= Interior Schwarzschild metric
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= Interior Schwarzschild solution
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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 $0\leq r\leq R$, define
$$
s=\sqrt{1-\frac{2M}{R}},\qquad q(r)=\sqrt{1-\frac{2Mr^2}{R^3}}.
$$
The metric and <pressure> are
$$
ds^2=-\frac{[3s-q(r)]^2}{4}\,dt^2+\frac{dr^2}{q(r)^2}+r^2d\Omega^2,\qquad
p(r)=\rho\frac{q(r)-s}{3s-q(r)}.
$$
At $r=R$, $p=0$ and the lapse equals $s$, matching the exterior <Schwarzschild metric>. The mass profile is $m(r)=Mr^3/R^3$. The lapse can also be recovered from hydrostatic balance $p'=-(\rho+p)N'/N$: constant $\rho$ makes $(\rho+p)N$ constant, and the surface condition fixes $N=(3s-q)/2$.
At the centre,
$$
p_c=\rho\frac{1-s}{3s-1}.
$$
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.
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