Schwarzschild geometry

Spherical symmetry

A metric that is time-reversal and time-translation invariant is said to be static.

Lemma 1: a static, spherically symmetric metric tensor \(\bm{g}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F\) must be of the form

\[ \bm{g} = A(r) dr \otimes dr + r^2 (\sin^2 (\varphi) d\theta \otimes d\theta + d\varphi \otimes d\varphi) - B(r) dt \otimes dt, \]

for all \((r, \theta, \varphi, t) \in \mathbb{R}^4\) with \(A,B: r \mapsto A(r),B(r)\).

Proof:

Will be added later.

Reducing the determination of the metric to only two functions \(A\) and \(B\).

Exterior solution

Outside of the mass distribution the energy-momentum tensor vanishes, so we can impose \(\mathbf{W} = \mathbf{0}\). Then, by imposing the weak field limit we have the following.

Principle 1: a metric outside a static, spherically symmetric mass distribution is described by the Schwarzschild metric

\[ \bm{g} = \Big(1 - \frac{2 G M}{c^2 r}\Big)^{-1} dr \otimes dr + r^2 (\sin^2 (\varphi) d\theta \otimes d\theta + d\varphi \otimes d\varphi) - c^2 \Big(1 - \frac{2 G M}{c^2 r} \Big) dt \otimes dt, \]

for all \((r, \theta, \varphi, t) \in \mathbb{R}^4\) with \(G\) the gravitational constant and \(M\) the mass of the spherically symmetric mass distribution.

Derivation:

Will be added later.

Notice that for \(r_s = \frac{2 G M}{c^2}\) the metric with these coordinates is not defined. This radius is called the Schwarzschild radius.

Theorem 1 (Birkhoff's theorem): the Schwarzschild metric is the only spherically symmetric solution, outside a spherical mass distribution.

Proof:

Will be added later.

Note that static is automatically implied by spherical symmetry. An important consequence of the theorem is that a purely radially pulsating star cannot emit gravitational radiation, because outside of this star such gravitational radiation would amount to a time-dependent spherically symmetric spacetime geometry in (approximate) vacuum, which, according to the Birkhoff’s theorem, cannot be consistent with Einstein’s field equations.