  We consider bulk quantum fields in AdS/CFT in the background of an eternal
black hole. We show that for black holes with finite entropy, correlation
functions of semiclassical bulk operators close to the horizon deviate from
their semiclassical value and are ill-defined inside the horizon. This is due
to the large-time behavior of correlators in a unitary CFT, and means the
region near and inside the horizon receives corrections. We give a prescription
for modifying the definition of a bulk field in a black hole background, such
that one can still define operators that mimic the inside of the horizon, but
at the price of violating microcausality. For supergravity fields we find that
commutators at spacelike separation generically ~ exp(-S/2). Similar results
hold for stable black holes that form in collapse. The general lesson may be
that a small amount of non-locality, even over arbitrarily large spacelike
distances, is an essential aspect of non-perturbative quantum gravity.
