  Recently, remarkably simple exact results were presented about the dynamics
of heat transport in the local Luttinger model for nonequilibrium initial
states defined by position-dependent temperature profiles. We present
mathematical details on how these results were obtained. We also give an
alternative derivation using only algebraic relations involving the
energy-momentum tensor which hold true in any unitary conformal field theory
(CFT). This establishes a simple universal correspondence between initial
temperature profiles and the resulting heat-wave propagation in CFT. We extend
these results to larger classes of nonequilibrium states. It is proposed that
such universal CFT relations provide benchmarks to identify nonuniversal
properties of nonequilibrium dynamics in other models.
