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
].
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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