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

TL;DR

This paper investigates when a bulk operator in Lorentzian AdS/CFT can be expressed as a smeared boundary operator. By reformulating the wave equation as a Schrödinger problem and analyzing mode sums, it shows that a convergent smearing function exists in pure AdS but fails in AdS-Schwarzschild due to an unlimited set of high-angular-momentum modes (l ≫ ω) whose boundary imprint is exponentially suppressed. A WKB analysis confirms this exponential suppression for large l, indicating a breakdown of the bulk-to-boundary dictionary in black-hole spacetimes. The work extends the analysis to general static spherically symmetric spacetimes, linking the nonexistence of a smearing function to radial potential barriers and to trapped null geodesics, and suggests that for such geometries the holographic encoding requires nonlocal boundary data or alternative representations beyond a simple smearing function.

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.

Figures (6)

• Figure 1: To construct the bulk operator $\Phi(B)$, the CFT operator $O(b')$ is smeared with the smearing function $K(B|b')$ as indicated in (\ref{['eq:Smearing']}). (a) The support of the pure AdS smearing function $K(B|b')$ is all boundary points $b'$ spacelike separated from $B$ (hatched region). (b) Had the AdS-Rindler smearing function existed, it would have only made use of the boundary region that overlaps with $J^+(q)\cap J^-(p)$ (the intersection of the causal future of $q$ and causal past of $p$), where $q$ and $p$ are chosen so that $J^+(q)\cap J^-(p)$ just barely contains $B$. Any changes outside this bulk region $J^+(q)\cap J^-(p)$ would have been manifestly irrelevant for computing $\Phi(B)$.
• Figure 2: The wave equation can be recast as a Schrödinger equation (\ref{['eq:Schro']}). We plot the global AdS$_4$ potential (\ref{['eq:VGlob']}) for $l=3$ for a massless field. The plot on the left is in terms of the radial coordinate $r$ appearing in the AdS metric (\ref{['eq:Global']}). The plot on the right is in terms of the tortoise coordinate $r_*$, and is the one relevant for solving (\ref{['eq:Schro']}). The two are related through $r = \tan r_*$. The tortoise coordinate has the effect of compressing the potential at large $r$, while leaving small $r$ unaffected. The AdS barrier occurs at $r_*$ very close to $\pi/2$; its narrowness allows the modes to decay only as a power law: $\phi \sim r^{-\Delta}$.
• Figure 3: A plot of the $AdS_4$-Schwarzschild ($r_0=1$) potential (\ref{['eq:VBH']}) as a function of the radial coordinate $r$. The plot on the left is for $l=4$, and the one on the right for $l = 10$. Unlike for pure AdS, $\omega$ is not bounded from below by $l$; for a fixed $\omega$, $l$ can be arbitrarily high. The barrier an $\omega$ mode must pass through grows as $l$ increases. This results in the $\omega \ll l$ modes having exponential behavior in $l$. Intuitively, these modes become ever more confined near the horizon with increasing $l$.
• Figure 4: We are interested in finding for which static spherically spacetimes without horizons a smearing function exists. The smearing function involves the sum (\ref{['eq:Ksphere']}) over modes, which can be grouped into $3$ different regimes. Only $A$ posses a threat to the convergence of (\ref{['eq:Ksphere']}). At large $r$ the metric, and consequently the potential (\ref{['eq:Pot']}), looks like that of pure AdS (\ref{['eq:Vbigr']}). At smaller $r$, in regime $A$ the angular momentum $l$ is so large that all terms in the potential except for the centrifugal barrier (\ref{['eq:Vbigl']}) are irrelevant.
• Figure 5: The wave equation can be recast as a Schrödinger equation (\ref{['eq:Schro']}) with a potential $V(r)$ and an energy $\omega^2$. Here we sketch a possible potential (\ref{['eq:Pot']}) for which a smearing function doesn't exist. At large $r$, $r>R$, the the potential looks like that of pure AdS (the figure has been compressed; the distance between $r_2$ and $R$ is really much larger). At smaller $r$ the potential, for large $l$, is approximated by (\ref{['eq:Vbigl']}). If $f(r)/r^2$ ever has positive slope, as shown above, some of the modes $\omega$ (dashed line) will have to tunnel through the barrier. Consequently, the sum (\ref{['eq:Ksphere']}) will diverge for $r<r_2$.