Notes on nonsingular models of black holes

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 ℓ, 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 ∼ℓ-². We also make a "more technical" assumption that the metric coefficients are rational functions of r . In particular, the invariant (∇r )² has the form P_n(r )/P_∼n(r ), where P_n and P_∼n are polynomials of the order of n . We discuss first the case of four dimensions. We show that when n ≤2 such a metric cannot describe a nonsingular black hole. For n =3 we find a suitable metric, which besides M and ℓ contains a dimensionless numerical parameter. When this parameter vanishes, the obtained metric coincides with Hayward's one. The characteristic property of such spacetimes is -ξ²=(∇r )², where ξ² 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.