“ […] 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 […]”. (, p. 327)293
We will now describe theories with a different geometrical background than affine or mixed geometry and its linearized versions.
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: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 : .
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” (, p. 133; (, p. 154).
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  and suggested an earlier attempt by Stephenson & Kilmister  starting from the line element [cf. (428*)]:
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 , p. 113). Eddington had spoken of the space-time interval depending “on a general quartic function of the ’s” (, p. 11). Thus it was not unnatural that K. Tonooka from Japan looked at Finsler spaces with fundamental form , 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  and by Horváth & Moór, . He mentioned the thesis of E. Schaffhauser-Graf  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 :, the starting point is the line element:296 297 Equation (555*) can also be written as
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:298 560*) 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” (, 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.
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). […].