This question is about Schwarzschild black holes. (a) The Schwarzschild solution

Question

This question is about Schwarzschild black holes. (a) The Schwarzschild solution to the Einstein field equations is spherically symmetric around a point mass M. It is also asymptotically flat, stationary and static. (i) Explain what is meant by 'asymptotically flat' and demonstrate that the Schwarzschild metric exhibits this property. (ii) Explain what is meant by 'stationary' and explain why the Schwarzschild metric exhibits this property. (iii) Explain what is meant by 'static' and explain why the Schwarzschild metric exhibits this property.

Explanation / Answer

(i) Asymptotically flast means that as r->, the space time manifold resembles that of Minkowski space. Minkowski space is a combination of 3-dimensional Euclidean space and time into a 4-dimensional manifold. The spacetime interval between any two events is independent of the inertial frame of reference. The solution for these spherically symmetric solution follow the Minkowski space in r-> limit.

(ii) Schwarzschild metric can also be derived considering the point mass is stationary, both in rotational as well as translational degree of freedom. The Schwarzschild metric follow this property due to the time dilation and properties of inertial frame of reference.

(iii) The static means that any pulsating star which is spherically symmetric cannot generate gravitational waves because the region exterior to the star must remain static. In general theory of relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution (the spacetime outside of a spherical, nonrotating, gravitating body) must be given by the Schwarzschild metric.

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