The Schwarzschild Solution

An important solution to the vacuum Einstein equations is given by the Schwarzschild metric, which in so-called Schwarzschild coordinates.

Schwarzschild Spacetime

The Schwarzschild spacetime is a static, spherically symmetric vacuum solution with , where in the standard global coordinates , the Schwarzschild metric takes the form where is the gravitational constant, the mass of the gravitating body, and the speed of light. For simplicity, we omit the physical constants and write

Eddington–Finkelstein Coordinates

The Schwarzschild spacetime is isometric to under a coordinate system , called the Eddington–Finkelstein coordinates, in which the metric takes the form where .

Spacelike Submanifold

A three dimensional submanifold of a spacetime is called a spacelike submanifold if the induced metric on is positive definite, i.e. is a Riemannian metric. Equivalently, this means for all , the vectors in which are orthogonal to the tangent space are causal.

Cauchy Development

The future (past) Cauchy development of a subset of a spacetime is the set of all points in that can be reached by a future (past) directed causal (i.e. non-spacelike) curve starting from . The future and past Cauchy development is denoted by and , respectively.

Cauchy Surface

If is a spacelike submanifold such that , then is called a Cauchy surface of the spacetime .

e.g. In Minkowski spacetime, the constant time surfaces are Cauchy surfaces.

Birkhoff’s Theorem

Lemma

For a spherically symmetric 4 dimensional Lorentzian manifold, there exists a coordinate such that the metric has the form with , where is the standard metric on a 2-sphere.

Proof See Willem’s proof.

Birkhoff's Theorem

Any spherically symmetric, asymptotically flat, vacuum spacetime is locally isometric to the Schwarzchild spacetime.

Proof Using the coordinate form of the metric from the above lemma, we compute the Ricci tensor components: with all other components vanishing. In vacuum, Einstein’s field equations reduce to so . Consider , yielding If , then , a contradiction. Hence , and we obtain implying for some function . Since , we solve for : Substituting into the metric yields:Introduce new coordinates defined by Then and , so the metric becomes with . The metric retains the same functional form, so we reuse the Ricci components. In particular, gives which rearranges to Integrating yields where depends only on . Thus, Hence the metric becomes which is the Schwarzschild metric in Eddington–Finkelstein coordinates.

References

• Willem van Oosterhout, Birkhoff’s theorem in general relativity