A family of three maximally symmetric boost-invariant flows in relativistic hydrodynamics

TL;DR

This paper develops a unified framework to construct boost-invariant, conformal, dissipative relativistic flows by applying the Gubser–Yarom geometric procedure on $dS_3 \times \mathbb{R}$, using three maximal slicings corresponding to ISO$(2)$, SO$(3)$, and SO$(2,1)$. After Weyl transforming to flat Minkowski space, it yields three distinct solutions: the well-known Bjorken flow ($\kappa=0$), the Gubser flow ($\kappa=+1$), and a novel hyperbolic flow ($\kappa=-1$) featuring a finite-radius droplet whose edge exhibits free-streaming-like expansion. The paper provides explicit energy-density profiles $\varepsilon_\kappa(\tau, r)$ and four-velocity fields, and introduces first-order viscous corrections with $\eta = H_0 \varepsilon^{3/4}$, giving temperature profiles $T_0$, $T_{+1}$, and $T_{-1}$ that incorporate hypergeometric functions. The new $\kappa=-1$ solution offers a causal, boost-invariant model of a finite plasma droplet with a shockwave-like edge, potentially relevant to late-time heavy-ion collision dynamics and glasma phenomenology, while also serving as a benchmark for numerical hydrodynamics and future kinetic or holographic studies.

Abstract

I discuss the constructions of boost-invariant dissipative conformal hydrodynamic flows by elaborating on the geometric procedure by Gubser and Yarom, which starts from a static, maximally symmetric flow on dS$_3\times\mathbb{R}$. Three foliations of dS$_3$ preserve three-dimensional non-Abelian isometry groups, namely, the flat ISO(2)-invariant, the spherical (closed) SO(3)-invariant, and the hyperbolic (open) SO(2,1)-invariant slicings. I show that the fluids that preserve these symmetries, after they have been Weyl transformed to flat spacetime, give rise to three physically distinct and boost-invariant solutions of the relativistic dissipative Navier-Stokes equations: the well-known and widely studied Bjorken and Gubser flows, and a seemingly thus far unexplored solution that arises from the hyperbolic slicing of dS$_3$. The new solution combines the radial expansion characteristic of the Gubser flow with the late-proper-time applicability of Bjorken's solution, and features a finite, radially bounded droplet whose expanding edge resembles a free-streaming shockwave.

Figures (3)

• Figure 1: Velocity vector $(u^r,u^\tau)$ of the Gubser's $\kappa = +1$ ( left panel) and the $\kappa = -1$ ( right panel) solutions shown in the $(r,\tau)$ plane. Colours indicate the norm of the velocity vector, with its magnitude increasing from blue to red. White arrows show the size and the direction of $u^\mu$. White lines show contours of equal transverse $v_\perp$, running from 0 to 1 in steps of 0.1. For the Gubser flow ($\kappa = +1$), $q=2$, and for the $\kappa = -1$ flow, $q=1$. The latter solution exists inside the $r < \tau - 1/q$ future lightcone, with $u^\mu$ diverging at the edge.
• Figure 2: Evolution of the radial temperature profiles for the $\kappa = +1$ (Gubser flow) and the $\kappa = -1$ solutions. Top:$T_{+1}$ from Eq. \ref{['T_G']} is plotted for increasing $\tau$ from $1$ (red) to $6$ (blue) in spacings of $0.2$. From left to right, the three plots show different values of the shear viscosity parameter $H_0 = \{0, 0.5, 1\}$, with the remaining parameters set to $\mathcal{T}_{+1} = 1$ and $q=2$. Bottom:$T_{-1}$ from Eq. \ref{['T_H']}, with $\tau$ running from $1.1$ (red) to $6.1$ (blue) in spacings of $0.2$. From left to right, $H_0 = \{0, 0.3, 0.5\}$, with $\mathcal{T}_{-1} = 1$ and $q=1$.
• Figure 3: Plots of $\tilde{K}_{\pm 1}$ that quantifies the behaviour of the temperature gradients. Left: Gubser's $\kappa = +1$ solution with $\mathcal{T}_{+1} = 1$, $q=2$ and $H_0 = 0.5$, plotted as a function of $\tau$, for $r$ running from 1 (red) to 9 (blue) in steps of 2. Right: The $\kappa = -1$ solution with $\mathcal{T}_{-1} = 1$, $q=1$ and $H_0 = 0.3$, shown for $r$ between 0 (red) to 9 (blue) in steps of 1.