Improved Upper Limit on the Photon Rest Mass

By standard arguments, observed hydromagnetic waves in the Crab Nebula imply an upper limit to the photon rest mass of $\mu \lesssim {10}^{-16\mathrm{}}-{10}^{-15\mathrm{}}$ ${\mathrm{cm}}^{-1\mathrm{}}$, $m_{\mathrm{ph}}\lesssim 3\times {10}^{-54\mathrm{}}-3\mathrm{}\times {10}^{-53\mathrm{}}$ g (uncertainty < factor 100), smaller than the previous best limits by a factor ${10}^4$-${10}^5$. However, the standard wave arguments used here and elsewhere are reexamined and found not to be rigorous, although the limits given are probably valid in order of magnitude.