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Lectures in (2+1)-Dimensional Gravity

Steven Carlip

TL;DR

These lectures survey classical and quantum gravity in three spacetime dimensions, emphasizing the quantum mechanics of closed universes and the BTZ black hole. They compare three classical descriptions (geometric structures, Chern-Simons gauge theory, and ADM variables) and three quantization routes (reduced phase space, Chern-Simons quantization, and covariant canonical quantization), plus additional approaches. The discussion highlights how holonomies, moduli, and the mapping class group organize the solution space and guide quantization, and it explores topics such as black hole thermodynamics and topology-changing amplitudes. The results illustrate that 2+1 gravity, though lacking local degrees of freedom, provides a valuable testing ground for conceptual issues in quantum gravity and may shed light on similar questions in 3+1 dimensions.

Abstract

These lectures briefly review our current understanding of classical and quantum gravity in three spacetime dimensions, concentrating on the quantum mechanics of closed universes and the (2+1)-dimensional black hole. Three formulations of the classical theory and three approaches to quantization are discussed in some detail, and a number of other approaches are summarized. An extensive, although by no means complete, list of references is included. (Lectures given at the First Seoul Workshop on Gravity and Cosmology, February 24-25, 1995.)

Lectures in (2+1)-Dimensional Gravity

TL;DR

These lectures survey classical and quantum gravity in three spacetime dimensions, emphasizing the quantum mechanics of closed universes and the BTZ black hole. They compare three classical descriptions (geometric structures, Chern-Simons gauge theory, and ADM variables) and three quantization routes (reduced phase space, Chern-Simons quantization, and covariant canonical quantization), plus additional approaches. The discussion highlights how holonomies, moduli, and the mapping class group organize the solution space and guide quantization, and it explores topics such as black hole thermodynamics and topology-changing amplitudes. The results illustrate that 2+1 gravity, though lacking local degrees of freedom, provides a valuable testing ground for conceptual issues in quantum gravity and may shed light on similar questions in 3+1 dimensions.

Abstract

These lectures briefly review our current understanding of classical and quantum gravity in three spacetime dimensions, concentrating on the quantum mechanics of closed universes and the (2+1)-dimensional black hole. Three formulations of the classical theory and three approaches to quantization are discussed in some detail, and a number of other approaches are summarized. An extensive, although by no means complete, list of references is included. (Lectures given at the First Seoul Workshop on Gravity and Cosmology, February 24-25, 1995.)

Paper Structure

This paper contains 17 sections, 87 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Classical Gravity in 2+1 Dimensions
  3. Geometric Structures
  4. The Chern-Simons Formulation
  5. The ADM Formalism
  6. The Mapping Class Group
  7. The Torus Universe
  8. The (2+1)-Dimensional Black Hole
  9. Quantum Gravity in 2+1 Dimensions
  10. Reduced Phase Space Quantization
  11. Chern-Simons Quantum Theory
  12. Covariant Canonical Quantization
  13. Other Approaches to Quantization
  14. Assorted Topics
  15. Black Hole Thermodynamics
  16. ...and 2 more sections

Figures (4)

  • Figure 1: A flat torus of modulus $\tau$ is represented as a parallelogram with opposite sides identified.
  • Figure 2: The curve $\gamma$ is covered by coordinate patches $U_i$, with transition functions $g_i\!\in\!G$. The composition $g_1{\circ}\dots {\circ} g_n$ is the holonomy of the curve.
  • Figure 3: A Dehn twist of a torus is obtained by cutting along one of the circumferences, rotating one end by $2\pi$, and regluing.
  • Figure 4: A fundamental region for the Eucidean black hole in the upper half-space representation is obtained by identifying the inner and outer hemispheres along radial lines such as $L$.