Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Big-bang limit of $2+1$ gravity and Thurston boundary of Teichmüller space

Puskar Mondal

Abstract

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type $Σ_{p}\times \mathbb{R}$, $p>1$, where $Σ_{p}$ is a closed Riemann surface of genus $p$, in the regime of $2+1$ dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space ($\mathcal{T}Σ_{p}\approx \mathbb{R}^{6p-6}$) of $Σ_{p}$. Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations ($\mathcal{PML}$ $\mathcal{PMF}$), the Thurston boundary of the Teichmüller space.

Big-bang limit of $2+1$ gravity and Thurston boundary of Teichmüller space

Abstract

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type , , where is a closed Riemann surface of genus , in the regime of dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space () of . Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations ( ), the Thurston boundary of the Teichmüller space.

Paper Structure

This paper contains 18 sections, 18 theorems, 158 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Background on Teichmüller space
  4. Homeomorphism between $\mathcal{ML}$, $\mathcal{MF}$, and $\mathcal{QD}$
  5. Harmonic Maps
  6. Einstein flow on $\Sigma_{p}\times \mathbb{R}$
  7. Well-posedness:
  8. Reduced Dynamics
  9. Proof of the Theorem 1
  10. Asymptotic behavior of the solution curve at big-bang and Thurston boundary
  11. Proof of theorem 2
  12. Proof of theorem 3
  13. Concluding Remarks
  14. Acknowledgement
  15. Space of projective laminations as the Thurston boundary of the Teichmüller space
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Sigma_{p}$ be a closed (compact without boundary) Riemann surface of genus $p>1$ and $(\Sigma_{p}\times I,\widetilde{g}), I\subset \mathbb{R}$ be a globally hyperbolic spacetime solving vacuum Einstein equations (eq:eom), which is the maximally globally hyperbolic development of the initial da

Figures (4)

  • Figure 1: The schematics of the conformal dynamics on the configuration space $\mathcal{T}\Sigma_{p}$ ($\approx \mathbb{R}^{6p-6}$). As the big bang ($\tau\to-\infty$) approaches, the physical universe ($\Sigma_{2}$ in the figure) degenerates along a homotopically non-trivial geodesic (red) to produce two connected components $U_{1}$ and $U_{2}$ in the limit. This could also be interpreted as the emergence of the physical universe through the coalescence of $U_{1}$ and $U_{2}$ if one moves away from the big bang in time.
  • Figure 2: Pants decomposition of the hyperbolic surface $\Sigma_{p}$: hyperbolic length of $\gamma_{i}$ together with the twist about the same geodesic $\gamma_{i}$ parametrizes the Teichmüller space.
  • Figure 3: The Straightening map which transforms a singular measured foliation (and the transverse one) to a measured lamination (and respective transverse one).
  • Figure 4: The schematics of the reduced dynamics on the configuration space $\mathcal{T}\Sigma_{p}$ ($\approx \mathbb{R}^{6p-6}$). Each solution curve starts at $\tau\to0$ and approaches the Thurston boundary of $\mathcal{T}\Sigma_{p}$ in $\mathbb{P}$Curr (or equivalently $\mathbb {P}\Omega$) in the limit of big-bang.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • Proposition 3.1
  • Lemma 3.2
  • Corollary 3.1
  • Lemma 4.1
  • Proposition 4.1
  • Proposition 4.2
  • ...and 10 more