Planckian Energy Scattering and Surface Terms in the Gravitational Action

This is a revised version of our previous paper by the same name and preprint number. It contains various changes, two figures and new results in sect.5. We propose a new approach to four-dimensional Planckian-energy scattering in which the phase of the ${\cal S}$-matrix is written---to leading order in $\hbar$ and to all orders in $R/b =Gs/J$---in terms of the surface term of the gravity action and of a boundary term for the colliding quanta. The proposal is checked at the leading order in $R/b$ and also against some known examples of scattering in strong gravitational fields.