Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein's Equations

TL;DR

This paper evaluates existing static, spherically symmetric perfect fluid solutions of Einstein's equations for physical plausibility, finding only a small subset meet all basic physical criteria.

Contribution

It systematically tests 127 solutions against fundamental physical conditions, identifying only 16 physically acceptable solutions, with 9 having decreasing sound speed profiles.

Findings

Only 16 solutions pass all physical acceptability tests.

Among these, 9 solutions exhibit monotonically decreasing sound speed.

The study highlights the scarcity of physically realistic perfect fluid solutions in the literature.

Abstract

We ask the following question: Of the exact solutions to Einstein's equations extant in the literature, how many could represent the field associated with an isolated static spherically symmetric perfect fluid source? The candidate solutions were subjected to the following elementary tests: i) isotropy of the pressure, ii) regularity at the origin, iii) positive definiteness of the energy density and pressure at the origin, iv) vanishing of the pressure at some finite radius, v) monotonic decrease of the energy density and pressure with increasing radius, and vi) subluminal sound speed. A total of 127 candidate solutions were found. Only 16 of these passed all the tests. Of these 16, only 9 have a sound speed which monotonically decreases with radius. The analysis was facilitated by use of the computer algebra system GRTensorII.