Generalized quasi-topological gravities: the whole shebang

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) equivalent to -g(tt) equivalent to g(rr)(-1), which satisfies a second-order differential equation. In (Bueno et al 2020 Class. Quantum Grav. 37 015002) 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 (as far as static and spherically symmetric solutions are concerned) of order -n GQTGs exist for D >= 5. Amongst these, we show that 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.

Automated summary

This paper proves the existence of multiple inequivalent classes of order -n GQTGs in D≥5, some of which are of the quasi-topological type. Additionally, by calculating the thermodynamic charges of these GQTGs, it is verified that they satisfy the first law and can be entirely expressed in terms of the embedding function.