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Mechanics of multidimensional isolated horizons

M. Korzynski, J. Lewandowski, T. Pawlowski

TL;DR

This work extends the four-dimensional isolated horizon framework to $N+2$ dimensions, providing a dimension–independent formulation of weakly isolated horizon mechanics and proving a first law for arbitrary dimensional WIHs in vacuum gravity. It defines horizon angular momentum via a rotational symmetry on the horizon and shows that the horizon area generates null translations, yielding a Hamiltonian function that depends on the horizon area and angular momentum. Under appropriate integrability conditions, the first law relates variations of horizon energy to changes in area and angular momentum, reproducing the Kerr–ADM mass relation in stationary cases and reducing to the generalized Schwarzschild energy in spherical symmetry. The paper also analyzes gauge invariance of the horizon quantities and discusses under which boundary choices the angular momentum is well-defined, outlining future extensions to broader settings and higher-dimensional black hole thermodynamics.

Abstract

Recently a multidimensional generalization of Isolated Horizon framework has been proposed by Lewandowski and Pawlowski (gr-qc/0410146). Therein the geometric description was easily generalized to higher dimensions and the structure of the constraints induced by the Einstein equations was analyzed. In particular, the geometric version of the zeroth law of the black hole thermodynamics was proved. In this work we show how the IH mechanics can be formulated in a dimension--independent fashion and derive the first law of BH thermodynamics for arbitrary dimensional IH. We also propose a definition of energy for non--rotating horizons.

Mechanics of multidimensional isolated horizons

TL;DR

This work extends the four-dimensional isolated horizon framework to dimensions, providing a dimension–independent formulation of weakly isolated horizon mechanics and proving a first law for arbitrary dimensional WIHs in vacuum gravity. It defines horizon angular momentum via a rotational symmetry on the horizon and shows that the horizon area generates null translations, yielding a Hamiltonian function that depends on the horizon area and angular momentum. Under appropriate integrability conditions, the first law relates variations of horizon energy to changes in area and angular momentum, reproducing the Kerr–ADM mass relation in stationary cases and reducing to the generalized Schwarzschild energy in spherical symmetry. The paper also analyzes gauge invariance of the horizon quantities and discusses under which boundary choices the angular momentum is well-defined, outlining future extensions to broader settings and higher-dimensional black hole thermodynamics.

Abstract

Recently a multidimensional generalization of Isolated Horizon framework has been proposed by Lewandowski and Pawlowski (gr-qc/0410146). Therein the geometric description was easily generalized to higher dimensions and the structure of the constraints induced by the Einstein equations was analyzed. In particular, the geometric version of the zeroth law of the black hole thermodynamics was proved. In this work we show how the IH mechanics can be formulated in a dimension--independent fashion and derive the first law of BH thermodynamics for arbitrary dimensional IH. We also propose a definition of energy for non--rotating horizons.

Paper Structure

This paper contains 14 sections, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weakly isolated horizons
  3. Non-expanding horizons
  4. The definition of weakly isolated horizons
  5. Good cuts foliation
  6. Symmetries and symmetric WIH's
  7. The phase space of Einstein vacuums admitting WIH
  8. Generating functions for space--time diffeomorphisms
  9. Angular momentum
  10. Area as a generator of null translations
  11. The first law
  12. Invariance and non-invariance
  13. Discussion and conclusions
  14. Acknowledgments

Figures (2)

  • Figure 3: The structure fixed in $\Delta=\hat{\Delta}\times [0,\,1]$: the Cartesian product structure, the natural coordinate $v$ on $[0,\,1]$ extended to $\Delta$, the vector field $\frac{d}{dv}$ extended to $\Delta$ and denoted by $l$, and a vector field $\phi$, the natural extension to $\Delta$ of a vector field $\hat{\phi}$ fixed on $\hat{\Delta}$.
  • Figure 4: The integration surface for the symplectic form