Asymptotic Symmetries of the Holst Action at Spatial Infinity: Including Supertranslations

TL;DR

This paper explores the asymptotic symmetries of General Relativity at spatial infinity using the Holst action, demonstrating the full BMS group including supertranslations and analyzing the Holst term's impact on conserved charges.

Contribution

It introduces relaxed boundary conditions in the Holst formalism that admit the complete BMS group, including supertranslations, and clarifies the Holst term's effect on conserved charges.

Findings

Full BMS group, including supertranslations, is realized at spatial infinity.

Logarithmic divergences in the symplectic structure are regularized without suppressing supertranslations.

Holst term shifts Lorentz charge values but leaves supertranslation charges invariant.

Abstract

We investigate the asymptotic symmetries of General Relativity at spatial infinity within the first-order formalism described by the Holst action. Employing the covariant phase space method, we propose a set of relaxed boundary conditions for the co-tetrad and Lorentz connection that admit the full Bondi-Metzner-Sachs (BMS) group, including non-trivial supertranslations, which are typically eliminated in standard treatments. We demonstrate that the logarithmic divergences appearing in the symplectic structure can be removed by imposing specific, symmetry-preserving parity conditions on the asymptotic fields without suppressing the supertranslation sector. A detailed analysis of the conserved charges reveals that the Holst term contributes non-trivially to the charge variations due to the linear growth of Lorentz generators. We show that the naive surface integrals for the Holst charges exhibit linear divergences arising from the rotation of the background tetrad. These divergences are successfully regularized by supplementing the asymptotic symmetry generator with a compensating internal Lorentz gauge transformation defined to preserve the background structure. The resulting charges are manifestly finite and integrable. Crucially, we prove that while the Holst modification shifts the charges associated with Lorentz boosts and rotations, it leaves the supertranslation charges identically invariant. This framework provides a consistent derivation of the full BMS algebra at spatial infinity in terms of Ashtekar-Barbero variables, offering new insights into the role of the Immirzi parameter in classical and quantum gravity.