An analogue first law for general closed marginally trapped surfaces

TL;DR

This paper develops a quasi-local thermodynamic law for closed marginally trapped surfaces in arbitrary spacetimes, extending black hole thermodynamics beyond traditional horizon-based approaches.

Contribution

It introduces a transverse first law for marginally trapped surfaces that is intrinsic and applicable in general spacetimes, not limited to horizons.

Findings

Reproduces expected results for spherical symmetry

Applies to Kerr black holes and non-spherical surfaces

Remains valid in non-unique or complex horizon scenarios

Abstract

We formulate an analogue transverse first law for general closed marginally trapped surfaces in arbitrary spacetimes. The construction is intrinsically quasi-local and is attached directly to an individual marginally trapped surface, rather than to a preferred horizon worldtube. Taking the Hawking energy as the internal energy and an invariant effective surface gravity associated with the marginally trapped surface as the quantity controlling the thermal term, we derive a balance law in which the variation of energy splits into a generalized heat contribution and a total work contribution. In this way, the resulting law provides a codimension-two, transverse counterpart to existing horizon-based formulations of black-hole thermodynamics. We show that the formalism reproduces the expected results for round spheres in spherically symmetric spacetimes. We then examine semiclassical equilibrium and evaporating regimes, and extend the analysis to non-spherically symmetric marginally trapped surfaces in Kerr. These examples indicate that the framework remains applicable in situations where a horizon-based treatment is either nonunique or technically cumbersome, and suggest that closed marginally trapped surfaces provide a natural arena for a genuinely quasi-local thermodynamics of black holes.