[Paper Review] Hamiltonian consistency of the gravitational constraint algebra under deformations

José T. Gálvez Ghersi, Michael J. Desrochers+2 more|arXiv (Cornell University)|Nov 12, 2017

Black Holes and Theoretical Physics32 references3 citations

TL;DR

This paper develops a general framework to deform the canonical variables of general relativity—such as the metric or tetrad—while preserving the first-class structure of the gravitational constraint algebra, ensuring consistent gauge dynamics. It derives consistency conditions under which the deformed theory maintains diffeomorphism invariance and stable gauge fixing, even when the deformation is non-canonical and introduces new degrees of freedom.

ABSTRACT

The importance of the first-class constraint algebra of general relativity is not limited just by its self-contained description of the gauge nature of spacetime, but it also provides conditions to properly evolve the geometry by selecting a gauge only once throughout the whole evolution of a gravitational system. This must be a property of all background independent theories. In this paper we consider gravitational theories which arise from deformations of the fundamental canonical variables of general relativity where the proposed deformations are inspired by modifications of gravity. These variable deformations result in new theories when the deformation is not a canonical transformation. The new theory must preserve the first-class structure of the algebra, which is a non-trivial restriction for generic deformations. In this vein we present a general deformation scheme along with consistency conditions, so that the algebra of constraints is still satisfied in the resulting theory. This is illustrated both in metric theory as well as in tetrad theory.

Motivation & Objective

• To ensure that deformed gravitational theories—arising from non-canonical modifications of canonical variables—retain the first-class structure of the constraint algebra, essential for consistent gauge dynamics.
• To address the challenge that generic deformations of canonical variables can break the closure of the constraint algebra, which would undermine the theory's consistency and gauge-fixing stability.
• To provide a systematic method applicable to both metric and tetrad formulations of gravity, including cases with or without additional degrees of freedom.
• To extend existing approaches like holonomy corrections in loop quantum gravity by ensuring algebraic consistency even in the presence of matter.
• To demonstrate that deformations can be constrained such that extra degrees of freedom either decouple or are reinterpreted as Goldstone modes, preserving physical consistency.

Proposed method

• Formalize a general deformation scheme for canonical variables (metric, tetrad, connection) in Hamiltonian gravity, treating the deformed fields as functions of the original variables.
• Derive new conjugate momenta from the deformed variables using canonical mechanics, ensuring the new variables form a valid Hamiltonian system.
• Evaluate the Poisson brackets of the deformed scalar, vector, and Gauss constraints to determine closure conditions for the algebra.
• Apply the formalism to the Einstein-Hilbert action, the Palatini action with tetrad variables, and Ashtekar-Barbero variables, identifying deformation constraints.
• Analyze two cases: (I) deformations without higher-derivative terms (leading to Lovelock’s theorem unless canonical), and (II) non-unitary deformations introducing new degrees of freedom.
• Use traceless, symmetric metric deformations as a concrete example to derive conditions under which scalar and vector constraints remain gauge-invariant.

Experimental results

Research questions

• RQ1 Under what conditions does a deformation of the canonical variables in general relativity preserve the first-class structure of the constraint algebra?
• RQ2 How can non-canonical deformations introduce new degrees of freedom while still maintaining consistent gauge dynamics and diffeomorphism invariance?
• RQ3 What constraints must be imposed on metric or tetrad deformations to ensure the scalar and vector constraints remain closed and gauge-invariant?
• RQ4 Can holonomy-type corrections in loop quantum gravity be consistently implemented within this framework without breaking the constraint algebra?
• RQ5 What happens to the constraint algebra when deformations alter the shift and lapse functions, and how can the system still support stable gauge fixing?

Key findings

• The paper derives explicit consistency conditions under which deformed gravitational theories preserve the first-class structure of the constraint algebra, even when the deformation is not a canonical transformation.
• For deformations without higher-derivative terms, Lovelock’s theorem implies that only canonical transformations preserve the original number of degrees of freedom unless the deformation is trivial.
• In the case of non-unitary deformations introducing new degrees of freedom, the scalar and vector constraints remain consistent only if the deformations satisfy specific algebraic constraints derived from Poisson bracket closure.
• The example of traceless, symmetric metric deformations shows that the new Hamiltonian requires additional constraints to maintain gauge invariance, which can either remove extra degrees of freedom or reinterpret them as Goldstone modes.
• The framework is successfully extended to tetrad-Palatini and Ashtekar-Barbero formulations, showing that similar consistency conditions apply and that holonomy corrections can be consistently embedded.
• The analysis confirms that the constraint algebra can be preserved in deformed theories by projecting extra terms into the Gauss constraint, enabling a consistent metric realization even when the original action is modified.