Spherically symmetric spacetime

A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO(3) and the orbits of this group are 2-spheres (ordinary 2-dimensional spheres in 3-dimensional Euclidean space). 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 $M$, there are precisely 3 rotational Killing vector fields. Stated in another way, the dimension of the Killing algebra is 3

$\dim K(M)=3$.

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.

See also

• Spacetime symmetries
• Deriving the Schwarzschild solution

References

• Wald, Robert M. (1984). General Relativity. Chicago: University of Chicago Press. ISBN 0-226-87033-2. See Section 6.1 for a discussion of spherical symmetry.