Vacuum solution (general relativity)

In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present.

More generally, a vacuum region in a Lorentzian manifold is a region in which the Einstein tensor vanishes.

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 ${\displaystyle R={R^{a}}_{a},\;\;G={G^{a}}_{a}=-R}$.

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, ${\displaystyle R_{abcd}=C_{abcd}}$, in some region if and only if it is a vacuum region.

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