Counterterms for static Lovelock solutions

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.

Criticality and extended phase space thermodynamics of AdS black holes in higher curvature massive gravity

“…Obviously, the swallow-tail behavior demonstrates the first order phase transition (exactly the same as vdW fluid). For the sake of completeness, in the left and middle panels of three terms, [71,132,133]. We add generic mass terms for the gravitons in the Lovelock theory of gravity by supplementing higher order interaction terms of massive gravitons (which first proposed in [23]).…”

“…For the black hole entropy, since the area law is generally not valid for higher curvature gravities, one should use another approach to calculate it. For asymptotically flat spacetimes, one can easily apply the Wald method [135] as [133], one can still use the Wald formula to calculate the entropy for asymptotically AdS spacetimes. Using this method, in TOL gravity the entropy may be written as…”

Criticality and extended phase space thermodynamics of AdS black holes in higher curvature massive gravity

“…Again, the total action consists of three terms, I G = I b + I s + I ct . The Lovelock boundary terms (I s ) to have a well-defined gravitational action and the counterterms for static solutions with Ricci flat and curved horizons (I ct ) to regulate divergences of the Lovelock buck action are constructed before and the relevant details can be found in [71,128,129]. We add generic mass terms for the gravitons in the Lovelock theory of gravity by supplementing higher order interaction terms of massive gravitons (which first proposed in [23]).…”

“…where g is the determinant of the induced metric gµν on (d − 2)-dimensional boundary, Lk−1 is the k-th order of Lovelock Lagrangian constructed from gµν and [d 1 /2] denotes the integer part of d 1 /2. Moreover, as shown in [129], one can still use the Wald formula to calculate the entropy for asymptotically AdS spacetimes. Using this method, in TOL gravity the entropy may be written as…”

Non-analyticity of holographic Rényi entropy in Lovelock gravity

“…S E,reg can be computed by extending holographic counter-term methods to general Lovelock theories, explicitly developed in [59,60] and initiated in [61][62][63]. 8 The final result can be written as…”