In
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, a vacuum solution is a
Lorentzian manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...
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 field e ...
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 Albert Einstein in 1915 in the ...
, 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 stress ...
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 ...
s, which take into account the
electromagnetic field
An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
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
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general rel ...
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 (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be a ...
.
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:
:
where the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album), by Nell
Other uses in arts and entertainment
* ...
s are
.
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. Th ...
of the
Riemann curvature tensor
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...
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 () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
. 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 ani ...
, 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:
*
Minkowski spacetime
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a s ...
(which describes empty space with no
cosmological constant
In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant,
is a coefficient that Albert Einstein initially added to his field equations of general rel ...
)
*
Milne model
The Milne model was a special-relativistic cosmological model of the universe 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 th ...
(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 gravitational mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a body, until the discovery of the atom and particle physics. It was found that different atoms and different elementary parti ...
"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, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
1971) (a simple
colliding plane wave
In physics, a collision is any event in which two or more bodies exert forces on each other in a relatively short time. Although the most common use of the word ''collision'' refers to incidents in which two or more objects collide with great for ...
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, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
) (the family of all static vacuum solutions),
*the
Beck vacua (
Guido Beck 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 group ...
) (the family of all cylindrically symmetric vacuum solutions),
*the
Szekeres vacua (
George Szekeres) (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 waves.
See also
*
Introduction to the mathematics of general relativity
The mathematics of general relativity is complicated. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be s ...
*
Topological defect
References
Sources
*
{{DEFAULTSORT:Vacuum Solution (General Relativity)
Exact solutions in general relativity