Holographic Complexity in Vaidya Spacetimes II

TL;DR

This work extends holographic complexity analyses to Vaidya spacetimes with shock waves in a two-sided AdS black hole, examining both complexity=action and complexity=volume. It shows that including a null-boundary counterterm is crucial for the CA proposal to reproduce the switchback effect and correct late-time growth, while CV consistently captures the qualitative features of complexity evolution and formation across dimensions. The paper provides detailed BTZ (d=2) analytics and higher-dimensional (e.g., d=4) numerical results, highlighting how shock strength and timing control plateau phases, scrambling times, and the approach to late-time growth proportional to the sum of the black hole masses. It also analyzes the complexity of formation, UV structure, and compares CA, CV, and CSV perspectives, offering insights into chaotic dynamics and potential field-theory interpretations of complexity.

Abstract

In this second part of the study initiated in arxiv:1804.07410, we investigate holographic complexity for eternal black hole backgrounds perturbed by shock waves, with both the complexity$=$action (CA) and complexity$=$volume (CV) proposals. In particular, we consider Vaidya geometries describing a thin shell of null fluid with arbitrary energy falling in from one of the boundaries of a two-sided AdS-Schwarzschild spacetime. We demonstrate how known properties of complexity, such as the switchback effect for light shocks, as well as analogous properties for heavy ones, are imprinted in the complexity of formation and in the full time evolution of complexity. Following our discussion in arxiv:1804.07410, we find that in order to obtain the expected properties of the complexity, the inclusion of a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is required for the CA proposal.

Figures (34)

• Figure 1: Penrose-like diagram for one shock wave on an eternal black hole geometry. At $v_s = - t_w$ a thin shock is injected at the right boundary which raises the mass of the black hole from $M_1$ to $M_2$. We identify three points in the geometry that depend on time, $r_m$, where the boundaries of the WDW patch cross behind the past horizon, $r_s$, where the boundary of the WDW patch crosses the collapsing shell in the right exterior, and $r_b$ where the boundary of the WDW patch crosses the shock wave inside the future black hole.
• Figure 2: Time derivative of complexity, evolving both boundaries as $t_\textrm{\tiny L}=t_\textrm{\tiny R}=\frac{t}{2}$ with $T_2 t_w = 2$ (left) and $T_2 t_w = 6$ (right). We have set $w=1+10^{-5}$ and $\tilde{\ell}_\textrm{\tiny ct}=1$. The condition on $z$ implies that the smaller black hole is at the Hawking-Page transition, for both cases. The lower horizontal dashed line (near zero) corresponds to $(M_2-M_1)/(M_2+M_1)$, and by construction, the late time limit approaches $1$ at the higher horizontal line. The horizontal axis in both figures starts from the respective $t_{c0}$ in eq. (\ref{['tcrit']}). The first vertical black dashed line appears at $t=0$, while the vertical red dashed lines appear at $t_{c1}$ (left) and $t_{c2}=2t_w$ (right), see also eq. (\ref{['tcrit']}). There is a negative spike right after $t_{c1}$, where $x_m$ is close to the past singularity. For the earlier shock wave in the right figure, there is a long regime where the rate of change is close to zero. In both cases, the late time limit is approached from above.
• Figure 3: Time derivative of complexity, evolving both boundaries as $t_\textrm{\tiny L}=t_\textrm{\tiny R}=\frac{t}{2}$, with $T_2 t_w = 2$ (left) and $T_2 t_w = 6$ (right). In both cases, we have set $w=2$ and $\tilde{\ell}_\textrm{\tiny ct} =1$. The lower horizontal black dashed line corresponds to the time derivative at early times, i.e.,$(M_2 - M_1)/\pi$ in eq. (\ref{['EarlyTimesRate']}), and the higher line to the late time limit, i.e.,$(M_2 + M_1)/\pi$ in eq. (\ref{['LateTimeEtShock']}). The horizontal axis starts at $t_{c0}$, and the critical times $t_{c1}$ and $t_{c2}$ are shown by the left and right vertical dashed red lines, respectively. There is a negative spike right after $t_{c1}$, where $x_m$ is close to the past singularity. Pushing the shock wave to the past increases the plateau where the time derivative is given roughly by the difference of the masses.
• Figure 4: Dependence of the critical time $t_{c1}$ on $\log\frac{w-1}{2}=\log(\epsilon/2)$, which parametrizes the energy in the shock wave. In the left panel, we show the behaviour of $t_{c1}$ for early shock waves: $T_2 t_w = 3$ in solid blue, and $T_2 t_w = 2$ in dashed red. In this case, we see the transition between the light shock behaviour (\ref{['lavla']}) and the heavy shock behavior (\ref{['tc1Largew']}). In the right panel, we show the behaviour of $t_{c1}$ when the shock wave is not sent very early, i.e., with $T_2 t_w = 0.1, 0.25, 0.5, 0.75, 1$ from bottom to top. For these parameters, the range of $w$ that has approximately a log dependence starts appearing as the shock wave is sent earlier (larger $t_w$). The horizontal thin dashed black line is just $1$ (for both panels).
• Figure 5: Dependence of the critical time $t_{c0}$ on the energy of the shock wave, parametrized by the temperature ratio $w$ for BTZ. In the left, we show the behaviour of $t_{c0}$ with respect to early shock waves with $T_2 t_w = 3$ in solid blue, $T_2 t_w = 2$ in dashed red. Similarly to $t_{c1}$ in figure \ref{['tc1Plots']}, as $w$ approaches one, there is a stretched range of $w$ such that $t_{c1}$ grows as a logarithm, and the earlier the shock wave the longer this log regime. Also, we see that it approaches $2 t_{w}$ in the large $w$ regime. In the right, we show the behaviour of $t_{c0}$ when the shock wave is not sent early enough, with $T_2 t_w = 0.1, 0.25, 0.5, 0.75, 1$ from bottom to top. As the shock wave is sent earlier, the region with log dependence becomes more pronounced.