A numerical relativity approach to the initial value problem in asymptotically Anti-de Sitter spacetime for plasma thermalization - an ADM formulation

TL;DR

This work develops an ADM-formulation numerical relativity framework for solving the initial-value problem in asymptotically AdS spacetimes, targeting the dual of boost-invariant, strongly coupled plasma undergoing Bjorken expansion. A Kruskal-like coordinate construction and a vanishing-lapse outer boundary provide a controllable bulk cutoff that enables horizon excision and stable evolution from Far-from-equilibrium initial data informed by Beuf's FG analysis, with the boundary stress tensor and non-equilibrium entropy extracted from near-boundary and horizon data. The authors implement a high-precision Chebyshev spectral method in the radial direction and adaptive Runge–Kutta time stepping, validate the approach across multiple lapse choices, and demonstrate consistent evolution that matches early-time hydrodynamic expectations while revealing a robust correlation between initial non-equilibrium entropy and the onset of hydrodynamics. The study yields detailed insights into the thermalization process, including cooling and reheating behaviors, and establishes a practical numerical pipeline for exploring holographic thermalization and its dependence on initial bulk data.

Abstract

This article studies a numerical relativity approach to the initial value problem in Anti-de Sitter spacetime relevant for dual non-equilibrium evolution of strongly coupled non-Abelian plasma undergoing Bjorken expansion. In order to use initial conditions for the metric obtained in arXiv:0906.4423 we introduce new, ADM formalism-based scheme for numerical integration of Einstein's equations with negative cosmological constant. The key novel element of this approach is the choice of lapse function vanishing at fixed radial position, enabling, if needed, efficient horizon excision. Various physical aspects of the gauge theory thermalization process in this setup have been outlined in our companion article arXiv:1103.3452. In this work we focus on the gravitational side of the problem and present full technical details of our setup. We discuss in particular the ADM formalism, the explicit form of initial states, the boundary conditions for the metric on the inner and outer edges of the simulation domain, the relation between boundary and bulk notions of time, the procedure to extract the gauge theory energy-momentum tensor and non-equilibrium apparent horizon entropy, as well as the choice of point for freezing the lapse. Finally, we comment on various features of the initial profiles we consider.

Figures (9)

• Figure 1: Schematic view of constant time foliations depending on the locus at which the lapse vanishes ($u_0$). Situation A) shows a simulation filling in a triangle covering only a finite interval of boundary time. By pushing the position of bulk cutoff inwards (until one passes the event horizon), one can recover the dynamics on larger and larger regions of boundary. In the situation B) the lapse vanishes at the position of the event horizon on the initial time hypersurface, which ensures maximally long simulation time (theoretically infinite) and is optimal for studying the transition to hydrodynamics in dual gauge theory. In the case C) the simulation penetrates into the black brane/black hole interior revealing the apparent horizon at the cost of breaking down when constant time slices start approaching the curvature singularity. This type of behavior can be seen explicitly in Fig. \ref{['fig:ingoingnullAH']} showing the results of a sample simulation.
• Figure 2: The apparent horizon (black curve) and a radial null geodesic (red curve) sent from the boundary (left edge of the plot) at $\tau=0$ into the bulk for a sample profile (no. 23 from table \ref{['tab:profs']}). This curve coincides with a curve of fixed Eddington-Finkelstein proper time $\tau_{EF}=0$. Background colors correspond to curvature with blue denoting small curvature and red large curvature. The red region inside the apparent horizon denote the neighborhood of the curvature singularity. One can see (and check the numerical factor explicitly) that curvature at the right edge of the plot remains the same during the evolution, in agreement with expectation that vanishing lapse freezes the time flow there. The curvature at the left edge of the plot also remains constant due to imposed AdS asymptotics.
• Figure 3: Determination of the cutoff point $u_0^{(EH)}$ ensuring long evolution time necessary to see thermalization in the boundary theory for a sample profile (no. 23 from table \ref{['tab:profs']}). In order to obtain $u_{0}^{(EH)}$, which is expected to approximate well the position of the event horizon at initial time hypersurface, we anticipate that for large enough time the position of the event horizon coincides with the position of the apparent horizon we consider and shoot backwards in time an outgoing null geodesic (almost) tangent to late time apparent horizon (plotted as thick red curve). Other outgoing null geodesics are plotted as arrowed curves. The apparent horizon is plotted as thick black curve. Background colors, as in Fig. \ref{['fig:ingoingnullAH']} correspond to curvature, from blue (small curvature) to red (large curvature).
• Figure 4: Plots of initial warp-factors as functions of square root of radial position. Thicker curves denote parts of initial data outside event horizon on initial time slices. Numbers in the legend match profile numbers collected in table \ref{['tab:profs']} (color online).
• Figure 5: Dimensionless initial non-equilibrium entropy given by \ref{['eq.dimentropyden']} as a function of the position of Fefferman-Graham singularity showing clear correlation. Both quantities are plotted in the units of the effective temperature at $\tau = 0$. Color code matches figures \ref{['fig:fig_ini_data']} and \ref{['fig:uEH_as_a_func_of_uFG']} (color online).