We derive the equations for the propagation of relativistic ionization fronts (I-fronts) in both static and moving gases, including the cosmologically expanding intergalactic medium (IGM). For the supersonic R-type phase that occurs right after a source turns on, relativistic corrections can be significant up until the light crossing time of the equilibrium Str?mgren sphere. When q, the ratio of this light-crossing time to the recombination time, exceeds unity, the time for the expanding I-front to reach the Str?mgren radius is delayed by a factor of q. For a static medium, we obtain exact analytical solutions and apply them to the illustrative problems of an O star in a molecular cloud and a starburst in a high-redshift cosmological halo. Relativistic corrections can be important both at early times when the H II regions are small and at later times if a density gradient causes the I-front to accelerate. An analytical solution is also derived for a steady source in a cosmologically expanding IGM. Here relativistic corrections are significant for short-lived, highly luminous sources such as QSOs at the end of reionization (z 6) but negligible for weaker or higher redshift sources. Finally, we numerically solve the equations for relativistic, cosmological I-fronts in the presence of evolving small-scale structure, for a large-galaxy starburst and a luminous QSO. For such strong and short-lived sources at z ~ 7, relativistic corrections are quite significant, and small-scale structure can decrease the size of the H?II region by up to an additional ~25%. However, most of the IGM was ionized by smaller, higher redshift sources. Thus, the effect of relativistic corrections on global reionization is small and can usually be neglected.
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