Let us consider a thin spherical shell expanding at a uniform velocity . At some moment (in the lab frame, where the centre of the explosion is stationary) all the matter on the shell emit photons. The spectrum of the photon, in the fluid frame, is assumed to be a power law . The fluid and lab frame frequencies can be related through the Lorentz transform

where is the Lorentz factor and is the cosine of the angle between the fluid element direction of motion and the line of sight. The flux received by an observer at a distance is given by

where is the emissivity in the lab frame and is the emissivity in the fluid rest frame, and we recall that is a Lorentz invariant. The emissivity in the rest frame is given by

The photon arrival time can be related to the photon emission time using

Substituting everything into the integral and solving it yields

So when