Spherically symmetric spacetime | Wikipedia audio article
Channel: Subhajit Sahu
Date: 29/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Spherically_symmetric_spacetime


00:01:57 See also



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SUMMARY
=======
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
      
    
    {\displaystyle 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
      
    
    {\displaystyle \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.