Non-perturbative zeroth-law proof for generic higher-curvature theories

Establish a general non-perturbative proof of the black hole zeroth law for generic higher-curvature gravitational theories that fall outside the null-null-degenerate and null-null-nondegenerate classes, without restricting to perturbative effective-field-theory expansions around the Einstein branch.

Background

The paper classifies diffeomorphism-invariant gravitational theories according to which horizon projection of the field equations enforces constancy of the surface gravity. Type I theories have an off-shell-degenerate null-null projection and rely on the mixed null-transverse equation, while Type II theories can obtain the zeroth law non-perturbatively from a nondegenerate null-null equation, as demonstrated for generalized quadratic gravity and a cubic example.

The remaining Type III theories are more general and need not fit either mechanism. For these theories, the paper states that the only broadly available result is a perturbative proof within an effective-field-theory expansion around solutions continuously connected to general relativity. A general non-perturbative proof, particularly one applicable to non-Einstein branches, remains unresolved.