A Hamiltonian Formulation of the BKL Conjecture

TL;DR

This paper reformulates the BKL conjecture within a Hamiltonian framework, aiming to facilitate analytical and numerical studies of spacetime singularities in general relativity, with potential applications in loop quantum gravity.

Contribution

It introduces a new Hamiltonian formulation of the BKL conjecture tailored for loop quantum gravity, enhancing analytical and numerical exploration of singularities.

Findings

New Hamiltonian variables suited for the BKL conjecture

Analytical framework compatible with loop quantum gravity

Potential for improved numerical simulations of singularities

Abstract

The Belinskii, Khalatnikov and Lifshitz conjecture \cite{bkl1} posits that on approach to a space-like singularity in general relativity the dynamics are well approximated by `ignoring spatial derivatives in favor of time derivatives.' In \cite{ahs1} we examined this idea from within a Hamiltonian framework and provided a new formulation of the conjecture in terms of variables well suited to loop quantum gravity. We now present the details of the analytical part of that investigation. While our motivation came from quantum considerations, thanks to some of its new features, our formulation should be useful also for future analytical and numerical investigations within general relativity.