Generalized boundary conditions for general relativity for the asymptotically flat case in terms of Ashtekar's variables

TL;DR

The paper addresses the lack of a complete asymptotic Poincaré framework for general relativity in Ashtekar variables by translating ADM boundary data, fixing the asymptotic internal frame, and constructing finite, differentiable generators for translations, boosts, and rotations. It introduces boundary-corrected, finite constraint functionals and a spin-connection counterterm to ensure finiteness while preserving polynomial bulk constraints; the resulting vector and scalar (energy) generators reproduce the ADM charges on shell. The boosts and rotations become nonpolynomial at the boundary, while the Gauss constraint eliminates any asymptotic SO(3) charge, and the Dirac observable generators satisfy the correct algebra, reproducing the Poincaré structure modulo supertranslations. This provides a coherent, ADM-compatible foundation for asymptotically flat gravity in the Ashtekar-variable formalism, with clear implications for observables and canonical quantum gravity approaches.

Abstract

There is a gap that has been left open since the formulation of general relativity in terms of Ashtekar's new variables namely the treatment of asymptotically flat field configurations that are general enough to be able to define the generators of the Lorentz subgroup of the asymptotical Poincaré group. While such a formulation already exists for the old geometrodynamical variables, up to now only the generators of the translation subgroup could be defined because the function spaces of the fields considered earlier are taken too special. The transcription of the framework from the ADM variables to Ashtekar's variables turns out not to be straightforward due to the a priori freedom to choose the internal SO(3) frame at spatial infinity and due to the fact that the non-trivial reality conditions of the Ashtekar framework reenter the stage when imposing suitable boundary conditions on the fields and the Lagrange multipliers.