Authors:
Piotr T. Chrusciel (Université de Tours),
Jacek Jezierski
(Uniwersytet Warszawski), Jerzy Kijowski
(Uniwersytet Warszawski)
Comments: Tours preprint 220/2000, 175 pages,
Springer Lecture Notes in Physics m70 (2001)
We develop a geometric Hamiltonian formalism adequate for a canonical description of field theories in the radiation regime, extending previous work of Kijowski and Tulczyjew. The formalism is applied to the massless scalar field in Minkowski space-time at null infinity, and to general relativity at null infinity. Formulae for energy, momentum, angular-momentum, as well as for the Hamiltonians for boosts and supertranslations are derived. Several natural phase spaces are constructed, one of those leads to time-dependent Hamiltonians, an appropriate limit of which is the Trautman-Bondi energy at Scri. Another choice of phase spaces leads to time-independent Hamiltonians; in that case some ambiguities arise in the Hamiltonian, and we give an exhaustive description of those. One of the Hamiltonians for the gravitational field, which occur in this context, is again the Trautman-Bondi mass. We argue that one can get rid of the ambiguity in the definition of energy by adding the condition of monotonicity of the Hamiltonian with respect to motion of the reference hypersurface to the future, again singling out the Trautman-Bondi mass. We show that the Hamiltonian approach gives naturally a definition of angular momentum at null infinity, which is free of supertranslation ambiguities.