On the Time Dependence of Holographic Complexity

TL;DR

This work analyzes the full time dependence of holographic complexity in eternal AdS black holes using both the complexity=action (CA) and complexity=volume (CV) proposals across neutral and charged backgrounds and various horizon geometries. CV shows a monotonic increase of the complexity rate that saturates to a positive constant at late times, while CA exhibits an initial plateau, a short negative burst after a critical time t_c, and then growth that overshoots the late-time bound. Charging the black holes removes the early-time peculiarities but does not eliminate the late-time behavior, and the complexity of formation diverges for extremal charged black holes, signaling infinite complexity at zero temperature with finite chemical potential. The study also discusses ambiguities in null-boundary terms, averaging schemes, and the broader implications for proposed complexity bounds in holography.

Abstract

We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. Adding a charge to the eternal black holes washes out the early time behaviour, i.e., complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.

Figures (34)

• Figure 1: Penrose diagram of the WDW patch of an eternal AdS black hole, moving forward in time in a symmetric way ($t_L=t_R$).
• Figure 2: Left panel: time derivative of the complexity for the BTZ black hole ($d=2$) from eq. (\ref{['btznew2']}) with $\alpha=L/R$. Right panel: 'total' complexity found by integrating $d\mathcal{C}_A/d\tau$. Results are shown for several values of the horizon radius --- $r_h/L=1$ (blue), $r_h/L=1.5$ (dashed red) and $r_h/L=3.5$ (dot-dashed green).
• Figure 3: Critical time $t_c$ as a function of the horizon radius for $d=4$ for the various horizon geometries, i.e., spherical $k=1$ (blue), planar $k=0$ (dashed-red) and large hyperbolic $k=-1$ (dot-dashed green). Note that we only consider $r_h>L$.
• Figure 4: Time derivative of complexity as a function of time for spherical ($k=+1$, left) and planar ($k=0$, right) horizons with $d=4$ boundary dimensions for various values of the horizon radius, i.e.,$r_h/L=1$ (blue), $r_h/L=1.5$ (dashed red) and $r_h/L=3.5$ (dot-dashed green). We present the plots as a function of $\delta \tau= \tau-\tau_c$ to allow for a meaningful comparison between the different cases. We stress again that each of the curves has a different value of $\tau_c$ --- see figure \ref{['fig:d4ActionEternaltcOrh']}.
• Figure 5: A representation of the maximal wormhole connecting the two boundaries anchored at times $t_{L}$ and $t_{R}$ (depicted at symmetric times in the figure). The bridge reaches the minimum distance inside the future horizon at $r_{min}$, and approaches each boundary tangent to constant time slices.