  Models for gamma-ray burst afterglow dynamics and synchrotron spectra are
known to exhibit various scale invariances, owing to the scale-free nature of
fluid dynamics and the power-law shape of synchrotron spectra. Since GRB
170817A, off-axis jet models including a lateral energy structure in the
initial outflow geometry have gained in prominence. Here we demonstrate how the
scale-invariance for arbitrary jet structure and dynamical stage can be
expressed locally as a function of jet temporal light curve slope. We provide
afterglow flux expressions and demonstrate their use to quickly assess the
physical implications of observations. We apply the scaling expressions to the
Swift XRT sample, which shows a spread in observed fluxes, binned by light
curve slope at time of observation, that increases with increasing light curve
slope. According to the scaling relations, this pattern is inconsistent with a
large spread in environment densities if these were the dominant factor
determining the variability of light curves. We further show how the late Deep
Newtonian afterglow stage remains scale-invariant but adds distinct spectral
scaling regimes. Finally, we show that for given jet structure a universal
curve can be constructed of the centroid offset, image size and ellipticity
(that can be measured using very-large baseline interferometry) versus observer
angle, in a manner independent of explosion energy and circumburst density. Our
results apply to any synchrotron transient characterized by a release of energy
in an external medium, including supernova remnants, kilonova afterglows and
soft gamma-repeater flares.
