On the Theory of Gravitational Field in Finsler Spaces

Some structural considerations are made on the Finslerian gravitational field: A Finslerian metrical structure such as g λχ ( x, y ) = γ λχ ( x ) + h λχ ( x, y ) is proposed, where γ λχ denotes the Riemann metric of Einstein's gravitational field, while h λχ the Finsler metric induced by the Riemann metric h ij ( y ) of the internal field; The intrinsic behaviour of the internal variable y , which is expressed as ȳ i = K   j i ( x, y ) y j in the internal field, is grasped by the Finslerian parallelism δ y i (=0), which is reflected in the spatial structure of the external gravitational field by the mapping relation δy χ = e   i x ( x ) δ y i . The whole metrical Finsler connection D for g λχ (i.e., Dg λχ = 0) is determined by taking account of the intrinsic behaviour δ y χ .