The Finsler Spacetime Condition for (α,β)-Metrics and Their Isometries

Abstract

For the general class of pseudo-Finsler spaces with (α,β)-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has a Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future-pointing time-like vectors is ensured. The identified (α,β)-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an (α,β)-metric and the isometries of the underlying pseudo-Riemannian metric a; in particular, we list all (α,β)-metrics which admit isometries that are not isometries of a.

Finsler gravitational waves of (α, β)-type and their observational signature

“…For each of these types of (α, β)-metrics certain conditions need to be fulfilled in order to satisfy the definition of a Finsler spacetime [51].…”

“…This leads to a Randers metric of the form F = |A| + β. An undesirable consequence of this definition, however, is that light rays can only propagate into one half of the tangent space, namely the half given by β < 0, which follows immediately from the null condition F = 0 (see also [51]). In fact, the light cone separates the tangent space into only two connected components 10 and there is consequently not a straightforward interpretation in terms of timelike, spacelike and lightlike directions, at least not in the conventional way 11 .…”

“…We point out in particular that this is true even when the 1-form is spacelike. On the other hand, the classical Randers metric can only be considered a physically reasonable Finsler spacetime if the 1-form is either null or timelike [51]. Finally, regarding the regularity of F, clearly F is smooth everywhere except when A = 0 or β = 0.…”