Notes on non-singular models of black holes

TL;DR

This paper explores static, spherically symmetric non-singular black hole models in various dimensions, imposing regularity, asymptotic, and curvature constraints, and introduces new metrics that generalize known solutions like Hayward's.

Contribution

It introduces a class of rational-function metrics for non-singular black holes, including higher-dimensional and charged cases, extending previous models with new parameters and properties.

Findings

Non-singular black hole metrics exist for polynomial order n=3.

A new metric generalizes Hayward's solution with an additional dimensionless parameter.

Higher-dimensional and charged non-singular black hole models are constructed.

Abstract

We discuss static spherically symmetric metrics which represent non-singular black holes in four- and higher-dimensional spacetime. We impose a set of restrictions, such as a regularity of the metric at the center $r=0$ and Schwarzschild asymptotic behavior at large $r$. We assume that the metric besides mass $M$ contains an additional parameter $\ell$, which determines the scale where modification of the solution of the Einstein equations becomes significant. We require that the modified metric obeys the limiting curvature condition, that is its curvature is uniformly restricted by the value $\sim \ell^{-2}$. We also make a "more technical" assumption that the metric coefficients are rational functions of $r$. In particular, the invariant $(\nabla r)^2$ has the form $P_n(r)/\tilde{P}_n(r)$, where $P_n$ and $\tilde{P}_n$ are polynomials of the order of $n$. We discuss first the case of four dimensions. We show that when $n\le 2$ such a metric cannot describe a non-singular black hole. For $n=3$ we find a suitable metric, which besides $M$ and $\ell$ contains a dimensionless numerical parameter. When this parameter vanishes the obtained metric coincides with Hayward's one. The characteristic property of such spacetimes is $-\xi^2=(\nabla r)^2$, where $\xi^2$ is a time-like at infinity Killing vector. We describe a possible generalization of a non-singular black-hole metric to the case when this equality is violated. We also obtain a metric for a charged non-singular black hole obeying similar restrictions as the neutral one, and construct higher dimensional models of neutral and charged black holes.