A note on Birkhoff's theorem · Anandh Krishna Murichatta

Birkhoff’s theorem is one of the most elegant results in classical general relativity. It asserts that the geometry outside any spherically symmetric matter distribution is uniquely fixed — regardless of whether the source is static, pulsating, or collapsing.

Statement#

Theorem (Birkhoff).

Every spherically symmetric solution of the vacuum Einstein equations $G_{\mu\nu} = 0$ is locally isometric to the Schwarzschild spacetime.

In other words, the exterior metric is

$$ \mathrm{d}s^2 = -\left(1 - \frac{2m}{r}\right)\mathrm{d}t^2 + \left(1 - \frac{2m}{r}\right)^{-1}\mathrm{d}r^2 + r^2\, \mathrm{d}\Omega^2 $$

for some mass parameter $m \geq 0$.Strictly, “locally isometric to a region of the maximally extended Schwarzschild spacetime” — inside the horizon the Killing field $\partial_t$ is spacelike, so the solution is homogeneous rather than static there.

Proof sketch#

Write the most general spherically symmetric metric as

$$ \begin{equation} \mathrm{d}s^2 = -e^{2\alpha(t,r)}\,\mathrm{d}t^2 + e^{2\beta(t,r)}\,\mathrm{d}r^2 + r^2\, \mathrm{d}\Omega^2, \label{eq:ansatz} \end{equation} $$

with $\alpha, \beta$ smooth functions to be determined. The vacuum equations $R_{\mu\nu} = 0$ reduce to a coupled PDE system. The $(t,r)$ component gives $\partial_t \beta = 0$, so $\beta = \beta(r)$. The $(r,r)$ and $(t,t)$ components then force $\partial_t\alpha$ to depend on $t$ alone, which a redefinition of $t$ removes — establishing staticity. What remains integrates to Schwarzschild.

Remark.

No time-independence is assumed at the outset: staticity is a consequence of spherical symmetry and the vacuum equations, as stated in .

Corollaries#

No monopole radiation. A radially pulsating star produces no gravitational waves: the exterior is Schwarzschild at all times, hence static.

Collapse. For a collapsing shell of dust, the exterior is already Schwarzschild before the shell crosses $r = 2m$; the event horizon forms there independently of the interior dynamics.

Corollary (The case $m = 0$).

A spherically symmetric vacuum spacetime with $m = 0$ is (a region of) Minkowski space — the relativistic analogue of Newton’s shell theorem.

References#

The classic sources are [Bir23], [Wal84, §6.1] and [HE73, §5].