### 14.2 On the Penrose mass

Penrose’s suggestion for the quasi-local mass (or, more generally, energy-momentum and angular
momentum) was based on a promising and far-reaching strategy to use twistors at the fundamental level.
The basic object of the construction, the so-called kinematical twistor, is intended to comprise both the
energy-momentum and angular momentum, and is a well-defined quasi-local quantity on generic spacelike
surfaces homeomorphic to . It can be interpreted as the value of a quasi-local Hamiltonian, and the
four independent 2-surface twistors play the role of the quasi-translations and quasi-rotations. The
kinematical twistor was calculated for a large class of special 2-surfaces and gave acceptable
results.
However, the construction is not complete. First, the construction does not work for 2-surfaces whose
topology is different from , and does not work even for certain topological 2-spheres for which the
2-surface twistor equation admits more than four independent solutions (‘exceptional 2-surfaces’). Second,
two additional objects, the so-called infinity twistor and a Hermitian inner product on the space of 2-surface
twistors, are needed to get the energy-momentum and angular momentum from the kinematical twistor and
to ensure their reality. The latter is needed if we want to define the quasi-local mass as a norm of
the kinematical twistor. However, no natural infinity twistor has been found, and no natural
Hermitian scalar product can exist if the 2-surface cannot be embedded into a conformally flat
spacetime. In addition, in the small surface calculations the quasi-local mass may be complex. If,
however, we do not want to form invariants of the kinematical twistor (e.g. the mass), but we
want to extract the energy-momentum and angular momentum from the kinematical twistor
and we want them to be real, then only a special combination of the infinity twistor and the
Hermitian scalar product, the so-called ‘bar-hook combination’ (see Equation (51)), would be
needed.

To save the main body of the construction, the definition of the kinematical twistor was modified.
Nevertheless, the mass in the modified constructions encountered an inherent ambiguity in the
small surface approximation. One can still hope to find an appropriate ‘bar-hook’, and hence
real energy-momentum and angular momentum, but invariants, such as norms, could not be
formed.