Severe Limits on Variations of the Speed of Light with Frequency

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