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Canonical Vielbeins for General Relativity: D + 1 Decomposition and Constraint Analysis

Joakim Flinckman, Daniel Blixt

Abstract

We provide a self-contained derivation of the Hamiltonian formulation of General Relativity in vielbein variables in $d=D+1$ dimensions. Starting from the Einstein--Hilbert action in a standard metric $D+1$ decomposition, we derive Lorentz- and $\mathrm{SO}(D)$-covariant phase-space actions, identify the primary Lorentz constraints, and obtain the Hamiltonian and momentum constraints. We compute the resulting first-class constraint algebra, relate the vielbein and metric phase-space formulations, and discuss the rotation/boost decomposition. In particular, we construct the boost generator in the $\mathrm{SO}(D)$-covariant formulation, thereby recovering full local Lorentz symmetry.

Canonical Vielbeins for General Relativity: D + 1 Decomposition and Constraint Analysis

Abstract

We provide a self-contained derivation of the Hamiltonian formulation of General Relativity in vielbein variables in dimensions. Starting from the Einstein--Hilbert action in a standard metric decomposition, we derive Lorentz- and -covariant phase-space actions, identify the primary Lorentz constraints, and obtain the Hamiltonian and momentum constraints. We compute the resulting first-class constraint algebra, relate the vielbein and metric phase-space formulations, and discuss the rotation/boost decomposition. In particular, we construct the boost generator in the -covariant formulation, thereby recovering full local Lorentz symmetry.
Paper Structure (22 sections, 188 equations)

This paper contains 22 sections, 188 equations.

Table of Contents

  1. Introduction
  2. D+1 Decomposition of the Vielbein and Einstein–Hilbert action
  3. D+1 Vielbein
  4. D+1 Decomposition of the Einstein–Hilbert action
  5. Lorentz-covariant Canonical Formulation
  6. Lorentz invariant phase-space action
  7. Hamiltonian Formulation
  8. Constraints and Symmetry Operators
  9. Lorentz Generator Decomposition
  10. SO(D) Covariant Canonical Formulation
  11. Internal D+1 split and time gauge
  12. SO(D) invariant phase-space action
  13. Hamiltonian formulation
  14. Constraints and Symmetry Operators
  15. Extending the Lorentz Algebra
  16. ...and 7 more sections