Expanding plasmas and quasinormal modes of anti-de Sitter black holes

TL;DR

This work computes gravitational quasinormal modes of the global $AdS_5$-Schwarzschild black hole and uses the AdS/CFT dictionary to describe finite-size, expanding plasmas on Minkowski space as holographic duals of these modes. By decomposing perturbations into tensor, vector, and scalar sectors via $S^3$ harmonics and solving the master equations, the authors extract the holographic stress-tensor perturbations of a conformal soliton flow, including explicit forms and conservation properties. They provide approximate frequency fits, explore case studies that mimic elliptic flow and rapid thermalization, and connect the late-time QNM dynamics to linearized hydrodynamics, highlighting a potential thermalization timescale $\tau_{\rm therm}$ of order $0.3$ fm/$c$ for temperatures relevant to RHIC. While the model relies on ${\cal N}=4$ SYM and conformal symmetry, its results offer qualitative insights into how strongly coupled, finite-size plasmas relax and how collective flow may emerge in a holographic setting.

Abstract

We compute the gravitational quasinormal modes of the global AdS_5-Schwarzschild solution. We show how to use the holographic dual of these modes to describe a thermal plasma of finite extent expanding in a slightly anisotropic fashion. We compare these flows with the behavior of quark-gluon plasmas produced in relativistic heavy ion collisions by estimating the elliptic flow coefficient and the thermalization time.

Figures (6)

• Figure 1: Four complementary views of the GAdSBH background (\ref{['GAdSBHmetric']}). The first three are cartoons based on the Penrose diagram for global $AdS_5$. The fourth is a true Penrose diagram of GAdSBH. See the main text for more detailed explanations.
• Figure 2: The zeroes of $s(y)$ are shown as crosses in the complex $y$ plane. The red cross at the origin is a single zero; all others are double zeroes. The region (i) is where a power series expansion around $y=0$ was used in computing $\psi(y)$. The line (ii) is where $\psi(y)$ was computed by numerical integration, seeded with initial conditions at the boundary with region (i). The region (iii) is where $\psi(y)$ was computed using a power series around $y=1$. The powers in this latter series are half-integral, so there is a branch cut which may be located as shown by (iv).
• Figure 3: The real (left) and imaginary parts (right) of the low-lying quasinormal frequencies for the scalar modes with $3 \leq n \leq 11$ (we have set $L=1$). The green diamond-shaped points correspond to $\rho_H = 6$, the red triangular ones to $\rho_H = 13$, and the blue star-shaped ones to $\rho_H = 20$. The dashed line shows the predictions from linearized hydrodynamics, as explained in section \ref{['HYDRO']}. The fit (\ref{['FitScalar']}) was done for the leftmost four points shown ($3 \leq n \leq 6$), but simultaneously for all three values of $\rho_H$---so $12$ data points altogether. These points are the ones inside the inset frame in each plot. The fit (\ref{['FitScalarAgain']}) comes from only the $\rho_H=20$ points.
• Figure 4: Snapshots of the energy density of the flow specified by (\ref{['ExpandEps']}) with $Q_1 = 8 \times 10^4$. The $z$ coordinate measures position along the beamline (where $\theta = 0$ or $\pi$), while the $x$ coordinate is transverse. The flow is azimuthally symmetric, which is to say symmetric around the $z$ axis. Note that the scales of the $x$ and $z$ axes change from frame to frame.
• Figure 5: Snapshots of the energy density of the flow specified by (\ref{['ExpandScalarEps']}) with $Q_2 = -10^4$. The $z$ coordinate measures position along the beamline (where $\theta = 0$ or $\pi$), while the $x$ coordinate is transverse. The flow is azimuthally symmetric, which is to say symmetric around the $z$ axis. Note that the scales of the $x$ and $z$ axes change from frame to frame.