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Spherically symmetric spacetime
 
 
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A spherically symmetric spacetime is one whose isometry group
Isometry group

In mathematics, the isometry group of a metric space is the Set of all isometry 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
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 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 Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
, especially the Schwarzschild solution.






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A spherically symmetric spacetime is one whose isometry group
Isometry group

In mathematics, the isometry group of a metric space is the Set of all isometry 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
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 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 Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
, 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....
 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 spacetime of the vacuum field equations must be stationary spacetime and asymptotically flat....
) 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...
 and asymptotically flat.

See also


  • Rotation group
    Rotation group

    In classical 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 functional composition....
    .


  • Spacetime symmetries
    Spacetime symmetries

    Spacetime symmetries refers to aspects of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important, for example, in simplifying solutions to many problems....
    .


  • Deriving the Schwarzschild solution
    Deriving the Schwarzschild solution

    The Schwarzschild solution is one of the simplest and most useful solutions of theEinstein field equations . It is worthwhile deriving this metric in some detail; the following is a reasonably rigorous derivation that is not always seen in the textbooks....
    .