  Astrophysical explosions are accompanied by the propagation of a shock wave
through an ambient medium. Depending on the mass and energy involved in the
explosion, the shock velocity $V$ can be non-relativistic ($V \ll c$, where $c$
is the speed of light), ultra-relativistic ($V \simeq c$), or moderately
relativistic ($V \sim few\times 0.1c$). While self-similar, energy-conserving
solutions to the fluid equations that describe the shock propagation are known
in the non-relativistic (the Sedov-Taylor blastwave) and ultra-relativistic
(the Blandford-McKee blastwave) regimes, the finite speed of light violates
scale invariance and self-similarity when the flow is only mildly relativistic.
By treating relativistic terms as perturbations to the fluid equations, here we
derive the $\mathcal{O}(V^2/c^2)$, energy-conserving corrections to the
non-relativistic, Sedov-Taylor solution for the propagation of a strong shock.
We show that relativistic terms modify the post-shock fluid velocity, density,
pressure, and the shock speed itself, the latter being constrained by global
energy conservation. We derive these corrections for a range of post-shock
adiabatic indices $\gamma$ (which we set as a fixed number for the post-shock
gas) and ambient power-law indices $n$, where the density of the ambient medium
$\rho_{\rm a}$ into which the shock advances declines with spherical radius $r$
as $\rho_{\rm a} \propto r^{-n}$. For Sedov-Taylor blastwaves that terminate in
a contact discontinuity with diverging density, we find that there is no
relativistic correction to the Sedov-Taylor solution that simultaneously
satisfies the fluid equations and conserves energy. These solutions have
implications for relativistic supernovae, the transition from ultra- to
sub-relativistic velocities in gamma-ray bursts, and other high-energy
phenomena.
