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.

我们研究了广义相对论中自引力完美流体的一类特殊奇异解的性质：奇异等温球。对于任意常数状态方程参数\(w=p/\rho \) ，存在密度分布为\(\propto 1/r^2\) 的静态球对称解，比例常数固定为是w的特殊函数。与黑洞一样，奇异等温球体具有与大小无关的固定质量半径比，但在\(r=0\)处没有隐藏曲率奇点的视界。对于\(w=1\) ，这些解可以从均匀膨胀背景构建，其中度量自发地打破空间均匀性。我们研究这些解的微扰结构，找到径向模态和潮汐乐夫数，并且还在该几何的测地线结构中找到有趣的属性。最后，讨论了这些几何形状和暗物质剖面、双副本和全息熵之间的联系，以及沼泽距离猜想如何掩盖裸奇点。