Exploration of a Singular Fluid Spacetime

TL;DR

This paper explores the properties, perturbations, and potential physical implications of singular isothermal sphere solutions in general relativity, revealing their unique features and connections to dark matter, holography, and quantum gravity conjectures.

Contribution

It introduces and analyzes a class of singular solutions with fixed mass-to-radius ratios, studying their perturbative structure and physical relevance in various theoretical contexts.

Findings

Singular isothermal spheres have a fixed mass-to-radius ratio similar to black holes.

These solutions possess a curvature singularity at the center without an event horizon.

Connections are made to dark matter profiles, the double copy, holography, and the swampland conjecture.

Abstract

We investigate the properties of a special class of singular solutions for a self-gravitating perfect fluid in general relativity: the singular isothermal sphere. For arbitrary constant equation-of-state parameter $w=p/\rho$, there exist static, spherically-symmetric solutions with density profile $\propto 1/r^2$, with the constant of proportionality fixed to be a special function of $w$. Like black holes, singular isothermal spheres possess a fixed mass-to-radius ratio independent of size, but no horizon cloaking the curvature singularity at $r=0$. For $w=1$, these solutions can be constructed from a homogeneous dilaton background, where the metric spontaneously breaks spatial homogeneity. We study the perturbative structure of these solutions, finding the radial modes and tidal Love numbers, and also find interesting properties in the geodesic structure of this geometry. Finally, connections are discussed between these geometries and dark matter profiles, the double copy, and holographic entropy, as well as how the swampland distance conjecture can obscure the naked singularity.