In the literature there is another modification of Hawking energy, due to Hayward . 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., ), 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 . In nonvacuum it gives the expected value . However, in vacuum it is , which is negative.
Living Rev. Relativity 12, (2009), 4
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