Further Contributions from the United States

12 Further Contributions from the United States

12.1 Eisenhart in Princeton

While Einstein pursued his research in UFT along the lines of mixed geometry at the Institute for Advanced Studies of Princeton, NJ, his colleague in the mathematics department of Princeton University, L. P. Eisenhart, kept quiet until the beginning of the 1950s. He had written a book on Non-Riemannian Geometry in 1927 [182 *], and since the twenties had had a long-standing interest in teleparallelism and UFT (cf. Section 6.4 of Part I and [181 ]). Being three years older than Einstein he had retired in 1945. Nevertheless, in the 1950s he wrote three further papers about UFT. He first introduced a non-symmetric metric $gij$ and a non-symmetric connection [184 ] in 1951. Unlike Einstein, whose papers he did not refer to, Eisenhart did not take the connection $k Lij$ as an independent variable but built it entirely from the metric tensor and its first derivatives such as the Japanese physicist K. Hattori²⁴⁸ had done in the 1920s [240 *]:

L ijk = Δ ijk + aijk, (432 ) 1 Δ ijk= -g(kl) [gli,j + gjl,i − gji,l], (433 ) 2

where $aijk$ is an arbitrary tensor. In fact, Eisenhart’s $Δ jik$ is exactly the same as Hattori’s connection $gρν ∗[μ λ,ρ]$ ([240 ], Eq. (1.6), p. 540). The tensor to be added was chosen by Eisenhart to be (indices are moved with the symmetric part of the metric):

From (433 *) we notice that

where the Christoffel symbol is formed with the symmetric part of the metric. The torsion tensor is given by:

Eisenhart used the curvature tensor $K i − jkl$ and its contraction $− K − jk$ After some manipulations, he obtained an equation for the Ricci tensor formed from $g(ij)$ which is contained already in Hattori’s paper (Eisenhart’s Eq. (26)).²⁴⁹ Eisenhart’s paper dealt only with differential geometry; no physical motivation or interpretation were given. This applies also to a subsequent publication in which, after formal manipulations, several expressions for possible curvature tensors and Einstein’s Hermitian-symmetrized Ricci tensor (196 *) were derived [185 ]. The 3rd edition of Einstein’s Meaning of Relativity [150 ] now was referred to.

Eisenhart’s second attempt, after the death of Einstein, presented a new unified theory of gravitation and electromagnetism within metric-affine geometry [186 *, 187 *, 188 *], and [189 *]. Although in a different geometrical setting, eventually the theory formally led to the Einstein–Maxwell field equations in Riemannian geometry. The main difference to Einstein’s approach was that Eisenhart kept the metric tensor $gij$ symmetric while embedding the electromagnetic field tensor $Fij$ into the connection by an ad-hoc ansatz [186 ]:

$Fij$ is derived from a 4-potential and supposed to satisfy $F is ;s = 0$ .²⁵⁰ Thus, vector torsion does vanish. In the first paper, Eisenhart’s field equations were:

With his ansatz (437 *), the equation for the auto-parallels of the connection $L$ read as:

These results were unphysical. Four months later, Eisenhart tried to find a remedy by postulating [187 ]:

with $k λ$ satisfying

All he arrived at was the field equation $Rij = const.λiλj$ . A slight generalization of (441 *):

did not help much. Because he did not use a variational principle, Eisenhart always had to build his theories such that an identity in Riemannian geometry was guaranteed: the vanishing of the divergence of the Einstein tensor.

Again two months later, in his 3rd installment, Eisenhart finally arrived at the Einstein–Maxwell equation of general relativity [188 *]. This time, the ansatz (440 *) with (441 *) was changed substantially into:

This equation is interesting because the energy-momentum tensor of the electromagnetic field is built directly into the connection (curvature). Eisenhart was aware that the Einstein–Maxwell equations did not represent the union of the gravitational and electromagnetic field aimed at. He quoted Einstein as saying that “A theory in which the gravitational field and electromagnetic field do not enter as logically distinct structures would be much preferable.” ([188 ], p. 881.) This quotation was repeated in the last of this sequence of papers by Eisenhart, in which “the final result of my third paper” were derived “in a somewhat different manner.” ([189 ], p. 333.)

In his last contribution to UFT, when he was 83 years old, Eisenhart returned to mixed geometry [190 ]. Starting from Einstein’s condition (200 *) on the metric, he aimed at solving it for the connection as a function of the metric and its first derivatives. Unaware of the solutions given previously (cf. Section 10.2.3), he achieved this only with the help of the additional constraint for torsion:

From this, again the vanishing of vector torsion follows. In this special case, from (444 *) with the notation of (3 *), (4 *), Eisenhart’s solution is given by:

Seen in context, Eisenhart’s papers on UFT from the 1950s did bring neither a new development in geometry nor an advance for physics. Cf. also the paper by Horvath [285 *] in Section 15.1.

12.2 Hlavatý at Indiana University

Hlavatý is the fourth of the main figures in UFT besides Einstein, Schrödinger, and Tonnelat. His research was published first in a sizeable number of articles in the Journal of Rational Mechanics and Analysis of Indiana University²⁵¹ and in other journals; they were then transformed into a book [269 *]. According to its preface, his main intent was “to provide a detailed geometrical background for physical application of the theory”. As he was very optimistic with regard to its relation to physics, he went on: “It so happens that the detailed investigation of Einstein’s geometrical postulates opens an easy way to a physical interpretation”([269 *], p. X). We have noticed in Section 9.7 that this possibly could not have been the case. In the preface of his book, Hlavatý became more explicit; his program was to encompass: (1) an investigation of the structure of the curvature and torsion imposed on space-time by the field equations, equations which he claimed to be “of a purely geometrical nature” without physical interpretation being “involved in them a priori”. The two further points of his program, i.e., (2) an attempt to identify the gravitational field and the electromagnetic field by means of the field equations, and (3) an investigation of the physical consequences of his theory, were treated only in “an outline of the basic ideas” ([269 *], p. XVIII). In comparison with Einstein, Schrödinger and Tonnelat who followed their physical and mathematical intuition, Hlavatý’s investigations were much more systematical and directed first to what could be proven by mathematics; whether a relation to physics could be established became secondary to him. Although mostly working and publishing alone, he corresponded with about 40 scientists working on UFT. He also was a frequent reviewer for Mathematical Reviews (cf. Section 18.1). Hlavatý began by introducing a systematical classification of the non-symmetric metric $gij$ according to the non-vanishing eigenvalues of its skew-symmetric part $kij$ (remember $ij is is ˇk = h h kst)$ . He distinguished three classes:

class 1: k = det(k ) ⁄= 0; ij 2 ij2 class 2: k = det(kij) = 0 and K := (kijˇk ) ⁄= 0; class 3: k = det(kij) = 0 and (kijˇkij)2 = 0.

In contrast to presentations in his articles [260 , 261 *, 262 , 263 ], in his book this simple algebraic problem is spread out on many pages ([269 *], 11–41). His student A. W. Sáenz simplified Hlavatý’s proofs by looking at the algebraic structure of the electromagnetic field tensor [520 ].

Throughout his research, the symmetric part $g(ij) = hij$ is used for raising and lowering indices. From Eq. (30 *) he concluded that there are metrics $gij$ for which this “metric compatibility” equation does not admit any solution”, and cases in which (30 *) admits more than one solution [261 *]. According to him, the condition for uniqueness of the solution is $det (glm )∕ det(hrs) =: g ⁄= 0$ if $det(klm)∕ det(hrs) := k ⁄= 0$ (class 1), and $g(g − 2)) ⁄= 0$ if $k = 0$ (class 2, 3). The gravitational potential is identified with $g(ij) = hij$ while the electromagnetic field is taken to be²⁵² [261 *]:

1 ∘ --- 1∘ --- fij := √--[κ |k |kij − -- |h|𝜖ijrsˇkrs] g 2 ij -1-- ∘ --ˇij 1∘ --- ijrs f := √g--[κ |k |k − 2 |h|𝜖 krs] (446 )

with $κ = − sgn [ 𝜖ijrskijkrs ]$ . The above classification of $kij$ is thus valid also for the electromagnetic field. For classes 2 and 3, Hlavatý’s definition of the electromagnetic field is a variant of one we have already met in Section 9.7.

His approach was more direct than Tonnelat’s: mostly, he worked just with $gij,hij$ and the decomposition (363 *) met before:

where $Uijk = 2hkrS r(si kj)s$ . The removal of the connection from (30 *) first gave torsion as a functional of the symmetric and skew symmetric parts of the metric via the linear equation:

with ${}h {}h {}h Kijk := ∇ kkij + ∇ jkki + ∇ ikjk$ , and $pqr pqr b dt X ijk = X ijk (δa,kcth )$ ([264 *], p. 320). After torsion is inserted into the decomposition of $k L ij$ given before, the connection is known as a functional of $hij,kij$ and its first derivatives. Thus, after about 80 pages in his book including degenerate cases, in the end Mme. Tonnelat’s calculations were only simplified a bit by Hlavatý and made more transparent, with some details added.

Hlavatý used the Ricci-tensors $K− ij$ and $∗ P ij$ (cf. (75 *) of Section 2.3.2), and proved that:

He stressed that Einstein’s weak field equations for UFT were of a purely geometrical nature with no physical interpretation needed. Equation (30 *) was written in the form [261 ]:

where the covariant derivative is defined by (16 *) in Section 2.1.1. The conditions on curvature are subsumed in

with arbitrary $Xk$ . For vanishing $Xk$ , the strong field equations, for $Xk ⁄= 0$ , the weak field equations are following.

According to Hlavatý, the first two classes cannot be handled simultaneously with the third class ([266 *], p. 421). This makes it more involved to read his papers, because the results proven by him must now be distinguished according to the special class of $kij$ .

i) Fields of third class.

In the course of his investigations when he tried to interpret geometrical quantities in terms of physical variables, Hlavatý replaced the four equations $R {[ik],j} = 0$ following from (449 *) by four complicated looking equations:

where $ˇ ij H$ and $H$ are the Ricci tensor and curvature scalar calculated from the symmetric part of the metric $hij$ (i.e., from the Levi-Civita connection). $G$ is a scalar function for which, for the 2nd class $G = g h$ , and for the third class $G = -g= 1 h$ hold.

The tensor $Qij$ received its meaning from what Hlavatý called “the gravitational field equations”, i.e., Einstein’s equations with a geometrical energy-momentum tensor of matter:

where $Tij := vi vj + Qij$ . $vi$ describes both the charge density and the mass density of matter which thus are related. As charge density it is defined by: $√ -- √ -- vk h = ∂s(fsk h)$ , with $fij$ from (446 *); mass density by $√ -- √ -- vk h = ϕM huk$ with a scalar function $ϕ$ and the unit vector $uk$ ; $M$ is the mass of a particle ([269 *], p. 175–176). In Hlavatý’s theory, Maxwell’s equations were taken to be:

Charge conservation was expressed by $s√ -- ∂s(v h) = 0$ . As the first equation of (452 *) is equivalent to (449 *) plus $Si = 0$ , we also have $σ ps{}h ˇqr vi = − 2 𝜖pqrih ∇ sk$ . Of course, the electromagnetic field in (446 *), for the third class, reduces to $∘ --- fij = − 12 |h|𝜖ijlmˇklm$ .

For incoherent matter, $Qij = 0$ , and from (451 *) ${}h ∇ s(M usuk ) = 0$ . A somewhat disappointing consequence is that, in a manageable approximation, charged particles remain unaffected by the electromagnetic field: they move along geodesics in the gravitational field $hij$ ([264 ], p. 329; [269 *], p. 174, 187). Upon neglect of the cubic terms in $fij$ , i.e., for $Uij ≃ 0$ , the equation of motion coincides with the geodesic equation:

This is the more strange as Hlavatý claimed:

“In the unified theory the electromagnetic field is always present; hence we might look upon it as a primary field which […] creates the gravitational field. However, there is at least one known electromagnetic field which does not create a gravitational field (i.e., the field of the plane wave in the electromagnetic theory of light).” ([266 *], p. 420.)

ii) Fields of class 1 and 2.

Here, $G = (1 + 12frsfˇrs + fh), f = det(frs),$ and two new quantities were introduced:

The skew-symmetric part of the asymmetric metric $gij$ now reads as $σkij = 1𝜖ijklFkl + Lij 2$ . Tangent vectors $k w$ and $k W$ were defined by

related by $k √ -- k sk W = G (w + Gsf )$ with $1 Gs = 2∂slnG$ . The gravitational and Maxwell equations (451 *) and (452 *) remain the same except for an exchange of $vi$ by $wi$ and a different complicated expression for $GQij$ if $R {[ik],j} = 0$ is kept as a field equation; cf. [269 *], p. 204, Eq. (20.3a,b), p. 203 Eqs. (20.1), (20.2b). Hlavatý did present an exact spherically symmetric solution with $− 4 2 2 2 g00 = 1,g11 = B (1 − Ar ),g22 = r ,g33 = r sin 𝜃$ , $A, B$ constants which is obtained from Papapetrou’s solution (cf. Section 8.3) by setting there $m = λ = 0$ . The electromagnetic field in Hlavatý’s solution is $√-- f 23 = 4-A2-- rsin 𝜃$ . The gravitational function replacing the gravitational constant is $G = (1 − Ar −4)−1$ . But in this case, according to Hlavatý “we are unable to derive the second set of Maxwell’s equations from our field equations” ([269 *], p. 208). Therefore, as for classes 1, 2, this field equation again is replaced by (450 *). In consequence, for the motion of a particle Hlavatý obtained an improved result: A (massive) charged particle moving freely in the unified field $gij$ describes an auto-parallel of the unified connection $L k ij$ ([269 ], p. 211). Thus, two of the three effects in the planetary system were the same as in general relativity; the third (Perihelion shift) in his theory depended on the electrical field of the sun. Hlavatý did not get as far as to clearly show the experimental physicist how this electric field enters the formula for the perihelion shift.

For paths of photons Eq. (453 *) still holds. If gravitation is neglected, i.e., $h = η ij ij$ , Hlavatý found a discrepancy with the special relativistic explanation of the Michelson-Morley experiment. Although he referred to the judgment of Shankland et al. that Miller’s result is erroneous [575 ], he concluded: “From the point of view of the unified field theory Miller’s result, properly interpreted, is not necessarily at variance with the assumption of the constant velocity of light.” ([266 ], p. 471).

Hlavatý’s research will be appealing to some by its logical guideline concerning mathematical structures. His many special cases and set up “agreements” in proving results are somewhat bemusing for a physicist. An example is given by his publications dealing with the special case $h = 0, g ⁄= 0$ when the symmetric part $hij$ of the metric $gij$ is degenerated [267 , 268 ]. It is a purely mathematical exercise meant to fill a gap, but is without physical meaning. For the cases in which the theory could be applied to physical systems, in principle, Hlavatý was also forced to alter the original field equations in order to avoid objections against the unphysical results following from them. It is not unfair to conclude that he did not succeed in making a break-through in the sense of his physical interpretations being more convincing than those suggested by others.

The investigations of his doctoral student R. Wrede were directed to the mathematical structure of the theory: He partially extended Hlavatý’s theory to an n-dimensional space by adhering to the two principles: A.) The algebraic structure of the theory is imposed on the space by a general real tensor $gab$ ; B.) The differential geometrical structure is imposed on the space by the tensor $gab$ by means of a connection defined by (30 *). Hlavatý’s third principle, i.e., the existence of the constraints $r r Rab = ∂[aXb],Sar = L[ar] = 0$ with an arbitrary vector field $Xa$ is left out [708 ]. The paper solves (30 *) in $n$ dimensions for the various possible cases.

12.3 Other contributions

A theoretician of the younger generation and assistant at Princeton University, R. L. Arnowitt , tried to look at UFT from the point of view of the electromagnetic field forming a link between the description of microscopic charges by quantum field theory and macroscopic ray optics [5 *]. He introduced four postulates:

Any unified field theory should reduce to Einstein–Maxwell theory in a first approximation for weak electromagnetic fields.
First-order corrections to the Coulomb field of the electron should not become appreciable for $r ≥ 10−13 cm$ .
The affine connection has the form $Γ ijk = C ijk + Γ jδki$ , where $Γ j$ is related to the vector potential of the Maxwell field. Also $C ijk = C jik$ is assumed.
The Lagrangian must be invariant under the combined gauge transformation $Γ ijk → Γikj + Λ,jδki$ and (for the metric) $gij → exp[2Λ(x )] gij$ . The metric tensor is also symmetric.

The appearance of a microscopic length-parameter (and the cosmological constant) in his Lagrangian and the occurrence of two “metrical” tensors turned out to be a consequence of the postulates. The symmetrical first one $gij$ is supposed to “be measured by rods and clocks” and used to set up the Lagrangian; the second asymmetric one is derived from the Lagrangian: $ˆg ′ ij := δℒ∕ δR ij$ . It is an auxiliary device for the introduction of the electromagnetic field. Arnowitt chose the Lagrangian:

with two constants $α1,α2$ of dimension $2 (length )$ . The theory looks similar to Buchdahl’s gauge-invariant UFT published in the same year (cf. Section 13.1) but is different. Immediately,

In order to obtain the field equations, the quantities $C ijk,Γ j$ and $gij$ are to be varied. From the first two variations $′[is] ′ij ˆg ;s = 0, ˆg ;k = 0$ and $k 1 ′(ks) ′ ′ ′ C ij = 2h (h(is),j + h(js),i − h(ij),s)$ resulted where $′ij h$ was introduced by $√ ---- gˆ′(ij) = h′ij − h′$ and interpreted to be the “gravitational” metric tensor. The first of these equations was rewritten as sourceless Maxwell equation such that $gij$ is the “electromagnetic” metric tensor:

with $fij = − a− 12R [ij]$ , and the constant a is determined later in a weak field approximation to be $2 2 a ≃ − κ ρec$ with $2 e2 -e2 ρec = r41 ,r1 ∼ mc2$ the classical electron radius, $m$ the electron mass, and $κ$ the gravitational constant in Einstein’s equations.

After some manipulation, variation with respect to $gij$ led to:

with

$T ik$ was interpreted as ‘the electromagnetic stress-energy tensor”. After linearization, (459 *) formally became Einstein’s equations. Again, in a weak field approximation introduced by $gij = g(ij) + hij$ with small $hij$ , the free parameters $α1, α2$ were fixed to be $α1 ≃ κρec2$ and $(α2)−1 ≃ 4λ$ with the cosmological constant $λ$ . Thus, $α1$ is the microscopic length parameter mentioned above. In linear approximation, the author also has obtained a static, spherically symmetric Schwarzschild-like solution with an event horizon and finite electrical field (and field energy) for $r → 0$ .

A further contribution came from B. Kursunŏglu, whom we have met before in Section 9.3.3, now situated in Coral Gables, Florida. He continued to alter and study his variant of the Einstein–Schrödinger field equations [345 ]. In place of (300 *) – (302 *), he postulated the system:²⁵³

[il] ˆg ,l = 0, (461 ) K + p2(h − b ) = 0, (462 ) − (ik) ij ij K [ik] + p2(qkik − Fik) = 0, (463 ) −

where

---------hij +-kirˇkjr---------- bij := ˇrs ∗ rs 2 1 + 1∕2 krsk − (1∕4 k krs)

with

∗ rs 1 rspq k := -√-----η kpq. 2 − h

The denominator is related to $g$ .

The auxiliary field $Fij$ satisfies the vacuum Maxwell equations.

	Abstract
1	Introduction
2	Mathematical Preliminaries
	2.1	Metrical structure
	2.2	Symmetries
	2.3	Affine geometry
	2.4	Differential forms
	2.5	Classification of geometries
	2.6	Number fields
3	Interlude: Meanderings – UFT in the late 1930s and the 1940s
	3.1	Projective and conformal relativity theory
	3.2	Continued studies of Kaluza–Klein theory in Princeton, and elsewhere
	3.3	Non-local fields
4	Unified Field Theory and Quantum Mechanics
	4.1	The impact of Schrödinger’s and Dirac’s equations
	4.2	Other approaches
	4.3	Wave geometry
5	Born–Infeld Theory
6	Affine Geometry: Schrödinger as an Ardent Player
	6.1	A unitary theory of physical fields
	6.2	Semi-symmetric connection
7	Mixed Geometry: Einstein’s New Attempt
	7.1	Formal and physical motivation
	7.2	Einstein 1945
	7.3	Einstein–Straus 1946 and the weak field equations
8	Schrödinger II: Arbitrary Affine Connection
	8.1	Schrödinger’s debacle
	8.2	Recovery
	8.3	First exact solutions
9	Einstein II: From 1948 on
	9.1	A period of undecidedness (1949/50)
	9.2	Einstein 1950
	9.3	Einstein 1953
	9.4	Einstein 1954/55
	9.5	Reactions to Einstein–Kaufman
	9.6	More exact solutions
	9.7	Interpretative problems
	9.8	The role of additional symmetries
10	Einstein–Schrödinger Theory in Paris
	10.1	Marie-Antoinette Tonnelat and Einstein’s Unified Field Theory
	10.2	Tonnelat’s research on UFT in 1946 – 1952
	10.3	Some further developments
	10.4	Further work on unified field theory around M.-A. Tonnelat
	10.5	Research by and around André Lichnerowicz
11	Higher-Dimensional Theories Generalizing Kaluza’s
	11.1	5-dimensional theories: Jordan–Thiry theory
	11.2	6- and 8-dimensional theories
12	Further Contributions from the United States
	12.1	Eisenhart in Princeton
	12.2	Hlavatý at Indiana University
	12.3	Other contributions
13	Research in other English Speaking Countries
	13.1	England and elsewhere
	13.2	Australia
	13.3	India
14	Additional Contributions from Japan
15	Research in Italy
	15.1	Introduction
	15.2	Approximative study of field equations
	15.3	Equations of motion for point particles
16	The Move Away from Einstein–Schrödinger Theory and UFT
	16.1	Theories of gravitation and electricity in Minkowski space
	16.2	Linear theory and quantization
	16.3	Linear theory and spin-1/2-particles
	16.4	Quantization of Einstein–Schrödinger theory?
17	Alternative Geometries
	17.1	Lyra geometry
	17.2	Finsler geometry and unified field theory
18	Mutual Influence and Interaction of Research Groups
	18.1	Sociology of science
	18.2	After 1945: an international research effort
19	On the Conceptual and Methodic Structure of Unified Field Theory
	19.1	General issues
	19.2	Observations on psychological and philosophical positions
20	Concluding Comment
	Acknowledgements
	References
	Footnotes
	Biographies