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

Abstract

In this work we study the relativistic kinetic theory of a boost-invariant conformal gas on a static, maximally symmetric background $d{S}_3\times \mathbb{R}$, considering all constant-curvature slicings of $d{S}_3$ - 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 $d{S}_3\times \mathbb{R}$. 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.

v2: 27 pages + appendices, 2 figures. Expanded symmetry-reduction discussion in Sects. 2 and 3, including theorem/proposition statements and a complete new appendix; conclusions unchanged