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.
Highlights
The construction and detailed study of solutions to the Einstein equations has played an important role in a century of progress in physics
Work on discovering and characterizing spacetime geometries is of particular relevance in many areas of active research ranging from holography and quantum gravity, to scattering amplitudes and the double copy, to metrics for compact astrophysical objects
We have examined a set of remarkable singular solutions to the TOV equations—the singular isothermal sphere (SIS)—with metric given in Eq (7), describing a general relativistic perfect fluid with fixed equation-of-state parameter w = p/ρ
Summary
The construction and detailed study of solutions to the Einstein equations has played an important role in a century of progress in physics. The SIS possesses many interesting properties, including a naked singularity in curvature, fixed mass-to-radius ratio like black holes, and the ability to trap light Examples of these solutions can occur in string theory and are relevant for the gravity/YangMills double copy. While the dynamics of fluid sphere solutions has been well investigated in the context of collapse [6,7,8,12,13,14], a general treatment of the behavior of geodesics within the static, singular SIS geometry has been less studied Orbits in these geometries exhibit remarkable features, including equation-of-statedependent precession, photon spheres, and trapped spiraling null geodesics
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