Rational regular black holes in non-polynomial gravity

Abstract

We consider the $d\geq 4$ non-polynomial pure gravity theories defined so that their 2D reduction to spherical symmetry minimally modifies that of General Relativity at small distances. They satisfy Birkhoff theorem and admit exact black hole solutions for an arbitrary number of couplings. These solutions have a logarithmic leading correction to the entropy and a quantum-like correction to the Newtonian potential, both arising from the regularised critical order $d=2p$ theory, which is related in even dimensions to the Euler characteristic of the 2D orbit space. These black holes can be made regular, without the need for infinite towers of corrections, using invariants of any order $p>d/2$. In even dimensions, their smoothness is linear in the number of corrections and does not require fine-tuning. We construct a class of theories admitting as unique (massless) vacuum an extremal black hole deformation of (A)dS$_4$ or M$_4$ with vanishing entropy, so that Nernst's third law of thermodynamics is satisfied without discontinuity in the entropy. We show that a Maxwell field does not disrupt the regularity of the metric, and that simple non-minimal couplings in the action enable to regularise the electric field. Considering infinite towers of corrections, we construct quasi-regular black holes with one horizon as well as regular (A)dS-core ones with a near-extremal inner horizon for any mass. When charged, these two types of black holes prevent or tame the mass inflation provided the mass and charge satisfy an inequality of the form $M > α\ell + βQ^2 / \ell$, where $α$ and $β$ are numerical factors depending on the specific model, while $\ell$ is the length scale controlling the high-energy corrections. The quasi-regular black holes are sufficient to resolve the Coulomb singularity without the need for non-minimal couplings.