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An introduction to the mechanics of black holes

Geoffrey Compere

TL;DR

This work provides a self-contained derivation of the zero, first, and second laws of black hole mechanics by exploiting the null, horizon geometry and global conservation laws in gravity. It shows that event horizons are null hypersurfaces whose null generators and Raychaudhuri dynamics control area growth, leading to the area theorem under the null energy condition. The zero law establishes a constant surface gravity on Killing horizons, while the first law follows from matching horizon and infinity charges for nearby stationary black holes, with natural extensions to Einstein-Maxwell theory and to arbitrary diffeomorphism-invariant gravities via the Iyer-Wald formalism. The approach connects horizon dynamics to conserved charges, providing a thermodynamic-like framework (including entropy and temperature concepts) with broad applicability in higher dimensions and alternative theories. Overall, the notes synthesize horizon geometry, generalized Noether methods, and quasi-equilibrium thermodynamics to foundationally link gravity, causality, and thermodynamics.

Abstract

These notes provide a self-contained introduction to the derivation of the zero, first and second laws of black hole mechanics. The prerequisite conservation laws in gauge and gravity theories are also briefly discussed. An explicit derivation of the first law in general relativity is performed in appendix.

An introduction to the mechanics of black holes

TL;DR

This work provides a self-contained derivation of the zero, first, and second laws of black hole mechanics by exploiting the null, horizon geometry and global conservation laws in gravity. It shows that event horizons are null hypersurfaces whose null generators and Raychaudhuri dynamics control area growth, leading to the area theorem under the null energy condition. The zero law establishes a constant surface gravity on Killing horizons, while the first law follows from matching horizon and infinity charges for nearby stationary black holes, with natural extensions to Einstein-Maxwell theory and to arbitrary diffeomorphism-invariant gravities via the Iyer-Wald formalism. The approach connects horizon dynamics to conserved charges, providing a thermodynamic-like framework (including entropy and temperature concepts) with broad applicability in higher dimensions and alternative theories. Overall, the notes synthesize horizon geometry, generalized Noether methods, and quasi-equilibrium thermodynamics to foundationally link gravity, causality, and thermodynamics.

Abstract

These notes provide a self-contained introduction to the derivation of the zero, first and second laws of black hole mechanics. The prerequisite conservation laws in gauge and gravity theories are also briefly discussed. An explicit derivation of the first law in general relativity is performed in appendix.

Paper Structure

This paper contains 16 sections, 86 equations, 4 figures.

Table of Contents

  1. Event horizons
  2. Null hypersurfaces
  3. The Raychaudhuri equation
  4. Properties of event horizons
  5. The area theorem
  6. Equilibrium states
  7. Killing horizons
  8. Zero law
  9. Conservation laws
  10. The generalized Noether theorem
  11. The energy in general relativity
  12. Quasi-equilibrium states
  13. The first law for Einstein gravity
  14. Extension to electromagnetism
  15. Extension to any theory of gravity
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Penrose diagram of an asymptotically flat spacetime with spherically symmetric collapsing star. Each point is a $n-2$-dimensional sphere. Light rays propagate along $45^\circ$ diagonals. The star region is hatched and the black hole region is indicated in grey.
  • Figure 2: The null vector $n$ is defined with respect to $\xi$.
  • Figure 3: If two null generators of $\mathcal{H}$ cross, they may be deformed to a timelike curve.
  • Figure :