Gravitational field equations based on Finsler geometry

Abstract

The analysis of a previous paper (see Ref. 1), in which the possibility of a Finslerian generalization of the equations of motion of gravitational field sources was demonstrated, is extended by developing the Finslerian generalization of the gravitational field equations on the basis of the complete contractionK = Kljlj of the Finslerian curvature tensorKljhk(x, y). The relevant Lagrangian is constructed by the replacement of the directional variableyi inK by a vector fieldyi(x), so that the notion of osculation may be regarded as the key concept on which the approach is based. The introduction of the auxiliary vector fieldyi(x) is shown to be of physical significance, for the field equations refer not only to the proper field variables but also to a special coordinate system associated withyi(x) through the Clebsch representation of the latter. The status of the conservation laws proves to be similar to that in the theory of the Yang-Mills field. By choosing a special Finslerian metric function we elucidate in detail the structure of the field equations in the static case.