It is shown that special and general relativity can be derived as low-energy approximations from the nonrelativistic quantum chaos of a vortex sponge. If the average distance of separation between the filaments of the vortex sponge is chosen to be 10^–30cm, a value suggested by the grand unified scale of elementary particle theories, the vortex core radius becomes about equal to the Planck length. The model permits a simple explanation of the phenomenon of charge, and in conjunction with a hypothesis by A. D. Sakharov, can explain Dirac spinors.