Lambdavacuum solution
In general relativity, a lambdavacuum solution is an exact solution to the Einstein field equation in which the only term in the stress-energy tensor is a cosmological constant term. This can be interpreted physically as a kind of classical approximation to a nonzero vacuum energy.
Terminological note: this article concerns a standard concept, but there is apparently no standard term to denote this concept, so we have attempted to supply one for the benefit of Wikipedia.
Mathematical definition
Eigenvalues
Relation with Einstein manifolds
The definition of a lambdavacuum solution makes mathematical sense irrespective of any physical interpretation, and lambdavacuums are in fact a special case of a concept which is studied by pure mathematicians.
Einstein manifolds are Riemannian manifolds in which the Ricci tensor is proportional (by some constant, not otherwise specified) to the metric tensor. Such manifolds may have the wrong signature to admit a spacetime interpretation in general relativity, and may have the wrong dimension as well. But the Lorentzian manifolds which are also Einstein manifolds are precisely the Lambdavacuum solutions.
Examples
Noteworthy examples of lambdavacuum solutions include:
- de Sitter lambdavacuum, often referred to as the dS cosmological model,
- anti-de Sitter lambdavacuum, often referred to as the AdS cosmological model,
- Nariai lambdavacuum; this is the only solution in general relativity, other than the Bertotti-Robinson electrovacuum, which has a Cartesian product structure.