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Discrete gravity as a topological field theory with light-like curvature defects

Wolfgang Wieland

TL;DR

This work proposes a discrete gravity model as a topological SL(2,C) gauge theory where curvature is confined to three-dimensional null interfaces and two-dimensional corners, with bulk regions being locally flat. The dynamics are encoded entirely in boundary spinor data on null surfaces, through a boundary action that couples spinors and a U(1) gauge field to the bulk connection, yielding a rich set of first- and second-class constraints and a Hamiltonian structure that is topological in the bulk. Explicit solutions include plane-fronted gravitational waves with distributional curvature localized at interfaces, demonstrating a direct link to Einstein's equations in a distributional sense. The framework offers a continuum interpretation for loop quantum gravity boundary degrees of freedom and suggests a path toward a covariant, spinfoam-like quantum gravity formulated in terms of boundary CS-type dynamics and light-like defects.

Abstract

I present a model of discrete gravity, which is formulated in terms of a topological gauge theory with defects. The theory has no local degrees of freedom and the gravitational field is trivial everywhere except at a number of colliding null surfaces, which represent a system of curvature defects propagating at the speed of light. The underlying action is local and it is studied in both its Lagrangian and Hamiltonian formulation. The canonically conjugate variables on the null surfaces are a spinor and a spinor-valued two-surface density, which are coupled to a topological field theory for the Lorentz connection in the bulk. I discuss the relevance of the model for non-perturbative approaches to quantum gravity, such as loop quantum gravity, where similar variables have recently appeared as well.

Discrete gravity as a topological field theory with light-like curvature defects

TL;DR

This work proposes a discrete gravity model as a topological SL(2,C) gauge theory where curvature is confined to three-dimensional null interfaces and two-dimensional corners, with bulk regions being locally flat. The dynamics are encoded entirely in boundary spinor data on null surfaces, through a boundary action that couples spinors and a U(1) gauge field to the bulk connection, yielding a rich set of first- and second-class constraints and a Hamiltonian structure that is topological in the bulk. Explicit solutions include plane-fronted gravitational waves with distributional curvature localized at interfaces, demonstrating a direct link to Einstein's equations in a distributional sense. The framework offers a continuum interpretation for loop quantum gravity boundary degrees of freedom and suggests a path toward a covariant, spinfoam-like quantum gravity formulated in terms of boundary CS-type dynamics and light-like defects.

Abstract

I present a model of discrete gravity, which is formulated in terms of a topological gauge theory with defects. The theory has no local degrees of freedom and the gravitational field is trivial everywhere except at a number of colliding null surfaces, which represent a system of curvature defects propagating at the speed of light. The underlying action is local and it is studied in both its Lagrangian and Hamiltonian formulation. The canonically conjugate variables on the null surfaces are a spinor and a spinor-valued two-surface density, which are coupled to a topological field theory for the Lorentz connection in the bulk. I discuss the relevance of the model for non-perturbative approaches to quantum gravity, such as loop quantum gravity, where similar variables have recently appeared as well.

Paper Structure

This paper contains 29 sections, 168 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Boundary spinors in general relativity
  3. Self-dual area two-form on a null surface
  4. Boundary term on a null surface
  5. Discretised gravity with impulsive gravitational waves
  6. Glueing flat four-volumes along null surfaces
  7. Area-matching condition
  8. Reconstruction of $q_{ab}$ from $\boldsymbol{\eta}_A$ and $\ell^A$
  9. Shape-matching conditions
  10. Definition of the action
  11. General idea
  12. Construction of the action
  13. Equations of motion
  14. Three immediate observations: Integrability, geodesity and the expansion of the null surface
  15. Difference tensor
  16. ...and 14 more sections

Figures (1)

  • Figure 1: The spacetime manifold $\mathcal{M}$ splits into a union of four-dimensional cells $\mathcal{M}_1,\mathcal{M}_2,\dots$, which contain no local degrees of freedom inside. Non-trivial curvature is confined to three-dimensional interfaces $\mathcal{N}_{13}=\mathcal{M}_1\cap\mathcal{M}_3,\dots$, which intersect in two-dimensional corners $\mathcal{C}_{12}^{34},\dots$.