Holographic thought experiments

The Hamiltonian of classical anti–de Sitter gravity is a pure boundary term on-shell. If this remains true in nonperturbative quantum gravity then (i) boundary observables will evolve unitarily in time and (ii) the algebra of boundary observables is the same at all times. In particular, information available at the boundary at any one time $t_1$ remains available at any other time $t_2$. Since there is also a sense in which the equations of motion propagate information into the bulk, these observations raise what may appear to be potential paradoxes concerning simultaneous (or spacelike separated) measurements of noncommuting observables, one at the asymptotic boundary and one in the interior. We argue that such potentially paradoxical settings always involve a breakdown of semiclassical gravity. In particular, we present evidence that making accurate holographic measurements over short time scales radically alters the familiar notion of causality. We also describe certain less intrinsically paradoxical settings which illustrate the above boundary unitarity and render the notion more concrete.

Figure 1

A conformal diagram of global ${\mathrm{AdS}}_4$ with the $S^2$ suppressed. A signal leaves the boundary at time $t_1$. The information is still present in the CFT at time $t_2$ though no signal has returned to the boundary.

Figure 2

Our AdS system lives in a (conformal) box in Alice’s laboratory. Outside the AdS box are various ancilla. A clock and a measuring device with adjustable coupling are shown.