Maximally symmetric boost-invariant solutions of the Boltzmann equation in foliated geometries

In this work we study the relativistic kinetic theory of a boost-invariant conformal gas on a static, maximally symmetric background <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>d</a:mi> <a:msub> <a:mi>S</a:mi> <a:mn>3</a:mn> </a:msub> <a:mo>×</a:mo> <a:mi mathvariant="double-struck">R</a:mi> </a:math> , considering all constant-curvature slicings of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>d</d:mi> <d:msub> <d:mi>S</d:mi> <d:mn>3</d:mn> </d:msub> </d:math> —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 timelike coordinate are 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 <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>d</f:mi> <f:msub> <f:mi>S</f:mi> <f:mn>3</f:mn> </f:msub> <f:mo>×</f:mo> <f:mi mathvariant="double-struck">R</f:mi> </f:math> . 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.