Dynamical black holes and expanding plasmas

TL;DR

The paper analyzes time-dependent holographic duals to expanding plasmas, focusing on BF (Bjorken flow) and CS (conformal soliton) geometries. The BF dual is constructed perturbatively at late proper time in ingoing EF coordinates, revealing a regular horizon and transport coefficients like $rac{ ilde{ ext{eta}}}{ ilde{s}}=rac{1}{4\pi}$; for CS, the dual is exact and corresponds to a Poincaré patch of Schwarzschild-AdS, where the event horizon diverges while the apparent horizon remains finite, highlighting that CFT entropy is more naturally associated with the apparent horizon in dynamical settings. The CS example shows a sharp distinction between event and apparent horizons and argues that the boundary entropy tracks the apparent horizon rather than the event horizon, while BF illustrates a near-equilibrium, hydrodynamic regime where the two horizons track each other closely. The study thus advocates using the apparent horizon as the quasi-local entropy carrier in far-from-equilibrium holographic plasmas and clarifies how foliation and global structure influence the bulk-boundary entropy correspondence.

Abstract

We analyse the global structure of time-dependent geometries dual to expanding plasmas, considering two examples: the boost invariant Bjorken flow, and the conformal soliton flow. While the geometry dual to the Bjorken flow is constructed in a perturbation expansion at late proper time, the conformal soliton flow has an exact dual (which corresponds to a Poincare patch of Schwarzschild-AdS). In particular, we discuss the position and area of event and apparent horizons in the two geometries. The conformal soliton geometry offers a sharp distinction between event and apparent horizon; whereas the area of the event horizon diverges, that of the apparent horizon stays finite and constant. This suggests that the entropy of the corresponding CFT state is related to the apparent horizon rather than the event horizon.

Figures (4)

• Figure 1: Illustration of the horizons for the Bjorken flow metrics $g^{(k)}$ at various orders in the perturbation expansion. The event horizons are the solid curves while the apparent horizons are the dashed curves. The locations of the horizons of course should only be trusted at late times as indicated in the figure.
• Figure 2: Illustration of Poincaré coordinates superposed on conformally compactified AdS. The surfaces $t=0,1,5$ (left), $z=1,5$ (middle), and $x=0,1,5$ (right) are plotted, with colour-coding blue for 0, green for 1, and red for 5. To guide the eye, surperposed are also the boundary of the $\mathscr I_{CS}^+$ corresponding to $t= \pm \infty$ and the $t=0$ boundary slice (black curves).
• Figure 3: Plot of the event horizon of the CS spacetime. For ease of visualization we have also plotted the location of the event horizon of the global Schwarzschild-AdS$_{}$ black hole.
• Figure 4: Behaviour of the Poincaré time along the line of caustics $t_c(\ell)$. We have plotted here the situation for a sampling of BTZ horizon size. As explained in the text, as $r_+ > 1$ (recall that we normalize $L_{\text{AdS}} = 1$) the allowed domain in $\ell$ shrinks exponentially.