Hamiltonian formulation of the Belinskii-Khalatnikov-Lifshitz conjecture

The Belinskii, Khalatnikov, and Lifshitz conjecture [V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Adv. Phys. 19, 525 (1970)] posits that on approach to a spacelike singularity in general relativity the dynamics are well approximated by “ignoring spatial derivatives in favor of time derivatives.” In A. Ashtekar, A. Henderson, and D. Sloan, Classical Quantum Gravity 26, 052 001 (2009), 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.

Figure 1

Evolution of each of the three $C_I$ in vacuum, starting from a point near the Kasner fixed point surface. Initial data are $C_1=1\times {10}^{-7}$, $C_2=2\times {10}^{-7}$, $C_3=2.2\times {10}^{-7}$, $P_1=0.4$, $P_2=0.8$, $P_3=0.0686$ ($C_1$ in blue, $C_2$ in green, $C_3$ in red). Since none of the initial $C_I$ vanish, as expected from analytical considerations, there is a series of separate Taub transitions between Kasner states. Time has been rescaled by a power of $1/4$ to allow multiple transitions to be shown on a single plot.

Figure 2

Evolution of each of the three $P_I$ in vacuum, starting from a point near the Kasner fixed point surface, with the same initial data as in Fig. 1 ($P_1$ in blue, $P_2$ in green, $P_3$ in red). The largest eigenvalue, $P_2$, transits first. After this transition, $P_1$ becomes the largest eigenvalue, now making $C_1$ unstable. In time all three $P_I$ tend to zero. In terms of parameters $p_i$ used in the Kasner metric (5.14), the initially expanding direction $p_2$ starts contracting at the end of the transition and initially contracting $p_1$ starts expanding.

Figure 3

Evolution of each of the three $C_I$ in vacuum, starting from a point away from the Kasner fixed point surface. Initial data are $C_1=0.7431$, $C_2=0.3922$, $C_3=0.6555$, $P_1=0.1712$, $P_2=0.7060$, $P_3=-0.3140$ ($C_1$ in blue, $C_2$ in green, $C_3$ in red). Even though we start out far from the Kasner surface (where all $C_I$ vanish), dynamics drives the state to the Kasner surface. Again, time has been rescaled by a power of $1/4$ to allow multiple transitions to be shown on a single plot.