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

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 restriction 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 without using $\chi_\alpha=0$ is also explored, this strategy selecting some specific possibilities. 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.