On the exact analytic expressions for the equitemporal surfaces in gamma-ray burst afterglows

We have recently shown that marked differences exist between the equitemporal surfaces ( EQTSs) for the gamma- ray burst ( GRB) afterglows numerically computed by the full integration of the equations of motion and the ones found in the current literature expressed analytically on the grounds of various approximations. In this Letter, the exact analytic expressions of the EQTSs are presented both in the case of fully radiative and adiabatic regimes. The new EQTS analytic solutions validate the numerical results recently obtained by Bianco & Ruffini and offer a powerful tool to analytically perform the estimates of the physical observables in GRB afterglows.