Does Complexity Equal Anything?

Abstract

We present a new infinite class of gravitational observables in asymptotically Anti-de Sitter space living on codimension-one slices of the geometry, the most famous of which is the volume of the maximal slice. We show that these observables display universal features for the thermofield-double state: they grow linearly in time at late times and reproduce the switch-back effect in shock wave geometries. We argue that any member of this class of observables is an equally viable candidate as the extremal volume for a gravitational dual of complexity.

Figures (3)

• Figure 1: The time evolution of the extremal hypersurfaces from $\Sigma(\tau)$ to a nearby extremal hypersurface $\Sigma(\tau')$. At infinite time limit $\tau \to \infty$, the extremal surface approaches a constant-$r$ hypersurface at $r=r_f$ where the effective potential arrives at a local maximal value.
• Figure 2: The blue curve denotes a characteristic effective potential $\widetilde{U}(r)$ depending on the spacetime curvature and the black curve presents the potential from the volume. The turning point at the minimal radius $r_{\rm{min}}$ satisfies $\dot{r}=0$ which is equivalent to $P_v^2=\widetilde{U} (r_{\rm{min}})$ for a given conserved momentum. The critical value of the momentum $P_{\infty}$ is obtained at the infinite boundary time $\tau \to \infty$.
• Figure 3: Left: The effective potentials $\widetilde{U}$ defined in eq.\ref{['eq:ham3']} with $-1<\tilde{\lambda} < \tilde{\lambda}_{\rm{crt}1}$ present two extremal points inside the horizon which is located at $w=1$. Right: The effective potentials with $\tilde{\lambda} <-1$ or $\tilde{\lambda} > \tilde{\lambda}_{\rm{crt}1}$ have no extremal point inside the horizon. The dashed black curves in both plots denote the potential corresponding to the pure volume $V$.