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Maximally Symmetric Boost-Invariant Solutions of the Boltzmann Equation in Foliated Geometries

Mauricio Martinez, Christopher Plumberg

TL;DR

The paper develops a covariant kinetic-theory framework for a boost-invariant conformal gas on $dS_3\times\mathbb{R}$ and derives an exact Boltzmann solution valid for all constant-curvature foliations. By exploiting the cotangent-bundle formulation and the lifted isometries, it shows that the distribution function depends only on foliation Casimirs and the timelike coordinate, unifying Bjorken, Gubser, and Grozdanov flows as coordinate projections of a single geometric construction. The authors map the kinetic solution to macroscopic hydrodynamics, obtaining ideal, viscous, and second-order conformal hydrodynamics across foliations, as well as a genuine Grozdanov-flow solution on the hyperbolic foliation, plus a well-defined free-streaming limit. The work clarifies how symmetry and Casimir invariants govern the evolution, enabling a coherent interpretation of different expanding systems in Minkowski space through a common geometric lens. These results provide new benchmarks for kinetic theory in curved, highly symmetric backgrounds and pave the way for exploring far-from-equilibrium dynamics and hydrodynamization in novel boost-invariant settings.

Abstract

In this work we study the relativistic kinetic theory of a boost-invariant conformal gas on a static, maximally symmetric background $dS_3\times \mathbb{R}$, considering all constant-curvature slicings of $dS_3$ - flat, spherical, or hyperbolic- and their associated symmetry groups. Using a symmetry-driven cotangent-bundle approach, we show that the isometry group of each slicing acts on phase space in such a way that only its Casimir invariants and the time-like coordinate unconstrained, so the distribution function depends solely on these quantities. This yields a unified boost-invariant exact solution of the Boltzmann equation valid for each constant-curvature foliation of \ds. Specializing this general solution to the flat and spherical foliations reproduces the Bjorken and Gubser flows, respectively, while its restriction to the hyperbolic foliation produces a genuinely new analytic solution (`Grozdanov flow'). Hydrodynamics and free streaming emerge naturally as limiting regimes of this novel exact solution. We further comment on several relevant aspects of the new boost-invariant solution on the hyperbolic slicing and on their interpretation once mapped back to Minkowski space.

Maximally Symmetric Boost-Invariant Solutions of the Boltzmann Equation in Foliated Geometries

TL;DR

The paper develops a covariant kinetic-theory framework for a boost-invariant conformal gas on and derives an exact Boltzmann solution valid for all constant-curvature foliations. By exploiting the cotangent-bundle formulation and the lifted isometries, it shows that the distribution function depends only on foliation Casimirs and the timelike coordinate, unifying Bjorken, Gubser, and Grozdanov flows as coordinate projections of a single geometric construction. The authors map the kinetic solution to macroscopic hydrodynamics, obtaining ideal, viscous, and second-order conformal hydrodynamics across foliations, as well as a genuine Grozdanov-flow solution on the hyperbolic foliation, plus a well-defined free-streaming limit. The work clarifies how symmetry and Casimir invariants govern the evolution, enabling a coherent interpretation of different expanding systems in Minkowski space through a common geometric lens. These results provide new benchmarks for kinetic theory in curved, highly symmetric backgrounds and pave the way for exploring far-from-equilibrium dynamics and hydrodynamization in novel boost-invariant settings.

Abstract

In this work we study the relativistic kinetic theory of a boost-invariant conformal gas on a static, maximally symmetric background , considering all constant-curvature slicings of - flat, spherical, or hyperbolic- and their associated symmetry groups. Using a symmetry-driven cotangent-bundle approach, we show that the isometry group of each slicing acts on phase space in such a way that only its Casimir invariants and the time-like coordinate unconstrained, so the distribution function depends solely on these quantities. This yields a unified boost-invariant exact solution of the Boltzmann equation valid for each constant-curvature foliation of \ds. Specializing this general solution to the flat and spherical foliations reproduces the Bjorken and Gubser flows, respectively, while its restriction to the hyperbolic foliation produces a genuinely new analytic solution (`Grozdanov flow'). Hydrodynamics and free streaming emerge naturally as limiting regimes of this novel exact solution. We further comment on several relevant aspects of the new boost-invariant solution on the hyperbolic slicing and on their interpretation once mapped back to Minkowski space.

Paper Structure

This paper contains 29 sections, 105 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Cotangent bundle formulation of kinetic theory
  3. Symmetry Constraints on the Distribution Function
  4. Boltzmann equation and its general solution on foliated $dS_3\times\mathbb{R}$
  5. From Minkowski $\mathbb{R}^{3,1}$ to foliated $dS_3\times \mathbb{R}$
  6. Isometries of $dS_3\times \mathbb{R}$ foliations
  7. Symmetry Constraints on the Distribution Function in Foliated $dS_3\times \mathbb{R}$
  8. Casimirs of the foliated $dS_3\times \mathbb{R}$ geometry
  9. Casimir of the 2d constant curvature foliation $\Sigma_\kappa$
  10. Casimir for $E_1$
  11. Reduced Boltzmann equation in a $dS_3\times \mathbb{R}$ foliation
  12. From exact kinetics to macroscopic dynamics in $dS_3\times \mathbb{R}$ slicings
  13. Energy–momentum tensor from the exact solution on foliated $dS_3\times \mathbb{R}$ geometry
  14. Landau matching conditions
  15. Emergent Hydrodynamics in foliated $dS_3\times \mathbb{R}$
  16. ...and 14 more sections

Figures (2)

  • Figure 1: Comparison of viscous hydrodynamical evolution schemes for the Grozdanov flow. Panel (a): Energy density $\hat{\varepsilon}_{-1}(\rho)$ and panel (b) effective shear $\bar{\pi}_{-1}(\rho)$ for four dynamical schemes: the full second-order hydrodynamic equation (solid blue line), Eqs. \ref{['eq:2ndordervischydro']} , its cold-limit approximation (orange long dashed), Eqs. \ref{['eq:cold-sol']}-\ref{['eq:energyds-coldsol']}, the Israel–Stewart (IS) equation (green dotted), Eq. \ref{['eq:IS-barpi']}, and the IS cold-limit solution (red dot-dashed), Eqs.\ref{['eq:IS-cold']}- \ref{['eq:energyds-coldISsol']}. Panel (b) displays the relaxation of the effective shear panel (a) highlights the distinct decay rates of the energy density associated with each approximation. The initial conditions for the energy density where $\hat{\varepsilon}_{-1}(\rho)$ while for the effective shear $\bar{\pi}_{-1}(\rho_0)=\bar{\pi}_-(c)$ (where $\bar{\pi}_-(c)$ is given by Eq. \ref{['eq:fixedpoints']}) with $\rho_0=0.1$, $c=5$ and $\eta/S=0.2$.
  • Figure 2: Different choices of foliation of $dS_3\times \mathbb{R}$ , parameterized by $(\rho,\theta)$, and the corresponding relations to Milne coordinates $(\tau,r)$. Left panels: The Gubser solution (top row; $\kappa = +1$) corresponds to foliations at $X^0 = \text{constant}$; the Bjorken solution (middle row; $\kappa = 0$) is defined by foliating with $X^3 - X^0 = \text{constant}$; and the Grozdanov solution (bottom row; $\kappa = -1$) reflects foliations with respect to $X^3 = \text{constant}$. Intersections between the chosen level and the hyperboloid are displayed as solid red contours, while other levels are displayed as dashed red contours. Black dashed contours at constant $X^0$ are included to help visualize the hyperboloid. Right panels: relations between $(\rho,\theta)$ and $(\tau,r)$ for each foliation. The ranges shown for $\rho$ and $\theta$ in each panel are chosen solely for aesthetic purposes; the full ranges are given in \ref{['table:ds3defs']}.