Holonomy and the Einstein equations

CARLOS KOZAMEH; PEDRO W. LAMBERTI; EZRA NEWMAN

ANNALS OF PHYSICS (NEW YORK)

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 1991 vol. 206 p. 193 - 220

0003-4916

A new point of view towards the vacuum Einstein equations on asymptotically flat space-times is presented. This new perspective involves the introduction of two different non-local geometric objects as the basic variables?rather than the usual (local) metric and connection?and then to find equations, equivalent to the Einstein field equations, satisfied by these variables. The first of these objects is the holonomy operator, i.e., the parallel propagator around closed curves, associated with a subclass of loops that arises naturally on asymptotically simple space-times. We establish the relationship between the holonomy operator and both the curvature tensor and connection of the space. The second non-local variable is the ?light-cone cut function,? which analytically describes the intersection of the light cone from an arbitrary space-time point xa, with null infinity, the conformal boundary of the space-time. The field equations for the holonomy operator and the cut function are derived and several implications of this formalism are discussed.