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.
Contents
- Mathematical definition
- Physical interpretation
- Einstein tensor
- Eigenvalues
- Relation with Einstein manifolds
- Examples
- References
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
The Einstein field equation is often written as
with a so-called cosmological constant term
is a lambdavacuum. An equivalent formulation in terms of the Ricci tensor is
Physical interpretation
A nonzero cosmological constant term can be interpreted in terms of a nonzero vacuum energy. There are two cases:
The idea of the vacuum having an energy density might seem outrageous, but this does make sense in quantum field theory. Indeed, nonzero vacuum energies can even be experimentally verified in the Casimir effect.
Einstein tensor
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
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.
Eigenvalues
The characteristic polynomial of the Einstein tensor of a lambdavacuum must have the form
Using Newton's identities, this condition can be re-expressed in terms of the traces of the powers of the Einstein tensor as
where
are the traces of the powers of the linear operator corresponding to the Einstein tensor, which has second rank.
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 metric 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 individual examples of lambdavacuum solutions include: