Trapping, chaos and averaging in bubbling AdS spaces

Abstract

We discuss chaos and ensemble averaging in 1/2 BPS bubbling $AdS$ spaces of Lin, Lunin and Maldacena (LLM) by studying trapped and escaping null geodesics and estimating their decay rates. We find typical chaotic scattering behavior and confirm the Pesin relation between escape rates, Lyapunov exponents and Kolmogorov-Sinai entropy. On the other hand, for geodesics in coarse-grained (grayscale) LLM geometries (which exhibit a naked singularity) chaos is strongly suppressed, which is consistent with orbits and escape rates averaged over microscopic backgrounds. Also the singularities in these grayscale geometries produce an attractive potential and have some similarities to black hole throats trapping geodesics for a long time. Overall, averaging over the ensembles of LLM geometries brings us closer toward the typical behavior of geodesics in black hole backgrounds, but some important differences remain, in particular the existence of a threshold timescale when the averaging fails.

Figures (20)

• Figure 1: Disk+ring configuration in the LLM plane (A) and the null geodesic (magenta) in the whole 3D space ($x,y,\xi$) (B) in the same background.
• Figure 2: The 3-disk configuration in the LLM plane (A) and a null geodesic (magenta) in the whole 3D space ($x,y,\xi$) (B) in the same background. Notice how the trajectory can jump between disks after being trapped near one for a while to become trapped by another one.
• Figure 3: Poincare section $P_{\xi} = 0$ in $P_r-r$ plane for two sets of null geodesic orbits in the LLM geometry sourced by disk + ring 'source' pattern in a plane. Integrals of motion are $\{ J_- = 0.01, J_+ = 0.01, E=0.1 \}$ and $P_{\phi} = 0.002$. The panels (A) and (B) differ just by boundary conditions: in (B) we see trapping, where orbits explore a smaller volume in phase space, while remaining equally chaotic.
• Figure 4: Positive principal Lyapunov exponents for the disk+ring system, with the same initial conditions and integrals of motion as in Fig. \ref{['figPS_disc_ring_PrRr']}. The sticky orbits of Fig. \ref{['figPS_disc_ring_PrRr']}(B) have about the same Lyapunov exponent (even somewhat larger in this case) as the orbits in Fig. \ref{['figPS_disc_ring_PrRr']}(A) which are obviously chaotic.
• Figure 5: Decay of the number of trapped orbits as a function of proper time $N(\tau)$ (red full lines) for the disk+ring geometry, for four cells of initial conditions of decreasing size, given by $r^{(0)}=20,P_r^{(0)}=-0.02$, $P_\theta^{(0)}=-0.001$, $\phi^{(0)}=\pi/8$ and $\theta^{(0)}$ from the interval $\theta^{(00)}\pm\Delta\theta$, with $\Delta\theta=10^{-3},10^{-7},10^{-8},10^{-9}$ (A to D respectively). The black dashed lines yield the best linear fit, the slope defining the escape rate $\gamma$ from $N(\tau)\sim\exp(-\gamma\tau)$. We detect four distinct populations of orbits, corresponding to the four escape rates on the fits: $\gamma\approx 0.010,0.0038,0.0043,0.0057$.