Metal halide perovskites exhibit a materials physics that is distinct from traditional inorganic and organic semiconductors. While materials such as CH3NH3PbI3 are non-magnetic, the presence of heavy elements (Pb and I) in a non-centrosymmetric crystal environment result in a significant spin-splitting of the frontier electronic bands through the Rashba-Dresselhaus effect. We show, from a combination of textit{ab initio} molecular dynamics, density-functional theory, and relativistic quasi-particle textit{GW} theory, that the nature (magnitude and orientation) of the band splitting depends on the local asymmetry around the Pb and I sites in the perovskite structure. The potential fluctuations vary in time as a result of thermal disorder and a dynamic lone pair instability of the Pb(II) 6s$^{2}$6p$^{0}$ ion. We show that the same physics emerges both for the organic-inorganic CH3NH3PbI3 and the inorganic CsPbI3 compound. The results are relevant to the photophysics of these compounds and are expected to be general to other lead iodide containing perovskites.
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