  For leptogenesis with heavy sterile neutrinos above the electroweak scale,
asymmetries produced at early times (in the relativistic regime) are relevant,
if they are protected from washout. This can occur for weak washout or when the
asymmetry is partly protected by being transferred to spectator fields. We thus
study the relevance of relativistic effects for leptogenesis in a minimal
seesaw model with two sterile neutrinos in the strongly hierarchical limit.
Starting from first principles, we derive a set of momentum-averaged fluid
equations to calculate the final $B-L$ asymmetry as a function of the washout
strength and for different initial conditions at order one accuracy. For this,
we take the leading fluid approximation for the relativistic $CP$-even and odd
rates. Assuming that spectator fields remain in chemical equilibrium, we find
that for weak washout, relativistic corrections lead to a sign flip and an
enhancement of the asymmetry for a vanishing initial abundance of sterile
neutrinos. As an example for the effect of partially equilibrated spectators,
we consider bottom-Yukawa and weak-sphaleron interactions in leptogenesis
driven by sterile neutrinos with masses $\gtrsim 5\times10^{12}$ GeV. For a
vanishing initial abundance of sterile neutrinos, this can give rise to another
flip and an absolute enhancement of the final asymmetry in the strong washout
regime by up to two orders of magnitude relative to the cases either without
spectators or with fully equilibrated ones. These effects are less pronounced
for thermal initial conditions for the sterile neutrinos. The $CP$-violating
source in the relativistic regime at early times is important as it is
proportional to the product of lepton-number violating and lepton-number
conserving rates, and therefore less suppressed than an extrapolation of the
nonrelativistic approximations may suggest.
