Lambdavacuum solution

Einstein field equation 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. These are discussed here as distinct from the vacuum solutions in which the cosmological constant is vanishing. 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.

The components of a tensor computed with respect to a frame field rather than the coordinate basis are often called physical components, because these are the components which can (in principle) be measured by an observer. A frame consists of four unit vector fields

Here, the first is a timelike unit vector field and the others are spacelike unit vector fields, and is everywhere orthogonal to the world lines of a family of observers (not necessarily inertial observers). Remarkably, in the case of lambdavacuum, all observers measure the same energy density and the same (isotropic) pressure. That is, the Einstein tensor takes the form

Saying that this tensor takes the same form for all observers is the same as saying that the isotropy group of a lambdavacuum is SO(1,3), the full Lorentz group.

The definition of a lambdavacuum solution makes sense mathematically irrespective of any physical interpretation, and lambdavacuums are a special case of a concept that is studied by pure mathematicians. Einstein manifolds are pseudo Riemannian manifolds in which the Ricci tensor is proportional to the metric tensor. The Lorentzian manifolds that are also Einstein manifolds are precisely the lambdavacuum solutions.