Lorentzian manifold with vanishing Einstein tensor
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
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It is a mathematical fact that the Einstein tensor vanishes if and only if the
Ricci tensor
vanishes. This follows from the fact that these two second rank tensors stand in a kind of dual relationship; they are the
trace reverse
of each other:
![{\displaystyle G_{ab}=R_{ab}-{\frac {R}{2}}\,g_{ab},\;\;R_{ab}=G_{ab}-{\frac {G}{2}}\,g_{ab}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1133bff5d201bedf7574f21221ed7f0bbbc90c6d)
where the
traces
are
.
A third equivalent condition follows from the
Ricci decomposition
of the
Riemann curvature tensor
as a sum of the
Weyl curvature tensor
plus terms built out of the Ricci tensor: the Weyl and Riemann tensors agree,
, in some region if and only if it is a vacuum region.
Gravitational energy
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Since
in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no
energy
. But the gravitational field can do
work
, so we must expect the gravitational field itself to possess energy, and it does. However, determining the precise location of this gravitational field energy is technically problematical in general relativity, by its very nature of the clean separation into a universal gravitational interaction and "all the rest".
The fact that the gravitational field itself possesses energy yields a way to understand the nonlinearity of the Einstein field equation: this gravitational field energy itself produces more gravity. (This is described as "the gravity of gravity",
[1]
or by saying that "gravity gravitates".) This means that the gravitational field outside the Sun is a bit
stronger
according to general relativity than it is according to Newton's theory.
Examples
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Well-known examples of explicit vacuum solutions include:
These all belong to one or more general families of solutions:
Several of the families mentioned here, members of which are obtained by solving an appropriate linear or nonlinear, real or complex partial differential equation, turn out to be very closely related, in perhaps surprising ways.
In addition to these, we also have the vacuum
pp-wave spacetimes
, which include the
gravitational plane waves
.
See also
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References
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Sources
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