This works out very nicely for the non-rotating case, where we can match the Schwarzschild vacuum exterior to a Schwarzschild fluid interior, and indeed to more general static spherically symmetric perfect fluid solutions.
22.
According to the Einstein field equation, the gravitational field is locally coupled not only to the distribution of non-gravitational mass-energy, but also to the distribution of momentum and perfect fluid ).
23.
It can be shown that the G�del solution is, up to local isometry, the " only " perfect fluid solution of the Einstein field equation admitting a five-dimensional Lie algebra of Killing vectors.
24.
Given this success, it may come as a nasty shock that it seems to be very difficult, mathematically speaking, to construct " rotating " stellar models in which a perfect fluid interior is matched to an asymptotically flat vacuum exterior.
25.
But in working with general relativity, it turns out to be very useful to admit solutions which are not everywhere smooth; examples include many solutions created by matching a perfect fluid interior solution to a vacuum exterior solution, and impulsive plane waves.
26.
His 1943 theory of gravitation is also puzzling, since Birkhoff knew ( but didn't seem to mind ) that his theory allows as sources only matter which is a perfect fluid in which the speed of sound must equal the speed of light.
27.
In astrophysics, fluid solutions are often employed as "'stellar models "'. ( It might help to think of a perfect gas as a special case of a perfect fluid . ) In cosmology, fluid solutions are often used as cosmological models.
28.
There is no difficulty whatsoever in obtaining models of " static " spherically symmetric stellar models, in which a perfect fluid interior is matched across a spherical surface, the surface of the star, to a vacuum exterior which is in fact a region of the Schwarzschild vacuum.
29.
The figure shows a perfect fluid " interior region " ( gold ) matched ( with continuous metric and extrinsic curvature tensor ) across a spherical surface ( represented here as a circle ) to a vacuum " exterior region " ( khaki ), which is a portion of the Schwarzschild vacuum solution.
30.
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.
