A Periodic Analog of the Schwarzschild Solution

TL;DR

This paper presents a new exact vacuum solution to Einstein's equations representing a black hole in a space-time with one periodic dimension, smoothly transitioning to Schwarzschild as the period increases.

Contribution

It introduces a novel periodic analog of the Schwarzschild solution, expanding the set of known exact solutions in general relativity.

Findings

Solution is free of singularities outside the horizon.

Approaches Schwarzschild solution as the period L tends to infinity.

Describes a black hole in a space-time with one periodic dimension.

Abstract

We construct a new exact solution of Einstein's equations in vacuo in terms of Weyl canonical coordinates. This solution may be interpreted as a black hole in a space-time which is periodic in one direction and which behaves asymptotically like the Kasner solution with Kasner index equal to $4M L^{-1}$, where $L$ is the period and $M$ is the mass of the black hole. Outside the horizon, the solution is free of singularities and approaches the Schwarzschild solution as $L \rightarrow \infty$.