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Vacuum solution (general relativity)

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Lorentzian manifold with vanishing Einstein tensor
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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

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:

G a b = R a b − R 2 g a b , R a b = G a b − G 2 g a b {\displaystyle G_{ab}=R_{ab}-{\frac {R}{2}}\,g_{ab},\;\;R_{ab}=G_{ab}-{\frac {G}{2}}\,g_{ab}}

where the traces are R = R a a , G = G a a = − R {\displaystyle R={R^{a}}_{a},\;\;G={G^{a}}_{a}=-R} .

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, R a b c d = C a b c d {\displaystyle R_{abcd}=C_{abcd}} , in some region if and only if it is a vacuum region.

Gravitational energy

Since T a b = 0 {\displaystyle T^{ab}=0} 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", 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

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

References

  1. Markus Pössel (2007), "The gravity of gravity", Einstein Online, Max Planck Institute for Gravitational Physics
  2. Beck, Guido (1925-12-01). "Zur Theorie binärer Gravitationsfelder". Zeitschrift für Physik (in German). 33 (1): 713–728. doi:10.1007/BF01328358. ISSN 0044-3328.

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