AdS black holes, the bulk-boundary dictionary, and smearing functions

TL;DR

This paper investigates the limitations of the bulk-boundary mapping in Lorentzian AdS/CFT, showing that certain black hole geometries prevent the construction of smearing functions due to trapped modes with negligible boundary imprint.

Contribution

It demonstrates that the presence of horizons and potential barriers in AdS black hole backgrounds obstructs the existence of smearing functions, challenging assumptions about the bulk-boundary dictionary in such spacetimes.

Findings

Modes with exponentially small near boundary imprint exist in AdS-Schwarzschild.

Barriers preventing null geodesics from reaching the boundary hinder smearing function construction.

The bulk-boundary dictionary is not straightforward in black hole backgrounds with trapping modes.

Abstract

In Lorentzian AdS/CFT there exists a mapping between local bulk operators and nonlocal CFT operators. In global AdS this mapping can be found through use of bulk equations of motion and allows the nonlocal CFT operator to be expressed as a local operator smeared over a range of positions and times. We argue that such a construction is not possible if there are bulk normal modes with exponentially small near boundary imprint. We show that the AdS-Schwarzschild background is such a case, with the horizon introducing modes with angular momentum much larger than frequency, causing them to be trapped by the centrifugal barrier. More generally, we argue that any barrier in the radial effective potential which prevents null geodesics from reaching the boundary will lead to modes with vanishingly small near boundary imprint, thereby obstructing the existence of a smearing function. While one may have thought the bulk-boundary dictionary for low curvature regions, such as the exterior of a black hole, should be as in empty AdS, our results demonstrate otherwise.