The paper by Re'em Sari, Tsvi Piran, and Ramesh Narayan discusses the spectra and light curves of gamma-ray burst (GRB) afterglows, which are believed to be primarily synchrotron emissions from relativistic electrons colliding with an external medium. The authors calculate the broad-band spectrum and light curve of synchrotron radiation from a power-law distribution of electrons in an expanding relativistic shock. They consider two limiting models for the hydrodynamical evolution of the shock: fully adiabatic and fully radiative. The light curve is constructed under these models, and the results are compared with observations of GRB afterglows. The paper details the synchrotron spectrum, which consists of several power-law segments, and the hydrodynamical evolution of the shock, which affects the break frequencies and peak flux. The authors derive expressions for the break frequencies and peak flux in both adiabatic and radiative cases, and show that the peak flux remains constant even during fast cooling if the evolution is adiabatic. They also derive scalings for the light curve, which depend on the critical times and the initial conditions of the shock. The main findings include the distinction between fast and slow cooling, the dependence of the peak flux and break frequencies on time, and the potential implications for the spectral index and decay rate of the light curve. The authors suggest that future observations of GRB afterglows could confirm the shock model and the adiabatic assumption by showing consistent decay rates and spectral indices.The paper by Re'em Sari, Tsvi Piran, and Ramesh Narayan discusses the spectra and light curves of gamma-ray burst (GRB) afterglows, which are believed to be primarily synchrotron emissions from relativistic electrons colliding with an external medium. The authors calculate the broad-band spectrum and light curve of synchrotron radiation from a power-law distribution of electrons in an expanding relativistic shock. They consider two limiting models for the hydrodynamical evolution of the shock: fully adiabatic and fully radiative. The light curve is constructed under these models, and the results are compared with observations of GRB afterglows. The paper details the synchrotron spectrum, which consists of several power-law segments, and the hydrodynamical evolution of the shock, which affects the break frequencies and peak flux. The authors derive expressions for the break frequencies and peak flux in both adiabatic and radiative cases, and show that the peak flux remains constant even during fast cooling if the evolution is adiabatic. They also derive scalings for the light curve, which depend on the critical times and the initial conditions of the shock. The main findings include the distinction between fast and slow cooling, the dependence of the peak flux and break frequencies on time, and the potential implications for the spectral index and decay rate of the light curve. The authors suggest that future observations of GRB afterglows could confirm the shock model and the adiabatic assumption by showing consistent decay rates and spectral indices.