general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

, a vacuum solution is a

Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

whose

Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein fie ...

vanishes identically. According to the

Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the for ...

, this means that the

stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the str ...

also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the

electrovacuum solution In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (c ...

s, which take into account the

electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...

in addition to the gravitational field. Vacuum solutions are also distinct from the

lambdavacuum solution 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 ...

s, where the only term in the stress–energy tensor is the

cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...

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 In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical sh ...

Equivalent conditions

It is a mathematical fact that the Einstein tensor vanishes if and only if the

Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...

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_ = R_ - \frac \, g_, \; \; R_ = G_ - \frac \, g_

where the traces are

R = _a, \; \; G = _a = -R

. A third equivalent condition follows from the

Ricci decomposition In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. ...

of the

Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the 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,

R_=C_

, in some region if and only if it is a vacuum region.

Gravitational energy

Since

T^ = 0

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

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...

. But the gravitational field can do

work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal t ...

, 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 In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...

(which describes empty space with no

) *

Milne model The Milne model was a special-relativistic cosmological model proposed by Edward Arthur Milne in 1935. It is mathematically equivalent to a special case of the FLRW model in the limit of zero energy density and it obeys the cosmological prin ...

(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 (a famous counterexample describing the exterior gravitational field of an isolated object with strange properties), * Kerns–Wild vacuum (Robert M. Kerns and Walter J. Wild 1982) (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), * Khan–Penrose vacuum (K. A. Khan and

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...

1971) (a simple colliding plane wave model), * Oszváth–Schücking vacuum (the circularly polarized sinusoidal gravitational wave, another famous counterexample). * Kasner metric (An anisotropic solution, used to study gravitational chaos in three or more dimensions). These all belong to one or more general families of solutions: *the Weyl vacua (

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...

) (the family of all static vacuum solutions), *the Beck vacua (

Guido Beck Guido Beck (29 August 1903 in Liberec – 21 October 1988 in Rio de Janeiro) was an Argentinian physicist of German Bohemian origin. Biography Beck studied physics in Vienna and received his doctorate in 1925, under Hans Thirring. He worked in ...

1925) (the family of all cylindrically symmetric nonrotating vacuum solutions), *the Ernst vacua (Frederick J. Ernst 1968) (the family of all stationary axisymmetric vacuum solutions), *the Ehlers vacua (

Jürgen Ehlers Jürgen Ehlers (; 29 December 1929 – 20 May 2008) was a German physicist who contributed to the understanding of Albert Einstein's theory of general relativity. From graduate and postgraduate work in Pascual Jordan's relativity research gro ...

) (the family of all cylindrically symmetric vacuum solutions), *the Szekeres vacua (

George Szekeres George Szekeres AM FAA (; 29 May 1911 – 28 August 2005) was a Hungarian–Australian mathematician. Early years Szekeres was born in Budapest, Hungary, as Szekeres György and received his degree in chemistry at the Technical University of ...

) (the family of all colliding gravitational plane wave models), *the Gowdy vacua (Robert H. Gowdy) (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 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 ds^2= (u)(x^2-y^2)+2b(u)xyu^2+2dudv+dx^2+dy^2 Here, a(u), b(u) can be any smooth functions ...

References

* H. Stephani, ''et al.'',
Exact solutions of Einstein's field equations
(2003) Cambridge University Press, 690 pages. {{DEFAULTSORT:Vacuum Solution (General Relativity) Exact solutions in general relativity

Equivalent conditions

Gravitational energy

Examples

See also

References