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Dirac Observables for Gowdy Cosmologies regular at the Big Bang

Max Niedermaier, Mahdi Sedighi Jafari

TL;DR

The paper constructs infinite families of Dirac observables for Gowdy cosmologies with two commuting spacelike Killing vectors, valid off-shell and without gauge fixing, and proves their regularity at the Big Bang. These observables are built from a gauge-free Lax pair and a transition matrix, yielding nonlocal yet gauge-invariant quantities that can be matched to simpler Carroll-type observables. In the $T^3$ Gowdy case, a periodic extension with a gauge-invariant trace resolves subtleties from the quasi-periodic structure, while in the $ eal imes T^2$ case a direct construction suffices. The authors introduce a velocity-dominated (VD) gravity system as the leading-order limit and show that full Gowdy observables admit an anti-Newtonian expansion whose leading terms reproduce the VD observables, thus providing an off-shell generalization of the Asymptotic Velocity Domination property. Together, these results establish a concrete framework for gauge-invariant off-shell analyses in Gowdy cosmologies with potential for quantum extensions.

Abstract

Gowdy cosmologies are exact, spatially inhomogeneous solutions of the vacuum Einstein equations which describe nonlinear gravitational waves coalescing at the Big Bang singularity. With toroidal spatial sections they provenly have the Asymptotic Velocity Domination property, in that close to the Big Bang dynamical spatial gradients fade out and the dynamics is governed by a Carroll-type gravity theory. Here we construct an infinite set of Dirac observables for Gowdy cosmologies, valid off-shell, strongly, and without gauge fixing. These observables stay regular at the Big Bang and can be matched to much simpler Dirac observables of the Carroll-type gravity theory. Conversely, in an adapted foliation there is a systematic anti-Newtonian expansion (in inverse powers of the reduced Newton constant) of the full Dirac observables whose leading terms are the Carroll ones. In particular, this provides an off-shell generalization of the Asymptotic Velocity Domination property.

Dirac Observables for Gowdy Cosmologies regular at the Big Bang

TL;DR

The paper constructs infinite families of Dirac observables for Gowdy cosmologies with two commuting spacelike Killing vectors, valid off-shell and without gauge fixing, and proves their regularity at the Big Bang. These observables are built from a gauge-free Lax pair and a transition matrix, yielding nonlocal yet gauge-invariant quantities that can be matched to simpler Carroll-type observables. In the Gowdy case, a periodic extension with a gauge-invariant trace resolves subtleties from the quasi-periodic structure, while in the case a direct construction suffices. The authors introduce a velocity-dominated (VD) gravity system as the leading-order limit and show that full Gowdy observables admit an anti-Newtonian expansion whose leading terms reproduce the VD observables, thus providing an off-shell generalization of the Asymptotic Velocity Domination property. Together, these results establish a concrete framework for gauge-invariant off-shell analyses in Gowdy cosmologies with potential for quantum extensions.

Abstract

Gowdy cosmologies are exact, spatially inhomogeneous solutions of the vacuum Einstein equations which describe nonlinear gravitational waves coalescing at the Big Bang singularity. With toroidal spatial sections they provenly have the Asymptotic Velocity Domination property, in that close to the Big Bang dynamical spatial gradients fade out and the dynamics is governed by a Carroll-type gravity theory. Here we construct an infinite set of Dirac observables for Gowdy cosmologies, valid off-shell, strongly, and without gauge fixing. These observables stay regular at the Big Bang and can be matched to much simpler Dirac observables of the Carroll-type gravity theory. Conversely, in an adapted foliation there is a systematic anti-Newtonian expansion (in inverse powers of the reduced Newton constant) of the full Dirac observables whose leading terms are the Carroll ones. In particular, this provides an off-shell generalization of the Asymptotic Velocity Domination property.
Paper Structure (20 sections, 220 equations, 2 figures)

This paper contains 20 sections, 220 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two spacelike Killing vector reduction of Einstein gravity
  3. The Hamiltonian action and its gauge symmetries
  4. Linear system without gauge fixing and conserved currents
  5. Dirac observables: strong, off-shell, and without gauge-fixing
  6. $\mathbb{R} \times T^2$ Gowdy
  7. $T^3$ Gowdy
  8. The velocity dominated Gowdy System
  9. Hamiltonian action and its gauge symmetries
  10. Dirac observables for VD Gowdy
  11. Anti-Newtonian expansion of Dirac observables
  12. On-shell specialization and relation to AVD
  13. Conclusions
  14. Two Killing vector reduction and its VD counterpart
  15. Reduced Lagrangian action and its symmetries
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Branch cut in the complex $\theta$-plane as seen by $L_\mu^H(\theta)$ and ${\cal J}_\mu^H(\theta)$, $\mu=0,1$.
  • Figure 2: Illustration of the commutativity of the operations "taking the anti-Newtonian limit" and "taking the two-Killing vector reduction".