As in the case of the Schwarzschild solution for empty space with an isotropic static gravitational field, the case of a static gravitating spherical body of constant density can also be found exactly, as was first done by Schwarzschild (1916).Let M be the total mass, R the radius, the density, the mass inside radius r,P the pressure,andthe gravitational potential, then




For a spherically symmetric metric,


With constant,

and (6) becomes

Integrating,

Expanding the denominator of the left side,

Using the integral

(for q < 0) with


so the indefinite integral of the left side is


and the left side of (9) becomes



Use the integral

with


so solve the right side of (9),



Equating (15) and (19) gives


Solving for P,

Using

and solving

For spherical symmetry, the metric coefficients are then


(Weinberg 1972, p. 331).
According to (24),when



which is unphysical, so it must be the case that