Alternative Geometries

17 Alternative Geometries

Although a linear theory of gravitation can be derived as a first approximation of Einstein–Schrödinger-type theories ([641 *], pp. 441–446), the results may be interpreted not only in UFT, but also in the framework of “alternative” theories of gravitation. Nevertheless, M.-A. Tonnelat remained defensive with respect to her linear theory:

“ […] this theory does not pursue the hidden aim of substituting general relativity but of exploring in a rather heuristic way some specifically tough and complex domains resulting from the adoption of the principles of a non-Euclidean theory […]”. ([382 ], p. 327)²⁹³

We will now describe theories with a different geometrical background than affine or mixed geometry and its linearized versions.

17.1 Lyra geometry

In Lyra geometry, the notion of gauge transformation is different from its use in Weyl geometry. Coordinate and gauge transformations are given the same status: they are defined without a metric or a connection [386 , 531 ]. A reference system now consists of two elements: Besides the usual coordinate transformations $xi → xi′ = xi′(x1, x2,...,xn)$ a gauge transformation $x0′ = f0(x1,x2, ...,xn,x0(x1,x2,...,xn ))$ with gauge function $f$ is introduced. The subgroup of coordinate transformations is given by those transformations for which $0 1 2 n 0 0 f (x ,x ,...,x ;x ) = x$ . A change of the reference system implies both coordinate and gauge transformations. Tangent vectors $ξs$ under a change of coordinates and gauge transform like:

where $0′ λ = xx0-$ is the gauge factor (Lyra’s “Eichverhältnis”); a basis of tangential space is given by $-10 ∂i- x ∂x$ ; a 1-form basis would be $x0dxi$ . The metric then is introduced by $ds2 = grs(x0dxr )(x0dxs)$ , and the asymmetric connection $i 1 i Γ rs − 2δrϕs$ is defined via

Here, similar to Weyl’s theory, an arbitrary 1-form $ϕk$ appears and the demand that the length of a transported vector be conserved leads to

with the Christoffel symbol calculated from $gij$ . The curvature tensor is defined by

Hence the curvature scalar becomes $K = --R- + -3ϕ;s + 3ϕsϕ + 20ϕ ϕi (x0)2 x0 ;s 2 s i$ where the semicolon denotes covariant derivation with regard to $gij$ , and $ϕ0 = -1 ∂log(x0)2- i x0 ∂xi$ .²⁹⁴ In the thesis of D. K. Sen, begun with G. Lyra in Göttingen and finished in Paris with M.-A. Tonnelat, the field equations are derived from the Lagrangian $0 4√ -- ℒ = (x ) gK$ . In the gauge $0 x = 1$ , they are given by [571 ]:

Weyl’s field equations in a special gauge are the same – apart from the cosmological term $Λg ij$ . The problem with the non-integrability of length-transfer does not occur here. For further discussion of Lyra geometry cf. [572 ].

In relying on a weakened criterion for a theory to qualify as UFT suggested by Horváth (cf. Section 19.1.1), after he had added the Lagrangian for the electromagnetic field, Sen could interpret his theory as unitary. In later developments of the theory by him and his coworkers in the 1970s, it was interpreted just as an alternative theory of gravitation (scalar-tensor theory) [573 , 574 , 310 ]. In both editions of his book on scalar-tensor theory, Jordan mentioned Lyra’s “modification of Riemannian geometry which is close to Weyl’s geometry but different from it” ([319 ], p. 133; ([320 ], p. 154).

17.2 Finsler geometry and unified field theory

Already one year after Einstein’s death, G. Stephenson expressed his view concerning UFT:

“The general feeling today is that in fact the non-symmetric theory is not the correct means for unifying the two fields.”²⁹⁵

In pondering how general relativity could be generalized otherwise, he criticized the approach by Moffat [439 ] and suggested an earlier attempt by Stephenson & Kilmister [591 ] starting from the line element [cf. (428 *)]:

the geodesics of which correctly describe the Lorentz force.

Since Riemann’s habilitation thesis, the possibility of more general line elements than those expressed by bilinear forms was in the air. One of Riemann’s examples was “the fourth root of a quartic differential expression” (cf. Clifford’s translation in [329 ], p. 113). Eddington had spoken of the space-time interval depending “on a general quartic function of the $dx$ ’s” ([139 ], p. 11). Thus it was not unnatural that K. Tonooka from Japan looked at Finsler spaces with fundamental form $∘ --------------- 3 aαβγdxαdx βdxγ$ [648 ], except that only $3∘ -------α--β---γ | aαβγdx dx dx |$ is an acceptable distance. With (551 *), Stephenson went along the route to Finsler (or even a more general) geometry which had been followed by O. Varga [669 *], by Horváth [283 ] and by Horváth & Moór, [286 ]. He mentioned the thesis of E. Schaffhauser-Graf [530 ] to be discussed below. In Finsler geometry, the line element is dependent on the direction of moving from the point with coordinates $xi$ to the point with $xi + dxi$ :

with $u$ an arbitrary parameter. In the approach to Finsler geometry by Cartan [74 ], the starting point is the line element:²⁹⁶

with $L$ a homogeneous function of the velocity $m x˙$ , and the Finsler metric defined by:

Quantities $k C ij$ and $k Lij$ acting as connections are introduced through the change a tangent 4-vector $k X$ is experiencing:

where the totally symmetric object

transforms like a tensor. The asymmetric affine connection is defined by

with $γisγjs = δji$ . The “geodesic coefficient” $2 2 2 Gp = 14γpr[∂∂˙xr(L∂x)sx˙s − ∂(∂Lxr)]$ is resulting from the Euler–Lagrange equation $d2xk k dt2 + 2G = 0$ for $L$ reformulated with the Finsler metric.²⁹⁷ Equation (555 *) can also be written as

with Cartan’s connection $L∗ijk= Lijk− γktCtjpGpi$ .

Edith Schaffhauser-Graf at the University of Fribourg in Switzerland hoped that the various curvature and torsion tensors of Cartan’s theory of Finsler spaces would offer enough geometrical structure such as to permit the building of a theory unifying electromagnetism and gravitation. She first introduced the object met before , also known as “Cartan” torsion:

and, by contraction with $γkj$ , the “torsion” vector $Si = 1L ∂-ln|iγ| 2 ∂x˙$ . With its help and its covariant derivative taken to be:

the electromagnetic field tensor is defined by:²⁹⁸

The form of the covariant derivative (560 *) follows from the supplementary demand that the Finsler spaces considered do allow an absolute parallelism of the line elements. Schaffhauser-Graf used the curvature tensor 0. Varga had introduced in a “Finsler space with absolute parallelism of line elements” ([669 ], Eq. (37)), excluding terms from torsion. The main physical results of her approach are that a charged particle follows a “geodesic”, i.e., a worldline the tangent vectors of which are parallel. The charge experiences the Lorentz force, yet this force, locally, can be transformed away like an inertial force. Also, charge conservation is guaranteed. For vanishing electromagnetic field, Einstein’s gravitational theory follows. It seems to me that the prize paid, i.e., the introduction of Finsler geometry with its numerous geometric objects, was exorbitantly high.

Stephenson’s paper on equations of motion [590 *], from which the quotation above is taken, was discussed at length by V. Hlavatý in Mathematical Reviews [ External Link MR0098611] reproduced below:

The paper consists of three parts. In the first part, the author points out the difference between Einstein–Maxwell field equations of general relativity and Einstein’s latest unified field equations. The first set yields the equations of motion in the form

{ } d2xν ν dxλ dxμ νdxμ ---2-+ λ μ --------+ F μ ----= 0 ds ds ds ds

( $ν μ F μdx ∕ds =$ Lorentz’ force vector). Callaway’s application of the EIH method to the second set does not yield any Lorentz force and therefore the motion of a charged particle and of an uncharged particle would be the same.

In the second part, the author discusses the attempt to describe unified field theory by means of a Finsler metric

∘ ----------- ds = A λdxλ + gλμdx λdxμ,

which leads by means of $∫ δ ds = 0$ to (1) with $Fμλ = 2∂[λA μ]$ . However the tensor $R μλ$ does not yield any appropriate scalar term which could be taken as Lagrangian. […]

Remarks of the reviewer: 1) If $F μλ$ is of the second or third class (which is always the case in Callaway’s approximation) we could have $Fν (dxμ∕ds ) = 0 μ$ in (1) for $F ⁄= 0 μλ$ . 2) The clue to Callaway’s result is that four Einstein’s equations $∂{ωRμλ} = 0$ do not contribute anything to the equations of motion of the considered singularities. If one replaces these four differential equations of the third order by another set of four differential equations of the third order, then the kinematical description of the motion results in the form

d2xν dx λdx μ ---2-+ Γ λνμ--------= 0. dq dq dq

There is such a coordinate system for which the first approximation of (2) is the classical Newton gravitational law and the second approximation acquires the form (1). […].

Stephenson dropped his plan to calculate the curvature scalar $R$ from (557 *) and (551 *), and to use it then as a Lagrangian for the field equations, because he saw no possibility to arrive at a term $R (g) + FstFst$ functioning as a Finslerian Lagrangian. His negative conclusion was: “It so seems that this particular generalization of Riemannian geometry is not able to lead to a correct implementation of the electromagnetic field.”²⁹⁹

	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