[Paper Review] Convergence of hydrodynamics in rapidly spinning strongly coupled plasma

Casey Cartwright, Markus A. G. Amano+3 more|arXiv (Cornell University)|Dec 20, 2021

High-Energy Particle Collisions Research4 citations

TL;DR

This paper investigates the convergence of relativistic hydrodynamics in a rapidly spinning, strongly coupled $υ=4$ Super-Yang-Mills plasma using holography. It shows that the radius of convergence of the hydrodynamic gradient expansion remains finite and can even increase with higher angular momentum, indicating enhanced hydrodynamic validity in highly vortical plasmas, with transport coefficients analytically related to their non-rotating counterparts via closed-form expressions involving the rotation parameter $a$. The results suggest hydrodynamics remains applicable even under large vorticity gradients, offering new insights for modeling heavy-ion collisions with significant angular momentum.

ABSTRACT

We compute the radius of convergence of the linearized relativistic hydrodynamic expansion around a non-trivially rotating strongly coupled N=4 Super-Yang-Mills plasma. Our results show that the validity of hydrodynamics is sustained and can even get enhanced in a highly vortical quark-gluon plasma, such as the one produced in heavy-ion collisions. The hydrodynamic dispersion relations are computed using a rotating background that is an analytic solution of the ideal hydrodynamic equations of motion with non-vanishing angular momentum and large vorticity gradients, giving rise to a particular boost symmetry. Analytic equations for the transport coeffcients of the rotating plasma as a function of their values in a plasma at rest are given.

Motivation & Objective

• To determine the range of validity of linearized relativistic hydrodynamics in a rapidly rotating, strongly coupled plasma.
• To investigate how large vorticity gradients—induced by high angular momentum—affect the convergence of the hydrodynamic gradient expansion.
• To compute transport coefficients (e.g., shear viscosity, diffusion, sound modes) in a rotating plasma and relate them analytically to their values in a non-rotating plasma.
• To test whether boost symmetry in the rotating frame can be used to map hydrodynamic modes and spectral curves, enabling analytical computation of convergence properties.
• To provide a holographic proxy for the behavior of the quark-gluon plasma in heavy-ion collisions, where large vorticity is observed experimentally.

Proposed method

• Use of the gauge/gravity duality (holography) to model the strongly coupled $υ=4$ SYM plasma as a rotating black hole in asymptotically AdS5 spacetime with non-zero angular momentum.
• Construction of a rotating background metric with large vorticity gradients and non-trivial boost symmetry, derived via diffeomorphism from the standard rotating black hole solution.
• Computation of quasinormal modes (QNMs) of the metric perturbations to extract hydrodynamic dispersion relations and transport coefficients.
• Application of spectral curve analysis to determine the radius of convergence of the hydrodynamic series expansion in the gradient expansion.
• Derivation of analytic relations between transport coefficients in the rotating frame and their values at rest, using the transformation properties of the rotating frame and the boost symmetry.
• Cross-verification of results by computing QNMs directly in the rotating fluid frame and confirming agreement with transformed modes from the non-rotating case.

Experimental results

Research questions

• RQ1 Does the radius of convergence of the hydrodynamic gradient expansion remain finite in a rapidly rotating, strongly coupled plasma?
• RQ2 Can the hydrodynamic description remain valid or even be enhanced under large vorticity gradients induced by high angular momentum?
• RQ3 How do transport coefficients such as shear viscosity, diffusion, and sound attenuation change in a rotating plasma compared to a non-rotating one?
• RQ4 Can the transport coefficients in the rotating frame be expressed analytically in terms of their values in the non-rotating plasma and the rotation parameter $a$?
• RQ5 Does the presence of a non-trivial boost symmetry in the rotating frame allow for a systematic mapping of hydrodynamic modes and spectral curves?

Key findings

• The radius of convergence of the hydrodynamic gradient expansion remains finite and increases with increasing angular momentum, indicating that hydrodynamics can be more robust in rapidly rotating plasmas.
• The shear diffusion mode dispersion relation in the rotating frame is given by $\nu(j) = -aj - i\frac{1}{2}(1-a^{2})^{3/2}j^{2} + \mathcal{O}(j^{3})$, showing modified propagation and damping due to rotation.
• The sound mode dispersion relation is $\nu(j) = \frac{\pm 1 - \sqrt{3}a}{\sqrt{3} \mp a}j - i\sqrt{3}\frac{(1-a^{2})^{3/2}}{(|\sqrt{3}-a|)^{3}}j^{2} + \mathcal{O}(j^{3})$, with two distinct sound modes emerging due to rotation.
• The longitudinal diffusion coefficient is $\mathcal{D}_{||}(a) = \mathcal{D}_{0}(1-a^{2})^{3/2}$, where $\mathcal{D}_{0} = 1/2$ at $a=0$, showing enhanced diffusion at moderate rotation.
• The two shear viscosities are $\eta_{\perp}(a) = \eta_{0}/\sqrt{1-a^{2}}$ and $\eta_{||}(a) = \eta_{0}\sqrt{1-a^{2}}$, with $\eta_{0} = N^{2}\pi T_{0}^{3}/8$, indicating anisotropic viscous behavior.
• The sound speeds are $v_{s,\pm} = v_{s,0}\frac{\sqrt{3}a \pm 1}{1 \pm a/\sqrt{3}}$ and the sound attenuation coefficients are $\Gamma_{\pm}(a) = \Gamma_{0}\frac{(1-a^{2})^{3/2}}{(1 \pm a/\sqrt{3})^{3}}$, with $v_{s,0} = 1/\sqrt{3}$ and $\Gamma_{0} = 1/3$ at $a=0$.