Regge-Teitelboim analysis of the symmetries of electromagnetic and gravitational fields on asymptotically null spacelike surfaces

We present a new application of the Regge-Teitelboim method for treating symmetries which are defined asymptotically. It may be regarded as complementary to the one in their original 1974 paper. The formulation is based on replacing an asymptotic plane by the two--sheeted ``hourglass" shaped surface obtained by joining smoothly an incoming hyperboloid with an outgoing one. The hyperboloids have a fixed radius, and as one moves the center of the hourglass along the time axis one covers the whole of spacetime. The motivation is to study radiation, and the hourglass is well suited to the task because it is asymptotically null, and thus is able to register the details of the process. A simple parity condition for the fields on the hyperboloid is given. It specifies that as much radiation as is coming in as it is going out. With it, a Hamiltonian formulation of the symmetry of Bondi, van der Burg, Metzner and Sachs is developed fir both electromagnetism and gravitation. It is indispensable for the construction to have electric--magnetic duality asymptotically. For gravitation, a formulation for the linearized theory on the hourglass has not been explicitly constructed; but enough rudiments of it are given so that the main results can be established. A definition for angular momentum wish is conserved (for which the ``magnetic sector'' is essential) is given. It incorporates an interrelationship between spin and charge. For the gravitational field, Taub-NUT space appears as the analog of a magnetic pole.