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
These anomalies lead Penrose to modify slightly [423]. 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 [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.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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