  Synchrotron emission is ubiquitous in explosive astrophysical events -- it is
a natural byproduct of shocks formed when matter expelled by the explosion
collides with ambient material. This emission is well-observed in various
classes of transients, and is often interpreted within a canonical
`equipartition' framework that allows physical properties of the shock to be
inferred from the frequency and luminosity at which the observed spectral
energy distribution (SED) peaks. This framework has been remarkably successful
in explaining observations of radio supernovae. It has also been used for
trans-relativistic explosions, where the shock velocities approach the speed of
light. However, the conventional framework does not incorporate relativistic
effects. Neither does it account for thermal electrons, which have been shown
to be important for high-velocity shocks. In this paper we describe a revised
framework that accounts for these two effects, and is applicable to
non-relativistic, trans-relativistic, and ultra-relativistic explosions. We
show that accounting for these effects can dramatically change the inferred
parameters of high-velocity shocks, and in particular -- that the shock
velocity, ambient density, and total energy are overestimated by the
conventional non-relativistic framework. We delineate the phase-space where
such modifications are important in terms of observationally measurable
parameters. We also find a novel upper limit on the peak synchrotron luminosity
of shock-powered transients, which is remarkably consistent with existing
observations. Finally, we discuss a prediction of the model -- that the SED
will qualitatively change as a function of shock velocity -- and show that this
is broadly consistent with data for representative events (e.g., SN1998bw,
AT2018cow, CSS161010, AT2020xnd).
