### 6.3 The Hayward energy

We saw that can be nonzero, even in the Minkowski spacetime. This may motivate us to consider
the following expression
(Thus, the integrand is , where is given by Eq. (4.4).) By the Gauss equation, this is zero
in flat spacetime, furthermore, it is not difficult to see that its limit at spatial infinity is still the ADM
energy. However, using the second expression of , one can see that its limit at the future null infinity
is the Newman–Unti, rather than the Bondi–Sachs energy.
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.