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.