Spacetime geometry of static fluid spheres

TL;DR

This paper presents a simple, explicit formula for the metric of static spherically symmetric perfect fluid spacetimes, enabling analysis without assuming a specific equation of state, and introduces a new exact solution that unifies several known models.

Contribution

The authors derive a general metric formula depending on a single generating function, applicable without specifying the equation of state, and introduce a new exact solution unifying multiple known solutions.

Findings

Derived a simple explicit metric formula for static fluid spheres.

Introduced a new three-parameter exact solution unifying six known solutions.

Analyzed regularity conditions and constraints for the metric.

Abstract

We exhibit a simple and explicit formula for the metric of an arbitrary static spherically symmetric perfect fluid spacetime. This class of metrics depends on one freely specifiable monotone non-increasing generating function. We also investigate various regularity conditions, and the constraints they impose. Because we never make any assumptions as to the nature (or even the existence) of an equation of state, this technique is useful in situations where the equation of state is for whatever reason uncertain or unknown. To illustrate the power of the method we exhibit a new form of the ``Goldman--I'' exact solution and calculate its total mass. This is a three-parameter closed-form exact solution given in terms of algebraic combinations of quadratics. It interpolates between (and thereby unifies) at least six other reasonably well-known exact solutions.