Notes on nonsingular models of black holes

Abstract

We discuss static spherically symmetric metrics which represent nonsingular 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 $\ensuremath{\ell}$, 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 $\ensuremath{\sim}{\ensuremath{\ell}}^{\ensuremath{-}2}$. We also make a ``more technical'' assumption that the metric coefficients are rational functions of $r$. In particular, the invariant $(\ensuremath{\nabla}r{)}^{2}$ has the form ${P}_{n}(r)/{\stackrel{\texttildelow{}}{P}}_{n}(r)$, where ${P}_{n}$ and ${\stackrel{\texttildelow{}}{P}}_{n}$ are polynomials of the order of $n$. We discuss first the case of four dimensions. We show that when $n\ensuremath{\le}2$ such a metric cannot describe a nonsingular black hole. For $n=3$ we find a suitable metric, which besides $M$ and $\ensuremath{\ell}$ contains a dimensionless numerical parameter. When this parameter vanishes, the obtained metric coincides with Hayward's one. The characteristic property of such spacetimes is $\ensuremath{-}{\ensuremath{\xi}}^{2}=(\ensuremath{\nabla}r{)}^{2}$, where ${\ensuremath{\xi}}^{2}$ is a timelike at infinity Killing vector. We describe a possible generalization of a nonsingular black-hole metric to the case when this equality is violated. We also obtain a metric for a charged nonsingular black hole obeying similar restrictions as the neutral one and construct higher dimensional models of neutral and charged black holes.