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

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.

• The paper establishes a finite, integrable realization of the full BMS₄ algebra at spatial infinity by imposing symmetry-compatible parity conditions for non-trivial supertranslations.
• It employs a covariant phase space analysis of the Holst action and introduces a compensating internal Lorentz transformation to regularize divergences in conserved charges.
• The study demonstrates that the Barbero-Immirzi parameter controls shifts in asymptotic Lorentz charges while remaining decoupled from the supertranslation sector.

Asymptotic Symmetries of the Holst Action at Spatial Infinity: Covariant BMS Analysis with Supertranslations

Introduction and Motivation

This work rigorously analyzes the asymptotic symmetry algebra of General Relativity at spatial infinity within the canonical first-order formalism governed by the Holst action. The approach is grounded in a covariant phase space analysis, with particular attention given to boundary conditions that admit the complete BMS$_4$ group, including non-trivial supertranslation generators. The analysis addresses long-standing issues encountered in Hamiltonian treatments, notably the incompatibility between supertranslation non-triviality and the finiteness/integrability of Lorentz charges in Ashtekar-Barbero variables, and resolves divergences in the symplectic structure and associated charges arising from relaxed parity conditions.

Holst Action, Covariant Phase Space, and Boundary Structures

The Holst action extends the Palatini action for General Relativity by a term proportional to the Hodge dual of the curvature two-form, weighted by the Barbero-Immirzi parameter $\beta$. This modification, while leaving the classical bulk dynamics untouched (yielding Einstein's vacuum equations on-shell), has non-trivial implications at the boundary. The fundamental configuration variables are the co-tetrad $e^I_\mu$ and the independent Lorentz connection $\omega_\mu^{IJ}$. The pre-symplectic structure, derived via the Lee-Wald-Kovelman covariant phase space method, contains a Palatini bulk term and a pure boundary contribution from the Holst term.

The phase space is defined on solutions asymptotically approaching Minkowski spacetime in a radial-hyperbolic chart, with metric fall-off specified by an even mass aspect $\sigma(\Phi)$ and unconstrained angular perturbations ${}^1h_{AB}(\Phi)$. Unlike prior treatises that imposed strong gauge-fixing (e.g., ${}^1h_{AB} \propto \sigma h_{AB}$) to restrict the symmetry group to the Poincaré algebra, this framework leaves ${}^1h_{AB}$ independent, enabling inclusion of the full BMS group, with supertranslations parameterized by both global time/spatial translations and angle-dependent (higher spherical harmonic) modes.

Regularization of Divergences: Parity Assignments and Boundary Conditions

A central technical challenge arises from logarithmic divergences in the symplectic structure driven by the non-trivial transformation of the angular perturbations and the unbounded growth of Lorentz generators. The power-counting of the pre-symplectic current unveils terms potentially divergent as $\int (d\rho/\rho)$, requiring elimination for physical viability.

This is resolved by imposing parity conditions on the asymptotic data: $\sigma(\Phi)$ is restricted to be even under the antipodal map, whereas trace-free parts of the angular metric perturbation and related variables are required to be odd. These assignments are proved compatible with the transformation properties under the BMS algebra, including field-dependent modifications to supertranslation parameters necessary for charge integrability. Critically, these conditions allow for non-trivial supertranslations in contrast to the vanishing charges produced by the standard Regge-Teitelboim setup.

Construction and Finiteness of Conserved Charges

Palatini Sector

The construction of conserved charges proceeds via contraction of the presymplectic form with symmetry generators, rendered as surface integrals over the sphere at spatial infinity. For the Palatini term, the parity conditions ensure the cancellation of naive linear and logarithmic divergences. The charge variation associated with Lorentz and supertranslation generators is separated into integrable and non-integrable pieces, with non-integrable terms canceled by promoting the time-translation parameter $\beta$0 to a specific field-dependent form involving parity-odd metric perturbations. The resulting Hamiltonians are manifestly finite and integrable for the full algebra, and supertranslation charges do not vanish.

Formally, the supertranslation charge takes the structure: $\beta$1 Here $\beta$2 and $\beta$3 are even and odd parameters, and the nonzero value is a direct consequence of the relaxed parity and field assignments.

Holst Sector: Compensating Internal Lorentz Gauge Transformation

The Holst contribution introduces subtleties absent in metric or ADM analyses. The naive computation of the surface charge yields a linearly divergent result for Lorentz transformations, reflecting the unbounded rotation/boost of the background tetrad induced by large diffeomorphisms. This is regularized by supplementing the asymptotic generator $\beta$4 with a compensating internal Lorentz transformation $\beta$5, chosen to rigidly fix the background frame and cancel the unphysical flux. This transformation is determined such that $\beta$6 (the background) remains invariant up to $\beta$7 in the asymptotic expansion.

After regularization, the Holst term makes a nontrivial, finite, and integrable contribution to the Lorentz (angular momentum, center-of-mass) charges, proportional to $\beta$8, but is proven to be exactly vanishing for all supertranslation generators. This is a robust geometric statement, relying on the absence of a net $\beta$9 rotation induced by supertranslations on the background frame and the exact cancellation between the Lie derivative and gauge variation in the Holst boundary integral.

Closure of the Extended BMS Algebra

To address algebraic closure, the field-dependent modifications of symmetry generators lead to a nontrivial structure in the asymptotic symmetry algebra, requiring the use of the Barnich-Troessaert modified Lie bracket. This bracket incorporates both the transformation of field-dependent parameters and the internal Lorentz sector. The analysis confirms exact closure: non-integrable contributions to the bracket are removed by the compensating gauge action, and all field-dependent terms arising from the parity prescription and supertranslation redefinition vanish after bracket commutation, yielding a faithful representation of BMS$e^I_\mu$0 in the covariant first-order phase space.

Broader Implications: Edge Modes, Holography, and the Self-Dual Limit

The parity condition prescription can be relaxed in the presence of dynamical boundary edge modes, as in the extended phase space picture (Donnelly-Freidel). The immunity of the supertranslation sector from Holst/Immirzi dependence is a geometric feature preserved in both strict and extended phase space formulations. The derived structure provides a natural interpretation for the distinct roles of the Barbero-Immirzi parameter: as a regulator of quantum area flux at isolated horizons (inner boundaries) and as a classical coupling of asymptotic Lorentz charges at spatial infinity. The analysis of complex self-dual limits ($e^I_\mu$1) demonstrates a mechanism by which the reality conditions and phase space reductions can be absorbed into the internal gauge sector, preserving the classical radiative degrees without unphysical truncations.

Conclusion

This work resolves a longstanding ambiguity in the description of gravitational asymptotic symmetries at spatial infinity in the first-order (Ashtekar-Barbero/Holst) formalism. By imposing symmetry-compatible parity conditions and incorporating a compensating internal Lorentz gauge transformation into the definition of asymptotic generators, the analysis constructs a finite, integrable, and nontrivial realization of the full BMS$e^I_\mu$2 algebra in the covariant phase space, with explicit formulas for all conserved charges. The Holst/Immirzi parameter, previously believed to be dynamically inert, is shown to control shifts in asymptotic Lorentz charges while being completely decoupled from supertranslations. These developments provide a robust foundation for quantization, the study of edge modes, and a unified treatment of boundary symmetries in quantum gravity frameworks, including Loop Quantum Gravity and holographic perspectives.

Reference:

"Asymptotic Symmetries of the Holst Action at Spatial Infinity: Including Supertranslations" (2604.02052).