Entropy function from the gravitational surface action for an extremal near horizon black hole

TL;DR

This paper demonstrates that the surface term of the gravitational action, specifically the Gibbons-Hawking-York term, alone can be used to derive the entropy of extremal near horizon black holes, highlighting a holographic aspect of gravity.

Contribution

It shows that the entropy function for extremal black holes can be constructed solely from the surface term, providing new insights into holography in gravitational theories.

Findings

Surface term alone yields exact horizon entropy.

Extremization of the surface-based entropy function reproduces known results.

Supports the holographic nature of gravitational action.

Abstract

It is often argued that {\it all the information of a gravitational theory is encoded in the surface term of the action}; which means one can find several physical quantities just from the surface term without incorporating the bulk part of the action. This has been observed in various instances; e.g. derivation of the Einstein's equations, surface term calculated on the horizon leads to entropy, etc. Here I investigate the role of it in the context of entropy function and entropy of extremal near horizon black holes. Considering only the Gibbons-Hawking-York (GHY) surface term to define an entropy function for the extremal near horizon black hole solution, it is observed that the extremization of such function leads to the exact value of the horizon entropy. This analysis again supports the previous claim that there exists a ``{\it holographic}'' nature in the gravitational action -- surface term contains the information of the bulk.