Holographic black hole formation and scrambling in time-ordered correlators

TL;DR

The paper provides a holographic description of black hole formation in AdS$_3$ as the collision of two shockwaves, represented in the dual CFT by two boosted precursor operators. By analyzing the cross-channel Virasoro mean-field spectrum, the authors identify a wave packet of light double-twist exchanges whose mean dimension grows exponentially with time, and show that crossing into heavy, BTZ-like states occurs when this mean reaches the BTZ threshold, i.e., at twice the scrambling time. This framework reveals scrambling physics within an in-time-order four-point function, rather than a standard OTOC, by tracking operator growth across Virasoro blocks and their descendants. The results connect bulk gravitational backreaction to precise CFT data and offer a microscopic mechanism for black hole formation, with implications for operator growth and gravitational dressing in holographic theories.

Abstract

We describe a holographic mechanism for black hole formation via the collision of two shockwaves in three-dimensional anti-de Sitter spacetime. In the dual conformal field theory (CFT), a two-shockwave state corresponds to the insertion of two boosted precursor operators in complementary Rindler patches. Their operator product expansion is initially described by a universal mean field spectrum of exchanged states, which is dominated by operator dimensions that grow exponentially in the boost parameter. We propose their mean value as diagnosing the mass of the collision product in the bulk. It crosses the CFT heavy state threshold after two scrambling times, in accordance with expectations about black hole formation in general relativity. Our analysis also allows us to identify the scrambling characteristics usually associated with out-of-time-order correlation functions, using only the internal composition of thermal in-time-order correlators.

Figures (5)

• Figure 1: Kinematic setup: each shockwave originates in one AdS$_3$ Rindler patch. Information about the collision product is contained in the distribution of exchanged operators in the cross-channel OPE, which depends strongly on the boost.
• Figure 2: The (discrete) probability distribution $P_m(h_v,h_w,z)$ on the space of possible s-channel exchange dimensions. The distribution is peaked and moves to higher weights exponentially with time $t$. We set $h_v=h_w=1,\, \delta=0.1,\,b=0$.
• Figure 3: The mean value of exchanged double-twist dimensions as a function of time. Different lines correspond to different external operator weights $h_v = h_w = 2^n$; $\delta=0.1$, $b=0$.
• Figure 4: The joint distribution $P^{(V)}_{m,n}(h_v,h_w,z)$ for $h_v=h_w=1$ at different times $t$. Darker colors indicate more support. The dashed lines indicate peak values in the vertical direction for fixed double-twist primary $h_m$. The dots mark the point $(\mathbb{E}[m],\mathbb{E}^{(V)}[n])$, which determines the global energy $\mathbb{E}^{(V)}[L_0]$.
• Figure 5: Probability distributions for exchange dimension $\Delta_s = \Delta_v+\Delta_w +\bar{m} + m$ and spin $\ell_s = \ell_v + \ell_w + \bar{m} - m$ as functions of time. We set $h_v=h_w=\bar{h}_v=\bar{h}_w =1$ and $b=0$, thus producing a symmetric distribution of spins with $\mathbb{E}[\ell_s]=0$. A non-zero $\mathbb{E}[\ell_s]$ can be produced either by using spinning external operators, or by setting $b \neq 0$.