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

where is a constant multiple of the determinant in Eq. (7.7). Since on noncontorted two-surfaces the determinant is constant, for such surfaces reduces to , and hence, all the nice properties proven for the original construction on noncontorted two-surfaces are shared by . The quasi-local mass calculated from Eq. (7.16) 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 (7.15) with vanishing and but with nonvanishing 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 and and the constant part of the Hermitian scalar product, and the fifth-order and the still ambiguous -order part of the Hermitian metric.

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

The quasi-local mass on small spheres coming from is positive [516].

A beautiful property of the original construction was its connection with the Hamiltonian formulation of the theory [357]. Unfortunately, such a simple Hamiltonian interpretation is lacking for the modified constructions. Although the form of Eq. (7.17) is that of the integral of the Nester–Witten 2-form, and the spinor fields and could still be considered as the spinor constituents of the ‘quasi-Killing vectors’ of the two-surface , their structure is not so simple, because the factor 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, 358]. He considers the four solutions , , of the two-surface twistor equations, and uses these solutions in the integral (7.14) of the Nester–Witten 2-form. Since 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 Hermitian matrix . However, it is not clear how the four ‘quasi-translations’ of should be found among the 16 vector fields (called ‘quasi-conformal Killing vectors’ of ) 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.

Living Rev. Relativity 12, (2009), 4
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