On the renormalization of the Gibbons-Hawking boundary term

TL;DR

This paper investigates how the boundary and bulk terms in gravitational actions are renormalized differently due to quantum effects, using heat kernel methods for various fields and boundary conditions, with implications for black hole entropy.

Contribution

It provides a detailed analysis of the renormalization behavior of Gibbons-Hawking boundary terms for different quantum fields and boundary conditions using heat kernel techniques.

Findings

Renormalization affects boundary and bulk terms differently depending on boundary conditions.

Certain boundary conditions preserve the ratio of boundary to bulk terms.

Implications for black hole entropy calculations are discussed.

Abstract

The bulk (Einstein-Hilbert) and boundary (Gibbons-Hawking) terms in the gravitational action are generally renormalized differently when integrating out quantum fluctuations. The former is affected by nonminimal couplings, while the latter is affected by boundary conditions. We use the heat kernel method to analyze this behavior for a nonminimally coupled scalar field, the Maxwell field, and the graviton field. Allowing for Robin boundary conditions, we examine in which cases the renormalization preserves the ratio of boundary and bulk terms required for the effective action to possess a stationary point. The implications for field theory and black hole entropy computations are discussed.