Interior Schwarzschild metric

The Interior Schwarzschild metric describes a static spherical perfect fluid of constant energy density. Use geometrized units $G=c=1$, mass $M>0$, surface areal radius $R$ and $\rho =3M/(4\pi R^3)$. For $0\le r\le R$, defineThe metric and pressure areAt $r=R$, $p=0$ and the lapse equals $s$, matching the exterior Schwarzschild metric. The mass profile is $m(r)=M{r}^3/R^3$. The lapse can also be recovered from hydrostatic balance $p^{\prime }=-(\rho +p)N^{\prime }/N$: constant $\rho$ makes $(\rho +p)N$ constant, and the surface condition fixes $N=(3s-q)/2$.

At the centre,Finite nonnegative central pressure requires $s>1/3$, equivalently $R>9M/4$. At the equality limit the central pressure diverges even though $R>2M$. This makes the model a useful demonstration of relativistic pressure support and stellar compactness. Constant density is an idealization, rather than a realistic universal equation of state for dense matter.

The Interior Schwarzschild metric has central pressureIt diverges when $M/R=4/9$, the limiting compactness in Buchdahl's theorem.

The dominant energy condition for the Interior Schwarzschild metric reduces to its largest pressure satisfying $p_c\le \rho$. Substitution of the exact central pressure gives $\sqrt{1-2M/R}\ge 1/2$, hence $M/R\le 3/8$. Equality has finite central pressure and is allowed by the non-strict condition.