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 $\ell$, 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 nonsingular 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 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.

• Received 30 August 2016

DOI:https://doi.org/10.1103/PhysRevD.94.104056