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First order gravity on the light front

Sergei Alexandrov, Simone Speziale

TL;DR

This work develops a canonical formulation of general relativity on a null (light-front) foliation using a real first-order tetrad/connection action. The analysis reveals a rich constraint structure in which the Hamiltonian constraint becomes second class on the light front and two tertiary constraints reproduce the dynamical Einstein equations, yielding two local gravitational degrees of freedom. A non-covariant reformulation clarifies the role of the two propagating modes and the degenerate null-induced metric, while the stabilization procedure ensures that no essential Einstein equation is lost despite partial gauge fixing. The framework provides a path toward quantizing null hypersurfaces within loop quantum gravity or spin foam approaches and highlights subtle zero-mode issues that depend on boundary conditions and asymptotics, linking to familiar double-null and Newman–Penrose constructions.

Abstract

We study the canonical structure of the real first order formulation of general relativity on a null foliation. We use a tetrad decomposition which allows to elegantly encode the nature of the foliation in the norm of a vector in the fibre bundle. The resulting constraint structure shows some peculiarities. In particular, the dynamical Einstein equations propagating the physical degrees of freedom appear in this formalism as second class tertiary constraints, which puts them on the same footing as the Hamiltonian constraint of the Ashtekar's connection formulation. We also provide a framework to address the issue of zero modes in gravity, in particular, to study the non-perturbative fate of the zero modes of the linearized theory. Our results give a new angle on the dynamics of general relativity and can be used to quantize null hypersurfaces in the formalism of loop quantum gravity or spin foams.

First order gravity on the light front

TL;DR

This work develops a canonical formulation of general relativity on a null (light-front) foliation using a real first-order tetrad/connection action. The analysis reveals a rich constraint structure in which the Hamiltonian constraint becomes second class on the light front and two tertiary constraints reproduce the dynamical Einstein equations, yielding two local gravitational degrees of freedom. A non-covariant reformulation clarifies the role of the two propagating modes and the degenerate null-induced metric, while the stabilization procedure ensures that no essential Einstein equation is lost despite partial gauge fixing. The framework provides a path toward quantizing null hypersurfaces within loop quantum gravity or spin foam approaches and highlights subtle zero-mode issues that depend on boundary conditions and asymptotics, linking to familiar double-null and Newman–Penrose constructions.

Abstract

We study the canonical structure of the real first order formulation of general relativity on a null foliation. We use a tetrad decomposition which allows to elegantly encode the nature of the foliation in the norm of a vector in the fibre bundle. The resulting constraint structure shows some peculiarities. In particular, the dynamical Einstein equations propagating the physical degrees of freedom appear in this formalism as second class tertiary constraints, which puts them on the same footing as the Hamiltonian constraint of the Ashtekar's connection formulation. We also provide a framework to address the issue of zero modes in gravity, in particular, to study the non-perturbative fate of the zero modes of the linearized theory. Our results give a new angle on the dynamics of general relativity and can be used to quantize null hypersurfaces in the formalism of loop quantum gravity or spin foams.

Paper Structure

This paper contains 24 sections, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generic features of the light front formalism
  3. Null foliation via tetrads
  4. Decomposition of the tetrad
  5. Field equations
  6. Cartan equations
  7. Tetrad equations
  8. Canonical analysis
  9. Hamiltonian
  10. Primary constraints
  11. Secondary constraints
  12. Tertiary constraints
  13. Non-covariant analysis
  14. Peculiarities on the light front
  15. Origin of the tertiary constraints
  16. ...and 9 more sections

Figures (1)

  • Figure 1: The past light cone of an event in spacetime. All world-lines intersect the light front hypersurface in a finite time, except the one parallel to it. The latter corresponds to the global zero mode.