How Universal Is the Black Hole Zeroth Law?

Rajes Ghosh; Sudipta Sarkar

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arXiv · 2609.24648

How Universal Is the Black Hole Zeroth Law?

Rajes Ghosh·Sudipta Sarkar

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Abstract

The black hole zeroth law asserts the thermal equilibrium of a stationary Killing horizon, yet its status beyond general relativity remains limited. We demonstrate that its validity is not governed by a universal mechanism: distinct gravitational theories enforce constancy of the surface gravity through different horizon-projections of their field equations. In general relativity, $f(R)$-gravity, and Lovelock theories, the zeroth law is known to follow from the mixed (null-transverse) horizon equations. By contrast, we show that in non-Lovelock theories the zeroth law is enforced by qualitatively different mechanisms. In generalized quadratic gravity, for instance, the null-null component of the field equations dictates its validity. Exploiting this novel diagnostic, we obtain the first exact proof of the zeroth law in the broader landscape of higher-curvature gravity, without relying on a perturbative expansion around general relativity. Our result establishes the zeroth law as a sharp probe of the underlying thermodynamic structure of gravity, and points toward a classification of modified theories according to the mechanism responsible for its validity.

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Rajes Ghosh, Sudipta Sarkar. 2026-09-21. How Universal Is the Black Hole Zeroth Law?. https://arxiv.org/abs/2609.24648

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