7.3 The modified constructions

Independently of the results of the small-sphere calculations, Penrose claims that in the Schwarzschild spacetime the quasi-local mass expression should yield the same zero value on two-surfaces, contorted or not, which do not surround the black hole. (For the motivations and the arguments, see [422Jump To The Next Citation Point].) Thus, the original construction should be modified, and the negative results for the small spheres above strengthened this need. A much more detailed review of the various modifications is given by Tod in [516Jump To The Next Citation Point].

7.3.1 The ‘improved’ construction with the determinant

A careful analysis of the roots of the difficulties lead Penrose [422, 426] (see also [511, 313Jump To The Next Citation Point, 516Jump To The Next Citation Point]) to suggest the modified definition for the kinematical twistor

∮ A ′ Z αW β := --i-- ηλA ωBRABcd, (7.16 ) αβ 8 πG 𝒮
where η is a constant multiple of the determinant ν in Eq. (7.7View Equation). Since on noncontorted two-surfaces the determinant ν is constant, for such surfaces ′ Aα β reduces to A αβ, and hence, all the nice properties proven for the original construction on noncontorted two-surfaces are shared by A ′αβ. The quasi-local mass calculated from Eq. (7.16View Equation) for small spheres (in fact, for small ellipsoids [313]) in vacuum is vanishing in the fifth order. Thus, apparently, the difficulties have been resolved. However, as Woodhouse pointed out, there is an essential ambiguity in the (nonvanishing, sixth-order) quasi-local mass [560]. In fact, the structure of the modified kinematical twistor has the form (7.15View Equation) with vanishing ′ P A B and ′ PAB but with nonvanishing λAB in the fifth order. Then, in the quasi-local mass (in the leading sixth order) there will be a term coming from the (presumably nonvanishing) sixth-order part of P A′B and PAB ′ and the constant part of the Hermitian scalar product, and the fifth-order λAB and the still ambiguous 𝒪 (r)-order part of the Hermitian metric.

7.3.2 Modification through Tod’s expression

These anomalies lead Penrose to modify A ′αβ slightly [423]. This modified form is based on Tod’s form of the kinematical twistor:

∮ ′′ α β 1 A′B ′ [ ( √-- A)] [ (√ --B )] AαβZ W := 4πG-- ¯γ iΔA ′A η λ iΔB ′B ηω d𝒮. (7.17 ) 𝒮
The quasi-local mass on small spheres coming from A ′′αβ is positive [516].

7.3.3 Mason’s suggestions

A beautiful property of the original construction was its connection with the Hamiltonian formulation of the theory [357Jump To The Next Citation Point]. Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified constructions. Although the form of Eq. (7.17View Equation) is that of the integral of the Nester–Witten 2-form, and the spinor fields √ ηλA and iΔA ′A(√ ηλA ) could still be considered as the spinor constituents of the ‘quasi-Killing vectors’ of the two-surface 𝒮, their structure is not so simple, because the factor η itself depends on all four of the independent solutions of the two-surface twistor equation in a rather complicated way.

To have a simple Hamiltonian interpretation, Mason suggested further modifications [357, 358Jump To The Next Citation Point]. He considers the four solutions λA i, i = 1,...,4, of the two-surface twistor equations, and uses these solutions in the integral (7.14View Equation) of the Nester–Witten 2-form. Since H 𝒮 is a Hermitian bilinear form on the space of the spinor fields (see Section 8), he obtains 16 real quantities as the components of the 4 × 4 Hermitian matrix Eij := H 𝒮[λi,λ¯j ]. However, it is not clear how the four ‘quasi-translations’ of 𝒮 should be found among the 16 vector fields λAi ¯λAj′ (called ‘quasi-conformal Killing vectors’ of 𝒮) for which the corresponding quasi-local quantities could be considered as the components of the quasi-local energy-momentum. Nevertheless, this suggestion leads us to the next class of quasi-local quantities.

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