Boost invariant flow, black hole formation, and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory

TL;DR

Using gauge/gravity duality, the paper studies how a boost-invariant, strongly coupled N=4 SYM plasma is produced and relaxes when the boundary geometry is time-dependent and anisotropic. The authors solve the five-dimensional Einstein equations with a negative cosmological constant under a boost-invariant ansatz, and compute the field theory stress tensor from the near-boundary expansion. They find that the entire production and relaxation process completes within a time comparable to the inverse local temperature, and that the onset of hydrodynamics is controlled by exponentially decaying non-hydrodynamic modes rather than the breakdown of the hydrodynamic gradient expansion. This work provides a controlled holographic framework for genuinely far-from-equilibrium dynamics and offers a path toward more complex, less symmetric numerical relativity problems in AdS/CFT.

Abstract

Using gauge/gravity duality, we study the creation and evolution of boost invariant anisotropic, strongly coupled N = 4 supersymmetric Yang-Mills plasma. In the dual gravitational description, this corresponds to horizon formation in a geometry driven to be anisotropic by a time-dependent change in boundary conditions.

Figures (6)

• Figure 1: A spacetime diagram depicting several stages of the evolution of the field theory state in response to the changing spatial geometry. At proper time $\tau = \tau_i$, the $4d$ spacetime geometry starts to deform. The region of spacetime where the geometry undergoes time-dependent deformation is shown as the red region, labeled I. After proper time $\tau = \tau_f$, the deformation in $4d$ spacetime geometry turns off and the field theory state is out of equilibrium. From proper time $\tau_f$ to $\tau_*$, shown as the yellow region labeled II, the system is significantly anisotropic and not yet close to local equilibrium. After time $\tau_*$, shown in green and labeled III, the system is close to local equilibrium and the evolution of the stress tensor is well-described by hydrodynamics.
• Figure 2: The congruence of outgoing radial null geodesics. The surface coloring displays $A/r^2$. Before time $\tau_i = 1/4$ this quantity equals one. The excised region lies inside the apparent horizon, which is shown by the dashed magenta line. The geodesic shown as a solid blue line is the event horizon; it separates geodesics which escape to the boundary from those which cannot escape.
• Figure 3: Area of the event horizon and apparent horizon, per unit rapidity, as a function of proper time $\tau$. The growth of the apparent horizon area, shown by the magenta dotted line, is causally connected to the changing boundary geometry. In contrast, the growth of the event horizon area, shown as a solid blue line, is non-zero at arbitrarily early times, long before the boundary geometry has started to change.
• Figure 4: Close-up view of the event horizon and apparent horizon areas, per unit rapidity, as a function of proper time $\tau$, together with their corresponding asymptotic expressions (\ref{['asmarea']}). Both horizon areas are very well approximated by their asymptotic expansions, at second order in gradients, after time $\tau_f = 2.25$ when the boundary geometry becomes flat. Note the rather abrupt change in the growth of the apparent horizon area at $\tau_f$.
• Figure 5: Energy density, longitudinal and transverse pressure, all divided by $N_{\rm c}^2/2\pi^2$, as a function of time for $c = -1$ (left) and $c = + 1$ (right). The energy density and pressures start off at zero at time $\tau_i = 1/4$ when the system is in the vacuum state. During the interval of time $\tau \in (\tau_i,\tau_f) = (0.25,2.25)$, the $4d$ geometry is changing and doing work on the field theory state. After time $\tau_f$ the deformation in the geometry turns off and the field theory state subsequently relaxes onto a hydrodynamic description. The smooth tails in both plots occur during this regime. At late times, from top to bottom, the three curves (in both plots) correspond to the energy density $\mathcal{E}$, transverse pressure $\mathcal{P}_\perp$, and longitudinal pressure $\mathcal{P}_\|$.