Black bounces, wormholes, and partly phantom scalar fields
Abstract
Simpson and Visser recently proposed a phenomenological way to avoid some kinds of space-time singularities by replacing a parameter whose zero value corresponds to a singularity (say, $r$) with the manifestly nonzero expression $r(u)=\sqrt{{u}^{2}+{b}^{2}}$, where $u$ is a new coordinate, and $b=\text{const}>0$. This trick, generically leading to a regular minimum of $r$ beyond a black hole horizon (called a ``black bounce''), may hopefully mimic some expected results of quantum gravity, and was previously applied to regularize the Schwarzschild, Reissner-Nordstr\"om, Kerr, and some other metrics. In this paper it is applied to regularize the Fisher solution with a massless canonical scalar field in general relativity (resulting in a traversable wormhole) and a family of static, spherically symmetric dilatonic black holes (resulting in regular black holes and wormholes). These new regular metrics represent exact solutions of general relativity with a sum of stress-energy tensors of a scalar field with nonzero self-interaction potential and a magnetic field in the framework of nonlinear electrodynamics with a Lagrangian function $\mathcal{L}(\mathcal{F})$, $\mathcal{F}={F}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}$. A novel feature in the present study is that the scalar fields involved have ``trapped ghost'' properties, that is, are phantom in a strong-field region and canonical outside it, with a smooth transition between the regions. It is also shown that any static, spherically symmetric metric can be obtained as an exact solution to the Einstein equations with the stress-energy tensor of the above field combination.