### 12.1 The Komar integral for spacetimes with Killing vectors

Although the Komar integral (and, in general, the linkage (16) for some ) 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 reduces to the Komar integral
for any .) One of its most important properties is that in vacuum depends only
on the homology class of the 2-surface (see for example [387]): If and are any two
2-surfaces such that for some compact 3-dimensional hypersurface on which the
energy-momentum tensor of the matter fields is vanishing, then . In particular, the Komar
integral for the static Killing field in the Schwarzschild spacetime is the mass parameter
of the solution for any 2-surface surrounding the black hole, but it is zero if does
not.
On the other hand [371], the analogous integral in the Reissner-Nordström spacetime
on a metric 2-sphere of radius is , which deviates from the generally accepted
round-sphere value . Similarly, in Einstein’s static universe for the spheres of
radius in a hypersurface is zero instead of the round sphere result
, where is the energy density of the matter and is the cosmological
constant.