Probing Black Hole Thermal Effects in the Dual CFT via Wave Packets

Abstract: We investigate how the gravitational effects of a black hole manifest themselves as thermal behavior in the dual finite-temperature conformal field theory (CFT). In the holographic framework of AdS/CFT, we analyze a wave packet propagating into a black hole geometry in the bulk by computing three-point functions of a scalar primary operator in the boundary CFT. Our setup captures thermal signatures such as exponential damping of the expectation value, which are absent at zero-temperature. This provides a concrete and analytically tractable example of how black hole physics can be probed from the CFT side.

• The paper demonstrates how localized bulk wave packets uncover exponential damping of CFT primary operator expectation values in a BTZ black hole background.
• The methodology involves analytic computations of smeared three-point functions in both zero and finite temperature settings, highlighting clear thermal transitions.
• The study underscores the distinct behaviors of non-conserved primaries versus conserved energy density, offering new insights for bulk reconstruction in holography.

Probing Thermal Effects of Bulk Black Holes in Dual CFTs via Wave Packet Analysis

Introduction

The paper "Probing Black Hole Thermal Effects in the Dual CFT via Wave Packets" (2601.04647) provides an analytic study of how gravitational effects in a bulk AdS black hole are encoded holographically as thermal signatures in the dual finite-temperature CFT. The approach employs the construction of bulk wave packets and computes their imprints on boundary correlators, specifically three-point functions of scalar primaries. The analysis is carried out both in pure AdS ($T=0$) and in the BTZ black hole background ($T>0$), focusing on the behavior of CFT observables that are sensitive to thermalization and dissipation.

Wave Packets in the AdS/CFT Correspondence

The authors develop a precise construction of localized bulk wave packet states dual to boundary CFT excitations. The prototypical wave packet is a smeared state in the CFT, constructed from a Gaussian envelope in $(t,x)$, and characterized by sharply defined averages of energy and momentum. The duality is formulated via the BDHM dictionary, mapping bulk scalar field excitations near the boundary to insertions of primary CFT operators. This construction ensures that the bulk state corresponds to a single quasi-particle localized in both space and time, with deep control over its geometric propagation.

For a free scalar in AdS$_{d+1}$, the bulk wave packet is represented by

$$|p,\omega\rangle_{\mathrm{bulk}} = \lim_{z\to 0} \frac{1}{z^\Delta} \int dt\, d^{d-1}x\, e^{-\frac{t^2 + {\bf x}^2}{2a^2} + i \vec{p}\cdot\vec{x} - i\omega t} \phi(t, z, \vec{x}) |0\rangle$$

which is mapped to a corresponding operator in the CFT via $\mathcal O(t,x)$.

Boundary Correlators as Probes of Bulk Thermality

The core technical advancement is an explicit evaluation of the expectation value of a primary operator,

$$\langle O(t,x)\rangle_{\mathrm{wp}} = \langle p,\omega| O(t,x) |p,\omega \rangle$$

in both $T=0$ (pure AdS) and $T>0$ (BTZ) backgrounds. The key procedure is the computation of three-point functions,

$\langle O(u_1, v_1) O(u, v) O(u_3, v_3) \rangle$

smeared appropriately with the Gaussian wave packet, where the correlators are known analytically in 2D CFT and at finite temperature via a conformal map from the plane to the cylinder.

At $T=0$, the resulting behavior is dominated by light-cone propagation: the expectation value is strongly peaked along the loci $u, v \approx 0$ and decays as a power law away from the light cone, reflecting ballistic spreading in the dual CFT. The energy density remains chiral and independently conserved for left and right movers.

At finite $T$ (bulk BTZ), after mapping the correlators to the cylinder and analytic continuation to Lorentzian signature (with careful $i\epsilon$ prescription), the authors show that the expectation value of the primary exhibits exponential decay,

$$\langle O(t,x)\rangle_{\beta, \mathrm{wp}} \sim e^{-2\pi |t|/\beta}, \quad |t| \gg a$$

along and inside the light cone, in sharp contrast to the zero-temperature case. This exponential damping is a signature of the underlying black hole geometry and its thermal nature. The analytic framework captures the crossover between low- and high-temperature limits, showing explicitly how thermal effects enter even for finite-$N$ CFTs, not just in the semiclassical gravity limit.

Analysis of Energy Density and Lack of Thermal Decay

In 2D CFTs, the authors compute the expectation value of the energy density ($T_{00}$) in the wave packet excited state at finite temperature and find it factorizes into left- and right-moving components:

$\mathcal E(t,x) = f(t-x) + g(t+x)$

The energy density propagates strictly along the light cone, without any thermal attenuation or delay—even in the presence of the dual bulk black hole. This is traced to the chirality and conservation of $T_{00}$ in $d=2$, which obstructs bulk time delay effects from affecting this observable. This absence of causal modification is non-generic and expected to be lifted in higher dimensions, as indicated by previous studies [Terashima:2021klf].

Discussion and Theoretical Implications

The analytic results provide a concrete demonstration that specific CFT observables serve as sensitive probes for bulk black hole thermalization. The exponential damping of the expectation value of primary operators is presented as a direct dual to the presence of black hole horizons and their associated Hawking radiation in the bulk. The absence of exponential decay in the energy density quantifies the selectivity of CFT observables in encoding bulk causal structure.

The framework clarifies that not every boundary observable captures gravitational time delays or thermalization. The exponential damping in the primary operator expectation value is a robust marker for the black hole thermal scale $\beta$, while more conserved quantities (like energy) remain blind to such effects in two dimensions.

The set of results further underscores the utility of wave packet constructions for studying bulk locality and thermal propagation in AdS/CFT, offering a toolkit that extends to finite-$N$ and arbitrary CFTs with a holographic interpretation.

Future Outlook

The formalism delineated in this work lays the groundwork for a more systematic classification of which CFT observables encode which aspects of bulk geometry and thermodynamics—in particular, causal structure, time delay, and information about horizons. Extending this analytic framework to higher dimensions, different operator insertions, or more general black hole geometries will likely elucidate the limits of holographic detection of bulk phenomena.

Moreover, the distinction between exponential damping of non-conserved primaries and the robustness of energy density propagation may offer insight for out-of-equilibrium and nonequilibrium studies in holography, especially for questions related to equilibration, chaos, and information loss.

Conclusion

This paper rigorously identifies and computes the thermal signatures induced by a BTZ black hole in the expectation values of scalar primary operators in the dual finite-temperature CFT. The analytic approach using smeared wave packets provides explicit results for both zero and finite temperatures, revealing that while energy density remains unaffected due to 2D CFT chirality, the expectation values of non-conserved primaries display temperature-dependent exponential decay directly attributable to bulk gravitational effects. These results sharpen our understanding of how and when specific CFT observables reveal the presence and thermal nature of black holes in the holographic bulk, with significant implications for bulk reconstruction and the study of quantum gravitational phenomena in AdS/CFT.