In the literature there is another modification of Hawking energy, due to Hayward [248]. His suggestion
is essentially
with the only difference being that the integrands of Eq. (6.5
) above contain an
additional term, namely the square of the anholonomicity
(see Sections 4.1.8 and 11.2.1).
However, we saw that
is a boost-gauge–dependent quantity, thus, the physical significance of this
suggestion is questionable unless a natural boost gauge choice, e.g., in the form of a preferred
foliation, is made. (Such a boost gauge might be that given by the mean extrinsic curvature
vector
and
discussed in Section 4.1.2.) Although the expression for the Hayward
energy in terms of the GHP spin coefficients given in [81
, 83] seems to be gauge invariant, this
is due only to an implicit gauge choice. The correct, general GHP form of the extra term is
. If, however, the GHP spinor dyad is fixed, as in the large or small sphere
calculations, then
, and hence, the extra term is, in fact, the gauge invariant
.
Taking into account that near the future null infinity (see, e.g., [455
]), it is obvious from
the remark on the asymptotic behavior of
above that the Hayward energy tends to the
Newman-Unti, instead of the Bondi–Sachs, energy at the future null infinity. The Hayward energy
has been calculated for small spheres both in nonvacuum and vacuum [81]. In nonvacuum it
gives the expected value
. However, in vacuum it is
, which is
negative.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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