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New boundary variables for classical and quantum gravity on a null surface

Wolfgang Wieland

TL;DR

This work develops an SL(2,C) covariant Hamiltonian framework for general relativity in a region bounded by an inner null surface, revealing that the natural boundary degrees of freedom are a boundary spinor ℓ^A and a spinor-valued two-form η_Aab that encode the null boundary's intrinsic geometry. The authors introduce a boundary action and derive a complete covariant phase-space structure, including a pre-symplectic form with corner contributions and canonical brackets for bulk and boundary data, along with a quasi-local first law for boundary-dragging diffeomorphisms. They identify a set of gauge symmetries (U(1) on the boundary, SL(2,C) frame rotations) and genuine Hamiltonian corner diffeomorphisms with corresponding charges, connecting local boundary dynamics to ADM quantities at infinity. The framework provides a concrete route to quantum gravity by enabling corner-based quantization of boundary spinors, yielding a discrete area spectrum and shedding light on edge modes and soft hair in bounded regions.

Abstract

The covariant Hamiltonian formulation for general relativity is studied in terms of self-dual variables on a manifold with an internal and lightlike boundary. At this inner boundary, new canonical variables appear: a spinor and a spinor-valued two-form that encode the entire intrinsic geometry of the null surface. At a two-dimensional cross-section of the boundary, quasi-local expressions for the generators of two-dimensional diffeomorphisms, time translations, and dilatations of the null normal are introduced and written in terms of the new boundary variables. In addition, a generalisation of the first-law of black-hole thermodynamics for arbitrary null surfaces is found, and the relevance of the framework for non-perturbative quantum gravity is stressed and explained.

New boundary variables for classical and quantum gravity on a null surface

TL;DR

This work develops an SL(2,C) covariant Hamiltonian framework for general relativity in a region bounded by an inner null surface, revealing that the natural boundary degrees of freedom are a boundary spinor ℓ^A and a spinor-valued two-form η_Aab that encode the null boundary's intrinsic geometry. The authors introduce a boundary action and derive a complete covariant phase-space structure, including a pre-symplectic form with corner contributions and canonical brackets for bulk and boundary data, along with a quasi-local first law for boundary-dragging diffeomorphisms. They identify a set of gauge symmetries (U(1) on the boundary, SL(2,C) frame rotations) and genuine Hamiltonian corner diffeomorphisms with corresponding charges, connecting local boundary dynamics to ADM quantities at infinity. The framework provides a concrete route to quantum gravity by enabling corner-based quantization of boundary spinors, yielding a discrete area spectrum and shedding light on edge modes and soft hair in bounded regions.

Abstract

The covariant Hamiltonian formulation for general relativity is studied in terms of self-dual variables on a manifold with an internal and lightlike boundary. At this inner boundary, new canonical variables appear: a spinor and a spinor-valued two-form that encode the entire intrinsic geometry of the null surface. At a two-dimensional cross-section of the boundary, quasi-local expressions for the generators of two-dimensional diffeomorphisms, time translations, and dilatations of the null normal are introduced and written in terms of the new boundary variables. In addition, a generalisation of the first-law of black-hole thermodynamics for arbitrary null surfaces is found, and the relevance of the framework for non-perturbative quantum gravity is stressed and explained.

Paper Structure

This paper contains 15 sections, 147 equations, 1 figure.

Table of Contents

  1. Introduction: Why spinors?
  2. Action and variational principle for inner null boundaries
  3. Variables in the bulk
  4. Kinematical spin structures on a null surface
  5. Spin connection coefficients, extrinsic curvature
  6. Boundary term and boundary equations of motion
  7. Null surfaces as Hamiltonian subsystems
  8. Pre-symplectic potential
  9. Gauge symmetries
  10. Quasi-local first law
  11. Conclusion, relevance for quantum gravity
  12. Summary
  13. Relevance for quantum gravity
  14. Reconstruction of the intrinsic null geometry from the boundary spinors
  15. ADM energy from the corner term at infinity

Figures (1)

  • Figure 1: We study the gravitational field in a subregion $\mathcal{M}$ as a Hamiltonian system on the covariant phase of the theory (for asymptotically flat spacetimes). The four-volume $\mathcal{M}$ is bounded by an inner and expanding null surface $\mathcal{N}$ reaching future null infinity, by partial Cauchy hypersurfaces $\varSigma_0$ and $\varSigma_1$, and by an outer time-like cylinder $\mathcal{B}$, which will be sent to spacelike infinity $i^o$.