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Unfashionable observations about three dimensional gravity

E. Buffenoir, K. Noui

TL;DR

This work reexamines the program of 2+1 gravity that recasts dynamics as a gauge theory, highlighting conceptual issues around metric non-degeneracy, causality, and matter coupling. It develops a covariant, participant-observer framework where self-gravitating point masses in a closed universe define a complete set of observables and a Dirac algebra, linking the dynamics to combinatorial and Poisson-Lie structures. By explicitly computing Dirac brackets for a comprehensive set of observables (including generalized spin-networks) and contrasting Chern-Simons and true gravity, the paper lays a foundation for a covariant quantization approach and a possible second-quantization of gravity-matter. The results provide a concrete, covariant pathway to describe the reduced phase space via combinatorial data, suggesting robust algebraic structures amenable to quantization and deeper understanding of gravity’s observable content.

Abstract

It is commonly accepted that the study of 2+1 dimensional quantum gravity could teach us something about the 3+1 dimensional case. The non-perturbative methods developed in this case share, as basic ingredient, a reformulation of gravity as a gauge field theory. However, these methods suffer many problems. Firstly, this perspective abandon the non-degeneracy of the metric and causality as fundamental principles, hoping to recover them in a certain low-energy limit. Then, it is not clear how these combinatorial techniques could be used in the case where matter fields are added, which are however the essential ingredients in order to produce non trivial observables in a generally covariant approach. Endly, considering the status of the observer in these approaches, it is not clear at all if they really could produce a completely covariant description of quantum gravity. We propose to re-analyse carefully these points. This study leads us to a really covariant description of a set of self-gravitating point masses in a closed universe. This approach is based on a set of observables associated to the measurements accessible to a participant-observer, they manage to capture the whole dynamic in Chern-Simons gravity as well as in true gravity. The Dirac algebra of these observables can be explicitely computed, and exhibits interesting algebraic features related to Poisson-Lie groupoids theory.

Unfashionable observations about three dimensional gravity

TL;DR

This work reexamines the program of 2+1 gravity that recasts dynamics as a gauge theory, highlighting conceptual issues around metric non-degeneracy, causality, and matter coupling. It develops a covariant, participant-observer framework where self-gravitating point masses in a closed universe define a complete set of observables and a Dirac algebra, linking the dynamics to combinatorial and Poisson-Lie structures. By explicitly computing Dirac brackets for a comprehensive set of observables (including generalized spin-networks) and contrasting Chern-Simons and true gravity, the paper lays a foundation for a covariant quantization approach and a possible second-quantization of gravity-matter. The results provide a concrete, covariant pathway to describe the reduced phase space via combinatorial data, suggesting robust algebraic structures amenable to quantization and deeper understanding of gravity’s observable content.

Abstract

It is commonly accepted that the study of 2+1 dimensional quantum gravity could teach us something about the 3+1 dimensional case. The non-perturbative methods developed in this case share, as basic ingredient, a reformulation of gravity as a gauge field theory. However, these methods suffer many problems. Firstly, this perspective abandon the non-degeneracy of the metric and causality as fundamental principles, hoping to recover them in a certain low-energy limit. Then, it is not clear how these combinatorial techniques could be used in the case where matter fields are added, which are however the essential ingredients in order to produce non trivial observables in a generally covariant approach. Endly, considering the status of the observer in these approaches, it is not clear at all if they really could produce a completely covariant description of quantum gravity. We propose to re-analyse carefully these points. This study leads us to a really covariant description of a set of self-gravitating point masses in a closed universe. This approach is based on a set of observables associated to the measurements accessible to a participant-observer, they manage to capture the whole dynamic in Chern-Simons gravity as well as in true gravity. The Dirac algebra of these observables can be explicitely computed, and exhibits interesting algebraic features related to Poisson-Lie groupoids theory.

Paper Structure

This paper contains 34 sections, 158 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gravity in vacuo
  3. Basic notions on 2+1 deSitter pure gravity
  4. basic notations
  5. equations of motion and action principles
  6. The symmetries of general relativity
  7. first digression : the status of general covariance
  8. off-shell symmetries in hamiltonian gravity
  9. Overview on the reduced phase space of Chern-Simons deSitter gravity in vacuo
  10. Point masses in deSitter gravity
  11. Free relativistic particle on deSitter spacetime
  12. Basic notions on deSitter spacetime
  13. The relativistic particle on $dS_{3}$
  14. Coupling gravity to point masses
  15. the Kerr-deSitter solution
  16. ...and 19 more sections

Figures (3)

  • Figure 1: notations
  • Figure 2: decomposition of $\Sigma _{t}^{*}$ in two connected simply connected closed subsets
  • Figure :