Dirichlet boundary value problem for Chern-Simons modified gravity

TL;DR

This paper derives a boundary term for Chern-Simons modified gravity to ensure well-posed Dirichlet boundary conditions and explores its implications for black hole thermodynamics.

Contribution

It introduces a boundary Chern-Simons action for the extrinsic curvature in Chern-Simons modified gravity, extending the Gibbons-Hawking-York formalism.

Findings

Derived the boundary term for well-posedness of the boundary value problem.

Connected the boundary term to black hole thermodynamics.

Provided a framework for analyzing boundary effects in modified gravity theories.

Abstract

Chern-Simons modified gravity comprises the Einstein-Hilbert action and a higher-derivative interaction containing the Chern-Pontryagin density. We derive the analog of the Gibbons-Hawking-York boundary term required to render the Dirichlet boundary value problem well-defined. It turns out to be a boundary Chern-Simons action for the extrinsic curvature. We address applications to black hole thermodynamics.