Simulation of Asymptotically AdS5 Spacetimes with a Generalized Harmonic Evolution Scheme

TL;DR

This work develops a generalized harmonic evolution scheme to solve the Einstein equations in asymptotically AdS$_5$ spacetimes, enforcing a timelike AdS boundary via regularized variables and constraint-damping dynamics. The authors demonstrate stable, convergent Cauchy evolution with $SO(3)$ symmetry, generating strongly distorted black holes from scalar collapse and analyzing their quasi-normal ringdown, while extracting a boundary CFT stress tensor that behaves like a viscous, conformal fluid with $\\ ext{ε}=3P$. They further map bulk solutions to a Minkowski boundary patch to model heavy-ion collision-like flows. Collectively, the results establish a robust link between nonlinear bulk dynamics and boundary hydrodynamics, including higher-order transport, and demonstrate the potential to explore far-from-equilibrium holographic phenomena in AdS/CFT.

Abstract

Motivated by the gauge/gravity duality, we introduce a numerical scheme based on generalized harmonic evolution to solve the Einstein field equations on asymptotically anti-de Sitter (AdS) spacetimes. We work in global AdS5, which can be described by the (t,r,χ,θ,φ) spherical coordinates adapted to the R{\times}S3 boundary. We focus on solutions that preserve an SO(3) symmetry that acts to rotate the 2-spheres parametrized by θ,φ. In the boundary conformal field theory (CFT), the way in which this symmetry manifests itself hinges on the way we choose to embed Minkowski space in R{\times}S3. We present results from an ongoing study of prompt black hole formation via scalar field collapse, and explore the subsequent quasi-normal ringdown. Beginning with initial data characterized by highly distorted apparent horizon geometries, the metrics quickly evolve, via quasi-normal ringdown, to equilibrium static black hole solutions at late times. The lowest angular number quasi-normal modes are consistent with the linear modes previously found in perturbative studies, whereas the higher angular modes are a combination of linear modes and of harmonics arising from non-linear mode-coupling. We extract the stress energy tensor of the dual CFT on the boundary, and find that despite being highly inhomogeneous initially, it nevertheless evolves from the outset in a manner that is consistent with a thermalized N=4 SYM fluid. As a first step towards closer contact with relativistic heavy ion collision physics, we map this solution to a Minkowski piece of the R{\times}S3 boundary, and obtain a corresponding fluid flow in Minkowski space.

Figures (22)

• Figure 1: The conformal diagram of Minkowski space. The boundary consists of the point at spatial infinity $i^0$, and the null surfaces at future null infinity $\mathcal{J}^+$ and past null infinity $\mathcal{J}^-$. In this compactification, future time-like infinity $i^+$ and past time-like infinity $i^-$ are represented by points. Dashed lines are constant $t$ surfaces, and solid lines are constant $r$ surfaces.
• Figure 2: The conformal diagram of anti-de Sitter space. The boundary consists of the time-like surface $\mathcal{J}$; past and future time-like infinity are represented by the points $i^-$ and $i^+$, respectively. Conventions used here are the same as those in Fig. \ref{['fig:minkowski']}.
• Figure 3: Convergence factors (\ref{['eq:qconv']}) for the $\bar{g}_{\rho\rho}$ grid function, constructed from a simulation run at 4 different resolutions; the highest resolution run has mesh spacing $h/2$. Here the $L^2$-norm of the convergence factors are taken over the entire grid. The trends indicate that this grid function is converging to second-order. Other grid functions exhibit similar trends.
• Figure 4: Convergence factors for the independent residual (\ref{['eq:qires']}), constructed from simulations run at 4 different resolutions; the highest resolution run has mesh spacing $h/2$. At each point on the grid an $L^\infty$ norm is taken over all components of the independent residual, and what is shown here is then the $L^2$-norm of this over the entire grid. The trends in this plot indicate second-order convergence.
• Figure 5: Maximum conformal factor vs. maximum amplitude of initial scalar matter distribution, with Gaussian profile (\ref{['eqn:timesymmetryenergydensity']}) with $A=\phi_{\text{max}}$, $\delta=0.2$ (with AdS scale $L=1$), $w_x=w_y=1$ and $R_0=0$. We differentiate between "strong-field" and "weak-field" data based on whether there is a trapped surface present on the initial slice or not, respectively. (Though of course this distinction is somewhat arbitrary, particularly since subsequent evolution of weak-field data could eventually result in black hole formation, as argued in Bizon:2011gg even for arbitrarily small amplitude initial data.) The value of $\phi_{max}$ beyond which trapped surfaces are found in the initial data is indicated by the dashed vertical line. The open circles denote numerical solutions, while the solid lines are fits to the data, as shown.