  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.
