Severe Limits on Variations of the Speed of Light with Frequency

Explosive astrophysical events at high red shift can be used to place severe limits on the fractional variation in the speed of light ($\Delta c/c$), the photon mass ($m_{\gamma}$), and the energy scale of quantum gravity ($E_{QG}$). I find $\Delta c/c < 6.3 \times 10^{-21}$ based on the simultaneous arrival of a flare in GRB 930229 with a rise time of $220 \pm 30 \mu s$ for photons of 30 keV and 200 keV. The limit on $m_{\gamma}$ is $4.2 \times 10^{-44} g$ for GRB 980703 from radio to gamma ray observations. The limit on $E_{QG}$ is $8.3 \times 10^{16}$ GeV for GRB 930131 from 30 keV to 80 MeV photons.