Spherically symmetric spacetime
Encyclopedia
A spherically symmetric spacetime is a spacetime
whose isometry group
contains a subgroup which is isomorphic to the (rotation) group
and the orbits of this group are 2-dimensional spheres (2-spheres). The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is "invariant under rotations". The spacetime metric induces a metric on each orbit 2-sphere (and this induced metric must be a multiple of the metric of a 2-sphere).
Spherical symmetry is a characteristic feature of many solutions of Einstein's field equations of general relativity
, especially the Schwarzschild solution. A spherically symmetric spacetime can be characterised in another way, namely, by using the notion of Killing vector fields, which, in a very precise sense, preserve the metric. The isometries referred to above are actually local flow diffeomorphisms
of Killing vector fields and thus generate these vector fields. For a spherically symmetric spacetime
, there are precisely 3 rotational Killing vector fields. Stated in another way, the dimension of the Killing algebra is 3
.
It is known (see Birkhoff's theorem
) that any spherically symmetric solution of the vacuum field equations is necessarily isometric to a subset of the maximally extended Schwarzschild solution. This means that the exterior region around a spherically symmetric gravitating object must be static
and asymptotically flat.
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
whose isometry group
Isometry group
In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
contains a subgroup which is isomorphic to the (rotation) group
and the orbits of this group are 2-dimensional spheres (2-spheres). The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is "invariant under rotations". The spacetime metric induces a metric on each orbit 2-sphere (and this induced metric must be a multiple of the metric of a 2-sphere).Spherical symmetry is a characteristic feature of many solutions of Einstein's field equations of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, especially the Schwarzschild solution. A spherically symmetric spacetime can be characterised in another way, namely, by using the notion of Killing vector fields, which, in a very precise sense, preserve the metric. The isometries referred to above are actually local flow diffeomorphisms
Local diffeomorphism
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a function between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below....
of Killing vector fields and thus generate these vector fields. For a spherically symmetric spacetime
, there are precisely 3 rotational Killing vector fields. Stated in another way, the dimension of the Killing algebra is 3.It is known (see Birkhoff's theorem
Birkhoff's theorem (relativity)
In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric....
) that any spherically symmetric solution of the vacuum field equations is necessarily isometric to a subset of the maximally extended Schwarzschild solution. This means that the exterior region around a spherically symmetric gravitating object must be static
Static
Static has several meanings:* Static electricity, a net charge of an object** The triboelectric effect, e.g. from shoes rubbing carpet* White noise, a random signal with a flat power spectral density** Noise , a type of radio interference...
and asymptotically flat.
See also
- Rotation groupRotation groupIn mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
.
- Spacetime symmetriesSpacetime symmetriesSpacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field...
.
- Deriving the Schwarzschild solutionDeriving the Schwarzschild solutionThe Schwarzschild solution is one of the simplest and most useful solutions of theEinstein field equations . It describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object...
.