  We elucidate the nature of the electron-phonon interaction in the archetypal
hybrid perovskite CH$_3$NH$_3$PbI$_3$ using ab initio many-body calculations
and an exactly solvable model. We demonstrate that electrons and holes near the
band edges primarily interact with three distinct groups of
longitudinal-optical vibrations, in order of importance: the stretching of the
Pb-I bond, the bending of the Pb-I-Pb bonds, and the libration of the organic
cations. These polar phonons induce ultrafast intraband carrier relaxation over
timescales of 6-30 fs and yield polaron effective masses 28% heavier than the
bare band masses. These findings allow us to rationalize previous experimental
observations and provide a key to understanding carrier dynamics in halide
perovskites.
