Counterterms for Static Lovelock Solutions

TL;DR

This paper develops counterterms for third order Lovelock gravity to remove divergences in static spacetimes, enabling finite action and mass calculations, and explores black hole thermodynamics and stability.

Contribution

It introduces a method to remove divergences in Lovelock gravity using Einstein gravity counterterms, extending the analysis of black hole thermodynamics and stability.

Findings

Counterterms remove non-logarithmic divergences in Lovelock gravity.

Finite action and mass are computed for static black holes.

Small black holes exhibit phase transitions and stability in Lovelock gravity.

Abstract

In this paper, we introduce the counterterms that remove the non-logarithmic divergences of the action in third order Lovelock gravity for static spacetimes. We do this by defining the cosmological constant in such a way that the asymptotic form of the metric have the same form in Lovelock and Einstein gravities. Thus, we employ the counterterms of Einstein gravity and show that the power law divergences of the action of Lovelock gravity for static spacetimes can be removed by suitable choice of coefficients. We find that the dependence of these coefficients on the dimension in Lovelock gravity is the same as in Einstein gravity. We also introduce the finite energy-momentum tensor and employ these counterterms to calculate the finite action and mass of static black hole solutions of third order Lovelock gravity. Next, we calculate the thermodynamic quantities and show that the entropy calculated through the use of Gibbs-Duhem relation is consistent with the obtained entropy by Wald's formula. Furthermore, we find that in contrast to Einstein gravity in which there exists no uncharged extreme black hole, third order Lovelock gravity can have these kind of black holes. Finally, we investigate the stability of static charged black holes of Lovelock gravity in canonical ensemble and find that small black holes show a phase transition between very small and small black holes, while the large ones are stable.