Entropy function from the Einstein boundary term

TL;DR

This paper demonstrates that the boundary term in the Einstein-Hilbert action alone can reproduce the Bekenstein-Hawking entropy for static extremal black holes, but bulk contributions are essential for AdS and rotating cases.

Contribution

It reveals the sufficiency of the boundary term for entropy calculation in static extremal black holes and highlights the importance of bulk terms in more complex scenarios.

Findings

Boundary term alone yields entropy for static extremal black holes.

Bulk term is crucial for AdS black holes due to the cosmological constant.

Both boundary and bulk terms contribute for extremal rotating black holes.

Abstract

We show using the entropy function formalism developed by Sen \cite{Sen:2005wa} that the boundary term which arises from the Einstein-Hilbert action is sufficient to yield the Bekenstein-Hawking entropy of a static extremal black hole which is asymptotically flat. However, for asymptotically $AdS$ black holes, the bulk term also plays an important role due to the presence of the cosmological constant. Further, we show that for extremal rotating black holes, both the boundary and the bulk terms contribute non-vanishing pieces to the entropy.