Regular black holes from pure gravity in four dimensions

Abstract

We derive static spherically symmetric regular black holes as vacuum solutions to purely gravitational theories in four dimensions. To that end, we construct four-dimensional non-polynomial gravities starting from subclasses of two-dimensional Horndeski actions. By construction, these theories possess second-order equations of motion on spherically symmetric backgrounds. We show that a subset of these non-polynomial gravities, referred to as non-polynomial quasi-topological gravities, admit single-function static spherically symmetric solutions whereby the metric function is determined by an algebraic equation. Solutions to these theories include the Hayward regular black-hole spacetime, for which a corresponding gravitational action is stated explicitly.

Figures (1)

• Figure 1: Landscape of $d\geq 4$ dimensional second-order spherical gravities, by which we mean here gravitational theories $\mathcal{L}\qty(g^{\alpha\beta},R_{\mu\nu\rho\sigma})$ with second-order equations of motion on spherically symmetric backgrounds. All of these theories feature reduced actions representing two-dimensional Horndeski actions of the form \ref{['eq:SHorndeskiSubclass']}. Polynomial gravities of this type only exist in $d>4$ dimensions and these are equivalent to polynomial quasi-topological gravities Bueno:2025qjk. In this case, the functions $H_i(\psi)$ in the reduced action are analytic functions which can be parametrized in terms of one analytic function $H(\psi)$, and the unique static spherically symmetric solutions are single-function spacetimes (i.e. $n' = 0$) with metric function $f$ determined through an algebraic equation $H(\psi) \sim M/r^{d-1}$, where $M$ is proportional to the ADM mass of the solution Bueno:2025qjk. By contrast, non-polynomial gravities with second-order equations of motion on spherically symmetric backgrounds can be constructed in general dimension $d\geq 4$ by an application of the procedure described in this work. In this case, the functions $H_i(\psi)$ in the reduced action can be arbitrary, not necessarily analytic, functions. In this case, one may further distinguish between non-polynomial quasi-topological gravities satisfying the integrability constraint on the functions $H_i(\psi)$\ref{['eq:HConstraint']}, understood prior to performing any partial integrations following the reduction, which guarantees that static spherically symmetric solutions are single-function spacetimes, whereby the metric function $f$ is determined by an algebraic equation involving two functions $H(\psi)$ and $G(\psi)$ in general, cf. \ref{['eq:EqMasterHG']} in $d=4$. Generic static spherically symmetric solutions to non-polynomial second-order gravities which are not of quasi-topological type are in general spacetimes with two distinct metric functions $n$ and $f$.