Hamiltonian General Relativity and the Belinskii, Khalatnikov, Lifshitz Conjecture

TL;DR

This paper reformulates the Belinskii, Khalatnikov, Lifshitz conjecture within a Hamiltonian framework tailored for non-perturbative quantization, aiming to analyze the behavior of space-like singularities in loop quantum gravity.

Contribution

It provides a Hamiltonian formulation of the BKL conjecture suited for non-perturbative quantum analysis, advancing the study of singularities in loop quantum gravity.

Findings

Evidence supports the BKL conjecture near singularities.

A Hamiltonian framework compatible with loop quantum gravity is developed.

This formulation facilitates future quantum analysis of space-like singularities.

Abstract

The Belinkskii, Khalatnikov and Lifshitz conjecture says that as one approaches space-like singularities in general relativity, 'time derivatives dominate over spatial derivatives' so that the dynamics at any spatial point is well captured by an ordinary differential equation. By now considerable evidence has accumulated in favor of these ideas. Starting with a Hamiltonian framework, we provide a formulation of this conjecture in terms of variables that are tailored to non-perturbative quantization. Our formulation serves as a first step in the analysis of the fate of generic space-like singularities in loop quantum gravity.