Quasilocal quantities for GR and other gravity theories

TL;DR

This paper develops covariant boundary expressions for quasilocal quantities in general relativity and other gravity theories, enabling calculation of energy, momentum, and angular momentum, with applications to black hole thermodynamics.

Contribution

It introduces a covariant Hamiltonian approach to define quasilocal quantities in gravity theories, unifying different boundary conditions and applying to black hole thermodynamics.

Findings

Expressions yield good ADM and Bondi quantities for Einstein gravity.

Applied formalism to black hole thermodynamics, deriving the first law and entropy.

Compared results with previous studies on static spherically symmetric solutions.

Abstract

From a covariant Hamiltonian formulation, by using symplectic ideas, we obtain certain covariant boundary expressions for the quasilocal quantities of general relativity and other geometric gravity theories. The contribution from each of the independent dynamic geometric variables (the frame, metric or connection) has two possible covariant forms associated with the selected type of boundary condition. The quasilocal expressions also depend on a reference value for each dynamic variable and a displacement vector field. Integrating over a closed two surface with suitable choices for the vector field gives the quasilocal energy, momentum and angular momentum. For the special cases of Einstein's theory and the Poincar\'e Gauge theory our expressions are similar to some previously known expressions and give good values for the total ADM and Bondi quantities. We apply our formalism to black hole thermodynamics obtaining the first law and an associated entropy expression for these general gravity theories. For Einstein's theory our quasilocal expressions are evaluated on static spherically symmetric solutions and compared with the findings of some other researchers. The choices needed for the formalism to associate a quasilocal expression with the boundary of a region are discussed.