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

TL;DR

This paper develops generalized boundary conditions for asymptotically flat spacetimes in general relativity using Ashtekar's variables, enabling a comprehensive treatment of the Lorentz group generators at infinity.

Contribution

It introduces a new framework for boundary conditions in Ashtekar's variables that includes the full Lorentz subgroup, extending previous formulations limited to translations.

Findings

Successfully defines Lorentz generators at spatial infinity.

Adapts boundary conditions to accommodate Ashtekar's variables.

Bridges the gap between ADM and Ashtekar formulations for asymptotically flat spacetimes.

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 Poincar\'e 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.