General relativity MOC

Einstein field equation

The Einstein field equation is the extremal to the Einstein-Hilbert action. We have

where is the Einstein tensor, is the Stress-energy tensor, is the Cosmological constant, and

is the Einstein gravitational constant.

Proof

proof

Einstein field equation in vacuum

In a vacuum , so the Einstein field equation becomes . On a manifold of dimension this becomes

Proof

We have

Contracting with , we have

so the Einstein field equation becomes

giving the claimed equation.

Thus solutions to the Einstein field equation in a vacuum for arbitrary are precisely those for which the Ricci curvature is proportional to the metric tensor. Such a manifold is called an Einstein manifold. In particular, for , we see the solution is precisely a Ricci flat metric.

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