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) 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 . 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 . This modified form is based on Tod’s form of the kinematical twistor:.
A beautiful property of the original construction was its connection with the Hamiltonian formulation of the theory . 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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