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

Update
Although the Komar integral (and, in general, the linkage (3.15) for some ) does not satisfy our
general requirements discussed in Section 4.3.1, and 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 two-surface (see, e.g., [534]). This follows directly from the explicit form of Komar’s canonical Noether
current: . In fact, if and
are any two two-surfaces such that for some compact three-dimensional hypersurface on
which the energy-momentum tensor of the matter fields is vanishing and is a Killing vector, then
. (Note that, as we already stressed, the structure of the Noether current above dictates
that the numerical coefficient in the definition (3.15) of the linkage would have to be rather than
, i.e., the one that gives the correct value of angular momentum (rather than the mass) in Kerr
spacetime.) In particular, the Komar integral for the static Killing field in the Schwarzschild
spacetime is the mass parameter of the solution for any two-surface surrounding the black
hole, but it is zero if does not surround it. The explicit form of the current shows that,
for timelike Killing field , the small sphere expression of Komar’s quasi-local energy in
the first non-trivial order is , i.e., it does not reproduce the expected result
(4.9); moreover, in vacuum it always gives zero rather than, e.g., the Bel–Robinson ‘energy’ (see
Section 4.2.2).

Furthermore [510], the analogous integral in the Reissner–Nordström spacetime on a metric two-sphere
of radius is , which deviates from the generally accepted round-sphere value .
Similarly, in Einstein’s static universe for spheres of radius on a hypersurface, is
zero instead of the round sphere result , where is the energy density of the matter
and is the cosmological constant.

Accurate numerical calculations show that in stationary, axisymmetric asymptotically flat spacetimes
describing a black hole or a rigidly-rotating dust disc surrounded by a perfect fluid ring the Komar energy
of the black hole or the dust disc could be negative, even though the conditions of the positive energy
theorem hold [21]. Moreover, the central black hole’s event horizon can be distorted by the ring
so that the black hole’s Komar angular momentum is greater than the square of its Komar
energy [20].