Four-dimensional black holes in Einsteinian cubic gravity

Abstract

We construct static and spherically symmetric generalizations of the Schwarzschild- and Reissner-Nordstr\"om-(Anti-)de Sitter (RN-(A)dS) black-hole solutions in four-dimensional Einsteinian cubic gravity (ECG). The solutions are determined by a single blackening factor which satisfies a non-linear second-order differential equation. Interestingly, we are able to compute independently the Hawking temperature $T$, the Wald entropy $\mathsf{S}$ and the Abbott-Deser mass $M$ of the solutions analytically as functions of the horizon radius and the ECG coupling constant $\lambda$. Using these we show that the first law of black-hole mechanics is exactly satisfied. Some of the solutions have positive specific heat, which makes them thermodynamically stable, even in the uncharged and asymptotically flat case. Further, we claim that, up to cubic order in curvature, ECG is the most general four-dimensional theory of gravity which allows for non-trivial single-blackening factor generalizations of Schwarzschild- and RN-(A)dS which reduce to the usual Einstein gravity solutions when the corresponding higher-order couplings are set to zero.Comment: 13 pp, 5 figs; v3: minor modifications to match published versio

Generalized quasitopological gravity

“…In particular, in four dimensions the new term S d reduces to Einsteinian cubic gravity term plus an additional term which vanishes on a VSSS ansatz (1.1). This confirms the claim made in [21] that the Einsteinian cubic gravity is the most general four-dimensional theory with this property.…”

“…Note that this condition is not sufficient, as there are theories that are equivalent to Einstein's gravity at the linear level but do not admit single metric function VSSS: for example, Einsteinian cubic gravity in dimensions other than four [20,21]. One way to see that this is not a sufficient condition is to note that imposing the absence of the massive graviton and scalar modes leads to two constraint equations on the c i , however we found that (in d ≥ 5) there are in fact five constraint equations imposed by demanding E t t = E r r .…”

“…We are considering additional higher curvature terms supplementing the Einstein-Hilbert action in a manner such that these terms can be "turned off" by a suitable adjustment of parameters in the action. Our conditions are the same as those mentioned in [21]:…”

“…This can be achieved without loss of generality by reparametrizing the time coordinate t.) Here dΣ 2 (d−2),k is the line element on a surface of constant scalar curvature k = +1, 0, −1 corresponding to spherical, flat, and hyperbolic topologies. As remarked in [21], Einsteinian cubic gravity in d = 4 dimensions is the most general theory (up to cubic order) which admits a Schwarzschild-like solution, that is, a SSS solution characterized by a single metric function in vacuum or in the presence of suitable matter. However, this is no longer true in higher dimensions where the choice N = 1 leads to inconsistent field equations [20].…”

“…Contrary to the cubic Lovelock or quasi-topological terms, this new term is non-trivial in four-dimensions. Studies of black holes of the theory [20,21] revealed that, in four dimensions, the theory admits static spherically symmetric (SSS) solutions, in vacuum (VSSS) or in the presence of suitable matter (e.g. a Maxwell field), of the form 1) with N = const., i.e., the solution is characterized in terms of a single metric function f .…”