Black Hole in the Expanding Universe with Arbitrary Power-Law Expansion

TL;DR

This paper constructs an exact, time-dependent solution describing a charged black hole embedded in an expanding universe with arbitrary power-law behavior, extending previous models and satisfying energy conditions.

Contribution

It introduces a new exact solution of Einstein-Maxwell-dilaton equations that interpolates between extremal Reissner-Nordström black holes and FLRW universes with arbitrary expansion rates.

Findings

The solution describes a spherically symmetric charged black hole in an expanding universe.

It involves two harmonic functions on a Ricci-flat base space.

The spacetime admits a nondegenerate Killing horizon under certain parameter conditions.

Abstract

We present a time-dependent and spatially inhomogeneous solution that interpolates the extremal Reissner-Nordstr\"om (RN) black hole and the Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe with arbitrary power-law expansion. It is an exact solution of the $D$-dimensional Einstein-"Maxwell"-dilaton system, where two Abelian gauge fields couple to the dilaton with different coupling constants, and the dilaton field has a Liouville-type exponential potential. It is shown that the system satisfies the weak energy condition. The solution involves two harmonic functions on a $(D-1)$-dimensional Ricci-flat base space. In the case where the harmonics have a single-point source on the Euclidean space, we find that the spacetime describes a spherically symmetric charged black hole in the FLRW universe, which is characterized by three parameters: the steepness parameter of the dilaton potential $n_T$, the U$(1)$ charge $Q$, and the "nonextremality" $\tau $. In contrast with the extremal RN solution, the spacetime admits a nondegenerate Killing horizon unless these parameters are finely tuned. The global spacetime structures are discussed in detail.