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Concept
Vacuum solution (general relativity)
Summary
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. These are distinct from the electrovacuum solutions, which take into account the electromagnetic field in addition to the gravitational field. Vacuum solutions are also distinct from the lambdavacuum solutions, where the only term in the stress–energy tensor is the cosmological constant term (and thus, the lambdavacuums can be taken as cosmological models).
More generally, a vacuum region in a Lorentzian manifold is a region in which the Einstein tensor vanishes.
Vacuum solutions are a special case of the more general exact solutions in general relativity.
Equivalent conditions
It is a mathematical fact that the Einstein tensor vanishes if and only if the Ricci tensor vanishes. This follows from
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