Hamiltonian general relativity and the Belinskii-Khalatnikov-Lifshitz conjecture

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 spacelike singularities in loop quantum gravity.