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| In [[general relativity]], a '''lambdavacuum solution''' is an [[exact solutions in general relativity|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]].
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| ''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]].
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| ==Mathematical definition==
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| The Einstein field equation is often written, with a so-called ''cosmological constant term'', as | |
| :<math> G^{ab} + \Lambda \, g^{ab} = 8 \pi \, T^{ab}</math>
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| However, it is possible to move the extra term to the right hand side and absorb it into the [[stress-energy tensor]], so that the cosmological constant term becomes just another contribution to the stress-energy tensor. When other contributions vanish,
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| :<math> G^{ab} = -\Lambda \, g^{ab} </math>
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| we have a lambdavacuum. Equivalently, we can write this, in terms of the [[Ricci tensor]], in the form
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| <math> R^{ab} = \left( R/2 - \Lambda \right) \, g^{ab}</math>
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| ==Physical interpretation==
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| A nonzero cosmological constant term can be interpreted in terms of a nonzero [[vacuum energy]]. There are two cases:
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| *<math>\Lambda > 0</math>: positive vacuum energy density and negative vacuum pressure (isotropic suction), as in [[de Sitter space]],
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| *<math>\Lambda < 0</math>: negative vacuum energy density and positive vacuum pressure, as in [[anti-de Sitter space]].
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| 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]].
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| ==Einstein tensor==
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| The components of a tensor computed with respect to a [[frame fields in general relativity|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
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| :<math> \vec{e}_0, \; \vec{e}_1, \; \vec{e}_2, \; \vec{e}_3 </math>
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| Here, the first is a [[timelike]] unit vector field and the others are [[spacelike]] unit vector fields, and <math> \vec{e}_0</math> is everywhere orthogonal to the world lines of a family of observers (not necessarily inertial observers).
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| 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
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| :<math> G^{\hat{a}\hat{b}} = -\Lambda \, \left[ \begin{matrix} -1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix} \right] </math>
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| 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]].
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| ==Eigenvalues==
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| The [[characteristic polynomial]] of the Einstein tensor of a lambdavacuum must have the form
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| :<math> \chi(\zeta) = \left( \zeta + \Lambda \right)^4 </math>
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| Using [[Newton's identities]], this condition can be re-expressed in terms of the [[trace (linear algebra)|trace]]s of the powers of the Einstein tensor as
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| :<math> t_2 = t_1^2/4, \; t_3 = t_1^3/16, \; t_4 = t_1^4/64 </math> | |
| where
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| :<math> t_1 = {G^a}_a, \; t_2 = {G^a}_b \, {G^b}_a, \; t_3 = {G^a}_b \, {G^b}_c \, {G^c}_a, \; t_4 = {G^a}_b \, {G^b}_c \, {G^c}_d \, {G^d}_a</math>
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| are the traces of the powers of the linear operator corresponding to the Einstein tensor, which has second rank.
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| ==Relation with Einstein manifolds==
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| 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.
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| [[Einstein manifold]]s are [[Riemannian manifold]]s 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.
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| ==Examples==
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| Noteworthy individual examples of lambdavacuum solutions include:
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| *[[de Sitter space|de Sitter lambdavacuum]], often referred to as the '''dS cosmological model''',
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| *[[anti-de Sitter space|anti-de Sitter lambdavacuum]], often referred to as the ''AdS cosmological model'',
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| *[[de Sitter–Schwarzschild metric|Schwarzschild–dS lambdavacuum]], which models a spherically symmetric massive object immersed in a de Sitter universe (and likewise for AdS),
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| *[[Kerr–de Sitter metric|Kerr–dS lambdvacuum]], the rotating generalization of the latter,
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| *[[Nariai spacetime|Nariai lambdavacuum]]; this is the only solution in general relativity, other than the [[Bertotti–Robinson electrovacuum]], which has a Cartesian product structure.
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| ==See also== | |
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| *[[Exact solutions in general relativity]]
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| {{DEFAULTSORT:Lambdavacuum Solution}}
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| [[Category:Exact solutions in general relativity]]
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