Totally geodesic null hypersurfaces and constancy of surface gravity in Finsler spacetimes

TL;DR

This paper extends the theory of totally geodesic null hypersurfaces to Finsler spacetimes, demonstrating conditions under which surface gravity is constant, thus generalizing key Lorentzian results and linking to thermodynamic principles.

Contribution

It introduces a method to analyze null hypersurfaces in Finsler spacetimes, proving constancy of surface gravity under certain conditions, and connects these results to gravitational equations and thermodynamics.

Findings

Connected compact totally geodesic null hypersurfaces have constant surface gravity.

The null convergence condition and a gravitational equation imply Ricci 1-form vanishing.

Topological classification results for these hypersurfaces are obtained.

Abstract

We define and study totally geodesic null hypersurfaces in Finsler spacetimes. We prove that the null convergence condition and a certain mild gravitational equation $\chi_\alpha=0$, imply the vanishing of the Ricci 1-form on the hypersurface. This makes it possible to extend to the Lorentz-Finsler setting essentially all notable results for compact totally geodesic null hypersurfaces that hold in the Lorentzian case. In fact, we introduce a trick that reduces the Lorentz-Finsler analysis to a purely Lorentzian study. As a result, it follows that, under the stated conditions, connected compact totally geodesic null hypersurfaces admit constant surface gravity. Further topological classification results are also obtained. The possibility of deriving these results from the dominant energy condition is also explored, this strategy selecting an elegant unifying equation. In any case the vanishing of the Ricci 1-form is selected as a vacuum gravitational equation. Since surface gravity can be interpreted as temperature in some contexts, and its constancy expresses the zeroth law of thermodynamics, the present work provides a compelling physical argument in favour of some special Finslerian gravitational equations.