Holographic Complexity in Vaidya Spacetimes I

TL;DR

The paper studies holographic complexity in time-dependent AdS-Vaidya spacetimes formed by a thin shell of null fluid, comparing CV and CA proposals. It demonstrates that the null fluid action vanishes on-shell and that including a null-boundary counterterm is necessary for CA to yield physically sensible complexity in dynamical settings. With the counterterm, the late-time growth rate of complexity for a one-sided black hole matches the universal rate 2M/π found for eternal black holes, in both CV and CA, while finite-curvature effects modify early- and late-time behaviors in spherical geometries. The results highlight the importance of properly treating null boundaries in CA and reveal a consistent holographic picture of complexity growth across dynamical black-hole formation processes.

Abstract

We examine holographic complexity in time-dependent Vaidya spacetimes with both the complexity$=$volume (CV) and complexity$=$action (CA) proposals. We focus on the evolution of the holographic complexity for a thin shell of null fluid, which collapses into empty AdS space and forms a (one-sided) black hole. In order to apply the CA approach, we introduce an action principle for the null fluid which sources the Vaidya geometries, and we carefully examine the contribution of the null shell to the action. Further, we find that adding a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is essential if the gravitational action is to properly describe the complexity of the boundary state. For both the CV proposal and the CA proposal (with the extra boundary counterterm), the late time limit of the growth rate of the holographic complexity for the one-sided black hole is precisely the same as that found for an eternal black hole.

Figures (8)

• Figure 1: The null shell has a finite thickness $2\varepsilon$ around the null ray $v=v_s$. The portion enclosed by the WDW patch is shaded in orange. The contribution of the two joints indicated by red dots exactly cancels the surface term for the portion of the null boundary connecting the joints, where we have a time dependent $\kappa (v)$.
• Figure 2: Penrose-like diagrams for the thin shell collapsing geometries, we represent spherical horizon collapse from global AdS (left) and planar horizon from Poincaré patch (right). In order to not distort the diagrams, we represent the discontinuity in the outgoing coordinate $u$ by a jump while crossing the collapsing shell, e.g., the dashed blue line indicates the extension of the event horizon into the region before the collapsing shell. We use $r_s$ to denote the radial position where the null boundary of the WDW patch crosses the shock wave.
• Figure 3: The growth rate for the complexity, evaluated without (left) and with (right) the boundary counterterm in $d=2$. In both plots we have the collapse from Neveu-Schwarz vacuum ( i.e.,$k=+1$) with temperatures $LT = 0.16$ (blue, solid), $LT = 0.25$ (orange dashed) and $LT = 1.0$ (green dot-dashed). The collapse from Ramond vacuum ( i.e.,$k=0$) is shown in red. For the NS vacuum, the growth rate always starts at different values for different temperatures, as given by eq. \ref{['earlyRateBTZ_NoCT']} (left) and eq. \ref{['earlyRateBTZ']} (right). In both cases, the high temperature limit of the NS collapse approaches the Ramond collapse. At late times, independent of the temperature, the rate of change approaches zero on the left, and $2M/\pi$ on the right.
• Figure 4: The growth rate for the complexity in $d=3$ (left) and $d=4$ (right) and spherical geometry ($k=+1$), evaluated without (red and orange curves) and with (blue and cyan curves) the boundary counterterm (\ref{['counter']}). In both case, we evaluate the growth rate for temperatures $TL = 0.35$ (solid), $TL = 0.5$ (dashed) and $TL = 2.0$ (dot-dashed) in the left and $T L = 0.5$ (solid), $T L = 0.8$ (dashed) and $T L = 1.5$ (dot-dashed) in the right figure. In both dimensions, $d{\cal C}_\textrm{\tiny A}/dt_0$ (without the counterterm) starts at the value of the planar rate of change given by eq. \ref{['EarlyTimenoCT']} and approaches the late time limit from below in eq. \ref{['LateTimenoCT']}. The late time growth rate in this case is smaller than the one for the eternal black hole, and it depends on the temperature. With the inclusion of the counterterm, $d{\cal C}_\textrm{\tiny A}'/dt_0$ starts at half of its late time limit, then it grows at times of the order of the thermal length, and approaches the eternal black hole bound from below.
• Figure 5: Penrose-like diagram of maximal volume surfaces at different times embedded in the Vaidya AdS spacetime. Constant time slices are indicated by thin dashed gray lines and the maximal volume surfaces asymptote them near the boundary. The event horizon extends past the shell, as we have indicated by a thick dashed gray line. Since the momentum (\ref{['ener22']}) of the surfaces is positive, they evolve towards decreasing time outside the horizon. Surfaces lie on constant time slices in the vacuum part of spacetime to avoid a conical singularity at $r=0$.