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An analogue first law for general closed marginally trapped surfaces

Ramon Torres

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.

An analogue first law for general closed marginally trapped surfaces

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.
Paper Structure (11 sections, 81 equations, 1 figure)

This paper contains 11 sections, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometrical Preliminaries
  3. Physical Preliminaries
  4. The analogue transverse first law for closed MTSs
  5. Example: Round Spheres in Spherically Symmetric space-times
  6. Equilibrium case
  7. Evaporating case
  8. Example: Non-spherically symmetric MTS
  9. Kerr black hole in semiclassical equilibrium
  10. Evaporating Kerr black hole
  11. Conclusions

Figures (1)

  • Figure 1: A schematic representation showing some surfaces from the tube. The lightlike geodesic with tangent vector $\vec{l}^-$ is described by a dashed line. For the purposes of this figure we are considering the case of compact surfaces $\mathcal{S} (t)$.