## 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 a gauge transformation with gauge function is introduced. The subgroup of coordinate transformations is given by those transformations for which . A change of the reference system implies both coordinate and gauge transformations. Tangent vectors under a change of coordinates and gauge transform like:

where is the gauge factor (Lyra’s “Eichverhältnis”); a basis of tangential space is given by ; a 1-form basis would be . The metric then is introduced by , and the asymmetric connection is defined via Here, similar to Weyl’s theory, an arbitrary 1-form appears and the demand that the length of a transported vector be conserved leads to with the Christoffel symbol calculated from . The curvature tensor is defined by Hence the curvature scalar becomes where the semicolon denotes covariant derivation with regard to , and .^{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 . In the gauge , they are given by [571]: Weyl’s field equations in a special gauge are the same – apart from the cosmological term . 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 ’s” ([139], p. 11). Thus it was not unnatural that K. Tonooka from Japan looked at Finsler spaces with fundamental form [648], except that only 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 to the point with :

with an arbitrary parameter. In the approach to Finsler geometry by Cartan [74], the starting point is the line element:^{296}with a homogeneous function of the velocity , and the Finsler metric defined by: Quantities and acting as connections are introduced through the change a tangent 4-vector is experiencing: where the totally symmetric object transforms like a tensor. The asymmetric affine connection is defined by with . The “geodesic coefficient” is resulting from the Euler–Lagrange equation for reformulated with the Finsler metric.

^{297}Equation (555*) can also be written as with Cartan’s connection .

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 , the “torsion” vector . 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 [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 ( 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 which leads by means of to (1) with . However the tensor does not yield any appropriate scalar term which could be taken as Lagrangian. […] Remarks of the reviewer: 1) If is of the second or third class (which is always the case in Callaway’s approximation) we could have in (1) for . 2) The clue to Callaway’s result is that four Einstein’s equations 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 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). […]. |

^{299}