A general theory of fibre spaces with structure sheaf by A Grothendieck

By A Grothendieck

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Now, one knows from field theories in Minkowski space that such a program can be successful only if suitable boundary conditions are imposed on the fall-off of physical fields. In the present case, the choice of realistic boundary conditions is a delicate matter since the space-time geometry, in terms of which these notions are normally formulated, is itself the dynamical field of interest. Therefore, if one adopts the 'asymptotics' avenue one's task is first to find suitable boundary conditions for gravitational fields of isolated systems, then to introduce physically interesting notions using these boundary conditions and to prove theorems relating them, and finally, to apply the framework so obtained to physical problems.

Natl. Acad. USA,20,169. ASYMPTOTIC PROPERTIES OF ISOLATED SYSTEMS: RECENT DEVELOPMENTS Abhay Ashtekar* Laboratoire de Physique Theorique Institut H. Poincare, 11,rue P. Curie, 75 231 Paris Cedex OS, France and Physics Department, Syracuse University Syracuse, N:Y. 13210, USA 1. INTRODUCTION Since this report belongs to the general plenary session, I thought it would be best to present a review of the current status of the field in broad terms. Thus, I will omit all proofs and try to convey the essential ideas instead.

This leads to the following definition of the total 4-momentum, Pa' of the isolated system under consideration: Pa is a 4-vector at i O given by (3) where Va is any vector at i O, and, C, any 2-sphere cross-section of i ° (non-contractable to zero). Equation (2) implies that (3) is independent of the choice of C; P a is absolutely conserved. There are a number of theorems concerning the behavior of Pa. First of all, one can show (Ashtekar and Magnon-Ashtekar, 1983) that P is the same as the ADM 4-momentum defined in terms o~ the initial data on a 3-surface.

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