The hybrid perovskite CH3NH3PbI3 (MAPI) exhibits long minority-carrier lifetimes and diffusion lengths. We show that slow recombination originates from a spin-split indirect-gap. Large internal electric fields act on spin-orbit-coupled band extrema, shifting band-edges to inequivalent wavevectors, making the fundamental gap indirect. From a description of photoluminescence within the quasiparticle self-consistent GW approximation for MAPI, CdTe, and GaAs, we predict carrier lifetime as a function of light intensity and temperature. At operating conditions we find radiative recombination in MAPI is reduced by a factor of more than 350 compared to direct gap behavior. The indirect gap is retained with dynamic disorder.