6.3 The Hayward energy

We saw that EH can be nonzero, even in the Minkowski spacetime. This may motivate us to consider the following expression
∘ ---------( ∮ ) Area(𝒮-) -1- ′ ′ ′ I (𝒮 ) := 16 πG2 1 + 4π (2ρρ − σ σ − σ¯¯σ )d𝒮 ∘ --------- ∮ 𝒮 Area(𝒮-)-1- ( ¯ ) = 16 πG2 4π − ψ2 − ψ2′ + 2 ϕ11 + 2 Λ d 𝒮. (6.5 ) 𝒮
(Thus, the integrand is 14(F + F¯), where F is given by Eq. (4.4View Equation).) 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 I (𝒮), 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 [248Jump To The Next Citation Point]. His suggestion is essentially I(𝒮 ) with the only difference being that the integrands of Eq. (6.5View Equation) above contain an additional term, namely the square of the anholonomicity − ωa ωa (see Sections 4.1.8 and 11.2.1). However, we saw that ω a 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 Qa and Q&tidle;a discussed in Section 4.1.2.) Although the expression for the Hayward energy in terms of the GHP spin coefficients given in [81Jump To The Next Citation Point, 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 a ¯′ ¯ ′ − ωaω = 2(β − β )(β − β ). If, however, the GHP spinor dyad is fixed, as in the large or small sphere calculations, then β − β¯′ = τ = − ¯τ ′, and hence, the extra term is, in fact, the gauge invariant 2τ ¯τ.

Taking into account that τ = 𝒪 (r−2) near the future null infinity (see, e.g., [455Jump To The Next Citation Point]), it is obvious from the remark on the asymptotic behavior of I(𝒮 ) 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 4π 3 a b 3 r Tabt t. However, in vacuum it is -8- 5 ab cd − 45Gr Tabcdtt t t, which is negative.

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