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Improved and Perfect Actions in Discrete Gravity

Benjamin Bahr, Bianca Dittrich

Abstract

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they - although being defined on a triangulation - reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of General Relativity. We construct the perfect action in three dimensions with cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge Calculus with curved simplices, which arises naturally in this context.

Improved and Perfect Actions in Discrete Gravity

Abstract

We consider the notion of improved and perfect actions within Regge calculus. These actions are constructed in such a way that they - although being defined on a triangulation - reproduce the continuum dynamics exactly, and therefore capture the gauge symmetries of General Relativity. We construct the perfect action in three dimensions with cosmological constant, and in four dimensions for one simplex. We conclude with a discussion about Regge Calculus with curved simplices, which arises naturally in this context.

Paper Structure

This paper contains 18 sections, 3 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discretized actions in 1D
  3. Regaining reparametrization invariance
  4. Improving the discrete action $S_d$
  5. Regge Calculus
  6. Continuous preliminaries
  7. Discrete action
  8. Gauge invariance in Regge Calculus
  9. Improved and perfect action in $3D$
  10. Refinement of the Regge action
  11. Perfect action in $3D$
  12. Higher dimensions
  13. Improving the curved Regge action
  14. Perfect actions with flat and curved simplices
  15. Constantly curved subsector
  16. ...and 3 more sections

Key Result

Lemma A.1

For a simplex $\sigma$ of dimension $D$ and constant curvature we have

Figures (2)

  • Figure 1: Coarse triangulation $\mathcal{T}$ consisting of edged $E$, triangles $T$ and tetrahedra $\Sigma$.
  • Figure 2: Fine triangulation $\tau$ consisting of edged $e$, triangles $t$ and tetrahedra $\sigma$.

Theorems & Definitions (3)

  • Lemma A.1
  • Corollary A.1
  • Lemma A.2