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Relaxing the Parity Conditions of Asymptotically Flat Gravity

Geoffrey Compère, François Dehouck

TL;DR

Relaxing parity conditions in asymptotically flat gravity reveals a translation anomaly that must be accounted for to define a consistent variational principle. By adding a translation-breaking, logarithmic counter-term, the authors obtain finite charges for a broader set of asymptotic symmetries, including supertranslations and logarithmic translations, while standard Poincaré charges are recovered under parity. They construct both covariant phase spaces with parity constraints and an extended phase space without parity restrictions, showing the former preserves the usual Poincaré group and the anomaly does not affect classical dynamics. The work clarifies how asymptotic diffeomorphisms act on the phase space in a way reminiscent of holographic renormalization, with potential implications for quantum gravity in asymptotically flat spacetimes.

Abstract

Four-dimensional asymptotically flat spacetimes at spatial infinity are defined from first principles without imposing parity conditions or restrictions on the Weyl tensor. The Einstein-Hilbert action is shown to be a correct variational principle when it is supplemented by an anomalous counter-term which breaks asymptotic translation, supertranslation and logarithmic translation invariance. Poincaré transformations as well as supertranslations and logarithmic translations are associated with finite and conserved charges which represent the asymptotic symmetry group. Lorentz charges as well as logarithmic translations transform anomalously under a change of regulator. Lorentz charges are generally non-linear functionals of the asymptotic fields but reduce to well-known linear expressions when parity conditions hold. We also define a covariant phase space of asymptotically flat spacetimes with parity conditions but without restrictions on the Weyl tensor. In this phase space, the anomaly plays classically no dynamical role. Supertranslations are pure gauge and the asymptotic symmetry group is the expected Poincaré group.

Relaxing the Parity Conditions of Asymptotically Flat Gravity

TL;DR

Relaxing parity conditions in asymptotically flat gravity reveals a translation anomaly that must be accounted for to define a consistent variational principle. By adding a translation-breaking, logarithmic counter-term, the authors obtain finite charges for a broader set of asymptotic symmetries, including supertranslations and logarithmic translations, while standard Poincaré charges are recovered under parity. They construct both covariant phase spaces with parity constraints and an extended phase space without parity restrictions, showing the former preserves the usual Poincaré group and the anomaly does not affect classical dynamics. The work clarifies how asymptotic diffeomorphisms act on the phase space in a way reminiscent of holographic renormalization, with potential implications for quantum gravity in asymptotically flat spacetimes.

Abstract

Four-dimensional asymptotically flat spacetimes at spatial infinity are defined from first principles without imposing parity conditions or restrictions on the Weyl tensor. The Einstein-Hilbert action is shown to be a correct variational principle when it is supplemented by an anomalous counter-term which breaks asymptotic translation, supertranslation and logarithmic translation invariance. Poincaré transformations as well as supertranslations and logarithmic translations are associated with finite and conserved charges which represent the asymptotic symmetry group. Lorentz charges as well as logarithmic translations transform anomalously under a change of regulator. Lorentz charges are generally non-linear functionals of the asymptotic fields but reduce to well-known linear expressions when parity conditions hold. We also define a covariant phase space of asymptotically flat spacetimes with parity conditions but without restrictions on the Weyl tensor. In this phase space, the anomaly plays classically no dynamical role. Supertranslations are pure gauge and the asymptotic symmetry group is the expected Poincaré group.

Paper Structure

This paper contains 26 sections, 1 theorem, 187 equations, 1 figure.

Table of Contents

  1. Two puzzles in asymptotically flat spacetimes
  2. Main results and outline
  3. Specification of boundary conditions
  4. Boundary conditions in the hyperbolic representation of spatial infinity
  5. Boundary conditions in Hamiltonian formalism
  6. Lagrangian dynamics in the hyperbolic representation
  7. Action principle
  8. Equations of motion
  9. Linearization stability constraints
  10. Symplectic structure
  11. Covariant phase space charges
  12. Algebra of conserved charges
  13. Hamiltonian dynamics
  14. Summary and Discussion
  15. Comparison of 3+1 and covariant boundary conditions
  16. ...and 11 more sections

Key Result

Lemma 1

On the three dimensional hyperboloid, any scalar $\Phi$ satisfying $\square \Phi + 3\Phi = 0$ defines a symmetric, traceless, curl-free and divergence-free tensor $T_{ab} = D_a D_b \Phi + h^{(0)}_{ab}\Phi$ which can be written as where $P_{ab}$ is a symmetric, traceless tensor of the form where $P_{ab}^{reg}$ is regular and $k^{(\mu)}_{ab}$ are four singular tensors listed here below.

Figures (1)

  • Figure 1: The variational principle is defined in the spacetime delimited by initial and final hyperbolic temporal slices $\Sigma_\pm$ and the hyperbolic radial cut-off $\mathcal{H}$.

Theorems & Definitions (1)

  • Lemma 1