Black Holes in a Compactified Spacetime

TL;DR

This work analyzes a four-dimensional Schwarzschild black hole in a spacetime with one compactified spatial dimension, revealing horizon distortion due to compactification. The authors formulate the problem in the Weyl framework and derive a compactified Schwarzschild metric via two representations: a Green's-function integral form and a Fourier-series form, both validated against each other. They characterize the large-distance behavior, horizon size and shape, and provide embedding diagrams to visualize the distorted horizon, identifying an angular deficit at infinity and a deformation that grows with the mass parameter. The study offers analytic tools for black holes in compactified or KK-like spacetimes and discusses potential instabilities related to the fixed compactification radius, highlighting avenues for perturbative stability analyses and implications for brane-world scenarios.

Abstract

We discuss properties of a 4-dimensional Schwarzschild black hole in a spacetime where one of the spatial dimensions is compactified. As a result of the compactification the event horizon of the black hole is distorted. We use Weyl coordinates to obtain the solution describing such a distorted black hole. This solution is a special case of the Israel-Khan metric. We study the properties of the compactified Schwarzschild black hole, and develop an approximation which allows one to find the size, shape, surface gravity and other characteristics of the distorted horizon with a very high accuracy in a simple analytical form. We also discuss possible instabilities of a black hole in the compactified space.

Figures (6)

• Figure 1: Compactified Schwarzschild black hole solutions for $\mu=0.5$ (left) and $\mu=2.0$ (right). The surface plots show the gravitational potential $U(\rho,z)$ (top) and the function $V(\rho,z)$ (bottom); red and blue contours represent equipotential surfaces of $U$ and $V$ correspondingly.
• Figure 2: Redshift factor $u$ (left) and the irreducible mass $\mu_0=\mu\, \exp(-u)$ as functions of $\mu$.
• Figure 3: $l/(2\pi)$ as a function of $\mu$.
• Figure 4: The shape function $\exp({\cal F}(z))$ (left) and the Gaussian curvature of the horizon $K(z)$ (right) for different values of $\mu$.
• Figure 5: Embedding diagrams for the surface of the black hole horizon. By rotating a curve from a family shown at the plot around a horizontal axis one obtains surface isometric to the surface of a black hole described by the metric $d\sigma^2$. Different curves correspond to different values of $\mu$. The larger $\mu$ the more oblate is the form of the curve.