Toward the AdS/CFT Gravity Dual for High Energy Collisions. 3.Gravitationally Collapsing Shell and Quasiequilibrium

Abstract

The equilibration of matter and onset of hydrodynamics can be understood in the AdS/CFT context as a gravitational collapse process, in which "collision debris" create a horizon. In this paper we consider the simplest geometry possible, a flat shell (or membrane) falling in the holographic direction toward the horizon. The metric is a combination of two well known solutions: thermal AdS above the shell and pure AdS below, while motion of the shell is given by the Israel junction condition. Furthermore, when the shell motion can be considered slow, we were able to solve for two-point functions of all boundary stress tensor and found that an observer on the boundary sees a very peculiar $quasiequilibrium$: while the average stress tensor $<T_{μν>}$ contains the equilibrium plasma energy and pressure at all times, the spectral densities of the correlators(related with occupation probabilities of the modes) reveal additional oscillating terms absent in equilibrium. This is explained by the "echo" phenomenon, a partial return of the field coherence at certain "echo" times.

Figures (7)

• Figure 1: The shell trajectory as a function of time. It starts at rest at $z=z_0$ with a constant acceleration, followed by a constant falling and eventually approaches the horizon in a exponential fashion. The parameter we choose are ${\kappa}_5^2 p=1$ and $z_h=1.6$
• Figure 2: (color online)The spectral density of transverse stress $\chi_{xy,xy}$ in unit of $\pi^2N_c^2T^4$, at $q=0$ left, $q=1.5$ middle and $q=4.5$ right. Plotted are spectral densities at different stages of thermalization: black asterisk($u_m=0.1$), red box($u_m=0.3$), blue circle($u_m=0.5$), green cross($u_m=0.7$), brown diamond($u_m=0.9$). The thermal spectral density is also included(pink point) for comparison. The parameter we will keep using from here on is ${\kappa}_5^2 p=1$
• Figure 3: (color online)The relative deviation $R$ at $q=0$ left, $q=1.5$ middle and $q=4.5$ right. Different stages of thermalization are indicated by: black asterisk($u_m=0.1$), red box($u_m=0.3$), blue circle($u_m=0.5$), green cross($u_m=0.7$), brown diamond($u_m=0.9$). As $u_m$ approaches 1, i.e. the medium evolves to equilibrium, the oscillation decreases in amplitude and increases in frequency, thus the spectral density relaxes to thermal one
• Figure 4: (color online)The spectral density of momentum density $\chi_{tx,tx}$ in unit of $\pi^2N_c^2T^4$ at $q=1.5$ left and $q=4.5$ right. Plotted are spectral densities at different stages of thermalization: black asterisk($u_m=0.1$), red box($u_m=0.3$), blue circle($u_m=0.5$), green cross($u_m=0.7$), brown diamond($u_m=0.9$). The thermal spectral density is also included(pink point) for comparison
• Figure 5: (color online)The relative deviation $R$ at $q=1.5$ left and $q=4.5$ right. Different stages of thermalization are indicated by: black asterisk($u_m=0.1$), red box($u_m=0.3$), blue circle($u_m=0.5$), green cross($u_m=0.7$), brown diamond($u_m=0.9$). As $u_m$ approaches 1, i.e. the medium evolves to equilibrium, the oscillation decreases in amplitude and increases in frequency, thus the spectral density relaxes to thermal one