Static Spherically Symmetric Perfect Fluid

In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as ssspf) is a spacetime equipped with suitable tensor fields which models a static round ball of a fluid with isotropic pressure.

Such solutions are often used as idealized models of stars, especially compact objects such as white dwarfs and especially neutron stars. In general relativity, a model of an isolated star (or other fluid ball) generally consists of a fluid-filled interior region, which is technically a perfect fluid solution of the Einstein field equation, and an exterior region, which is an asymptotically flat vacuum solution. These two pieces must be carefully matched across the world sheet of a spherical surface, the surface of zero pressure. (There are various mathematical criteria called matching conditions for checking that the required matching has been successfully achieved.) Similar statements hold for other metric theories of gravitation, such as the Brans-Dicke theory.

In this article, we will focus on the construction of exact ssspf solutions in our current Gold Standard theory of gravitation, the theory of general relativity. To anticipate, the figure at right depicts (by means of an embedding diagram) the spatial geometry of a simple example of a stellar model in general relativity. The euclidean space in which this two-dimensional Riemannian manifold (standing in for a three-dimensional Riemannian manifold) is embedded has no physical significance, it is merely a visual aid to help convey a quick impression of the kind of geometrical features we will we encounter.