The imaginary part of the gravitational action at asymptotic boundaries and horizons

TL;DR

This paper investigates the imaginary part of the Lorentzian gravitational action at boundaries and horizons, linking it to black hole entropy and exploring its topological aspects in Lovelock gravity.

Contribution

It provides a detailed comparison between Lorentzian and Euclidean calculations of the imaginary action and clarifies its topological structure in Lovelock gravity.

Findings

The imaginary part of the action matches black hole entropy in various setups.

Black hole entropy and conserved charges contribute distinctly to the action.

The imaginary part is consistent across different geometric configurations.

Abstract

We study the imaginary part of the Lorentzian gravitational action for bounded regions, as described in arXiv:1301.7041. By comparing to a Euclidean calculation, we explain the agreement between the formula for this imaginary part and the formula for black hole entropy. We also clarify the topological structure of the imaginary part in Lovelock gravity. We then evaluate the action's imaginary part for some special regions. These include cylindrical slabs spanning the exterior of a stationary black hole spacetime, 'maximal diamonds' in various symmetric spacetimes, as well as local near-horizon regions. In the first setup, the black hole's entropy and conserved charges contribute to the action's imaginary and real parts, respectively. In the other two setups, the imaginary part coincides with the relevant entropy.