Vacuum Solution (general Relativity) - Equivalent Conditions

Equivalent Conditions

It is a mathematical fact that the Einstein tensor vanishes if and only if the Ricci tensor vanishes. This follows from the fact that these two second rank tensors stand in a kind of dual relationship; they are the trace reverse of each other:

where the traces are .

A third equivalent condition follows from the Ricci decomposition of the Riemann curvature tensor as a sum of the Weyl curvature tensor plus terms built out of the Ricci tensor: the Weyl and Riemann tensors agree, in some region if and only if it is a vacuum region.

Read more about this topic:  Vacuum Solution (general Relativity)

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