In the literature there is another modification of the Hawking energy, due to Hayward [183]. His
suggestion is essentially
with the only difference that the integrands 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 main 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 [63
, 65] 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 sphere or in
the small sphere calculations, then
, and hence the extra term is, in fact,
.
Taking into account that near the future null infinity (see for example [338
]), it is
immediate from the remark on the asymptotic behaviour 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 non-vacuum and vacuum [63]. In non-vacuum it
gives the expected value
. However, in vacuum it is
, which is
negative.
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