Shockwaves and the Butterfly Effect

An innocuous boundary perturbation inserted far in the past is exponentially blueshifted relative to an outgoing near-horizon probe. Its gravitational field becomes a null shock that shifts the conjugate Kruskal coordinate. The shift reaches order one at the scrambling time and spreads transversely with the butterfly velocity. This derivation fixes the exponential, the logarithmic time scale, and the geometric regime in which they are trustworthy.

Required background. Gauge Theory and String-Theory Preparation provides stress-tensor and eikonal conventions; Quantum Information provides scrambling and correlator language.

Helpful background. Quantum Gravity supplies the semiclassical scattering regime, while Black-Hole Perturbation Theory provides horizon-regular coordinates and linearized constraints.

Near-horizon boost and the null shift

For a nonextremal horizon, introduce Kruskal coordinates with surface gravity $\kappa=2\pi T$. A quantum of boundary energy $E$ inserted at time $-t_w$ has near-horizon momentum

$$p^V\sim E\,e^{\kappa t_w}.$$

The relative center-of-mass energy with an outgoing probe therefore grows exponentially even though the injected boundary energy stays fixed. In the eikonal window—large boost, small local curvature, and impact parameter larger than the Planck/string resolution—the backreaction is represented by

$$U\longrightarrow U+h(\mathbf{x})\,\Theta(V),$$

with a null stress tensor localized at $V=0$. The sign and normalization of $h$ depend on which horizon and Kruskal orientation are chosen; physical time delay and the field equation fix them.

Shenker and Stanford 2014, §§3.1–3.3, pp. 7–11 construct the BTZ shock and compute the resulting geodesic shift. The essential boost is kinematic, while the transverse profile depends on the background geometry.

Transverse profile, scrambling, and butterfly speed

For a homogeneous black brane, the horizon constraint takes the schematic form

$$\left(-\nabla_\perp^2+\mu^2\right)h(\mathbf{x}) =\mathcal C\,G_N E\,e^{2\pi Tt_w}\delta^{(d)}(\mathbf{x}).$$

Let $\varepsilon_{\rm sh}\ll1$ denote the dimensionless initial shift amplitude after fixing Kruskal normalization and wavepacket width. At large separation, the transverse Green function gives, up to an algebraic spatial prefactor,

$$h(t_w,\mathbf{x})\sim \varepsilon_{\rm sh}\exp\!\left[2\pi T\left(t_w-\frac{\lvert\mathbf{x}\rvert}{v_B}\right)\right], \qquad v_B=\frac{2\pi T}{\mu}.$$

The perturbation becomes order one at

$$t_*=\frac{1}{2\pi T}\log\frac{1}{\varepsilon_{\rm sh}}$$

For a thermal-scale quantum spread over a thermal cell of a planar brane, $\varepsilon_{\rm sh}$ scales parametrically as $(E/T)/S_{\rm cell}$, where $S_{\rm cell}=sT^{-d}$ and $d$ counts boundary spatial dimensions. Thus $t_*\sim(\beta/2\pi)\log S_{\rm cell}$. A finite black hole can instead supply its total entropy in the corresponding homogeneous problem. An infinite brane’s divergent total entropy is not the appropriate quantity. Writing $\log(1/G_NE)$ without the required length scales would also take a logarithm of a dimensionful quantity. Additive constants depend on packet and operator normalization.

For the neutral planar AdS–Schwarzschild brane with $d$ boundary spatial dimensions in two-derivative Einstein gravity, direct substitution yields $v_B^2=(d+1)/(2d)$ in units where the boundary light speed is one. Charge or a nontrivial matter profile can change this value even at two-derivative order. The independent check is the pole-skipping value $k_0=2\pi T/v_B$ extracted from the linearized horizon equation.

From the shift to an OTOC

In a two-sided thermal geometry, the shock changes the geodesic length or eikonal phase connecting probes. Before saturation, a normalized out-of-time-order correlator has the schematic behavior

$$F(t,\mathbf{x})\simeq 1-\frac{1}{N_{\rm eff}} e^{\lambda_L(t-\lvert\mathbf{x}\rvert/v_B)}+\cdots , \qquad \lambda_L=2\pi T.$$

The equality $\lambda_L=2\pi T$ is the Einstein eikonal result in this regime, not a definition of chaos. The prefactor and even the front shape depend on the operators and transverse geometry. Shenker and Stanford 2014, §§3.4–4, pp. 11–17 show how mutual information and correlators diagnose the shock and also discuss string- and Planck-scale modifications.

The geometric spread describes a high-energy scattering front. It should not be identified automatically with every possible OTOC velocity, diffusion constant, or signal velocity. Causality requires $v_B$ not to exceed the boundary light cone in a consistent relativistic theory, but it need not equal that speed.

Operator choice enters twice. The early insertion determines the shock energy and transverse profile, while the late probe determines how the coordinate shift is converted into a phase or correlator change. Heavy operators admit a geodesic approximation; light operators require the appropriate bulk propagator. Agreement of the extracted exponent across suitable probes is a universality check, whereas different prefactors are expected. A conserved or symmetry-protected operator can have additional slow structure that should not be mistaken for a change in the gravitational boost.

There is also a useful energy check. The boundary energy remains $E$; the exponential is a relative boost, not energy creation. Computing the stress flux in regular horizon coordinates confirms that the source is finite, while its Kruskal momentum grows because the comparison frame changes with $t_w$. This distinction prevents the scrambling calculation from being misread as a violation of conservation.

Regime checks and failure modes

The calculation assumes a stationary nonextremal horizon, weak gravitational backreaction until near $t_*$, and an eikonal collision localized compared with curvature scales. At times much earlier than dissipation, the near-horizon approximation does not isolate the process; after $t_*$, multiple exchanges and nonlinear backreaction invalidate the first correction.

String spreading changes the impact-parameter dependence before Planckian gravity does, while higher-derivative terms can change $\mu$ and $v_B$. Rotation and anisotropy make the front direction-dependent. An extremal horizon has zero surface gravity, so the simple $e^{2\pi Tt}$ boost and logarithmic formula require a different near-AdS2 analysis.

Exercises

Derive the scrambling time from the shock amplitude and verify how the same transverse mass determines the butterfly speed and pole-skipping momentum.

Solution

For a resolved wavepacket the shift scales as $h\sim\varepsilon_{\rm sh}e^{2\pi Tt_w}$, so $t_*=(2\pi T)^{-1}\log(1/\varepsilon_{\rm sh})$. Away from the source the transverse Green function contributes $e^{-\mu\lvert\mathbf{x}\rvert}$, giving $v_B=2\pi T/\mu$ and $k_\star=i\mu$. For the neutral planar metric in AdS-radius units, $h(r)=r^2$ and $T=(d+1)r_0/(4\pi)$. Pole skipping gives $k_0^2=d\pi T(2r_0)$, so $(2\pi T)^2/k_0^2=(d+1)/(2d)$. This checks the speed with a separate linearized constraint and keeps the scrambling logarithm dimensionless.

References

• Shenker, Stephen H., and Douglas Stanford. “Black Holes and the Butterfly Effect.” Journal of High Energy Physics 03 (2014) 067. arXiv:1306.0622.