---
title: Shock-Like Defects in AdS3 Gravity
url: https://www.emergentmind.com/topics/shock-like-defects-in-ads-_3-gravity
type: topic
---

# Shock-Like Defects in AdS3 Gravity

Shock-like (thin-shell) defects in AdS$_3$ gravity represent non-conformal line defects in the bulk, realized as dynamically backreacting codimension-one shells composed of dust particles, possibly carrying intrinsic spin. These objects provide a concrete holographic dual to smeared nonlocal operator insertions in CFT$_2$, yielding a precise correspondence between gravitational solutions, vacuum Virasoro blocks, and statistical properties of matrix elements in high-energy eigenstates. The chaotic nature of the high-energy sector in AdS$_3$/CFT$_2$ is illuminated by matching bulk shell actions, CFT$_2$ defect correlators, and Eigenstate Thermalization Hypothesis (ETH) analysis, with explicit treatment of spinning domain walls and higher-point correlators of multiple defects [2506.12711].

## 1. Bulk Setup and Shell Stress–Energy

The gravitational action for AdS$_3$ with a codimension-one shell $\mathcal{W}$, in units with AdS radius $\ell_\mathrm{AdS}=1$, is given by:
\[
S = -\frac{1}{16\pi G}\int_{\mathcal M}(R+2)\sqrt{g}\,d^3x + \int_{\mathcal W}\sqrt{h}\,\sigma\,d^2y + S_\mathrm{spin}
\]
where $G$ is Newton's constant, $h_{ab}$ is the induced metric on the shell, $\sigma$ is its surface energy density, and $S_\mathrm{spin}$ encodes the spin current for shells composed of spinning particles. For shells without spin, the localized bulk stress tensor reads:
\[
T^{\mu\nu}(x) = \sigma \int d^2y\, \sqrt{h} \, u^\mu u^\nu\,\delta^3(x - X(y))
\]
with $u^a$ tangent to the worldvolume and $n^a$ its spacelike normal. The total ADM mass of the shell is $m = \int d\psi\,\sigma(\psi)$, with $\psi \sim \psi + 2\pi$ labeling the angular coordinate on $\mathcal{W}$. For shells with continuous spin $s$, the spin-current $S^{\mu}{}_{ab} \propto s\,\delta(\mathcal W)$ supplements the defect data.

## 2. Junction Conditions: First-Order Formalism

Shells in AdS$_3$ obey junction (matching) conditions across their worldvolume, relating the intrinsic geometry and extrinsic curvature (in the metric formalism) or frame fields and connection (in the first-order formalism). For spinning defects, metric continuity fails and one must employ the vielbein $\{e^a\}$ and spin connection $\{\omega^{ab}\}$, leading to:
\[
\begin{aligned}
d\omega^{ab} + \omega^a{}_c \wedge \omega^{cb} &\sim T^{\mu}{}_c \\
de^c + \omega^c{}_d \wedge e^d &\sim S^{\mu}{}_{ab}
\end{aligned}
\]
Integrating over a thin "pillbox" enclosing the shell, one derives the junction conditions:
\[
\begin{aligned}
\iota [\omega^{ab} \wedge e^\mu]\epsilon_{abc} &= -2 T^\mu{}_d \mathrm{Vol} \\
\iota [e^a \wedge e^b]\epsilon_{abc} &= S^\mu{}_{ab} \mathrm{Vol}
\end{aligned}
\]
with $\mathrm{Vol}=\sqrt{h}\,d^2y$. The requirement $[\mathrm{Vol}]=0$ allows the induced metric $h_{ab}$ to jump, provided the area element is continuous. In the spinless case ($s=0$), standard Israel conditions are recovered:
\[
[g_{ab}] = 0 \qquad [K_{ab} - g_{ab} K] = -8\pi G\,T_{ab}
\]

## 3. Rotating BTZ Geometry and Matching Conditions

Both interior and exterior of the shell are described by regions of the rotating BTZ metric:
\[
ds^2_\pm = -f_\pm(r)dt_\pm^2 + \frac{dr^2}{f_\pm(r)} + r^2\left( d\phi_\pm - \frac{J_\pm}{2r^2}dt_\pm \right)^2
\]
where $f_\pm(r)=r^2 - M_\pm + \frac{J_\pm^2}{4r^2}$, with subscripts $\pm$ denoting exterior ($+$) and interior ($-$) patches. The shell's worldvolume, at radius $r(\ell)$, is characterized by the trajectory equations:
\[
\begin{cases}
(t')_\pm = \frac{4e_\pm r^2 + 2iJ_\pm j_\pm}{4(r^2-r_{o\pm}^2)(r^2-r_{i\pm}^2)} \\
(r')^2 = \frac{4r^4 - 4(M_\pm + e_\pm^2 + j_\pm^2) r^2 + J_\pm^2 + 4j_\pm^2 M_\pm - 4i e_\pm j_\pm J_\pm}{4r^2} \\
(\phi')_\pm = \frac{4j_\pm r^2 - 4j_\pm M_\pm + 2i e_\pm J_\pm}{4(r^2-r_{o\pm}^2)(r^2-r_{i\pm}^2)}
\end{cases}
\]
where $(e_\pm, j_\pm)$ are conserved energy and angular momentum per shell particle, and $r_{o\pm}, r_{i\pm}$ are horizon radii. Imposing conservation laws and continuity of the area element $r_+^2 - j_+^2 = r_-^2 - j_-^2$, the conserved quantities are explicitly fixed. For spinless shells:
\[
\boxed{
\begin{aligned}
j_+ = j_- &= \frac{i(J_+ - J_-)}{2m} \\
e_+ - e_- &= \frac{M_- - M_+}{2m} \\
e_+ + e_- &= m, \quad e_+ - e_- = -4Gm
\end{aligned}
}
\]
For spinning shells (total spin $s$):
\[
\boxed{
\begin{aligned}
e_\pm &= \frac{m(M_- - M_+) - s(J_- - J_+)}{2(m^2 - s^2)} \pm \frac{m}{2} \\
j_\pm &= \frac{i[m(J_- - J_+) - s(M_- - M_+)]}{2(m^2 - s^2)} \pm \frac{is}{2}
\end{aligned}
}
\]
The area-form continuity again constrains $r_+^2-j_+^2 = r_-^2-j_-^2$.

## 4. Bulk On-Shell Action: Static and Spinning Shells

The Euclidean on-shell action for a static shell is:
\[
\log \left[ Z(\beta) G_\beta(t) \right] = \frac{t M_+ + (\beta-t) M_-}{8G} + \frac{m}{2G} \ln \left( \frac{r_*}{2} \right)
\]
with $r_*$ the radial turning point specified by
\[
r_*^2 = r_\pm^2 + \left( \frac{M_\pm - M_\mp}{2m} - \frac{m}{2} \right)^2
\]
and the "gluing" equations
\[
\frac{r_\pm}{r_*} = \sin \left( \frac{r_\pm t_\pm}{2} \right), \quad t_- = \beta - t,\; t_+ = t
\]
For shells with spin, one includes an additional term:
\[
I_{\rm spin} = \frac{s}{4G}\int d\psi\,d\ell\, n_2^\mu v^\nu \nabla_\nu(n_1)_\mu \longrightarrow -\frac{s}{4G} \ln \frac{r_{*L}}{r_{*R}}
\]
with $r_{*L}, r_{*R}$ the left/right-moving turning points. The result factorizes into sectors echoing the Virasoro block decomposition.

## 5. Boundary CFT$_2$ Line Operators and Correlators

In the dual CFT$_2$, the shell corresponds to a "thin-shell operator" constructed by distributing a primary $\Psi$ of dimension $h_\Psi$ uniformly along a circle:
\[
D(t) = \lim_{n\rightarrow\infty}\prod_{k=1}^n \Psi(t, 2\pi(k-1)/n), \quad h_D = n h_\Psi \sim \frac{c}{6}m
\]
Inserted into a thermofield double of energy $E$, the two-point correlation functions are:
\[
G_E(t) = \langle E|D^\dagger(t) D(0)|E\rangle, \qquad G_\beta(t) = \frac{\mathrm{Tr}\,[e^{-\beta H} D^\dagger(t) D(0)]}{Z(\beta)}
\]
Using the monodromy method for the large-$c$ Virasoro block, the perturbed Fuchsian equation
\[
V''(z) + T(z) V(z) = 0, \quad T(z) = -\sum_{i=1}^2 \frac{6h_\Psi}{c}\int_0^{2\pi}\frac{d\theta}{2\pi} \frac{1}{(z - r_i e^{i\theta})^2} + \cdots
\]
and trivial monodromy leads to a transcendental equation for $\rho$:
\[
e^{2\rho t} = \frac{(\rho + \rho_E - 6h_D/c)(\rho - \rho_E - 6h_D/c)}{(\rho - \rho_E + 6h_D/c)(\rho + \rho_E + 6h_D/c)}
\]
with $\rho_E = \frac12\sqrt{1-24E/c}$. The vacuum block exponentiates:
\[
G_E(t) \approx \exp\left[ -\frac{c}{3}\mathcal{F}(\rho) \right]
\]
and one identifies
\[
\partial_t \ln G_E(t) = \frac{c}{3}(\rho^2 - \rho_E^2) \longleftrightarrow \frac{M_- - M_+}{8G}
\]
under the bulk/CFT dictionary $c=3/(2G)$, $h_D = m/(8G)$, $E = M/(8G)$. ETH-type ansatz for the defect matrix elements matches both gravitational and CFT blocks.

## 6. Higher-Point Defects and Correlator Order Dependence

The generalization to multi-defect correlators is explicit: higher-point functions factorize into products of two-point subblocks, with matching achieved via the monodromy method in CFT$_2$ and by gluing trajectories of multiple shells in AdS$_3$ bulk. Line operators, as codimension-one nonlocal objects, yield correlators that depend nontrivially on the insertion order, in contrast to Euclidean correlators for local operators. Explicit four-defect calculations reveal this noncommutativity, underlining the distinct operator algebra of shell defects.

## 7. Triangular Correspondence and Significance

A precise triangular equivalence is established among:
- (i) Bulk on-shell shell actions in AdS$_3$ (including spin contributions)
- (ii) CFT$_2$ vacuum Virasoro blocks for non-conformal line defects
- (iii) ETH-inspired statistical ansatz for defect matrix elements

This correspondence fully characterizes shock-like (thin-shell) defects in AdS$_3$ gravity and their dual non-conformal line operators in CFT$_2$, establishing a unified framework for analyzing backreacting, nonlocal defect dynamics, chaos, and operator algebra in holographic AdS$_3$/CFT$_2$ [2506.12711].

Source: https://www.emergentmind.com/topics/shock-like-defects-in-ads-_3-gravity