Volume forms for Finsler spacetimes

Abstract

The paper proposes extensions of the usual notions of Finslerian volume to Finsler metrics of Lorentzian signature. The basic idea is to replace, in the classical Busemann-Hausdorff and Holmes-Thompson definitions, integration on the indicatrices of the given metric (which are, in Lorentzian signature, non-compact, generally leading to infinite integrals) with integration on ellipsoids. These ellipsoids can be canonically attached to the given Lorentzian Finsler metric by means of a variational procedure. While the construction of the Holmes-Thompson volume form requires the determinant of the Finslerian metric tensor to be defined and smooth on the entire slit tangent bundle, the Busemann-Hausdorff-type volume form can be constructed even if the metric tensor is not defined or is degenerate along some directions - which is the case with the large majority of the known Lorentzian Finsler metrics. This feature makes it possible to build well-defined field-theoretical integrals having such metrics as a background.