In Riemannian geometry, the metric determines a unique notion of distance and angle at each point. Finsler geometry generalizes this: the metric depends not just on position but on direction. A Finsler manifold has different “speeds” in different directions at the same point — the indicatrix (unit sphere in the tangent space) is not an ellipsoid but a general convex body.
Finsler spacetimes generalize Lorentzian manifolds analogously. Light cones become direction-dependent, and the causal structure — which events can influence which — depends on both where you are and which direction you're looking.
The paper establishes that totally geodesic null hypersurfaces in Finsler spacetimes have constant surface gravity. Surface gravity measures the acceleration needed to hover at a horizon — it's the quantity that determines a black hole's Hawking temperature. In general relativity, the zeroth law of black hole thermodynamics states that surface gravity is constant on a stationary horizon. The paper proves this extends to Finsler spacetimes: the generalized surface gravity is constant on totally geodesic null hypersurfaces, without assuming stationarity.
The result is stronger than the Riemannian version in two ways: it holds for a broader class of geometries (Finsler, not just Lorentzian) and for a weaker class of horizons (totally geodesic null hypersurfaces, not just Killing horizons). The zeroth law generalizes because it depends on the hypersurface's intrinsic geometry, not on the spacetime's symmetry. The constancy of surface gravity is a geometric fact about null hypersurfaces, not a consequence of the particular metric theory.