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7.3 The modified constructions

Independently of the results of the small sphere calculations, Penrose claimed that in the Schwarzschild spacetime the quasi-local mass expression should yield the same zero value on 2-surfaces, contorted or not, which do not surround the black hole. (For the motivations and the arguments, see [309Jump 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 [377Jump 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 [309313] (see also [372235Jump To The Next Citation Point377Jump To The Next Citation Point]) to suggest the modified definition for the kinematical twistor

gf A' ZaW b := --i-- jcAwBRABcd, (57) ab 8pG S
where j is a constant multiple of the determinant n in Equation (49View Equation). Since on non-contorted 2-surfaces the determinant n is constant, for such surfaces ' Aab reduces to Aab, and hence all the nice properties proven for the original construction on non-contorted 2-surfaces are shared by A'ab too. The quasi-local mass calculated from Equation (57View Equation) for small spheres (in fact, for small ellipsoids [235]) 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 (non-vanishing, sixth order) quasi-local mass [398]. In fact, the structure of the modified kinematical twistor has the form (56View Equation) with vanishing ' P A B and ' PAB but with non-vanishing cAB in the fifth order. Then in the quasi-local mass (in the leading sixth order) there will be a term coming from the (presumably non-vanishing) sixth order part of P A'B and PAB' and the constant part of the Hermitian scalar product, and the fifth order cAB 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'ab slightly [310]. This modified form is based on Tod’s form of the kinematical twistor:

gf '' a b 1 A'B'[ ( V~ -- A)][ (V ~ - B)] A abZ W := 4pG-- g iDA'A jc iDB'B jw dS. (58) S
The quasi-local mass on small spheres coming from A''ab is positive [377].

7.3.3 Mason’s suggestions

A beautiful property of the original construction was its connection with the Hamiltonian formulation of the theory [265Jump To The Next Citation Point]. Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified constructions. Although the form of Equation (58View Equation) is that of the integral of the Nester-Witten 2-form, and the spinor fields V~ jcA and iDA'A(V ~ jcA) could still be considered as the spinor constituents of the ‘quasi-Killing vectors’ of the 2-surface S, their structure is not so simple because the factor j itself depends on all of the four independent solutions of the 2-surface twistor equation in a rather complicated way.

To have a simple Hamiltonian interpretation Mason suggested further modifications [265266Jump To The Next Citation Point]. He considers the four solutions cA i, i = 1,...,4, of the 2-surface twistor equations, and uses these solutions in the integral (55View Equation) of the Nester-Witten 2-form. Since HS is a Hermitian bilinear form on the space of the spinor fields (see Section 8 below), he obtains 16 real quantities as the components of the 4× 4 Hermitian matrix Eij := HS [ci,cj]. However, it is not clear how the four ‘quasi-translations’ of S should be found among the 16 vector fields cAi cA'j (called ‘quasi-conformal Killing vectors’ of S) for which the corresponding quasi-local quantities could be considered as the quasi-local energy-momentum. Nevertheless, this suggestion leads us to the next class of quasi-local quantities.

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