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 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 2 S. It can be interpreted as the value of a quasi-local Hamiltonian, and the four independent two-surface twistors play the role of the quasi-translations and quasi-rotations. The kinematical twistor was calculated for a large class of special two-surfaces and gave acceptable results.

However, the construction is not complete. First, the construction does not work for two-surfaces, whose topology is different from S2, and does not work even for certain topological two-spheres for which the two-surface twistor equation admits more than four independent solutions (‘exceptional two-surfaces’). Second, two additional objects, the infinity twistor and a Hermitian inner product on the space of two-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 two-surface cannot be embedded into a conformally flat spacetime. In addition, in 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 do 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 ‘bar-hook combination’ (see Eq. (7.9View Equation)), 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, cannot be formed.

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