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12.1 The Komar integral for spacetimes with Killing vectors

Although the Komar integral (and, in general, the linkage (16View Equation) for some a) does not satisfy our general requirements discussed in Section 4.3.1, and it does not always give the standard values in specific situations (see for example the ‘factor-of-two anomaly’ or the examples below), in the presence of a Killing vector the Komar integral, built from the Killing field, could be a very useful tool in practice. (For Killing fields the linkage LS[K] reduces to the Komar integral for any a.) One of its most important properties is that in vacuum LS [K] depends only on the homology class of the 2-surface (see for example [387Jump To The Next Citation Point]): If S and S' are any two 2-surfaces such that ' S - S = @S for some compact 3-dimensional hypersurface S on which the energy-momentum tensor of the matter fields is vanishing, then LS [K] = LS'[K]. In particular, the Komar integral for the static Killing field in the Schwarzschild spacetime is the mass parameter m of the solution for any 2-surface S surrounding the black hole, but it is zero if S does not.

On the other hand [371], the analogous integral in the Reissner-Nordström spacetime on a metric 2-sphere of radius r is m - e2/r, which deviates from the generally accepted round-sphere value m - e2/(2r). Similarly, in Einstein’s static universe for the spheres of radius r in a t = const. hypersurface LS [K] is zero instead of the round sphere result 4p 3 3 r[m + c/8pG], where m is the energy density of the matter and c is the cosmological constant.


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