Generalized quasi-topological gravities: the whole shebang

Abstract

Generalized quasi-topological gravities (GQTGs) are higher-curvature extensions of Einstein gravity in D-dimensions. Their defining properties include possessing second-order linearized equations of motion around maximally symmetric backgrounds as well as non-hairy generalizations of Schwarzschild's black hole characterized by a single function, f (r) ≡ −g tt = g −1 rr , which satisfies a second-order differential equation. In arXiv:1909.07983 GQTGs were shown to exist at all orders in curvature and for general D. In this paper we prove that, in fact, n − 1 inequivalent classes of order-n GQTGs exist for D ≥ 5. Amongst these, we show that one -and only one-type of densities is of the Quasi-topological kind, namely, such that the equation for f (r) is algebraic. Our arguments do not work for D = 4, in which case there seems to be a single unique GQT density at each order which is not of the Quasi-topological kind. We compute the thermodynamic charges of the most general D-dimensional order-n GQTG, verify that they satisfy the first law and provide evidence that they can be entirely written in terms of the embedding function which determines the maximally symmetric vacua of the theory.

Aspects of three-dimensional higher curvature gravities

“…In sum, we learn that in three dimensions there exist no non-trivial GQTs. This situation is very different from higher-dimensions: in D = 4 there exists one independent non-trivial GQT density for every n 3 whereas for D 5 there actually exist n − 1 independent inequivalent GQT densities for every n-namely, there exist n − 1 densities of order n each of which makes a functionally different contribution to the equation of f (r) [93]. As a matter of fact, the triviality of the three-dimensional case unveiled here is not so surprising given that all higher-curvature theories admit the BTZ solution-as opposed to non-trivial GQTs in higher dimensions, which admit modifications of Schwarzschild as solutions, but not Schwarzschild itself.…”