
[9], p. 159) Equations (33) and (34) are satisfied identically. Due
to this correspondence, these equations should be considered as valid as
the other three sets.
In all four cases, it has not been possible to ascertain if the
theory contains any information other than the simple reduction to the
gravitational equations. The reason for this is that a simple, general
b b
expression for the p .^ has not been found. True, the p .^ have been
expressed in special coordinates, but the differential equations have
not been solved using these values. In this chapter and the subsequent
b
ones an appealingly simple form for the p ^ in a special case, along
with new information from the field equations, will be presented.
b
The relation for P .. (31) is common to all four versions. It
I ,k b
is a set of sixtyfour equations from which the sixtyfour [""* .^ are to
be determined as functions of the sixteen g.. and their derivatives.
ik
A solution to these equations has been given for a system of
spherically symmetric coordinates. [17, 21] Kursunoglu [16] sets up
b
the differential equations, (2.31) and (2.32), using the j""1 expressed
in terms of the g.^, but can offer no explicit solution. In fact, no
solution has yet been given to equation (2.31) in these special coordi
nates. It is for this reason that a completely general (coordinate inde
pendent) solution to (31) is desired.
A little manipulation of (31) is enough to show that a general
solution is far from trivial. Nevertheless, several formalistic solu
tions have been offered. [4, 10, 22] Formalistic, in this context, is
b
meant to imply that an explicit expression for P ., is never written
lik b
down. What is given is a prescription for determining p A brief
demonstration of Einstein's formalistic solution [4] may illustrate this
are considered as individual
22
n
better. In (3.1), gsk Â¡b and gjs.p
bk
terms and named
9sk r ib = V* kb *
9isT bk Wbkf I
The V.kb and W.kjb are related to each other by
tb bt
9 Vikb 9 Wikb
This allows the Vjkjb to be expressed as a in (3.1)
9 i k, b Wbki gP 9tkWib(p = 0 '
b
Now, if the W.^jk are found, then the J .^ are known as
of (3.6)
r
bk
 fP,
9 Wbki *
Equation (38) implies another expression for W,
bkf
Wbki 9ik,b 9 9akWibk
which is reinserted in place of the last term in (38)
k 'a
9ik,b Ubk I "Â¡UkVb.l t sk a9akgb P9pbWkib' 0
If this procedure is repeated once more,
k1 a k1 a b 1 p
9ik,b 9 9ak9k'b,i 9 9ak9 9pb9p'i,k
ii ^ b 'a k'P *'c w
Wbki + 9 9ab9 9pk9 gciWb'k1{i' '
(3.5)
(3.6)
(37)
(3.8)
consequence
(3.9)
(3.10)
(3.11)
(3.12)
23
Einstein now defines
flikb=9bk,i gk\kVl,b + s'^a/'VVb.k' <3'l3)
and
i1k'b1 Ri1 Rk' Rb' i'a k'p b'c
U 5,.L = 5 .5 ,5 + g q_,g K
i kb
k b + 9 9ai9 9pk9 9cb
(314)
so that
i k' b '
U W. ,
ikb wi'k'b1 Aikb
The problem is to find U, the inverse of U, so that
(3.15)
Wikb =
i k' b 1
ikb
A.
k'  b *
(3.16)
and the knowledge of W.^j^ then gives the P by (3.9).
Einstein's prescription for finding the inverse is presented in
four pages and the answer becomes much too complicated to write down
explicitly here. To quote Einstein in this paper, Such a solution can
indeed by arrived at. .but it is cumbersome, and not of any practical
utility for solving the differential equations." This statement also
applies to the solutions obtained by Mishra [22] and Hlavaty[lO],
So, for all practical purposes, we are still left with the problem
of sixtyfour equations in sixtyfour unknowns. One additional reduction
is possible. The number of equations and unknowns can be reduced to
nb
twentyfour if we treat symmetric and ant isymmetric components of  .^
as separate quantities. [213 (This is a natural thing to do since they
transform separately. In fact, it can be seen from (2.10) that the anti
symmetric components transform like a tensor, whereas the symmetric
ones do not.) The fundamental tensor is also written as the sum of its
parts. Consider the two equivalent equations
24
9j_k,b + 9k,b 9sk
r5ib + gski
s s
l ib + 9sk i ib + 9sk \
s
v
V V ~~ V
V
s
n
s
p
s s
n n
(3.U)
+ r bk+9
'is 1 bk + 9
v
s
is i bk 9is I bk
V V V
S S
s
9ki ,b + 9ki,b 9sÂ¡ P kb *" 9s? T kb + 9siP kb + 9 si P kb
s s s
s
+ 9ksP bl+ 9ksr bi + kjT bi+ 9
ksP bi
V V
(3.17)
The sum and difference are two new equations
s s
. s
ts
9?kb 9sk P j_b + 9is P bk + 9sk
ib + 9isP
V V
bk
V
(3.18)
s s
S
s
9Â¡k,b = 9sk T ib + 9is! bk + 9sk [
V v
"n> + 9Â¡vsl
(3.19)
This procedure is repeated twice by permuting the indices i,
k, and b
Two more pairs of equations are obtained.
s s
S
s
9kb, i 9sb T kj. + 9ks P ib + 9sb [
n ki + 9ks [
V V
"ib
V
(3.20)
s s
s
S
9kb, i 9sb.P k + 9ks P ib + 9sb
r kÂ¡ + gks
V
r ib
(3.21)
and
s s
S
s
9bi k ~ 9s i P bk ^ 9bs P kj_ r 9sj
bk + 9bs1
V V
ki
(3.22)
s s
n tj
s
s
9bi,k 9si  bk 9bs ki + 9sil
V V v v
bk + 9bsP
V
' J
(3.23)
The following combinations are taken
r*s
S
s
Pk.b + 9kb,Â¡ + 9bi,k 2[gsb Ik + 9sil
1 bk+ 9ksf
V V
ib* .
V
(3.24)
25
Using (2.62) and (2.61) as definition,
r ik bkV ^ (9Â¡sr Pk+^rv 2s)
V v V n/
Substitution of (325) into (319) and use of definition (2.9) for a
covariant derivative gives the set of twentyfour equations,
P P
9ik;b = 9pk T ib + 9ip P bk + gE~ ^9pk^9mb T
m
m
V v
m
+ 9im P sb^ + 9ip^9mkP bs + 9bm P sk^
(326)
V
to tie solved for the twentyfour I .. When they are found, substi
b b
tution in (3.25) yields p .^ Therefore the sixtyfour j ^ are
known once (326) is solved. This still does not give a tractable
form for solving (32). Einstein's hypothesis cannot be tested unless
we find some way of solving the differential equations and these in
P*3
turn cannot be solved until a useful, general form for the J .^ is ob
tained.
We are not completely stymied by the formidabi1ity of the equations.
A further advancement can be made. It can be shown that, within the frame
work of the theory, the gravitational and electromagnetic fields are con
tained in the single tensor, g.^, and the field equations are those of
electrodynamics as well as gravitation. To see how this comes about,
an alteration in the fundamental tensor is made. It is chosen to be
a CliffordHermitian tensor. This means the antisymmetric part is
chosen as
9ik = ^ik
V
(3.27)
where e is so small that its squares and higher powers will be neglected.
Then the tensor,
26
9Â¡k 9Â¡k + ^ik
(328)
is called a CliffordHermitian tensor after W. K. Clifford.
For the CliffordHermitÂ¡an field, (326) becomes
ik;b = 9Â£k f^ib + bk '
v y
If i, k, and b are permuted, two more equations are obtained.
(3.29)
ev. kÂ¡ vr Â¡vb <3'3)
4,bÂ¡;k= Â£lp\k+ g^pPy (3.31)
The following combination of the three equations is taken
2 (ik;b 45 kb; ? ^b; k^ 9pb P fk
Which is immediately solved for
^ 1 pb
a* eaJ
V
This result causes (3.25) to become
b
r
ik i k l
(3.32)
r Ik 2 4>kp;i 0pi ;k> (3.33)
(3.3k)
and the displacement field is known
r ik = \i k^ + 2 gE^0ik;p 0kp; i ^pi ;k* (335)
This is a general form insofar as no particular coordinate system
has been specified. it is also in a useful form for the field equa
tions (32), (33), and (34).
27
First consider (3.2). If (34) is kept in mind, then R.^ is
given by
Rik 2 ^Rik + Rki>
^iPk),p + 2((iPp) ,k + ^kPp],i)
+ s.p0 q,{ > p
' q) cp k) (q p) I* k)
ck2 iJ<
, i 4 k
(336)
If or k take on the value "4", the second part of (32) is identically
satisfied. Equation (336) is a second order partial differential
equation in the g.^. It is easily verified that
'ogtTg]. ,
(3.37)
so that
. $ p,2 + $ p < + $ p K q 7
(i k),p ]f p),k (i q) (p k^
? p {\ q <
(q p) (i k)
(JC2 9ik
(338)
In analogy with general relativity, g.^ must be interpreted as the metric
in the space. If we choose the physical space to possess spherical
symmetry and the time dimension to be uncurved then the line element
g
may be brought into the form
ds2 = &.2cSZ + R2 sin^dO2 + G^2 sin^sinW#2 c2df2 (339)
In which case the metric is
8. "(3C" *s interpreted as the radius of this world.
28
9Â¡k
+ .
O
O
O
+(S?sn2lÂ£
O
O
O +sÂ¡n2'i's i n29
O O
O
O
O
1
(3.40)
The left hand side of (338) is calculated using (3.40) and indeed the
differential equations are satisfied by (3.40). The form (339) repre
sents what is called an Einstein cylindrical model of the universe.
The contravariant tensor to (3.40) is simply
ik
(3.41)
'ik
since g., is represented by a diagonal matrix, (340).
9
With the g.^ solved, we can move on to (3.4), using (333)
r ik = 0Â£3E!('>ik;pVÂ¡ 4,pt;k)
V
(3.42)
Since g^ is symmetric and is antisymmetric, the second term
vanishes. Furthermore, p and k are dummy indices so that the first
and last terms add.
. =0
L i k; p
(3.43)
In this form the field equation is significant since it is recognized
as one of the covariant electrodynamic field equations in the absence
of charges and currents [ 19 with the .. interpreted as components
K
of the electromagnetic tensor.
^ik ;p (3^)
9. The solution will be examined in more detail in the next
chapter.
29
The remaining set of field equations (33) may be written
Rik;b + Rkb;i + Rbi;k = ^2 (Â£f>i k;b + ^kb; i + ^bi;^ 0 M)
v v v ^
From definition (2.21), and (3.4) and the results of the Clifford
Hermitian field, it follows
P n p
Ik [ r lk,p + r Ik \q p'j P Â¡q,qp T qk iPp)'
The term in parentheses vanishes due to (338) and the remaining term
in brackets will be recognized as the definition of the covariant
pP
derivative of the tensor, ( .^ .
V
? k
v
 T Ik;,
(347)
Then by (3.45)
r
p
ik;pb 1 kb;pi + ^ bi ;pk^
(3.48)
It remains to be shown in the next chapter that a solution of
(347) is the second set of covariant electrodynamic equations.
So far, by use of a CliffordHermitian tensor, we have been able
b
to present a reasonable form for the P and a solution to field
I 1 k
equation (32). Also (33) and (3.4) have been put into a more
familiar form and are ready to be solved in the next chapter by use
of a special coordinate system which will make the equations especially
transparent.
CHAPTER 4
SPECIAL SOLUTIONS
In the preceding chapter the field equations were fashioned
into a suitable form for solution. The three equations which must
now be solved are
hLp+Â¡PpU+[.PWWW^^. (3.38,
QC 
i 4 k
g, =0
i k; p
(344)
P ik;pb*r kb;pi P bi;pk = ^2 (ik;b + 0kb;l + ^bi;^ (3,48)
V V V v
Minkowski Space
First, a very special situation will be considered. It is the
opposite of the limiting case where the antisymmetric field vanishes.
Now, the field equations will be examined when is chosen infinitely
large and the CliffordHermitian tensor is assumed to have Minkowski
form,
9ik sik + Â£ik <4J)
In light of this, all Christoffel symbols vanish and (338) is satis
fied Identically. There is no gravitational field.
It is to be expected, since there is no gravitational field, that
what is left of the tensor g.^ should represent the electromagnetic
30
31
field, and the field equations should be Maxwell's equations in a
Minkowski space. To see that this is so, let us investigate (3.44)
and (3.48) for ^
9k6!k (1*2)
First, it is seen that there is no distinction between contravariant
and covariant tensors. Furthermore, any covariant derivative may be
replaced by an ordinary derivative since all Christoffel symbols are
zero. Equation (344) can be expressed
0. = 0 .
>P.P
(4.3)
Indeed, If the 0.^ represents the electromagnetic field,
0
Ik
0
h3
"h2
ie
h3
0
h.
ie,
ie.
i e
ie
(4.4)
then the second set of field equations (344) becomes one set of the
Maxwell's equations in an empty space, as expected
Vxii = j; for 1 = 1, 2, 3
(4.5)
V e = 0
for i = 4
(4.6)
The third field equation, (3.48) can be written
10, We w?11 choose x =
i ct .
32
(["*,., + p., + p,. i) o (^ii, + l + i) (^>7)
' 1 ik,b I kb, i I bi,k',p fi\2 Ik,b kb,i bi ,ky '
v v v
The solutions, (3.33) for P are substituted, using 6!i< for g in
v
accord with (4.2).
ik
i[(<0
O.
J k + <*l

 4> )
21VTik,s ks,i si,k ,b kb,s bs, k sk,b,i
+ (,, ,,).] + ( + 0. ,)
v bi,s is,b sb,i ,k ,s ^2 ik,b kb, i bi,k/
(4.8)
or
oi^nu + ^ik'+^kii) + o (* i k + ct>ik + i) ~ 0 (49)
2 v ik,b kb,i bl,k ,ss Qt2 ik,b kb, i bi,k' '
A solution to this equation is
0 r
ik,b
) + (b
kb,l bi,k
(4.10)
which, in view of (4.4), represents the other set of Maxwell equations
in a Minkowski space.
V h* = 0 ;i,j,k4 (4.11)
V x e* = Â£ ; i, j, or k = 4 (4.12)
Therefore, as expected, the field equations In Minkowski space
become the Maxwell equations, while the fundamental tensor represents
both the metric and the components of the electromagnetic field.
These equations, (4.5), (4.6), (4.11) and (4.12) are well known
their solution and validity need no elaboration.
33
A More General Case
We now move to a more general case, where "(j^" is taken as finite.
A solution to (338) for the g^ has been given in chapter three, but a
slightly different version of the solution will be given here. First,
note must be taken that (3.38) involves only the symmetric part of the
fundamental tensor. By an extension of the example in Minkowski space,
we should anticipate that the g.. will depend upon the distribution of
l K
matter in the universe. At this point the following model is adopted
a static homogeneous universe. This means that all parts are considered
extrinsically and permanently alike. In this case, the line element
can be put in the general spherically symmetric form.
,2 X 2 2^2 2 2q ,.2 v 2 ,2 ...
ds =edr + r do + r sin 0d e c dt (H.13)
where
V = v(r) and X = X(r) (4.14)
Most of the g.^ are already known:
g,k = 0 i 4 k (4.15)
g22 = r2 (4.16)
g^3 = r2sin20 (4.17)
From the differential equations, (339). we need only determine
g,1 = e* (4.18)
V
g44 = e
and
(4.19)
34
The Christoffel symbols are computed according to (2.61) and substituted
in (338). It is seen that the solution is
\
e
0 '
(4.20)
(4.21)
Thus in matrix form, the g.. are
3 i k
0
0
0
0 0 0
r2 0 0
0 r2sin^0 0
0 01
(4.22)
This model is known as the Einstein cylindrical universe. (See [24],
page 359) A change of variables will serve to show more distinctly
the character of this universe. Let
*>,. r2
X] ~ ^ ~ (J(2
x2 = r cos 0
x^ = r s in0 cos
x^ = r sin0 sini (4.23)
In which case,
.2 ..2 2 ,2 .2. 2,2
ds = (dXj + dx? + dx^ + dx^ ) c dt
(4.24)
35
and
2 2 2 2 _2
x, +x2 +x3 +x4
(4.25)
This illustrates that the physical space of the Einstein universe may
be interpreted as the threedimensional bounding manifold of a sphere
of radius (ji^ in a fourdimensional Euclidean space with the cartesian
coordinates given above.^ The time dimension is uncurved. Hence the
name cylindrical universecurved space and straight time.
Even though the line element displays spherical symmetry, there
is no symmetry of form among the W and the R.^. To make the field
equations more transparent, we adopt a method used by Kronsbein [12].
The sphere represented by (4.25) is radially projected from its center
in the fourdimensional space, (x.), into the threedimensional space,
, with coordinates X (Greek letters take on values 1, 2, 3) by the
projection
12
^The solution shown
tained directly from (4.24)
in the previous chapter,
by the transformation
(3.41), may be ob
V
x2
X3
x4
cos Â¥
(Jt s inf cos
(SI sinf sin cos'F
(St sin Â¥sin0 sinY
which expresses the spatial part in fourdimensional spherical coordi
nates .
12
Some people prefer to call this a gnomonic projection.
36
a
% Al/2
x4 "
A1/2
(4.26)
where
A = (R2 + (x)2+ (X2)2 + (X3)2
(4.27)
In this space, X and X = ct t we wÂ¡]] denote the symmetric part of
the fundamental tensor by a^. It is computed by the standard method
to give
alk =
A2
[A (X1)2]
X1X2
x1X3
 X]X2
[A (X2)2]
xV
2 3
x r
 X2X3 [A (x3)2]
0
aL
(4.28)
The determinant is
hk1
a8
7
(4.29)
The contravariant tensor to (4.28) Is found to be
Ik A
a
[(R2 + (x1)2] xx2
xx3
xx2
x]x3
[(R.2 + (x2)2]
2 3
XXJ
x2x3 [? + (X3)2] o
(4.30)
37
With (4.28) and (4.30), the Christoffel symbols can be constructed.
They are very important because they are the symmetric part of the
13
displacement field.
Ia a) a
C )
6 3 = A (4.31)
There is one more quantity which we need to calculate in this space
for future use. It is the analog of the RiemannChristoffel tensor,
(2.20). It will be denoted by
kb
" jiPk],b + \\?b\,k ^k^b^ jiqb^q k'
(4.32)
Notice the antisymmetry.
(4.33)
The components will be listed here for future reference:
.r
xaxB
aBT .2
A
aar
[A (x)2]
A2
(4.34)
Al 1 others are zero
For the antisymmetric part of the fundamental tensor, the following
symbols will be used for the
13
Jn (4.31) and (4.34), the repeated index does not imply
summation.
38
Ik
O
h
3
O
hl
e.
1
0
e.
(4.35)
where the "e and "h are functions of (X^, X^, X3, t).
We now have the necessary material to calculate the antisymmetric
b
components of j in our special coordinates. The actual calculation
of all twentyfour  ..Is a long, tedious process. Only a sample
'v 3
calculation (the component, p w'11 be used as an example) and
V
the final results are given here. By (333),
P 12 ^12; 1 + 3
.32,
1 33
r ,2" a3 ^12;! + a3 012;2 + ^ S^12,3 S3,1 Sl.2^
V
S2 [l 3] *1! {3 2] ]}
(4.37)
Upon substitution of (4.30), (4.31), and (4.35), and consequent can
cellation and simplification, it is found that
^12 P + x3x'h3.1 + + [
v
 [(K. + (X^)21] [j(hj j + h2 Â£ + 3) + S (x'*1i +
2 /3 \ 21 rl, L l x 1 ,J, w2,
+ x3h3)3)
(4.38)
D
The other j are computed in the same manner and soon the following
39
,14
pattern is recognized
aU
X f a hA.,^ V\a (2 V.M + S A V
rV e^^(3Xah1 + aa\ J h + \ A,)l
(^39)
and
p" crA X an I aP., \
I B4 = ^ + X eS + A 9 eM, + 2 9 V.B^
X6 aM I #
 i VV ,1
2*
(4.40)
al so
r
V
6 \,4}
(4.41)
and
r
4?= e,4
(4.42)
These are the twentyfour P ., in the special coordinates. Combined
rib v
with the j .^ given by (4.31), the complete set of the sixtyfour
P jk 's known in this space. This completes the information needed
to set up the field equations, (344) and (348).
It has been pointed out that (34), which reduces to (345) is
identically a set of the covariant electrodynamic equations in the
presence of a gravitational field. To see what they look like in this
Pb
space, it will be easier to go back to (34), since the  ., have
ik
N/
already been computed. The equations represented are:
r
12
V
13
V
L
r
14
v
(4.43)
14c cbx
and are the usual permutation symbols whose value is
zero if any two indices are alike, +1 if (aBA) is an even permutation
of (123), 1 if (aBX) Is an odd permutation of (123).
40
r 3 p 4
21 + 23 +i 24 =
(4.44)
r2 p4
31 + 1 32 +l 34 = 0
(4.45)
4, P 42 1 43 
(4.46)
o
The I are substituted from equations (4.39) to (4.42).
simplification, the four equations are:
After
3(x2h3 x3h2) + [.2 + (x2)2]h3 2 ia2 + (x3)2]h2 3
4
 X2X3(h33 h2>2) x'(X2h3J X3h2J) 0 (4.47)
3(X3h 1 X1 h3) + [(SL2 + (X3)2Jh1 3 [iR.2 + (X1)2]h3jl
4
+ x3x'(h,,l h3 3> +. X2(X3h12 x'h3_2) 4e2,4=0 <4'43>
3 (X1 h2 X2h1) + [
4
+ X X (h2^2 hj^) + X (X h2^3 X hj ^) ^ = 0 (4.49)
2(X1e] + X2e2 + X3e3) + [ft2 + (x')2 e, ] + xVe, 2 + X,X3e] 3
+ xVe^, + [ft2 + (X2)2]e2j2 + xVe^
+ X3X1e3 j + X3X2e3 2 + [R.2 + (X3)2]e3 3 = 0 (4.50)
These equations are exceedingly complicated as they stand. They will
be left this way for the time being. After the other set of four field
equations (3.48), has been written in this space, a simplification and
solution will be presented for all eight equations.
A tensor identity which will be of great utility in the investi
gation of (348) is
41
Ik i k ik
A A = A
bm;pq bm;qp
_s ik
G + A Gs +
sm bpq bs mpq
. sk i .is* * k
 A G A G
bm spq bm spq
(4.51)
Ik*** s
where A is any tensor of arbitrary rank and G ^ is defined
by (4.32). This identity allows the left hand side of (3.48) to be
wr itten
r
ikjpb
v
pp ,+Pp (rp +Pp +pp )
kb;pi bi:Dk 'I ikh kb:i I bi:k .
v
bi ;pk
v
i k; b
v
kb; i
v
bi;k ;p
rpt/,pbnp,/kpb+rv
tkGtkpÂ¡+r kA,,*n
V
\k tbp
kt bp i
V
pK
kb tip
rv^rv.^r'
kt i pk
v
bi tkp
(4.52)
If G*3.^ is written out as prescribed in (4.32), then it is seen to be
Gk.. = R..
tkp i k
(453)
It is clear that if any of the i, k, b are equal, then (4.52)
vanishes, as does the right hand side of (348). Equation (3.48)
represents only four distinct equations. They are:
P U ;p2 ^ l1 32 ;pl +P y ;p3 ~ $2 (0 13 ;2 + ^32 ; 1 + *21 ;3}
P I4;p2  42;pl +n 21 ;p4 = ~^2 ^14,2 + ^42; 1 + *21 ;4^
V s/ V
P 43;p2+P 32;p4+P 24;p3 = ^2 (*43 ;2 + ^32 ;4 + *24;3*
P y3;p4+r 34;pl +P 41 ;P3 = ^2 (*13 ;4 + *34; 1 + *4l ;3}
(4.54)
(4.55)
(4.56)
(4.57)
The left hand side of (4.54) is expanded like (4.52).
42
H 13R12 +
r
321
1
r 32ri ip
V
17 1
M3
p P p p
21 ;p3 13 ;2 + P 32; 1 +P 21;
S/ v V V
p3 Gt
1 t3 132
V
n 3Gt332
2 n
3
,3R22 +
V
13R32
P3 Gt
' t2 331
V
+r 3tQt22i+
V
r3 g*
3t 231
V
i2 r
,3
32R2_L + 1
32R31
v
P2 G*
tr 223
V
r 2tQtU3 +Â¡
V
p 2 t
2tG 123
V
12 p
3
21R23 +l
31R33
V
(4.58)
If the index "4" appears in a G^pq* !* vanishes. For this reason, the
following type of combinations must be observed.
G 121 + g3123 R12
(4.59)
When the summation is carried out on the index "t" as indicated in
(4.58), and combinations like (4.59) heeded, a fortunate cancellation
occurs which leaves
P 13;p2 +P 32;pl +P 21;P3 ^ (T 13;2+P 32;1+P 21 ;3} ;p (4,60)
P +P P =(PP
3 2;p1 +l 21;p3 M
v v v v
The field equation (4.54) is then
,P P rr P
V
2e
(P 13 ;2 +P 32;1 +P 2J;3pp = ^2 (13;2 + 4,32;1 + $21;3) (4,6l)
b
At
this point, the set of ~1 .as given in equations (4.39)
through (4.42), could be substituted, but the left hand side of (4.6i)
43
can still be reduced. By (333),
,P
rr = aS ^ f + 0 +0 0 0
[I3;2];p 2 13; s2 32;sl 21 ;s3 3s ; 12 si;
* jf
32
?2s;31 ls;23 4>s2;13) ;p (4,62)
The summation Is carried out, as indicated, over s. Then (4.51) is
used to make the following type of rearrangement
0 , = <*> + $ .Gp j ^ cPk a (4.63)
ab;cd ab;dc pb acd ap bed
throughout each term in (4.62), giving
.P
[
v
P [I3;2l;p = 2 a P<>[ 13;2];sp + ^ P(Cl3G 112 + *23 112 + *12G 312
+ $31G 213 + ^l5 113 + 113^ ;p + 3 P^12G 321
+ ^l13 221 + ^2G 221 + ^l6 223 + ^l0 223 + 023G 123^ ;p
+ a P^23G 132 + 0I2G 332 + 01 3q3332 + <>12G 331
+ 0 C5 + 0 G }
32^ 331 31 23 r ;p
(4.64)
The covariant derivative of the G ... are all zero, since, according to
equations (4.34) and (4.28),
15
j kb
.7
1
G aBY = ^2 aaB
(4.65)
0tis a constant and
aB;p 0
(4.65a)
15
tions.
No summation is implied by repetition of the index 7 in these equa
44
therefore
^Wp =
(4.66)
The following equations are now substituted in (4.64): (4.65), (4.28),
and (4.30) Cancellation and simplification leave
,P
P [13;2];p = 2 a$P*[13;2] ;sp + $2 *[13;2]
(4.67)
This result reinserted in the left hand side of (4.54) gives finally
asP(4> + <}> + ) = o
** 1 9 9 991 91Vcn
(4.68)
13;2 32;1 21 ;3 ;sp
The same procedure and the same identities are used on equations
(4.55), (4.56), and (4.57), giving
F(*14;2 + *42;1 + *21;4*;sp =
0
(4.69)
F(*43;2 + *32;4 + *24;3);sp "
0
(4.70)
F(*13;4 + *34;] *4l;3);sp =
0 .
(^71)
Solutions to (4.67) through (4.70) are, respectively,
+ $ + =0
13,2 + 32,1 21,3
*24,1 + *41,2 +*12,4=
*34,2 + *42,3 + *23,4 =
*14,3 + *43,1 + *31,4 =
(4.72)
(^73)
(4.74)
(4.75)
Using ea and ha from (4.35) we see that this is the second set of
covariant electrodynamic equations.
45
hl.l + h2,2 + h3,3 =
0
(4.76)
e2,1 el,2 + h3,4 =
0
(4.77)
e3.2 e2,3 + hl,4 =
0
(4.78)
el ,3 e3,l + h2,4"
0
(4.79)
These four equations are simple compared with the other set, (4.47)
through (4.50). In order to simplify the latter, we are willing to
slightly complicate the former. The result will be a symmetry of
form for both sets, and moreover, a solution will follow easily. This
is accomplished by replacing the eff and h^ by the following quantities:
hE(x' ,x2,X3,t) = & [S.2Ha + xEhbx + iLeaB,fHp/7l (4.80)
8E(x';X2,X3,t) t(S.Ea + ealirEsXY] (4.81)
where
Ha Hfl(t) (4.82)
and
Ea
EW
(4.83)
are yet to be determined.
These values are substituted in the field equations in place of
ea and h In the first set, (4.50) is satisfied Identically by (4.81).
Equation (4.47), after simplification, reduces to
E^) + k(
2H, y2 2H,
T a. (T" E3.'t>
(4.84)
46
or
,2 > 1 3?. x X' ,
 c + (
Â§2
(L.
Â¡f.lS
H c SF;
(4.85)
which implies
2 >
(4.86)
This same result Is implied by (4.48) and (4.49).
In the second set of equations, (4.76) is satisfied identically
by (4.80). When (4.80) and (4.81) are substituted in (4.77), the
result is
[<&2 + (X3)2](2E3 +(RH3j4) + (X'X3 +(Rj<2)(2E1 +4)
+ (X2X3 (RX1)(2E2 + (R.H2>4) 0 (4.87)
or
[R2 + (X3)2] (2E* + f1 ^)3 + (xV +(RX2)( 2t+ ^),
+ (X2X3 (Rx1)(2E>+ f"f)2 = 0 (4.88)
which implies
1 ctf
c St *
2_
<5L
(4.89)
This result is also implied by (4.78) and (4.79).
These two sets of equations are written together to emphasize
the symmetry of form which has been brought about. In place of the
original sets, (4.47) through (4.50) and (4.76) through (4.79), we have
1
c dt (R. H
JL i s L. c*
c dt (R.
(4.86)
(4.89)
47
Substitution of these equations into one another allows E and H to be
eliminated from each equation respectively, giving two ordinary wave
equations,
c2 dt2
I d2?
c2 dt
2 2
These represent an electromagnetic wave of circular frequency
u =
2c
.(t
(4.90)
(4.90
(4.92)
where
Ea(t) = G sin ut (4.93)
Ha(t) = cos wt (4.94)
The is a constant.
So we finally have the solution to the two sets of equations,
(4.47) through (4.50) and (4.76) through (4.79). It is, in the special
coordinates,
hK [B.V + x\ Xa (H. XY]cos (Â£t) (4.95)
and
s [^ + W5*71 sl" (tk) (496)
With the given by (4.35), the above is the solution to the last two
unified field equations, (3.44) and (348), which we had set out to
solve at the beginning of this chapter.
In vector notation, the solutions are
48
? % [.V + (? fix* (H*x?U (4.97)
e = [<9[~ + (E*x >0 ] (4.98)
The reason for using vector notation will soon be apparent. Now that
the solutions are known, we want to visualize them in the spherical
space.
First, consider the "straight lines" in this space. The geodesic
equation,
becomes
dV
dX2
*
r
dxk dxb
kb dX dX
(4.99)
s i nee
and
d2X'
dX
C i ) dXk
1 k b) dX
k d
dX
r
. dXk dXb
kb dT* d"X*
(4.100)
(4.101)
(3.35)
This shows that the geodesics of a space are not altered by the presence
of electromagnetic fields in the space to the first order which we have
investigated. (A question which immediately comes to mind is, "Well,
do the properties of the space affect the electromagnetic fields?" Of
course they do. We have seen how the electrodynamic equations were
complicated. Exactly how they are affected remains to be seen below.)
49
From [12], the above equation (4.100) can be written
d2x 2 dX^ /Â£* dxt
(X 7) = 0
dA
2 A dA
dA'
2 4
JL.= 0
dA
An explicit integral of (4.102) is
yI s in p. sin (A + )
cos (A + e)
y2 sin M sin (A + 5)
cos (A + e)
X"5 = tan (A + e)
(4.102)
(4.103)
We have set (^ = 1 for simplicity. This is the mathematical form for
16
the "straight lines." A geometrical picture may be found in El21.
 &
Now when the sphere, (4.25) is rotated with angular velocity u,
ct
in E^, it induces transformations on X in ^. Points move with
elliptic velocities
&?
It can been seen that the geodesic equations fulfill (4.104) so that
the points move along straight lines In the space.
Now, replace 5* In (4.104) by H* and the result Is the same as
(4.97). We see that the magnetic vector, h*, has the form of the
associated contravariant elliptic velocity, (4.104), and lies along
the geodesics of the space. So not only is the magnetic vector altered
by the space, It Is altered in this very special way.
The geodesics will sometimes be referred to as Clifford lines.
50
If we tensor multiply the cont ravar I ant velocity by g^, we get
the covariant velocity vector, which turns out to be the same form
as the electric vector, (4.98)
2
X, [&., + (w x x}.] (4.105)
so that the electric vector lies along the covariant elliptic velocity
vector. The solutions for the unified field equations are, therefore,
standing waves lying along Clifford lines, and having fixed frequency,
2c
It may be pointed out, finally, that when the radius of the
Einstein universe becomes infinitely large, the field equations and
their solutions naturally go over to Maxwells equations and their
solutions in a Minkowski space.
CHAPTER 5
SUMMARY
There is a possibility that the main features of the theory pre
sented have been obscured by the preponderance of mathematics. For
this reason, it may be well to summarize briefly the results of this
dissertation.
The second chapter contains no new results. It is intended as
a background to familiarize the reader with the different versions
of the unified field theory. The similarities and differences are
pointed out. All three have the same goalto derive all of the fields
and field laws governing Nature from a single tensor, which need not
have symmetry. The main difference is the choice of integrand for
the variational principle. This, of course, leads to different
forms of the field equations; however, the same set of sixtyfour alge
braic equations appears in each version. The most important similarity
in the theories is that they are based on the same geometrical con
cepts, and they all go over into the gravitational equations of
general relativity in a limiting case. Notwithstanding all the beauti
ful mathematical formalism, this correspondence to the laws of gravi
tation is the only physical content which has been derived from any of
the theories up to now.
The reason for this difficulty was pointed out in the next chapter.
Before Einstein's hypothesis (which was discussed in the introduction)
can be tested, the differential field equations must be solved. Before
51
52
they can be solved, the sixtyfour algebraic equations must be solved
in a general case (without reference to any coordinates) for the
components of the displacement field. Since this general solution has
not been accomplished, there is no way of telling whether all the
field laws are included in the theories. The sixtyfour equations are
shown to be reducible to twentyfour, but this has not enabled the solu
tion to be found up to the present.
At this point, we present our version of the theory, in which the
fundamental tensor is modified. It must be noted that this is not a
linearization such as that used by Kursunoglu [14] or Einstein [3].
These linearizations exclude interaction between the symmetric and
antisymmetric components of g,^. The present work does not preclude
this possibility. In fact, the antisymmetric components of the dis
placement field were seen to be a combination of the symmetric and anti
symmetric parts of the fundamental tensor. So instead of a linearization,
we have a perturbation type of technique. It is basically a first order
antisymmetric perturbation of the gravitational field producing
symmetric tensor.
With this method, the algebraic equations are readily solved for
b
without the specification of any coordinate system. It is found
that the become the same expressions used in general relativity.
With this in mind, we adapt the set of field equations involving the
contracted curvature tensor to an Einstein model of the universe.
In the limit of an infinite radius of the Einstein universe, the
remaining differential equations go directly over to the Maxwell equa
tions of Minkowski space. For the general case (a finite radius), a
transformation was made to a symmetrical arrangement of coordinates.
Investigation of the remaining two differential field laws leads to the
53
important discovery that they are the covariant electrodynamic equations
in the absence of charges and currents.
As a consequence of the symmetry of the coordinates, we are able
to give an exact solution to these equations. This solution indicates
that the electromagnetic is bent along the geodesic caused by the gravi
tational field, while an investigation of the geodesic equations shows
that the gravitational field is unaffected by the electromagnetic field
in this case.
Perturbation to higher order terms in still makes it possible to
obtain the displacement field explicitly but this does not, up to the
present, Imply success in solving the associated differential equations.
In these cases, the presence of addition fields will distort the
geodesics of the pure gravitational field, but it is not known whether
the additional fields are only electromagnetic in Nature.
In summary, the most important result of this disseration is the
realization that, to the first order, the covariant electrodynamic field
equations, as well as the gravitational equations, are included in the
unified field theory. Heretofore this was conjectured but never shown.
It is clear that what has been done is far from an ultimate goal of the
theory. Nevertheless, we feel that our contributions should be an
impetus to further work in the field. The most general case must be pur
sued. Along these lines, Kursunoglu has made the greatest innovations
since the theory was formulated by Einstein. It would certainly be
desirable to study his plan in which there Is a possibility of deriving
nuclear fields together with those previously discussed. [16]
BIBLIOGRAPHY
1. Einstein, A. The Principle of Relativity. New York: Dover Publi
cations, 1923, pp 111164.
2. Einstein, A., Ann. Math. 46, 578 (1946).
3 Einstein, A. and Strauss, E. Ann. Math. 47, 731 (1946).
4. Einstein, A. and Kaufman, B., Ann. Math. 59, 230 (1954).
5. Einstein, A. and Kaufman, B., Ann. Math. 62, 128 (1955).
6. Einstein, A., Can. J. Math. 2, 120 (1950).
7. Einstein, A. The Meaning of Relativity. 5th ed. Princeton: Prince
ton University Press, 1955
8. Einstein, A., Revs. Modern Phys. 20, 35 (1948).
9. Eddington, A. S. The Mathematical Theory of Relativity. Cambridge:
Cambridge University Press, 1924"
10. Hlavaty, V. Geometry of Einstein's Unified Field Theory. Groningen
P. Noordhoff Ltd., 1957
11. Hlavaty, V., J. of Math, and Mech. 833 (1958).
12. Kronsbein, J., Phys. Rev. 109, 1815 (1958).
13. Kronsbein, J. Phys. Rev. H_2, 1384 (1958).
14. Kursunoglu, B., Phys. Rev. 88, 1369 (1952).
15. Kursunoglu, B., Revs. Modern Phys. 29, 412 (1957).
16. Kursunoglu, B., 1 Nuovo Cimento J_5, 729 (i960).
17 Papapetrou, A., Proc. Roy. Irish Ac. 52, A, 69 (1948).
18. Schr'edinger, E. SpaceTime Structure. Cambridge: Cambridge Uni
versity Press, i960.
19. Synge, J. L. and Schild, A. Tensor Calculus. Toronto: University
of Toronto Press, 1949
20. Tolman, R. C. Relativity, Thermodynamics, and Cosmology. Oxford:
Oxford University Press, 1934.
54
55
21. Tonnelat, M. A. La Theorie Du Champ Unifie1 D'Einstein. Paris:
GauthierVi1lars, 1955.
22. Mishra, R. S., J. of Math, and Mech. 7 877 (1958).
23. Landau, L. D. and Lifshltz, E. M. The Classical Theory of Fields,
2nd ed. Reading, Mass.: AddisonWesley Publishing Company, Inc.,
1962.
24. Miller, C. The Theory of Relativity. Oxford: Oxford University Press,
1952.
25. Kronsbein, J. Electromagnetic Fields in Einstein's Universe
[Unpublished].
BIOGRAPHICAL SKETCH
Joseph Francis Pizzo, Jr. was born on October 30, 1939, in Houston,
Texas. There he attended St. Thomas High School and received his B. A.
from the University of St. Thomas in 1961.
In September, 1961, Mr. Pizzo began graduate studies at the
University of Florida. He received his Ph.D. in August, 1964.
Mr. Pizzo is married to the former Paula Awtry of Dallas, Texas.
They have one child, a son.
56
This dissertation was prepared under the direction of the chairman
of the candidate's supervisory committee and has been approved by all
members of that committee. It was submitted to the Dean of the College
of Arts and Sciences and to the Graduate Council, and was approved as
partial fulfillment of the requirements for the degree of Doctor of
Philosophy.
August 8, 1964
Dean, Graduate School
Supervisory Committee:
ChaKrman
/l'
Cocha i rmarf
/
f> .s n
\, x\
'kjvOWVV HV
[T
g /S
9
Rik
rVp*r
ip3k
q
Pk
qp
r
I ik
(2.21)
which is named the Ricci tensor, or the contracted curvature tensor.
It is to play a centra.! role in the variational principle from which
the field equations are to be derived.
For a variational method, we postulate an invariant space integral
which involves the displacement field. Invariance of this integral pre
4
supposes the existence of a scalar density, L, which can be formed
by the contraction of the Ricci tensor with a contravariant tensor
density, Â£]j 1 The field laws are to be derived from
6/ LdT = 0 ,
(2.22)
where dt is an element of fourvolume and
(2.23)
ik
is considered as a function of the jj ^ and P which are to be
varied independently, their variations vanishing on the boundary of
integration.
This postulate would lead to the equations of the purely gravi
tational field if the condition of symmetry were imposed on and
Pb
I (This will be demonstrated in the next chapter,) However, the
present theory is an attempt to generalize these equations, and the
constructions which were essential for the setting up of covariant
4. A scalar density transforms like a scalar but with the inverse
of the Jacobian determinant included. This is necessary to cancel the
Jacobian which results from the transformation of the element of four
volume. The result is that the integral transforms like an invariant.
w
To Paula
46
or
,2 > 1 3?. x X' ,
 c + (
Â§2
(L.
Â¡f.lS
H c SF;
(4.85)
which implies
2 >
(4.86)
This same result Is implied by (4.48) and (4.49).
In the second set of equations, (4.76) is satisfied identically
by (4.80). When (4.80) and (4.81) are substituted in (4.77), the
result is
[<&2 + (X3)2](2E3 +(RH3j4) + (X'X3 +(Rj<2)(2E1 +4)
+ (X2X3 (RX1)(2E2 + (R.H2>4) 0 (4.87)
or
[R2 + (X3)2] (2E* + f1 ^)3 + (xV +(RX2)( 2t+ ^),
+ (X2X3 (Rx1)(2E>+ f"f)2 = 0 (4.88)
which implies
1 ctf
c St *
2_
<5L
(4.89)
This result is also implied by (4.78) and (4.79).
These two sets of equations are written together to emphasize
the symmetry of form which has been brought about. In place of the
original sets, (4.47) through (4.50) and (4.76) through (4.79), we have
1
c dt (R. H
JL i s L. c*
c dt (R.
(4.86)
(4.89)
39
,14
pattern is recognized
aU
X f a hA.,^ V\a (2 V.M + S A V
rV e^^(3Xah1 + aa\ J h + \ A,)l
(^39)
and
p" crA X an I aP., \
I B4 = ^ + X eS + A 9 eM, + 2 9 V.B^
X6 aM I #
 i VV ,1
2*
(4.40)
al so
r
V
6 \,4}
(4.41)
and
r
4?= e,4
(4.42)
These are the twentyfour P ., in the special coordinates. Combined
rib v
with the j .^ given by (4.31), the complete set of the sixtyfour
P jk 's known in this space. This completes the information needed
to set up the field equations, (344) and (348).
It has been pointed out that (34), which reduces to (345) is
identically a set of the covariant electrodynamic equations in the
presence of a gravitational field. To see what they look like in this
Pb
space, it will be easier to go back to (34), since the  ., have
ik
N/
already been computed. The equations represented are:
r
12
V
13
V
L
r
14
v
(4.43)
14c cbx
and are the usual permutation symbols whose value is
zero if any two indices are alike, +1 if (aBA) is an even permutation
of (123), 1 if (aBX) Is an odd permutation of (123).
18
This is precisely the gravitational field equation in empty space.
(See, for example, [9], page 81) A particular solution to these equa
tions corresponds to the field of an isolated particle continually at
rest. The famous explanation of the discrepancy in the advance of
Mercury's perihelion is a result of the solution to (2.63). There is
no question about the physical significance of the gravitational field
equations, so this gives a certain measure of confidence to the generali
zation, (2.30), (2.31), (2.32), and (2.40).
A similar situation exists with Schroedinger1s theory in the limit
of a symmetric g.,. The field equations, (2.45), become the same as
i K
Einstein's with the addition of a term involving X, which is now identi
fied with the cosmological constant. Actually, the limiting case of
Schroedinger's theory is the original form of the gravitational equa
tions. The cosmological constant is so small that it need not be in
cluded on a scale such as our solar system. (See [9], p. 100)
In Kursunoglu's theory, the correspondence to general relativity
is achieved by the vanishing of the fundamental length. Again we revert
to the equations of general relativity. This method is much like quantum
mechanical correspondence to classical mechanics when Planck's constant
vanishes.
Therefore all three theories have at least some basis, due to the
fact that they reduce in the limit to the wellknown and tested equations
of gravitation derived by Einstein in his general theory of relativity.
What has been presented so far is only the problem: a nonsymmetric,
fundamental tensor, g is chosen to represent the fields in Nature.
Ik b
The displacement field, p *s introduced to insure that the field
laws will not be dependent upon the choice of a coordinate system. From
44
therefore
^Wp =
(4.66)
The following equations are now substituted in (4.64): (4.65), (4.28),
and (4.30) Cancellation and simplification leave
,P
P [13;2];p = 2 a$P*[13;2] ;sp + $2 *[13;2]
(4.67)
This result reinserted in the left hand side of (4.54) gives finally
asP(4> + <}> + ) = o
** 1 9 9 991 91Vcn
(4.68)
13;2 32;1 21 ;3 ;sp
The same procedure and the same identities are used on equations
(4.55), (4.56), and (4.57), giving
F(*14;2 + *42;1 + *21;4*;sp =
0
(4.69)
F(*43;2 + *32;4 + *24;3);sp "
0
(4.70)
F(*13;4 + *34;] *4l;3);sp =
0 .
(^71)
Solutions to (4.67) through (4.70) are, respectively,
+ $ + =0
13,2 + 32,1 21,3
*24,1 + *41,2 +*12,4=
*34,2 + *42,3 + *23,4 =
*14,3 + *43,1 + *31,4 =
(4.72)
(^73)
(4.74)
(4.75)
Using ea and ha from (4.35) we see that this is the second set of
covariant electrodynamic equations.
26
9Â¡k 9Â¡k + ^ik
(328)
is called a CliffordHermitian tensor after W. K. Clifford.
For the CliffordHermitÂ¡an field, (326) becomes
ik;b = 9Â£k f^ib + bk '
v y
If i, k, and b are permuted, two more equations are obtained.
(3.29)
ev. kÂ¡ vr Â¡vb <3'3)
4,bÂ¡;k= Â£lp\k+ g^pPy (3.31)
The following combination of the three equations is taken
2 (ik;b 45 kb; ? ^b; k^ 9pb P fk
Which is immediately solved for
^ 1 pb
a* eaJ
V
This result causes (3.25) to become
b
r
ik i k l
(3.32)
r Ik 2 4>kp;i 0pi ;k> (3.33)
(3.3k)
and the displacement field is known
r ik = \i k^ + 2 gE^0ik;p 0kp; i ^pi ;k* (335)
This is a general form insofar as no particular coordinate system
has been specified. it is also in a useful form for the field equa
tions (32), (33), and (34).
36
a
% Al/2
x4 "
A1/2
(4.26)
where
A = (R2 + (x)2+ (X2)2 + (X3)2
(4.27)
In this space, X and X = ct t we wÂ¡]] denote the symmetric part of
the fundamental tensor by a^. It is computed by the standard method
to give
alk =
A2
[A (X1)2]
X1X2
x1X3
 X]X2
[A (X2)2]
xV
2 3
x r
 X2X3 [A (x3)2]
0
aL
(4.28)
The determinant is
hk1
a8
7
(4.29)
The contravariant tensor to (4.28) Is found to be
Ik A
a
[(R2 + (x1)2] xx2
xx3
xx2
x]x3
[(R.2 + (x2)2]
2 3
XXJ
x2x3 [? + (X3)2] o
(4.30)
45
hl.l + h2,2 + h3,3 =
0
(4.76)
e2,1 el,2 + h3,4 =
0
(4.77)
e3.2 e2,3 + hl,4 =
0
(4.78)
el ,3 e3,l + h2,4"
0
(4.79)
These four equations are simple compared with the other set, (4.47)
through (4.50). In order to simplify the latter, we are willing to
slightly complicate the former. The result will be a symmetry of
form for both sets, and moreover, a solution will follow easily. This
is accomplished by replacing the eff and h^ by the following quantities:
hE(x' ,x2,X3,t) = & [S.2Ha + xEhbx + iLeaB,fHp/7l (4.80)
8E(x';X2,X3,t) t(S.Ea + ealirEsXY] (4.81)
where
Ha Hfl(t) (4.82)
and
Ea
EW
(4.83)
are yet to be determined.
These values are substituted in the field equations in place of
ea and h In the first set, (4.50) is satisfied Identically by (4.81).
Equation (4.47), after simplification, reduces to
E^) + k(
2H, y2 2H,
T a. (T" E3.'t>
(4.84)
23
Einstein now defines
flikb=9bk,i gk\kVl,b + s'^a/'VVb.k' <3'l3)
and
i1k'b1 Ri1 Rk' Rb' i'a k'p b'c
U 5,.L = 5 .5 ,5 + g q_,g K
i kb
k b + 9 9ai9 9pk9 9cb
(314)
so that
i k' b '
U W. ,
ikb wi'k'b1 Aikb
The problem is to find U, the inverse of U, so that
(3.15)
Wikb =
i k' b 1
ikb
A.
k'  b *
(3.16)
and the knowledge of W.^j^ then gives the P by (3.9).
Einstein's prescription for finding the inverse is presented in
four pages and the answer becomes much too complicated to write down
explicitly here. To quote Einstein in this paper, Such a solution can
indeed by arrived at. .but it is cumbersome, and not of any practical
utility for solving the differential equations." This statement also
applies to the solutions obtained by Mishra [22] and Hlavaty[lO],
So, for all practical purposes, we are still left with the problem
of sixtyfour equations in sixtyfour unknowns. One additional reduction
is possible. The number of equations and unknowns can be reduced to
nb
twentyfour if we treat symmetric and ant isymmetric components of  .^
as separate quantities. [213 (This is a natural thing to do since they
transform separately. In fact, it can be seen from (2.10) that the anti
symmetric components transform like a tensor, whereas the symmetric
ones do not.) The fundamental tensor is also written as the sum of its
parts. Consider the two equivalent equations
2
structures would be much preferable." (See [73, page 115) This then is
the aim of Einstein's unified field theory: to derive all fields from
one, single nonsymmetric tensor.
Einstein formulated the unified field theory according to the
same pattern he had used for his theory of gravitation, with the excep
tion that now he did not require the fundamental tensor to be symmetric.
One of the resultant field equations turns out to be a set of sixty
four algebraic equations in sixtyfour unknown functions of the g.^.
These equations have to be solved before the other three field laws
(second order partial differential equations) can be set up. This pro
blem occupied much of Einstein's later years, and although he found a
solution to the sixtyfour equations, it was in his own expressed
opinion, too complicated to be of any further use.
This is essentially where the problem has stood until this time.
There has been no verification of Einstein's hypothesis, that other
fields are included in the fundamental tensor, because the differential
equations have never been solved. In this dissertation, we are able
to show that, by a special restriction on the fundamental tensor, the
unified field equations become those of gravitation and electrodynamics,
while the components of g.^ represent the gravitational and electro
magnetic fields. That which was introduced artificially before, now
comes about naturally from one single tensor, in this special case.
The geometrical concepts upon which the unified field theory is
based, are developed in the second chapter. Also the different versions
of the theory (Einstein's, Schroedinger's, and Kursunoglu's) are dis
cussed here. The correspondence of all of the theories to the gravita
tional equations of general relativity is shown for limiting cases.
CHAPTER 2
CURVATURE, DISPLACEMENT, AND FIELD EQUATIONS
There are three varieties of unified field theory: Einstein's,
Schroedinger's, and Kursunoglu's. Throughout each of these, two funda
mental entities are dominant; the displacement field and the curvature
tensor. These concepts require some elaboration before they are used
to derive the field equations.
In relativity, we deal with quantities known as tensors, which
have an appealing property; a tensor equation remains unchanged, re
gardless of the coordinate system in which it is expressed. That is,
tensor equations are covariant. When the laws of physics are written
as tensor equations, all reference frames are treated equally. The
idea of a preferred system no longer exists. This formulation breaks
down when we try to compare vectors at infinitesimally separated points.
Consider, for example, the vector A* at the point x' and the vector
A* + dA* at the point x' + dx'. The difference between the two is
(A* + dA1) A1 = dA* .
(2.1)
To illustrate the problem that has now arisen, suppose the vectors are
2
equal in a particular coordinate system.
dA = 0
= A* ,j dxJ* .
(2.2)
2. The comma indicates ordinary partial derivatives.
4
CHAPTER 3
REDUCTION AND SOLUTION
It is evident that the three versions of unified field theory,
which have been presented, are little more than postulates. Their
tenuous claim to validity comes from the correspondence to the known
equations of general relativity and the correct count of functions.
So far there has been no indication that they give a field description
of Nature in which all fields are united in a single tensor.
In keeping with this spirit, we choose to deal with yet another
form of the equations, which can be considered as an adaptation of
either Schroedinger's or Kursunoglu's equations to an Einstein model
of the universe.
s s
1.. ~
i k,b
9sk
r Â¡b 9s r bk
(3.1)
2_
g j k i 4 k
(32)
0
, otherwise
[ik.b]
V
= 
Â¡2 3[lk,b]
(3.3)
rk
1 ik
= 0
(3.4)
v
In the limit of a vanishing antisymmetric field, the antisymmetrJc part
b b
of j vanishes and the symmetric component, p .^, is the Chrlstoffel
symbol. Then (3.2) is seen to be exactly the set of differential equa
tions which describe the Einstein Cylindrical Model of the universe (See
20
33
A More General Case
We now move to a more general case, where "(j^" is taken as finite.
A solution to (338) for the g^ has been given in chapter three, but a
slightly different version of the solution will be given here. First,
note must be taken that (3.38) involves only the symmetric part of the
fundamental tensor. By an extension of the example in Minkowski space,
we should anticipate that the g.. will depend upon the distribution of
l K
matter in the universe. At this point the following model is adopted
a static homogeneous universe. This means that all parts are considered
extrinsically and permanently alike. In this case, the line element
can be put in the general spherically symmetric form.
,2 X 2 2^2 2 2q ,.2 v 2 ,2 ...
ds =edr + r do + r sin 0d e c dt (H.13)
where
V = v(r) and X = X(r) (4.14)
Most of the g.^ are already known:
g,k = 0 i 4 k (4.15)
g22 = r2 (4.16)
g^3 = r2sin20 (4.17)
From the differential equations, (339). we need only determine
g,1 = e* (4.18)
V
g44 = e
and
(4.19)
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38
Ik
O
h
3
O
hl
e.
1
0
e.
(4.35)
where the "e and "h are functions of (X^, X^, X3, t).
We now have the necessary material to calculate the antisymmetric
b
components of j in our special coordinates. The actual calculation
of all twentyfour  ..Is a long, tedious process. Only a sample
'v 3
calculation (the component, p w'11 be used as an example) and
V
the final results are given here. By (333),
P 12 ^12; 1 + 3
.32,
1 33
r ,2" a3 ^12;! + a3 012;2 + ^ S^12,3 S3,1 Sl.2^
V
S2 [l 3] *1! {3 2] ]}
(4.37)
Upon substitution of (4.30), (4.31), and (4.35), and consequent can
cellation and simplification, it is found that
^12 P + x3x'h3.1 + + [
v
 [(K. + (X^)21] [j(hj j + h2 Â£ + 3) + S (x'*1i +
2 /3 \ 21 rl, L l x 1 ,J, w2,
+ x3h3)3)
(4.38)
D
The other j are computed in the same manner and soon the following
14
are known, they are substituted in the other three field equations,^
k
r ,k
(2.30)
V
Â¡k=
(2.31)
Rik,b + Rkb,i + Rbi,k = = R[ik,b]
(2.32)
I IN W INU ,1 U I IN t IN
V V V V
for a solution of g.^ This is the formalism of Einstein's unified
field theory [5].
Schroedinger's unified field theory is quite similar to that of
Einstein's. Parallel transfer, displacement fields, and the curvature
tensor all form the basis for the theory. The difference appears in
the integrand of the variational principle. Whereas Einstein took
i k
j R.k as thescalar density, Schroedinger chooses [18]
L (det. R.k),/2 (2.41)
which is the simplest scalar density that can be built out of the
curvature tensor. The constants 2 and X have no influence on the result.
A contravariant tensor density is introduced here also:
r ik 3l
1)
(2.42)
but it can be eliminated in the end, leaving equations which contain
b
only the p .^.
When the variation,
6^LdT = 0RikdT = 0
i k
(2.43)
7. The square brackets in (2.32) indicate summation over the cyclic
permutations of the enclosed indices. This convention will be used
hereafter.
ACKNOWLEDGMENTS
The author extends his gratitude to the members of his committee.
In particular, he is deeply grateful to Dr. J. Kronsbein, who suggested
this problem and has given much of his time in helping to bring about
the solution.
The author would also like to thank both his wife, Paula for her
help and suggestions in the preparation of the first draft, and Miss
Nana Royer for the typing of the final copy.
ii 1
28
9Â¡k
+ .
O
O
O
+(S?sn2lÂ£
O
O
O +sÂ¡n2'i's i n29
O O
O
O
O
1
(3.40)
The left hand side of (338) is calculated using (3.40) and indeed the
differential equations are satisfied by (3.40). The form (339) repre
sents what is called an Einstein cylindrical model of the universe.
The contravariant tensor to (3.40) is simply
ik
(3.41)
'ik
since g., is represented by a diagonal matrix, (340).
9
With the g.^ solved, we can move on to (3.4), using (333)
r ik = 0Â£3E!('>ik;pVÂ¡ 4,pt;k)
V
(3.42)
Since g^ is symmetric and is antisymmetric, the second term
vanishes. Furthermore, p and k are dummy indices so that the first
and last terms add.
. =0
L i k; p
(3.43)
In this form the field equation is significant since it is recognized
as one of the covariant electrodynamic field equations in the absence
of charges and currents [ 19 with the .. interpreted as components
K
of the electromagnetic tensor.
^ik ;p (3^)
9. The solution will be examined in more detail in the next
chapter.
35
and
2 2 2 2 _2
x, +x2 +x3 +x4
(4.25)
This illustrates that the physical space of the Einstein universe may
be interpreted as the threedimensional bounding manifold of a sphere
of radius (ji^ in a fourdimensional Euclidean space with the cartesian
coordinates given above.^ The time dimension is uncurved. Hence the
name cylindrical universecurved space and straight time.
Even though the line element displays spherical symmetry, there
is no symmetry of form among the W and the R.^. To make the field
equations more transparent, we adopt a method used by Kronsbein [12].
The sphere represented by (4.25) is radially projected from its center
in the fourdimensional space, (x.), into the threedimensional space,
, with coordinates X (Greek letters take on values 1, 2, 3) by the
projection
12
^The solution shown
tained directly from (4.24)
in the previous chapter,
by the transformation
(3.41), may be ob
V
x2
X3
x4
cos Â¥
(Jt s inf cos
(SI sinf sin cos'F
(St sin Â¥sin0 sinY
which expresses the spatial part in fourdimensional spherical coordi
nates .
12
Some people prefer to call this a gnomonic projection.
CONTRIBUTIONS TO THE
EINSTEINKURSUNOGLU FIELD EQUATIONS
By
JOSEPH FRANCIS PIZZO JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1964
16
and
(a)
J/2eijkb *
j kb
(2.49)
,, or. 4 2n 2 8 4.2.
g = a(l t 2Gc rQ JL 4G c ro X. )
(2.50)
The action principle from which the equations are to be deduced is
6/[ Â£jj'kRJk. 2ro"2(^b ^a)]dT = 0 (2.51)
where b is the determinant of the tensor,
brk a'/23'l/2(aik **\\s*\) (252)
The variations carried out in (2.51) produces the Kursunoglu field
equations.
s
9Â¡k,b 9skT ib
s
(2.53)
(a
ik
(2.54)
R[Â¡k,b]
v
= r
2
(ik.!
kb, i
V J
b i k
= 0 .
(2.55)
(2.56)
The nomenclature, "fundamental length," is rationalized by examining
the field equations. The vanishing of r^ gives the general relativis
tic case in the absence of charges. This existence of free charges is
now associated with a finite fundamental length.
All three field theories discussed here have at least one thing in
common: they have been constructed so that there exists a correspond
ence principle which takes the unified field equations over the well
known gravitational equations of general relativity when the anti
symmetric field is absent. This is easy to see. First consider Einstein's
CHAPTER 4
SPECIAL SOLUTIONS
In the preceding chapter the field equations were fashioned
into a suitable form for solution. The three equations which must
now be solved are
hLp+Â¡PpU+[.PWWW^^. (3.38,
QC 
i 4 k
g, =0
i k; p
(344)
P ik;pb*r kb;pi P bi;pk = ^2 (ik;b + 0kb;l + ^bi;^ (3,48)
V V V v
Minkowski Space
First, a very special situation will be considered. It is the
opposite of the limiting case where the antisymmetric field vanishes.
Now, the field equations will be examined when is chosen infinitely
large and the CliffordHermitian tensor is assumed to have Minkowski
form,
9ik sik + Â£ik <4J)
In light of this, all Christoffel symbols vanish and (338) is satis
fied Identically. There is no gravitational field.
It is to be expected, since there is no gravitational field, that
what is left of the tensor g.^ should represent the electromagnetic
30
8
can be shown that the integral transforms as a tensor. Moreover, it is
an antisymmetric tensor since
= MnV) (2.17)
and the first term on the right hand side vanishes, because it is an
exact differential. The integral will be denoted by^
f^ = ^TfdT]1 (2.18)
In equation (2.16), the quantity in parentheses can be expressed as a
sum of parts which are symmetric and antisymmetric on the indices t
and q. The contraction of the symmetric part with the integral
vanishes leaving
W 
M pt,q 1 pq,t 1 stl pq sql pt '
It is know that Aa* is a vector since it is obtained by parallel dis
placement. The term A^f^ is a third rank tensor, therefore the
tensor character of
ptq
= rPt,q+r pq.t+r str pqr
r
sql pt
(2.20)
is established. This is the wellknown RiemannChristoffel tensor.
R* is also referred to as the curvature tensor, because whenever it
ptq
vanishes, there is no change in a vector after it is displaced parallel
to itself around a complete circuit. The space is then said to have
no curvature, or to be flat.
Contraction on the indices i and q gives the second rank tensor,
3. The hook under the indices is used to indicate antisymmetry.
32
(["*,., + p., + p,. i) o (^ii, + l + i) (^>7)
' 1 ik,b I kb, i I bi,k',p fi\2 Ik,b kb,i bi ,ky '
v v v
The solutions, (3.33) for P are substituted, using 6!i< for g in
v
accord with (4.2).
ik
i[(<0
O.
J k + <*l

 4> )
21VTik,s ks,i si,k ,b kb,s bs, k sk,b,i
+ (,, ,,).] + ( + 0. ,)
v bi,s is,b sb,i ,k ,s ^2 ik,b kb, i bi,k/
(4.8)
or
oi^nu + ^ik'+^kii) + o (* i k + ct>ik + i) ~ 0 (49)
2 v ik,b kb,i bl,k ,ss Qt2 ik,b kb, i bi,k' '
A solution to this equation is
0 r
ik,b
) + (b
kb,l bi,k
(4.10)
which, in view of (4.4), represents the other set of Maxwell equations
in a Minkowski space.
V h* = 0 ;i,j,k4 (4.11)
V x e* = Â£ ; i, j, or k = 4 (4.12)
Therefore, as expected, the field equations In Minkowski space
become the Maxwell equations, while the fundamental tensor represents
both the metric and the components of the electromagnetic field.
These equations, (4.5), (4.6), (4.11) and (4.12) are well known
their solution and validity need no elaboration.
BIOGRAPHICAL SKETCH
Joseph Francis Pizzo, Jr. was born on October 30, 1939, in Houston,
Texas. There he attended St. Thomas High School and received his B. A.
from the University of St. Thomas in 1961.
In September, 1961, Mr. Pizzo began graduate studies at the
University of Florida. He received his Ph.D. in August, 1964.
Mr. Pizzo is married to the former Paula Awtry of Dallas, Texas.
They have one child, a son.
56
47
Substitution of these equations into one another allows E and H to be
eliminated from each equation respectively, giving two ordinary wave
equations,
c2 dt2
I d2?
c2 dt
2 2
These represent an electromagnetic wave of circular frequency
u =
2c
.(t
(4.90)
(4.90
(4.92)
where
Ea(t) = G sin ut (4.93)
Ha(t) = cos wt (4.94)
The is a constant.
So we finally have the solution to the two sets of equations,
(4.47) through (4.50) and (4.76) through (4.79). It is, in the special
coordinates,
hK [B.V + x\ Xa (H. XY]cos (Â£t) (4.95)
and
s [^ + W5*71 sl" (tk) (496)
With the given by (4.35), the above is the solution to the last two
unified field equations, (3.44) and (348), which we had set out to
solve at the beginning of this chapter.
In vector notation, the solutions are
50
If we tensor multiply the cont ravar I ant velocity by g^, we get
the covariant velocity vector, which turns out to be the same form
as the electric vector, (4.98)
2
X, [&., + (w x x}.] (4.105)
so that the electric vector lies along the covariant elliptic velocity
vector. The solutions for the unified field equations are, therefore,
standing waves lying along Clifford lines, and having fixed frequency,
2c
It may be pointed out, finally, that when the radius of the
Einstein universe becomes infinitely large, the field equations and
their solutions naturally go over to Maxwells equations and their
solutions in a Minkowski space.
] 1
variation is carried out, the A. will be chosen to make the field
k
equations transposition invariant, and then they will be eliminated
from the system.
b
The contracted curvature tensor, with J""1 ^ replaced by (2.24)
becomes
*
Rik Rlk <2'2la>
where R.,(P)is the same as expression (2.21), except that a star is
b
put over all the p1 The variational integral is now
rk
^[Ulk[Rik+ <\k \)u dT (225)
Â¡k *b
After the variations in jj j1 .and A.^ are made, the integrand,
denoted by I, takes on the following appearance
I = [R
ik
*
+ (A..
i,k
+
k
s
(2.26)
The last term is a generalized divergence and since it appears in a four
volume integral, it can be converted to an integral over the three
manifold enclosing this volume. The variations vanish on the boundary,
hence the last term in equation (2.26) can be ignored. The variations
in the remaining terms are all independent of each other so each
coefficient must vanish separately, giving the three equations
R..
t k
*
(D
Vi} 
(2.27)
40
r 3 p 4
21 + 23 +i 24 =
(4.44)
r2 p4
31 + 1 32 +l 34 = 0
(4.45)
4, P 42 1 43 
(4.46)
o
The I are substituted from equations (4.39) to (4.42).
simplification, the four equations are:
After
3(x2h3 x3h2) + [.2 + (x2)2]h3 2 ia2 + (x3)2]h2 3
4
 X2X3(h33 h2>2) x'(X2h3J X3h2J) 0 (4.47)
3(X3h 1 X1 h3) + [(SL2 + (X3)2Jh1 3 [iR.2 + (X1)2]h3jl
4
+ x3x'(h,,l h3 3> +. X2(X3h12 x'h3_2) 4e2,4=0 <4'43>
3 (X1 h2 X2h1) + [
4
+ X X (h2^2 hj^) + X (X h2^3 X hj ^) ^ = 0 (4.49)
2(X1e] + X2e2 + X3e3) + [ft2 + (x')2 e, ] + xVe, 2 + X,X3e] 3
+ xVe^, + [ft2 + (X2)2]e2j2 + xVe^
+ X3X1e3 j + X3X2e3 2 + [R.2 + (X3)2]e3 3 = 0 (4.50)
These equations are exceedingly complicated as they stand. They will
be left this way for the time being. After the other set of four field
equations (3.48), has been written in this space, a simplification and
solution will be presented for all eight equations.
A tensor identity which will be of great utility in the investi
gation of (348) is
10
equations are independent of the assumption of symmetry.' Instead of
symmetry, an analogous condition is posited. it is referred to as
.. b
i k r~i
Transposition Invariance. Even though and  are nonsymmetric,
their transformation laws would be invariant if the indices of these
variables were to be transposed and then the free indices interchanged.
This is what is meant by transposition invariance. In addition to the
variational principle, it will be postulated that all field laws shall
be transposition invariant."'
If we proceed to substitute (2.21) for R.^ into equation (2.23)
and carry out the variation indicated in (2.22), we would find that the
resulting equations are not transposition invariant. To circumvent
such a distasteful result, four new arbitrary variables, are intro
duced by making a formal change in the description of the field. In
b
place of p the following substitution is made
U b
(2.24)
r ik r.
sb ..
., 4 V
i k i k
*b b
The  ^ is a displacement field just as the j The are treated
in the variational procedure as independent variables. After the
5. An example of an equation which is transposition invariant is
s s
9ik,b 9sk p ib 9isp bk ^
Transposing the indices on the variables yields
s s
9k,b 9ks P bi 9si P kb ~ ^
Interchanging the free indices (i and k) gives back the original
equation
s s
9ik,b 9is P bk 9sk ib ~ ^
37
With (4.28) and (4.30), the Christoffel symbols can be constructed.
They are very important because they are the symmetric part of the
13
displacement field.
Ia a) a
C )
6 3 = A (4.31)
There is one more quantity which we need to calculate in this space
for future use. It is the analog of the RiemannChristoffel tensor,
(2.20). It will be denoted by
kb
" jiPk],b + \\?b\,k ^k^b^ jiqb^q k'
(4.32)
Notice the antisymmetry.
(4.33)
The components will be listed here for future reference:
.r
xaxB
aBT .2
A
aar
[A (x)2]
A2
(4.34)
Al 1 others are zero
For the antisymmetric part of the fundamental tensor, the following
symbols will be used for the
13
Jn (4.31) and (4.34), the repeated index does not imply
summation.
PAGE 1
CONTRIBUTIONS TO THE EINSTEINKURSUNOGLU FIELD EQUATIONS t By JOSEPH FRANCIS PIZZO JR. y A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1964
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To Paula
PAGE 3
y ACKNOWLEDGMENTS The author extends his gratitude to the members of his committee. In particular, he is deeply grateful to Dr. J. Kronsbein, who suggested this problem and has given much of his time in helping to bring about the solution. The author would also like to thank both his wife, Paula for her help and suggestions in the preparation of the first draft, and Miss Nana Royer for the typing of the final copy. Â• Â• f I i I
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TABLE OF CONTENTS Page ACKNOWLEDGMENTS HI CHAPTER 1. INTRODUCTION 1 2. CURVATURE, DISPLACEMENT AND FIELD EQUATIONS .... k 3. REDUCTION AND SOLUTION 20 k. SPECIAL SOLUTIONS 30 Minkowski Space 30 A More General Case 33 5. SUMMARY 51 y BIBLIOGRAPHY 5h BIOGRAPHICAL SKETCH .... 56 Iv
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CHAPTER 1 INTRODUCTION In 1916, Einstein's classic paper, in which he formulated the general theory of relativity, appeared in Anna 1 en der Physik [l] In this paper he was able to describe the phenomena of gravitation in terms of geometrical concepts. The field equations of gravitation were shown to be derivable from a variational principle, using a symmetric second rank, covariant tensor. This tensor (which we will call the fundamental tensor and will d^iote by g.,) represents the gravitational potentials. The theory was beautiful, and moreover It worked.' Einstein was still not satisfied. He reasoned that there are other fields in Nature besides gravitation. How do they fit into this picture? An example is the electromagnetic field and equations. There is no natural way for them to be included. To express the covariant electrodynamlc equations. Maxwell's equati'ons are written, then covariant derivatives are taken In place of ordinary partial derivatives. In other words, electrodynamics must be introduced separately. This type of inclusion has been considered arbitrary and unsatisfactory by many theoreticians. Indeed, Einstein himself has stated, "A theory in which the gravitational field and the electromagnetic field do not enter as logically distinct 1. The numbers In square brackets refer to the bibliography at the end of the dissertation. I
PAGE 6
2 structures would be much preferable." (See [7], page 115) This then is the aim of Einstein's unified field theory: to derive all fields from one, single nonsymmetric tensor. Einstein formulated the unified field theory according to the same pattern he had used for his theory of gravitation, with the exception that now he did not require the fundamental tensor to be symmetric. One of the resultant field equations turns out to be a set of sixtyfour algebraic equations in sixtyfour unknown functions of the g.. These equations have to be solved before the other three field laws (second order partial differential equations) can be set up. This problem occupied much of Einstein's later years, and although he found a solution to the sixtyfour equations. It was In his own expressed opinion, too complicated to be of any further use. This is essentially where the problem has stood until this time. There has been no verification of Einstein's hypothesis, that other fields are included in the fundamental tensor, because the differential equations have never been solved. In this dissertation, we are able to show that, by a special restriction on the fundamental tensor, the unified field equations become those of gravitation and electrodynamics while the components of g.. represent the gravitational and electroI K magnetic fields. That which was introduced artificially before, now conies about naturally from one single tensor. In this special case. The geometrical concepts upon which the unified field theory is based, are developed In the second chapter. Also the different versions of the theory (Einstein's, Schroedinger 's and Kursunoglu 's) are discussed here. The correspondence of all of the theories to the gravitational equations of general relativity Is shown for limiting cases.
PAGE 7
J^F 3 Chapter three embodies our version of the theory. The sixtyfour equations are shown to be reducible to twentyfour, which still cannot be solved in a useful form. Nevertheless, they suggest a modification in the fundamental tensor which allows a tractable form for the solution of the equations to be obtained. The three sets of differential equations, referred to above, are constructed. One set is the same as the gravitational equations. The next is seen to be the first set of covariant electrodynamic equations in the absence of charges and currents. The last is actually a set of third order partial differential equations and needs further investigation. It turns out we are able to solve this problem in our version of the theory. In chapter four two cases are considered. First, the equations are examined when the radius of the Einstein universe is taken infinite, in which case the gravitational equations are satisfied identically. The other two sets of equations are shown to be Maxwell's equations in the absence of matter. Next we take the more general case of a finite radius. The first two sets of differential equations were investigated before. Now, a terra by term examination reveals that the last field law Implies the second set of covariant electrodynamic equations. The last part of this chapter Is devoted to an exact solution of these equations in a special coordinate system. Throughout this dissertation an elementary understanding of tensor algebra and a basic knowledge of general relativity are assumed.
PAGE 8
CHAPTER 2 CURVATURE, DISPLACEMENT, AND FIELD EQUATIONS There are three varieties of unified field theory: Einstein's, Schroedinger 's, and Kursunoglu 's. Throughout each of these, two fundamental entities are dominant; the displacement field and the curvature tensor. These concepts require some elaboration before they are used to derive the field equations. In relativity, we deal with quantities known as tensors, which have an appealing property; a tensor equation remains unchanged, regardless of the coordinate system in which it is expressed. That is, tensor equations are covariant. When the laws of physics are written as tensor equations, all reference frames are treated equally. The idea of a preferred system no longer exists. This formulation breaks down when we try to compare vectors at infinitesimal ly separated points. Consider, for example, the vector A at the point x and the vector Â• Â• Â• A + dA at the point x + dx The difference between the two is (a' + dA') a' = dA' (2.1) To illustrate the problem that has now arisen, suppose the vectors are 2 equal in a particular coordinate system. dA* = = a',j dx". (2.2) 2. The coiwna indicates ordinary partial derivatives. 4
PAGE 9
5 Requirements of covariance would demand that this relation be true in any coordinate system. Upon transformation " A. {Th AZi dA' = ^ (A ) dx" (2.3) Since A is a vector, use can be made of its transformation law. dA' =A(^a'^) ^'^'^' (2.4) dx Sx bK^ dA' = :^ dA*^ + a'^ $^ dx'' ^25^ ax^ ax'^ax^ The vanishing of the second term would insure the equality of the two vectors to be a covariant equation. Yet, in general, this is not so. An alternate way to require covariance of equation (2.2) is just to say that the difference between two vectors should transform like a vector. The difficulty in (2.5) is due to the fact that the vectors were compared at different points. It becomes necessary to find some prescription for translating vectors so they may be compared at the same point. This method is called parallel translation and is accomplished in such a manner as to make the equality of vectors a covariant relation. That is, the difference between the vectors, when compared at the same point, will be a vector. The vector A translated parallel from x to x + dx wi 1 1 be denoted i i i i by A + 5A Then at x + dx the difference between the two vectors wi 1 1 be da' = (A* + dA*) (a' + 5a') = dA* 5a' (2.6)
PAGE 10
6 The expression 5a' will depend on the vector A and dx and can be represented by 6a' = P'^^ a' dx* (2.7) where the] is called the Displacement Field Its components, which are to be determined, are functions of the x In this form, da' = la'.^ + r'st ^'^ '^^^ Â• ^^'^^ The term in brackets is denoted by a special symbol, %t = ^''t^r'stA' (29) and is ca 1 1 ed a cova riant derivative The difference, Da' between the two vectors at the same point is to be a vector itself. This requires the covariant derivative to be a tensor. Therefore the transformation law for the displacement field must be P =^^^r +^ijsl_ (2.10) ^*"ax^ a^ ^1 ^^' ax" ^a^' It is clear that ordinary derivatives do not form a tensor and covariant derivatives must be taken Instead. Before the displacement field can be determined beyond its transformation law, (2.10), we must know more about the distribution of matter and charges which dictate the structure of the space. It Is clear that the displacement field is somehow related to the curvature of the space. So far, in this presentation, the notion of curvature has been a vague one, at best. Using the idea of parallel displacement, a mathematical picture of curvature may be displayed.
PAGE 11
Let a vector, A be transferred parallel to itself along the boundary of an infinitesimal surface element and brought back to its starting f i point, X We propose to compute the change, Aa in the vector after one complete circuit. This change is given by Aa' = pA' (2.11) i Expression (2.7) is used for Sa'. The quantities j and A are expanded about their values at the Initial point, and terras kept to first order, since the curve is infinitely small. The expansions are s A^(xJ) (s) r pq A"(s) (X^X^(3)) (2.12) and r'st(^^''=r's. (s) r St,q J (x'' x^/,J (2.13) X ^5j (s; Let t t Â„t X X ^3j = n The change in the vector is now written v^r'^^^^A^fnw.r'strV^'^' (2.14) (2.15) With an appropriate renaming of dummy indices this becomes ^''[r'pt.q^r's.r'pq^*'^'''"'('Â•''' It is understood that the terms preceding the integral are the initial values. As usual, we are interested in the displacement field and want to investigate the term in brackets to determine tensor character. It
PAGE 12
8 can be shown that the Integral transforms as a tensor. Moreover, It is an antisymmetric tensor since ^n'^cn^ = f d(TiV) 'fn^dr^ (2.17) and the first term on the right hand side vanishes, because it is an 3 exact differential. The Integral will be denoted by f\^ ^TiW (2.18) In equation (2.16), the quantity In parentheses can be expressed as a sum of parts v^lch are symmetric and antisymmetric on the indices t and q. The contraction of the symmetric part with the Integral vanishes leaving ^' = .(p ^ p r T +r r jAPfir(2.i9) M pt,q I pq.t I stl pq > sql pV ^ ^' It is knoMi that Aa Is a vector since It is obtained by parallel displacement. The term A f <Â• is a third rank tensor, therefore the tensor character of '^ ptq ~ "J pt.q ^ 1 pq.t I stl pq 1 sq I pt V Â• ) Is established. This is the wel 1 known RiemannChristof fel tensor. R Is also referred to as the curvature tensor, because whenever it ptq vanishes, there is no change in a vector after it Is displaced parallel to itself around a complete circuit. The space Is then said to have no curvature, or to be flat. Contraction on the indices i and q gives the second rank tensor. 3. The hook under the Indices Is used to indicate antisymmetry.
PAGE 13
9 P q p pq rP pP pPpH ^PI'M .,+.,+ I ., (2.21) ik,p I ip,k iq pk qpl ik ^ which Is named the RfccI tensor, or the contracted curvature tensor. It is to play a centraJ role in the variational principle from which the field equations are to be derived. For a variational method, we postulate an invariant space integral which involves the displacement field. Invariance of this integral presupposes the existence of a scalar density, L, which can be formed by the contraction of the Ricci tensor with a contravariant tensor density, jj The field laws are to be derived from 6/LdT = (2.22) where dT is an element of fourvolunie and L f" R.,^ (2.23) ik n^ is considered as a function of the H\ and  ., which are to be varied independently, their variations vanishing on the boundary of integration. This postulate would lead to the equations of the purely gravitational field if the condition of symmetry were imposed on ^\ and pb ... (This will be demonstrated in the next chapter,) However, the present theory is an attanpt to generalize these equations, and the constructions which were essential for the setting up of covarJant 4. A scalar density transforms like a scalar but with the inverse of the Jacobian determinant included. This is necessary to cancel the Jacoblan which results from the transformation of the element of fourvolume. The result Is that the integral transforms like an invariant.
PAGE 14
equations are independent of the assmnption of symmetry .' Instead of symmetry, an analogous condition is posited. It is referred to as Transposition Invariance. Even though ^j and I .. are nonsytranetric, their transformation laws would be invariant if the indices of these variables were to be transposed and then the free indices interchanged. This Is what is meant by transposition invariance. In addition to the variational principle, it will be postulated that all field laws shall be transposition invariant. If we proceed to substitute (2.21) for R., into equation (2.23) I K and carry out the variation indicated in (2.22), we would find that the resulting equations are not transposition invariant. To circumvent such a distasteful result, four new arbitrary variables, K, are introduced by making a formal change in the description of the field. In b place of p ., the following substitution is made h h *b b The I ., is a displacement field just as the ] ... The K are treated in the variational procedure as independent variables. After the 5. An example of an equation which is transposition invariant is s s Sik.b "^skf ib ^isF bk= TranspKJSing the indices on the variables yields s s 9ki,b ^ksP bi 9sff i
PAGE 15
It variation is carried out, the X^ will be chosen to make the field equations transposition invariant, and then they will be eliminated from the system. b The contracted curvature tensor, with P replaced by (2.24) f^ikT) = Rik^r) M^.^k^ .) (2.21a) \^ere R.j^(p)is the same as expression (2.21), except that a star is b put over all the ["* .^. The variational integral is now /<5(^j''[R.,(r) M^j^k\^.)]} dT=0 (2.25) ik *^ After the variations in "^j p .^, and X^ are made, the integrand, denoted by I, takes on the following appearance :\, Â• I k i The last term is a generalized divergence and since it appears in a fourvolume integral, it can be converted to an integral over the threemanifold enclosing this volume. The variations vanish on the boundary, hence the last term in equation (2.26) can be ignored. The variations in the remaining terms are all independent of each other so each coefficient must vanish separately, giving the three equations ^Â•k^r) ^ (^i.k \,o = (227)
PAGE 16
:> ij .k 12 k = (2.28) The four extra variables, \ may be given any value. The three equations will be transposition invariant '^ ^^ is chosen such that P .. = (2.30) ik V Now V can be eliminated entirely from the equations by writing (2.27) as two equations. Rjj,= (2.31) R., + R, + R. = (2.32) ik,b kb,i bi,k ^ V V V b The star may just as well be excluded from above the P ., renaming them. By contraction on the indices k and s in (2.29), we can verify that the expression in parentheses vanishes and the equation can be written Before further simplification is made, it should be noted that the field law, (2.28) is already implied in this expression. This can be seen from the following consideration. The equation formed by exchanging free indices, k .. ? t jj .s^^i) Ptsi! Pst]) rst = (2.33a) 6. The dash under the indices indicates symmetry.
PAGE 17
is Just as valid, if both equations are contracted on the indices k and s and (2.33a) subtracted from (2.33), then clearly Â•s ^.si '11 .sl^'.s) = 'll^3Â• (^j.) The covariant form of a tensor density is defined so that DipF^^^^llpir^ (2.35) Equation (2.33) is multiplied through by Ij.^ and the summation carried out to give r St = 2I1 ikir\s (2.36) which is replaced in (2.33) t.sh]%^f.n)"'^\s*])"\'\^'o. ,..3;, The definition of a density, ^flik 1/2 ik ^ 1/2 Fk IJ = 9 g = (det. gj,^)'^ g"" (2.38) and the rule for the derivative of a determinant, ^s = 99^ 9ab.s = 99abg^^s (239) are used to bring (2.36) into the final form, s s ^ik.b ^skP ib "9.3P ^^ = (2.40) This set of sixtyfour equations gives the relations for the sixtyfour p .^ in terras of the g and their partial derivatives. Once the P I Ik
PAGE 18
14 are known, they are substituted in the other three field equations, r ik= (230) V Ri u + R. u Â• + RuI = = Rr., ., (2.32) ik,b kb, I bi ,k [ik,bj \ j i V V V v for a solution of g.^^. This is the formalism of Einstein's unified field theory [5]. Schroedinger 's unified field theory is quite similar to that of Einstein's. Parallel transfer, displacement fields, and the curvature tensor all form the basis for the theory. The difference appears in the integrand of the variational principle. Whereas Einstein took ik U Rji^ as the .scalar density, Schroedinger chooses [I8] L=^(det. R.^)^/2 (2.41) which is the simplest scalar density that can be built out of the curvature tensor. The constants 2 and ^have no influence on the result. A contravariant tensor density' is introduced here also: but it can be eliminated in the end, leaving equations which contain b only the P ... When the variation, 6/LdT = /^ SR.j^dT = (2.43) ik 7. The square brackets in (2.32) indicate summation over the cyclic permutations of the enclosed indices. This convention will be used hereafter.
PAGE 19
15 is carried out with (2.41) as integrand, the result is the following set of equations, 9ik.b9skP'ib9isr'bk=0 (2.Z,i) R., = Xg.^. (2.45) These can be taken as the field laws as they stand, but Schroedinger takes the theory further. Equation (2.45) can be substituted in (2.44) to eliminate the g., The field equations are now sixtyfour differential '" b equations involving nothing but the sixtyfour P ., R ik.bf^skP ib'^isThk^O Â• (2.46) This is known as the "purely affine theory." Kursunoglu's approach to the theory [16] retains the basic ideas of displacement field, curvature tensor, and the fundamental tensor ik g.j^ (or g ) but the scalar density for the variational integral is quite different and entails some new concepts. These will be defined before the variational principle and the field equations are displayed. The fundamental tensor, g., is expressed in Kursunoglu's theory, I K in terms of its synmetric part, a,. and its antisymmetric part, 4>., 9ik=^k^<2G)'/V\*., (2.47) where G is the gravitational constant and r is a "fundamental length." o The determinant, g, of g., is constructed in terms of what will be called the "Kursunoglu invariants," namely SI = Y*'''*ik (2.48)
PAGE 20
16 and ^ ] v1/2 ijkb^ ^ /~ iÂ„\ X=3 (a) eJ <>..(t.^^ (2.49) g = a(I + 2Gc ^r A 4g c r V) <2.50) o o The action principle from wliich the equations are to be deduced is *Â•^Ml'''^Â•<" 2'o'^'^b ^a)]dT= (2.51) where b is the determinant of the tensor, b., = a'''^g"'/^(a., + 2Gc'\ h. *^ ) (2.52) ik ^^ik oisk' \ ^ J The variations carried out in (2,51) produces the Kursunoglu field equations. 9ik,b SskP ib 9isr\k = ^2.53) '^ik='o"'(^ik^k^ (2.54) ^[i^k.bl = V'(*ik.b ^ *kb.i ^ *bi.k) (2.55) The nomenclature, "fundamental length," is rationalized by examining the field equations. The vanishing of r gives the general relativistic case in the absence of charges. This existence of free charges is now associated with a finite fundamental length. All three field theories discussed here have at least one thing in common: they have been constructed so that there exists a correspondence principle which takes the unified field equations over the wellknown gravitational equations of general relativity when the antisymmetric field is absent. This is easy to see. First consider Einstein's
PAGE 21
V 17 set of equations and require the g,, to be composed of only a symmetric part, a.^. g., = a = a (2.57) IK 1 k ki b The solution of equation (2.40) for the P is readily found by permuting the indices to get three eojuations s s a., a P .. a. P = (2.40) ik,b sk I lb IS 1 bk ^ ^sr ki ^ksT ib= (2.58) s s 'kb,i sbl ki ^ksl ib s % au^ a .P ,, a. P = (2.59) bi ,k SI 1 bk bs 1 ki ^ A combination of these three equations gives 5 Si u Â• + Su! I ~ ai u = 2a P (2.60) kb, I bi ,k I k,b sb I ki ^ Equation (2.35) applies here b P ,. = ~a''P(a. + a a., ) (2.61) I ki 2 ^ kp,i pi,k ik,p' b The symmetry of the P is a consequence of the synmetry of the a. The last notation is a matter of convention. Equation (2.61) is the "Christoffel symbol," where it is understood that only the symmetric part of the q., is used. The synmetric tensor, a is identified as the ^ik ik metric tensor of the space. Since equation (2.30) is satisfied identically, the X., may as well be chosen zero, in which case (2.31) and (2.32) could be recombined as
PAGE 22
18 This is precisely the gravitational field equation in empty space. (See, for example, [9], page 81) A particular solution to these equations corresponds to the field of an isolated particle continually at rest. The famous explanation of the discrepancy in the advance of Mercury's perihelion is a result of the solution to (2,63), There is no question about the physical significance of the gravitational field equations, so this gives a certain measure of confidence to the generalization. (2.30), (2.31), (2.32), and (2.40). A similar situation exists with Schroedinger 's theory in the limit of a symmetric g. The field equations, (2.45), become the same as Einstein's with the addition of a term involving X, which is now identified with the cosmological constant. Actually, the limiting case of Schroedinger 's theory is the original form of the gravitational equations. The cosmological constant is so small that it need not be included on a scale such as our solar system, (See [9], p. 100) In Kursunoglu's theory, the correspondence to general relativity is achieved by the vanishing of the fundamental length. Again we revert to the equations of general relativity. This method is much like quantum mechanical correspondence to classical mechanics when Planck's constant vanishes. Therefore all three theories have at least some basis, due to the fact that they reduce in the limit to the wellknown and tested equations of gravitation derived by Einstein in his general theory of relativity. What has been presented so far is only the problem: a nonsymmetric, fundamental tensor, g. is chosen to represent the fields in Nature. "^ b The displacement field, H .^, is introduced to insure that the field laws will not be dependent upon the choice of a coordinate system. From
PAGE 23
19 a postulatory basis, a variational principle is used to derive a set of field equations. A solution to these equations will presumably give a description of nature in which all fields are united in the single tensor, g.j^, in the same manner as the electric and magnetic fields are unified in the electromagnetic tensor. Now that the problem is laid before us, the next step is to provide a solution to these field equations.
PAGE 24
CHAPTER 3 REDUCTION AND SOLUTION It IS evident that the three versions of unified field theory, which have been presented, are little more than postulates. Their tenuous claim to validity comes from the correspondence to the known equations of general relativity and the correct count of functions. So far there has been no indication that they give a field description of Nature in which all fields are united in a single tensor. In keeping with this spirit, we choose to deal with yet another form of the equations, which can be considered as an adaptation of either Schroedinger 's or Kursunoglu's equations to an Einstein model of the universe. s s ^ik.b^skP ib SisP bk= (3.1) 2 9n, i ?^ 4 ?^ k ik=) '^ (3.2) otherwise R., = (Je ^'^ Rr. 2 [ik,b] ~ g2 9[lk,b] (33) k lik'O (3.^) V In the limit of a vanishing antisymnetric field, the antisynsnetric part of I .^ vanishes and the symmetric component, P .^, is the Christoffel symbol. Then (3.2) is seen to be exactly the set of differential equations which describe the Einstein Cylindrical Model of the universe (See 20
PAGE 25
[9], p. 159). Equations (33) and (34) are satisfied identically. Due to this correspondence, these equations should be considered as valid as the other three sets. In all four cases, it has not been possible to ascertain if the theory contains any information other than the simple reduction to the gravitational equations. The reason for this is that a simple, general b b expression for the P ,. has not been found. True, the P .. have been expressed in special coordinates, but the differential equations have not been solved using these values. In this chapter and the subsequent b ones an appeal ingly simple form for the P .. in a special case, along with new information from the field equations, will be presented. b The relation for P ., (31) is correnon to all four versions. It I Ik ^ b is a set of sixtyfour equations from which the sixtyfour 1 .. are to be determined as functions of the sixteen g., and their derivatives. ik A solution to these equations has been given for a system of spherically syrranetric coordinates. [17, 21] Kursunoglu [16] sets up b the differential equations, (2.31) and (2,32), using the P .. expressed in terms of the g,. but can offer no explicit solution. In fact, no solution has yet been given to equation (2.31) in these special coordinates. It Is for this reason that a completely general (coordinate independent) solution to (3.1) is desired. A little manipulation of (3.1) is enough to show that a general solution is far from trivial. Nevertheless, several formal istic solutions have been offered. [4, 10, 22] Formal istic, in this context, is b meant to Imply that an explicit expression for P ., is never written I Ik jj down. What is given is a prescription for determining P A brief danonstration of Einstein's formal istic solution [k] may illustrate this
PAGE 26
22 s better. In (3.1), ggi^P i. and g, P ,, are considered as individual terms and named 9skrib = ^ikb (3.5) The v., ,, and W., i, are related to each other by ikb ikb 9''Viklb = 9'^jÂ„b (3.7) This allows the V.. i, to be expressed as a W., In (3.1) 9ik.b^bki Ak^lbp=(3.8) b Now, if the W are found, then the H are known as a consequence of (3.6) r'bk^^Vn Â• (39) Equation (3.8) implies another expression for W.. i. bki = 9jk.b9^'\kWbk' Â• (3'0) which is reinserted in place of the last term in (3.8) 9k.b ^bki ^^'\k\'bri '^ ^^\k^^\b\'\\b' = (3'1^> If this procedure is repeated once more, k'a ^ k'a b'p 9k.b9 gak%'b.i*9 93^9 Qpt^pj^k b'3 k'p i 'c ., /, .Â„x = "bkii + 9 g^^g gp^g gci^b'k'M' Â• (3'2)
PAGE 27
23 Einstein now defines k'a i 'a k'p >, ,_ '^iklbSbk.i 9 gak%'i.b^9 9359 9pkgi,b.k' ^3.13) and I'k'b' .i' fk' ^b' ^ i 'a k'p b'c ,, ,.x so that "'''''' ikb^i'k'Ib=^ikb Â• ^3'5) The problem is to find U, the inverse of U, so that ikib = ii''''''ikb^'kÂ•b ^3.16) and the knowledge of W., 1, then gives the  ., by (3.9)Einstein's prescription for finding the inverse is presented in four pages and the answer becomes much too complicated to write down explicitly here. To quote Einstein in this paper, "Such a solution can indeed by arrived at. .but it is cumbersome, and not of any practical utility for solving the differential equations." This statement also applies to the solutions obtained by Hishra [22] and Hlavaty[lO]. So, for all practical purposes, we are still left with the problem of sixtyfour equations in sixtyfour unknowns. One additional reduction is possible. The number of equations and unknowns can be reduced to twentyfour if we treat synwietric and antisymmetric components of  ., as separate quantities. [21 ] (This Is a natural thing to do since they transform separately. In fact, it can be seen from (2.10) that the antisynwnetric components transform like a tensor, whereas the symmetric ones do not.) The fundamental tensor is also written as the sum of its parts. Consider the two equivalent equations
PAGE 28
24 s s s s + 9isrbk^9j3rb^ + g.3rbk^9.3rbk Â— Â— Â— V V V V s s s <; 9kL.b'5ki.b=93j_r kb^^siP kb^^siP kb^s^.p (3.1^) kb V Â— Â— V V Â— V V ^ksP'bl* ^ksP bi ^ ^sP'bi* "ksP'bi ^2 '7^ V V The sum and difference are two new equations s s s Â„s 9ik.b = 9skr ib*9.3Pbk + 9skr Fb^Sj^P bk (3.18) Sik.b = 9sk P ib ^isP bk ^skP ib 5;, r bk (3.19) This procedure is repeated twice by permuting the indices i, k, and b. Two more pairs of equations are obtained. s s s s %b.l = hb r ki ^ \s P lb ^ ^.bP ki \s P ib (3.20) Â— Â— Â— Â— Â— V V V V ^kb.l = ^sbP'kl 9ks P'ib ^sbp'kl ^ ^sP'ib (320 and > :> 3 9 9bl,k=9sirbk*^bsPki^93jP bk^gtsP ki (322) V V V V Sbi.k ^si r bk hsY ki 93. f bk ^ ^sP ki (3.23) The following combinations are taken s s s "^ik.b ^kb.i ^ ^bi.k = 2[g3bP ik SsiP bk ^ ^ksP ib^ (3.24) Â— Â— V V V V
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J 25 Using (2.62) and (2.6l) as definition, Substitution of (3.25) into (3.19) and use of definition (2.9) for a covariant derivative gives the set of twentyfour equations, P P f" 9ik;b = 9^r ib ^ ^i^r bi< 9^ ^W^mb P is m mm V V V ^ ^ V V b to he solved for tiie twentyfour ., When they are found, substi^ b tution in (325) yields P Therefore the sixtyfour r are known once (3.26) is solved. This still does not give a tractable form for solving (32). Einstein's hypothesis cannot be tested unless we find some way of solving the differential equations and these in b turn cannot be solved until a useful, general form for the J ., is obtained. We are not completely stymied by the formidabi 1 ity of the equations. A further advancement can be made. It can be shown that, within the framework of the theory, the gravitational and electromagnetic fields are contained in the single tensor, g., and the field equations are those of electrodynamics as well as gravitation. To see how this comes about, an alteration in the fundamental tensor is made. It is chosen to be a CI iffordHermitian tensor. This means the ant isytrmetric part is chosen as ^ik^^ik' <^27) V where e is so small that its squares and higher powers will be neglected. Then the tensor,
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26 9fk=9j_, + ^;k (3.28) is called a Cl iffordHermJtian tensor after W. K. Clifford. For the CnffordHermit ian field, (326) becomes ^^k;b = Vi^ib^'lEP^k Â• <3.29) V V If i, k, and b are permuted, two more equations are obtained. ^*bKk vr'br vr\i "Â•"' The following combination of the three equations is taken i^^*ik;b*kb;i "\i;k>'3Hbr'ik^^'^'^ V Which is immediately solved for b r ik=2^9^(*ik;p *kp;i %i;k) Â• (333) This result causes (3.25) to become \ \k (i k) Â• (3.34) and the displacement field is known r'ik' V^^^'^'iXip'V;! *Pi;k> Â• (3 This is a general form insofar as no particular coordinate system has been specified. It is also in a useful form for the field equations (3.2). (3.3). and (3.4).
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.J^' 27 First consider (3.2). If (3^) is kept in mind, then R., is given by R., = r (Rj, + R, .) ik 2 ^ jk ki '"ij'm 'Â•*'""' (3.36) If i or k take on the value "k", the second part of (32) is identically satisfied. Equation (336) is a second order partial differential equation in the g., It is easily verified that .Pp^= logfAgJj (3.37) so that In analogy with general relativity, g fnust be interpreted as the metric IK in the space. If we choose the physical space to possess spherical s^nmetry and the time dimension to be uncurved then the line element o may be brought into the form ds^ = (^^(S^ + (K sin^de^ + (^ sin^sin^ed*^ c^dt^ (339) in which case the metric is 8. "(5^" is interpreted as the radius of this world.
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2d 'jk + (C +(R?sin^f +afsin^sin^9 1 (3.^) The left hand side of (338) is calculated using (3.^0) and indeed the differential equations are satisfied by (340). The form (339) represents what is called an Einstein cylindrical model of the universe. The contravariant tensor to (3^0) is simply ?k (3.41) ik since g., is represented by a diagonal matrix, (3^0). With the g.,^ solved, we can move on to (3.4), using (3.33) 1 pk k r ., = = jeg^ (. is antisyrmietric, the second term Kp vanishes. Furthermore, p and k are dumny indices so that the first and last terms add. 1 ik;p (3.43) In this form the field equation is significant since it is recognized as one of the covariant electrodynamic field equations in the absence of charges and currents [19], with the 0., interpreted as components t K of the electromagnetic tensor 9^. ik: = (3.44) 9. The solution will be examined in more detail in the next chapter.
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29 The remaining set of field equations (33) may be written '^i^k;fa^\b;i "*\i;l<=^<*lk;b^\b:i **bi;k) (3^5) From definition (2.21), and (3.4) and the results of the CllffordHermitian field, it follows ,P rjPCrv") nP R 4(il'p],klk'pli) (3.46) The term in parentheses vanishes due to (338) and the remaining term in brackets wi 1 1 be recognized as the definition of the covariant p derivative of the tensor, \ ik V V V Then by (3.45) ^ik" T ik;p (3^7) P p p ^P l^k.pb^r kb;pi ^r bi;pk> ==T<^Â•k;b*\b;l ^^bijk^ V (3.48) It remains to be shown in the next chapter that a solution of (3.47) is the second set of covariant electrodynamic equations. So far, by use of a CI if fordHermit Ian tensor, we have been able b to present a reasonable form for the P ., and a solution to field I ik ^ equation (3.2). Also (33) and (3.4) have been put Into a more familiar form and are ready to be solved in the next chapter by use of a special coordinate system which will make the equations especially transparent.
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) CHAPTER k SPECIAL SOLUTIONS In the preceding chapter the field equations were fashioned into a suitable form for solution. The three equations which must now be solved are KlpMi^UMr^ft\i.W^,\i=^=H,. 0.38) \ <^ k ^ k J r" .pp Minkowski Space First, a very special situation will be considered. It is the opposite of the limiting case where the antisymmetric field vanishes. Now, the field equations will be examined when "(^' Js chosen Infinitely large and the Cl i f fordHermit i an tensor Is assumed to have Minkowski form, 9ik^^k^^*lk Â• ^^''> In light of this, all Christoffel symbols vanish and (3.38) is satisfied Identically. There is no gravitational field. It Is to be expected, since there is no gravitational field, that what is left of the tensor g.. should represent the electromagnetic 30
PAGE 35
i 31 field, and the field equations should be Maxwell's equations In a Minkowski space. To see that this Is so, let us Investigate {3.kk) and (3.48) for 10 ^Ik^^k (^Â•2) First, It Is seen that there Is no distinction between contravarlant and covarlant tensors. Furthermore, any covariant derivative may be replaced by an ordinary derivative since all Christoffel symbols are zero. Equation (3.M+) can be expressed J ) ^, = (^Â•3) Indeed, If the ^^^ represents the electromagnetic field, *lkle, le. ie, le. Ie, Â• e. (4.4) 1 2 3 then the second set of field equations (3.44) becomes one set of the Maxwell's equations in an empty space, as expected c Vx h = r g ; for i = 1, 2, 3 V e = for I a= 4 (^.5) (4.6) The third field equation, (3.48) can be written 10. We wl 1 1 choose x = let
PAGE 36
) 32 <'Â•'' The solutions, (3.33) for P j^ are substituted. usJng 6' for gÂ— In accord with (4.2). f'(*lk.S *k5.I sl.k'.b (kb.S bS.k *sk,b),l <*b>.s *ls.b sb.i'.kl.s ^ <*lk,b \b.i *bl,k' ("> or 2 ^ lk,b *kb,i bl,k\ss ^ (5^2 ^ ik.b kb,i bi.k' A solution to this equation Is Ik.b kb.i bl.k which, In view of {k.k) represents the other set of Maxwell equations In a Minkowski space. V Â• 1^= ; I, j, k 5^ ^ (^11) Vx?= .ij ; i. j. or k=4 (4.12) c ^ Therefore, as expected, the field equations in Minkowski space become the Maxwell equations, while the fundamental tensor represents ^ both the metric and the components of the electromagnetic field. These equations, (4.5), (4.6), (4.1!) and (4.12) are well known Â— their solution and validity need no elaboration.
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} 13 A More General Case We now move to a more general case, where "^" is taken as finite. A solution to (3.38) for the q^^ has been given in chapter three, but a slightly different version of the solution will be given here. First, note must be tal
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) ) 3k The Chrlstoffel symbols are computed according to (2,6l) and substituted in (338). It is seen that the solution is 2 \ f. r v1 e =('52) V e = Thus in matrix form, the g., are ih. 2 (4.20) (4.21) 'Ik 2 2^ sin o 1 (4.22) This model is known as the Einstein cylindrical universe. (See [24], page 359) A change of variables will serve to show more distinctly the character of this universe. Let x, = r cos 6 x_ = r sin cos* X. e r s i n 6 s i n4> (4.23) In which case, ds = (dx. + dx+ dx, + dx. ) c dt (4.24)
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35 and Xj + Xg + x^ + x^ = g^ (^.25) Thfs illustrates that the physical space of the Einstein universe may be interpreted as the threedimensional bounding manifold of a sphere % of radius (j\. in a fourdimensional Euclidean space with the cartesian coordinates given above. The time dimension is uncurved. Hence the name cylindrical universecurved space and straight time. Even though the line element displays spherical symmetry, there equations more transparent, we adopt a method used by Kronsbein 02]. 12 The sphere represented by (4,25) is radially projected from its center in the fourdimensional space, (x.), into the threedimensional space, \ Â€, with coordinates X (Greek letters take on values 1, 2, 3) by the projection Is no symmetry of form among the ^. ^\ and the R., To make the field The solution shown In the previous chapter, (3.4l)f may be obtained directly from {h.lk) by the transformation Xj = ^cos f x^ = (Jt s I n^ cos X=
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36 ^^ (R. A 1/2 (4.26) where A = (S(^ + (xV+ (X^)^ + (X^)^ (4.27) In this space, X^ and X = ct we wi 1 1 denote the symmetrFc part of the fundamental tensor by a., It Is computed by the standard method to give J Ik A [A(x')2] x'x^ x'x^ XX [A(X^)^] .x^x3 2 3 Â•X X^ xV [A(X^)^] (4.28) The determinant Is i^kf = 4 (4.29) The contravariant tensor to (4.28) Is found to be '} Ik A [(R? Â• (x')2] x'x^ x'x3 x'x^ [(Si} + (x^)^] xV x'x^ xV [(R + (X^)^l 9t A (4.30)
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37 With (4.28) and (4.30), the Chrlstoffel symbols can be constructed. They are very important because they are the symmetric part of the 13 displacement field. 21. [a ay (4.31) There is one more quantity which we need to calculate in this space for future use. It is the analog of the RiemannChr istoffel tensor, (2.20). It will be denoted by ll
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38 h. 3 (^.35) where the "e^^" and "h^" are functions of (x\ X^, X^ t). We now have the necessary material to calculate the antisymmetric b components of  ., in our special coordinates. The actual calculation of all twentyfour P ,, is a long, tedious process. Only a sample 'v 3 calculation (the component,  .j' ^' ^ ^ ^^ ^^^"^ ^^ =Â•" example) and the final results are given here. By (333), r 12= ^ta3'.,^^, .a32.,^.^^.la33(.^^^, .*,,.,..Â„. J] (4.36) ;3 23;1 31;2' r,2^^^'^^2;l* A2;2*^'^ti(<^12,3"'23J *31.2) *i2 {13)" II \3 23 ]} (^.37) Upon substitution of (4.30), (4.31), and (4.35), and consequent cancellation and simplification, It is found that le} + (X^)^][y(h,^, + h2^2 + SJ S ^^^^] + ^\ +x3h^)]) (4.38) b The other ., are computed In the same manner and soon the following
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\ ) pattern Is recognized 39 r e, = ^[^^(3X\ a"\_,) c^a<*(i V,. i V 1 (^33) and '' rA .A X^ aM1 oMi ^"^a^h^^^] {h.kO) also P V and These are the twentyfour jl'" ^^^ special coordinates. Combined b I K V with the n .. given by (^.30. the complete set of the sixtyfour is known in this space. This completes the information needed to set up the field equations, (3^) and (3.^8). It has been pointed out that (3.^). which reduces to (3^5) Is identically a set of the covariant electrodynamic equations in the presence of a gravitational field. To see what they look like in this space, It will be easier to go back to (3.^), since the  ., have ik already been computed. The equations represented are: r'i2*r'i3^rV= <"'"> '*^\ and Â€ are the usual permutation symbols whose value is zero if any two indices are alike, +1 if (<^BA.) is an even permutation of (123), 1 If (=BX) is an odd permutation of (123).
PAGE 44
40 } \ y ?\^^V\,V\k^ i'^) r\, r'32^r 34= ^^^5) 31 I 32 ^1 34 4] + I 42 I 43 r\, r\2r\3 = ^^Â•'^^^ ,b The n ,, are substituted from equations (4.39) to (4.42). After simplification, the four equations are: 3(X^h3 X^^) + f^ + (X^)^^'^3.2 ^^^ (X^)^^h2^3 ^xV(h3 3 h^^) ^X'(X% X^h^.,) ^e,^^=0 (4.47) 3(X^h, x'h^) + [(SL^ + (X^)^]h, 3 [(R.^ + (xV]h3 x3x'(h, h3 3) x2(x^,_2 >^'^^3,2) 4^ 2.4 = (^^S) 3(X^2 X^h,) + l^^ + (xV] h2 [(.^ + (X^)^]h, 2 *x'x^h2^2 ^j) Â•^X3(x'h2^3 X^h, 3) .^33^^=0 (4.49) 2(xU, ^ X^e^ + X^e3) + [(SL^ + (xV e, + x'x^e, 2 + X^X^e, 3 + X^x'e2 + [(Sl^ + (X^)^]e2^2 + ^^^^2,3 + X^x'e3 + X^X%3 2 + IS{} + (x^^]e3 3 = 0. (4.50) These equations are exceedingly complicated as they stand. They will be left this way for the time being. After the other set of four field equations (3.48), has been written in this space, a simplification and solution will be presented for all eight equations. A tensor identity which will be of great utility in the investigation of (3.48) is
PAGE 45
^1 .Ik.Ik. Ik^s ik^A A =A G,+A 1.GS + bnT;pq bmÂ• Â• ;qp smbpq bsmpq A^kÂ• Â„ isÂ„k /I ^,x A G A G Â• Â• (4.5 ) bmÂ• spq bmspq ^ "^ i kÂ• Â• s where A is any tensor of arbitrary rank and G is defined bmÂ• Â• tpq by (4.32). This identity allows the left hand side of (3.48) to be written I ik;pb*l kb;pi "^l bl;pkM ik;b"^' kbii"^' bi;k^p V V* V s^ V Â•Â• *l tk*^ Ipb *l it kpb *' U>''tbp Â• tk kpi kt bpl kb tip ^ If g'^j. Is written out as prescribed In (4.32), then it Is seen to be '^'ikp^k (^53) It Is clear that If any of the i, k, b are equal, then (4.52) vanishes, as does the right hand side of (3.48). Equation (3.48) represents only four distinct equations. They are: r'lj;p2r'3j;pl ^^'2J;p3I(^3;2^^2;l*^l;3) ^''''^ y r'l^;p2 ^ r\2;pl Â•^r'21;p/, = f2 (^4,2^^2;l *21 54^ ^^'^S) r 43;p2 *P 32;p4 ^P 24;p3 = % ^*43;2 ^32;4 + *24;3^ ^""'^^^ The left hand side of (4.54) is expanded like (4.52).
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^ Hi p I 13;p2'^n 3J;pl ^n 2J;p3"M 1^;2"^ 3J;1 ^l 21 ;3^ 1 ^3 ^ 1 tr 3 *n ,^^'uz^V t3^'l32*r 1/312 *P 1/332 >^ V V V ,1 n 2 r73 + 1 ..G^Â„, hI ..G\Â„ +1 ,.G\., +1 ^.. 231 I 13*^12 *l 13^22 U"^!?. "'V321*r\2^'331 *r 3t<^'221 ^P 3/ 1 2 3 "I 32^1 *n 32'^21 *l 32^^31 *r"t,s'2,3 r't,='2Â„ *r '2/113 ^r \/ 2 ri3 +r zi^^n^n 2i'^23*r 3i'^33 ^^^^^^ ^13 21 "23 I 31 33 .s If the index "4" appears In a G ^ it vanishes. For this reason, the tpq following type of combinations must be observed. g',2, .g3,,3 = R^ (4.59) Â• When the summation is carried out on the index "t" as Indicated In (4.58), and combinations like (4.59) heeded, a fortunate cancellation occurs which leaves r',3,p2 ^r'32,p, *r'2..p3 = f'^.a f '32;, ^r'2U3';p (^'' V V V V V V ^ The field equation (4.54) Is then 3.2*r'32;, *r'2.;3>;p&<*.3;2**32;, **2,,3'(^Â•^" b At this point, the set of M ., as given In equations (4.39) V through (4.42), could be substituted, but the left hand side of (4.61)
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i J i3 can still be reduced. By (3.33), p r1 s^Pr* +4) +d) d> * [I3;2];p 2^.M3;52* 32;sl '^21;s3 3s;12 *sl;32 The summation Is carried out, as indicated, over s. Tlien (4.51) is used to make tiie following type of rearrangement ab;cd ab;dc pb acd ap bed throughout each term in (4.62), giving r'[l3;2];p = f ^''^B^ahsp ^'="'(*13='n2 V'lU *I2='312 ^1^13 ''21^^l3 ^ V^13^P ^''^'l2'321 *3iG'221 V'221 ^ ^1^'223 ^ *31^^23 V^23^P a3^*23^'l32 *12'^'332 ^3'^^32 "^ ^2<^^31 *v'33l^^l^^3l^pÂ• ^''^^ The covarlant derivative of the G .., are all zero, since, according to J kb equations (4.34) and (4.28) J ^ J 1 is a constant and = B7:^=aB C^S) a^.p = (4.653) 1 5 No summation Is Implied by repetition of the index 7 In these equations,
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therefore The following equations are now substituted in {k.Gk): (4.65), (4.28), and (4.30), Cancellation and simplification leave 'P Â€ sp^ .2 .^:2];p 2 ^ '^[I3:2];sp "" This result reinserted in the left hand side of (4.54) gives finally r'[13;a].,pf^''*[,3.2].sp*i2^n3,2] Â• ''Â•"' aSP((j) + 4) + 4) ) =0 (4.68) ^ 13;2 32;1 21;3 ;sp The same procedure and the same Identities are used on equations (4.55), (4.56). and (4.57), giving ^''(^4;2*%2;l^^l;4>;sp = ^'''^ ^''^\3;2*S2;4**24;3^;sp" ^''^'^ ^ 13;4 34;1 4l;3 ,sp Solutions to (4.67) through (4.70) are, respectively, *.3,2**32., **21,3 = <"Â•''' *24.1*%, 2**12,4 C^' J V.2 \2.3 *23.4 = <"*' *I4,3*\3.1**31,4= Â• ^"^^ Using e^ and h^ from (4.35) we see that this is the second set of covariant electrodynamic equations.
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45 "2,2 "3,3 '"''' =3.2^2,3*"l,^= ''Â•Â•^' ; ^1,3 ^3,1* "2.4= f^^' These four equations are simple compared with the other set, (^.47) through (4.50). In order to simplify the latter, we are willing to slightly complicate the former. The result will be a symmetry of form for both sets, and moreover, a solution will follow easily. This is accomplished by replacing the e and h by the following quantities J e^(x';x^x^t) = ^ [.Â£Â„ + e^iJY^e^'^^ Â• ^^'^^^ where H^ = H^(t) (4.82) and E, = E^(t) (4.83) are yet to be determined. These values are substituted in the field equations in place of e and h In the first set, (4.50) is satisfied identically by (4.81) Equation (4.47), after simplification, reduces to 2H, y3 2H5 y2 2H.
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or 1 ht c5?^2 1 St ich implies 1 ^E* 2 > c ^t fit (^.85) (4.86) ^ This same result Is Implied by (4.48) and (4.49). In the second set of equations, (4.76) is satisfied identically by (4.80). When (4.80) and (4.81) are substituted in (4.77), the result Is [(%} + (X^)^](2E3 +
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^7 ^ Â— ^ Substitution of these equations Into one another allows E and H to be eliminated from each equation respectively, giving two ordinary wave equations, 7&'W^'' (MO) ^^+^r=o (^.91) c^ dt^ oJ^ These represent an electromagnetic wave of circular frequency '^=.a. 2c (4.92) where E^(t) = G sin wt (^93) H^(t) = G^ cos wt (^9^) The Gff ^s a constant. So we finally have the solution to the two sets of equations, (4.47) through (4.50) and (4.76) through (4.79). it is, in the special coordinates, and With the *,. given by (4.35), the above is the solution to the last two unified field equations, (3.44) and (348), which we had set out to solve at the beginning of this chapter. In vector notation, the solutions are
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48 h^% [aV+ (X*. m* (h'^x^] (^97) a2 e^^ [Â€^+ it^^lt)] (4.98) The reason for using vector notation will soon be apparent. Now that the solutions are known, we want to visualize them in the spherical space. First, consider the "straight lines" in this space. The geodesic equation, d^X* ll' dx"" dX*" 6X' + rkbfrjr" Cs^) becomes d^x' ^f f ) dxll dX^^ (k \00) s I nee r 'kbfrfr'O (""") and r kb k b\ Â• (3.35) This shows that the geodesies of a space are not altered by the presence of electromagnetic fields In the space to the first order which we have 1 Investigated. (A question which immediately comes to mind is, "Well, do the properties of the space affect the electromagnetic fields?" Of course they do. We have seen how the electrodynamic equations were complicated. Exactly how they are affected remains to be seen below.)
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From [12], the above equati'on (4.100) can be writt en ) A" 2 dX^ .dx! ^^2 A dA ^^ Â• d\^ 2 k ^^= (4.102) An explicit;' Integral of (4.102) Is j,l ^ sin Msin (X H6) cos (X + Â€) X^ B sin Msin (X + 5) cos (X + e) X = tan (X + Â€) (4.103) We liave set (J^ = 1 for simplicity. This is the mathematical form for 16 the "straight lines." A geometrical picture may be found in [l2]. Now when the sphere, (4.25) is rotated with angular velocity (^, In E^^, it Induces transformations on X in Â€_. Points move with el 1 ipt Ic velocities (R.(^ (S^a?+ (X*. (j5x*+ <5^(i?x X^ (4.104) It can been seen that the geodesic equations fulfill (4.104) so that the points move along straight lines In the space. Now, replace w^ln (4.104) by H*and the result Is the same as (4.97). We see that the magnetic vector, h*, has the form of the associated contravarlant elliptic velocity, (4.104), and lies along the geodesies of the space. So not only Is the magnetic vector altered by the space, It Is altered In this very special way. 16 The geodesies will sometimes be referred to as Clifford lines.
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) 50 If we tensor multiply the contravar lant velocity by g, we get the covariant velocity vector, which turns out to be the same form as the electric vector, (4.98) X. = s^ [
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; > CHAPTER 5 SUMMARY There is a possibility that the main features of the theory presented have been obscured by the preponderance of raathamatics. For this reason, It may be well to summarize briefly the results of this dissertation. The second chapter contains no new results. It is intended as a background to familiarize the reader with the different versions of the unified field theory. The similarities and differences are pointed out. All three have the same goalto derive all of the fields and field laws governing Nature from a single tensor, which need not have symmetry. The main difference is the choice of integrand for the variational principle. This, of course, leads to different forms of the field equations; however, the same set of sixtyfour algebraic equations appears in each version. The most Important similarity in the theories is that they are based on the same geometrical concepts, and they all go over into the gravitational equations of general relativity In a limiting case. Notwithstanding all the beautiful mathematical formalism, this correspondence to the laws of gravitation is the only physical content which has been derived from any of the theories up to now. The reason for this difficulty was pointed out in the next chapter. Before Einstein's hypothesis (which was discussed in the Introduction) can be tested, the differential field equations must be solved. Before 51
PAGE 56
_; 52 they can be solved, the sixtyfour algebraic equations must be solved in a general case (without reference to any coordinates) for the components of the displacement field. Since this general solution has not been accomplished, there is no way of telling whether all the field laws are Included in the theories. The sixtyfour equations are shown to be reducible to twentyfour, but this has not enabled the solution to be found up to the present. At this point, we present our version of the theory, in which the fundamental tensor is modified. It must be noted that this is not a linearization such as that used by Kursunoglu [\k] or Einstein [3]. These linearizations exclude interaction between the symmetric and antisymmetric components of g., The present work does not preclude this possibility. In fact, the antisymmetric components of the displacement field were seen to be a combination of the symmetric and antisymmetric parts of the fundamental tensor. So instead of a linearization, we have a perturbation type of technique. It is basically a first order antisymmetric perturbation of the gravitational field producing symmetric tensor. With this method, the algebraic equations are readily solved for j ., without the specification of any coordinate system. It is found that the R., become the same expressions used in general relativity. With this in mind, we adapt the set of field equations involving the contracted curvature tensor to an Einstein model of the universe. In the limit of an infinite radius of the Einstein universe, the remaining differential equations go directly over to the Maxwell equations of Minkowski space. For the general case (a finite radius), a transformation was made to a symmetrical arrangement of coordinates. Investigation of the remaining two differential field laws leads to the
PAGE 57
; 53 important discovery that tliey are tlie covariant electrodynamic equations in tlie absence of charges and currents. As a consequence of the synmetry of the coordinates, we are able to give an exact solution to these equations. This solution indicates that the electromagnetic is bent along the geodesic caused by the gravitational field, while an investigation of the geodesic equations shows that the gravitational field is unaffected by the electromagnetic field in this case^ Perturbation to higher order terms in Â€ still makes it possible to obtain the displacement field explicitly but this does not, up to the present, Imply success in solving the associated differential equations. In these cases, the presence of addition fields will distort the geodesies of the pure gravitational field, but it is not known whether the additional fields are only electromagnetic in Nature. In summary, the most important result of this disseration is the realization that, to the first order, the covariant electrodynamic field equations, as well as the gravitational equations, are Included in the unified field theory. Heretofore this was conjectured but never shown. It is clear that what has been done is far from an ultimate goal of the theory. Nevertheless, we feel that our contributions should be an impetus to further work in the field. The most general case must be pursued. Along these lines, Kursunoglu has made the greatest Innovations since the theory was formulated by Einstein. It would certainly be desirable to study his plan in which there is a possibility of deriving nuclear fields together with those previously discussed. [16]
PAGE 58
BIBLIOGRAPHY 1. Einstein, A. The principle of Relativity New York: Dover Publications, 1923, pp. i 11164. ^ 2. Einstein, A., Ann. Math. 46. 578 (1946). 3. Einstein, A. and Strauss. E. Ann. Math. 47. 73' (1946). 4. Einstein, A. and Kaufman, B., Ann. Math. 59, 230 (1954). 5. Einstein, A. and Kaufman, B. Ann. Math. 62, 128 (1955). 6. Einstein, A., Can. J. Math. 2, 120 (1950). 7. Einstein, A. The Meaning of Relativity 5th ed. Princeton: Princeton University Press, 1955a. Einstein, A., Revs. Modern Phys. iO, 35 (1948). 9. Eddington, A. S. The Mathematical Theory of Relativity Cambridge: ^ Cambridge University Press, 1924. 10. Hlavaty, V. Geometry of Einstein's Unified Field Theory Groningen; P. Noordhoff Ltd. 195711. Hlavaty, v., J. of Math, and Mech. 7. 833 (1958). 12. Kronsbein, J., Phys. Rev, JhOg, I8I5 (1958). 13. Kronsbein, J. Phys. Rev. J_12^, 1384 (1958). 14. Kursunoglu, B. Phys. Rev. 88, I369 (1952). 15. Kursunoglu, B. Revs. Modern Phys. 29, 412 (1957). 16. Kursunoglu, B., 1 Nuovo Cimento 21, 729 (I96O). 17Papapetrou, A., Proc. Roy. Irish Ac. 52, A, 69 (1948). 18. Schrd'edlnger, E. SpaceTime Structure Cambridge: Cambridge University Press, i960. 19. Synge, J. L. and Schlld, A. Tensor Calculus Toronto: University of Toronto Press, 1949. 20. Tolman, R. C. Relativity, Thermodynamics, and Cosmology Oxford: I Oxford University Press, 1934. 54
PAGE 59
; 55 21. Tonnelat, M. A. La Theorie Du Champ Unifie' D' Einstein Parts: GauthlerVMlars, 1955. 22. Mishra, R. S., J. of Math, and Mech. 7, 877 (1958). 23. Landau, L. D. and Lifshltz, E. M. The Classical Theory of Fields 2nd ed. Reading, Mass.: Add i sonWesley Publishing Company, Inc., 1962. 2k. M5ller, C. The Theory of Relativity Oxford: Oxford University Press, 1952. 25. Kronsbein, J. Electromagnetic Fields in Einstein's Universe [Unpubl ished]. ;
PAGE 60
BIOGRAPHICAL SKETCH Joseph Francis Pizzo, Jr. was born on October 30, 1939, In Houston, Texas. There he attended St. Thomas High School and received his B. A. from the University of St. Thomas in I96I. In September, I96I Mr. Pizzo began graduate studies at the University of Florida. He received his Ph.D. in August, 1964. Mr. Pizzo Is married to the former Paula Awtry of Dallas, Texas. They have one child, a son. 56
PAGE 61
This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Phi losophy. August 8, 1964 Dean, Col leg^'Xbf ,.A'rt& and Sciences Dean, Graduate School Supervisory Cormittee: Cha irman L^ FW^ Cochairma ^ f^.S.T Cm'Avv liv ei.v_.v C vU F '//J^c^/l.A
34
The Christoffel symbols are computed according to (2.61) and substituted
in (338). It is seen that the solution is
\
e
0 '
(4.20)
(4.21)
Thus in matrix form, the g.. are
3 i k
0
0
0
0 0 0
r2 0 0
0 r2sin^0 0
0 01
(4.22)
This model is known as the Einstein cylindrical universe. (See [24],
page 359) A change of variables will serve to show more distinctly
the character of this universe. Let
*>,. r2
X] ~ ^ ~ (J(2
x2 = r cos 0
x^ = r s in0 cos
x^ = r sin0 sini (4.23)
In which case,
.2 ..2 2 ,2 .2. 2,2
ds = (dXj + dx? + dx^ + dx^ ) c dt
(4.24)
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS Â¡II
CHAPTER
1. INTRODUCTION 1
2. CURVATURE, DISPLACEMENT AND FIELD EQUATIONS .... 4
3. REDUCTION AND SOLUTION 20
4. SPECIAL SOLUTIONS 30
Minkowski Space 30
A More General Case 33
5. SUMMARY 51
BIBLIOGRAPHY 54
BIOGRAPHICAL SKETCH 56
42
H 13R12 +
r
321
1
r 32ri ip
V
17 1
M3
p P p p
21 ;p3 13 ;2 + P 32; 1 +P 21;
S/ v V V
p3 Gt
1 t3 132
V
n 3Gt332
2 n
3
,3R22 +
V
13R32
P3 Gt
' t2 331
V
+r 3tQt22i+
V
r3 g*
3t 231
V
i2 r
,3
32R2_L + 1
32R31
v
P2 G*
tr 223
V
r 2tQtU3 +Â¡
V
p 2 t
2tG 123
V
12 p
3
21R23 +l
31R33
V
(4.58)
If the index "4" appears in a G^pq* !* vanishes. For this reason, the
following type of combinations must be observed.
G 121 + g3123 R12
(4.59)
When the summation is carried out on the index "t" as indicated in
(4.58), and combinations like (4.59) heeded, a fortunate cancellation
occurs which leaves
P 13;p2 +P 32;pl +P 21;P3 ^ (T 13;2+P 32;1+P 21 ;3} ;p (4,60)
P +P P =(PP
3 2;p1 +l 21;p3 M
v v v v
The field equation (4.54) is then
,P P rr P
V
2e
(P 13 ;2 +P 32;1 +P 2J;3pp = ^2 (13;2 + 4,32;1 + $21;3) (4,6l)
b
At
this point, the set of ~1 .as given in equations (4.39)
through (4.42), could be substituted, but the left hand side of (4.6i)
27
First consider (3.2). If (34) is kept in mind, then R.^ is
given by
Rik 2 ^Rik + Rki>
^iPk),p + 2((iPp) ,k + ^kPp],i)
+ s.p0 q,{ > p
' q) cp k) (q p) I* k)
ck2 iJ<
, i 4 k
(336)
If or k take on the value "4", the second part of (32) is identically
satisfied. Equation (336) is a second order partial differential
equation in the g.^. It is easily verified that
'ogtTg]. ,
(3.37)
so that
. $ p,2 + $ p < + $ p K q 7
(i k),p ]f p),k (i q) (p k^
? p {\ q <
(q p) (i k)
(JC2 9ik
(338)
In analogy with general relativity, g.^ must be interpreted as the metric
in the space. If we choose the physical space to possess spherical
symmetry and the time dimension to be uncurved then the line element
g
may be brought into the form
ds2 = &.2cSZ + R2 sin^dO2 + G^2 sin^sinW#2 c2df2 (339)
In which case the metric is
8. "(3C" *s interpreted as the radius of this world.
55
21. Tonnelat, M. A. La Theorie Du Champ Unifie1 D'Einstein. Paris:
GauthierVi1lars, 1955.
22. Mishra, R. S., J. of Math, and Mech. 7 877 (1958).
23. Landau, L. D. and Lifshltz, E. M. The Classical Theory of Fields,
2nd ed. Reading, Mass.: AddisonWesley Publishing Company, Inc.,
1962.
24. Miller, C. The Theory of Relativity. Oxford: Oxford University Press,
1952.
25. Kronsbein, J. Electromagnetic Fields in Einstein's Universe
[Unpublished].
15
is carried out with (2.4l) as integrand, the result is the following set
of equations,
9lk,b 9skP ib 9isP bk 0 (2.Mt)
Rik=X9lk <245)
These can be taken as the field laws as they stand, but Schroedinger
takes the theory further. Equation (2.45) can be substituted in (2.44)
to eliminate the g.,. The field equations are now sixtyfour differential
'k b
equations involving nothing but the sixtyfour
s s
R., R T *k ~ R P . = 0 (2.46)
ik,b ski ib is I bk '
This is known as the "purely affine theory."
Kursunoglu's approach to the theory [16] retains the basic ideas
of displacement field, curvature tensor, and the fundamental tensor
I k
g.^ (or g ) but the scalar density for the variational integral is
quite different and entails some new concepts. These will be defined
before the variational principle and the field equations are displayed.
The fundamental tensor, g.^, is expressed in Kursunoglu's theory,
in terms of its symmetric part, a.^, and its antisymmetric part, 4>.^.
gik aik + (2G)'/2cVik
where G is the gravitational constant and rQ is a "fundamental length."
The determinant, g, of g.^ is constructed in terms of what will be
called the "Kursunoglu invariants," namely
=Uik*
2 ik
(2.48)
CHAPTER 1
INTRODUCTION
In 1916, Einstein's classic paper, in which he formulated the
general theory of relativity, appeared in Annalen der Physik. [l]^
In this paper he was able to describe the phenomena of gravitation in
terms of geometrical concepts. The field equations of gravitation were
shown to be derivable from a variational principle, using a symmetric,
second rank, covariant tensor. This tensor (which we will call the
fundamental tensor, and will denote by g.^) represents the gravitational
potentials.
The theory was beautiful, and moreover it worked.' Einstein was
still not satisfied. He reasoned that there are other fields in Nature
besides gravitation. How do they fit into this picture? An example is
the electromagnetic field and equations. There is no natural way for
them to be included. To express the covariant electrodynamic equations,
Maxwell's equations are written, then covariant derivatives are taken in
place of ordinary partial derivatives. In other words, electrodynamics
must be introduced separately. This type of inclusion has been con
sidered arbitrary and unsatisfactory by many theoreticians. Indeed,
Einstein himself has stated, "A theory in which the gravitational field
and the electromagnetic field do not enter as logically distinct
1. The numbers in square brackets refer to the bibliography at the
end of the dissertation.
1
BIBLIOGRAPHY
1. Einstein, A. The Principle of Relativity. New York: Dover Publi
cations, 1923, pp 111164.
2. Einstein, A., Ann. Math. 46, 578 (1946).
3 Einstein, A. and Strauss, E. Ann. Math. 47, 731 (1946).
4. Einstein, A. and Kaufman, B., Ann. Math. 59, 230 (1954).
5. Einstein, A. and Kaufman, B., Ann. Math. 62, 128 (1955).
6. Einstein, A., Can. J. Math. 2, 120 (1950).
7. Einstein, A. The Meaning of Relativity. 5th ed. Princeton: Prince
ton University Press, 1955
8. Einstein, A., Revs. Modern Phys. 20, 35 (1948).
9. Eddington, A. S. The Mathematical Theory of Relativity. Cambridge:
Cambridge University Press, 1924"
10. Hlavaty, V. Geometry of Einstein's Unified Field Theory. Groningen
P. Noordhoff Ltd., 1957
11. Hlavaty, V., J. of Math, and Mech. 833 (1958).
12. Kronsbein, J., Phys. Rev. 109, 1815 (1958).
13. Kronsbein, J. Phys. Rev. H_2, 1384 (1958).
14. Kursunoglu, B., Phys. Rev. 88, 1369 (1952).
15. Kursunoglu, B., Revs. Modern Phys. 29, 412 (1957).
16. Kursunoglu, B., 1 Nuovo Cimento J_5, 729 (i960).
17 Papapetrou, A., Proc. Roy. Irish Ac. 52, A, 69 (1948).
18. Schr'edinger, E. SpaceTime Structure. Cambridge: Cambridge Uni
versity Press, i960.
19. Synge, J. L. and Schild, A. Tensor Calculus. Toronto: University
of Toronto Press, 1949
20. Tolman, R. C. Relativity, Thermodynamics, and Cosmology. Oxford:
Oxford University Press, 1934.
54
21
[9], p. 159) Equations (33) and (34) are satisfied identically. Due
to this correspondence, these equations should be considered as valid as
the other three sets.
In all four cases, it has not been possible to ascertain if the
theory contains any information other than the simple reduction to the
gravitational equations. The reason for this is that a simple, general
b b
expression for the p .^ has not been found. True, the p .^ have been
expressed in special coordinates, but the differential equations have
not been solved using these values. In this chapter and the subsequent
b
ones an appealingly simple form for the p ^ in a special case, along
with new information from the field equations, will be presented.
b
The relation for P .. (31) is common to all four versions. It
I ,k b
is a set of sixtyfour equations from which the sixtyfour [""* .^ are to
be determined as functions of the sixteen g.. and their derivatives.
ik
A solution to these equations has been given for a system of
spherically symmetric coordinates. [17, 21] Kursunoglu [16] sets up
b
the differential equations, (2.31) and (2.32), using the j""1 expressed
in terms of the g.^, but can offer no explicit solution. In fact, no
solution has yet been given to equation (2.31) in these special coordi
nates. It is for this reason that a completely general (coordinate inde
pendent) solution to (31) is desired.
A little manipulation of (31) is enough to show that a general
solution is far from trivial. Nevertheless, several formalistic solu
tions have been offered. [4, 10, 22] Formalistic, in this context, is
b
meant to imply that an explicit expression for P ., is never written
lik b
down. What is given is a prescription for determining p A brief
demonstration of Einstein's formalistic solution [4] may illustrate this
52
they can be solved, the sixtyfour algebraic equations must be solved
in a general case (without reference to any coordinates) for the
components of the displacement field. Since this general solution has
not been accomplished, there is no way of telling whether all the
field laws are included in the theories. The sixtyfour equations are
shown to be reducible to twentyfour, but this has not enabled the solu
tion to be found up to the present.
At this point, we present our version of the theory, in which the
fundamental tensor is modified. It must be noted that this is not a
linearization such as that used by Kursunoglu [14] or Einstein [3].
These linearizations exclude interaction between the symmetric and
antisymmetric components of g,^. The present work does not preclude
this possibility. In fact, the antisymmetric components of the dis
placement field were seen to be a combination of the symmetric and anti
symmetric parts of the fundamental tensor. So instead of a linearization,
we have a perturbation type of technique. It is basically a first order
antisymmetric perturbation of the gravitational field producing
symmetric tensor.
With this method, the algebraic equations are readily solved for
b
without the specification of any coordinate system. It is found
that the become the same expressions used in general relativity.
With this in mind, we adapt the set of field equations involving the
contracted curvature tensor to an Einstein model of the universe.
In the limit of an infinite radius of the Einstein universe, the
remaining differential equations go directly over to the Maxwell equa
tions of Minkowski space. For the general case (a finite radius), a
transformation was made to a symmetrical arrangement of coordinates.
Investigation of the remaining two differential field laws leads to the
48
? % [.V + (? fix* (H*x?U (4.97)
e = [<9[~ + (E*x >0 ] (4.98)
The reason for using vector notation will soon be apparent. Now that
the solutions are known, we want to visualize them in the spherical
space.
First, consider the "straight lines" in this space. The geodesic
equation,
becomes
dV
dX2
*
r
dxk dxb
kb dX dX
(4.99)
s i nee
and
d2X'
dX
C i ) dXk
1 k b) dX
k d
dX
r
. dXk dXb
kb dT* d"X*
(4.100)
(4.101)
(3.35)
This shows that the geodesics of a space are not altered by the presence
of electromagnetic fields in the space to the first order which we have
investigated. (A question which immediately comes to mind is, "Well,
do the properties of the space affect the electromagnetic fields?" Of
course they do. We have seen how the electrodynamic equations were
complicated. Exactly how they are affected remains to be seen below.)

