Everything about Spherically Symmetric Spacetime totally explained
A
spherically symmetric spacetime is one 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.
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