Local Operators in the Eternal Black Hole

We show that, in the AdS/CFT correspondence, states obtained by Hamiltonian evolution of the thermofield doubled state are also dual to an eternal black hole geometry, which is glued to the boundary with a time shift generated by a large diffeomorphism. We describe gauge invariant relational observables that probe the black hole interior in these states and constrain their properties using effective field theory. By adapting recent versions of the information paradox we show that these observables are necessarily described by state-dependent bulk-boundary maps, which we construct explicitly.