Metric inside a sphere of uniform density?

Question

Is an exact solution to Einstein's Field Equations known for the interior of a sphere of uniform density (to approximate a star or planet, for example?)

Answer 1

Yes, the exact solution is known. The general spherically symmetric metric is $$g=-B(r)\mathrm{d}t^2+A(r)\mathrm{d}r^2+r^2\mathrm{d}\Omega^2.$$

The solution for $A(r)$ is $$A(r)=\left[1-\frac{2G\mathcal{M}(r)}{r}\right]^{-1},\quad\mathcal{M}(r)=\int^r \rho \,\mathrm{d}V=\int_0^r 4\pi r'^2\rho(r')\,\mathrm{d}r.$$ The solution for $B(r)$ is $$B(r)=\exp\left\{-\int_r^\infty \frac{2G}{r'^2}[\mathcal{M}(r')+4\pi r'^3 P(r')]A(r')\,\mathrm{d}r'\right\}.$$ In these equations $\rho$ is the density of the star and $P$ its pressure.

The derivation can be found in e.g.

S. Weinberg, Gravitation and Cosmology (1973), Sect. 11.1

R.M. Wald, General Relativity (1984), Sect. 6.2

N. Straumann, General Relativity (2013), Sect. 7.4