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Vacuum solution (general relativity)
 
 
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In general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
, a vacuum solution is a Lorentzian manifold whose Einstein tensor
Einstein tensor

The Einstein tensor expresses spacetime curvature in the Einstein field equations for gravitation in the theory of general relativity. It is sometimes called the trace-reversed Ricci tensor....
 vanishes identically. According to the Einstein field equation, this means that the stress-energy tensor
Stress-energy tensor

The stress-energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress of Newtonian physics....
 also vanishes identically, so that no matter or non-gravitational fields are present.

More generally, a vacuum region in a Lorentzian manifold is a region in which the Einstein tensor vanishes.

Equivalent conditions
It is a mathematical fact that the Einstein tensor vanishes if and only if the Ricci tensor vanishes.






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In general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
, a vacuum solution is a Lorentzian manifold whose Einstein tensor
Einstein tensor

The Einstein tensor expresses spacetime curvature in the Einstein field equations for gravitation in the theory of general relativity. It is sometimes called the trace-reversed Ricci tensor....
 vanishes identically. According to the Einstein field equation, this means that the stress-energy tensor
Stress-energy tensor

The stress-energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress of Newtonian physics....
 also vanishes identically, so that no matter or non-gravitational fields are present.

More generally, a vacuum region in a Lorentzian manifold is a region in which the Einstein tensor vanishes.

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: where the trace
Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
s are .

A third equivalent condition follows from the Ricci decomposition
Ricci decomposition

In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties....
 of the Riemann curvature tensor
Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann?Christoffel tensor is the most standard way to express curvature of Riemannian manifolds....
 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


Since in a vacuum region, it might seem that according to general relativity, vacuum regions must contain no energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
. 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 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:

  • Minkowski spacetime (which describes empty space with no cosmological constant
    Cosmological constant

    In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a Einstein's universe....
    )
  • Milne model
    Milne model

    The Milne model was a special relativity physical cosmology Scientific modeling proposed by Edward Arthur Milne. It is a special case of the FLRW metric in the limit of zero mass density, and it obeys the cosmological principle....
     (which is a model developed by E. A. Milne describing an empty universe which has no curvature)
  • Schwarzschild vacuum (which describes the spacetime geometry around a spherical mass),
  • Kerr vacuum (which describes the geometry around a rotating object),
  • Taub-NUT vacuum
    Taub-NUT vacuum

    The Taub-NUT vacuum is an exact solutions in general relativity to Einstein's equations, a model universe formulated in the framework of general relativity that is Homogeneity but anisotropic....
     (a famous counterexample describing the exterior gravitational field of an isolated object with strange properties),
  • Kerns/Wild vacuum (a Schwarzschild object immersed in an ambient "almost uniform" gravitational field),
  • double Kerr vacuum (two Kerr objects sharing the same axis of rotation, but held apart by unphysical zero active mass "cables" going out to suspension points infinitely removed),
  • Penrose-Khan vacuum (a simple colliding plane wave model),
  • Oszváth-Schücking vacuum (the circularly polarized sinusoidal gravitational wave, another famous counterexample).
  • Kasner metric
    Kasner metric

    The Kasner metric is an Exact solutions in general relativity to Einstein's theory of general relativity. It describes an anisotropic universe without matter ....


These all belong to one or more general families of solutions:

  • the Weyl vacuums (the family of all static vacuum solutions),
  • the Beck vacuums (the family of all cylindrically symmetric nonrotating vacuum solutions),
  • the Ernst vacuums (the family of all stationary axisymmetric vacuum solutions),
  • the Ehlers vacuums (the family of all cylindrically symmetric vacuum solutions),
  • the Szekeres vacuums (the family of all colliding gravitational plane wave models),
  • the Gowdy vacuums (cosmological models constructed using gravitational waves),


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 wave
Gravitational plane wave

In general relativity, a gravitational plane wave is a special class of a vacuum pp-wave spacetime, and may be defined in terms of Brinkmann coordinates by...
s.

See also


  • vacuum solution
    Vacuum solution

    A vacuum solution is a solution of a field equation in which the sources of the field are taken to be identically zero.For example, in Maxwell's theory of electromagnetism, a vacuum solution would represent the electromagnetic field in a region of space where there are no electromagnetic sources , i.e....
    , an article about vacuum solutions in physics in general,
  • lambdavacuum solution
    Lambdavacuum solution

    In general relativity, a lambdavacuum solution is an exact solutions in general relativity to the Einstein field equation in which the only term in the stress-energy tensor is a cosmological constant term....
    , an article about a significant generalization of the notion of a vacuum solution in general relativity,
  • exact solutions in general relativity
    Exact solutions in general relativity

    In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor which are taken to model states of ordinary matter, such as a fluid, or classical classical field theory such as the electromagnetic field....
    , an article about all kinds of exact solutions to the Einstein field equation.