On volume subregion complexity in Vaidya spacetime

Abstract

We study holographic subregion volume complexity for a line segment in the AdS$_3$ Vaidya geometry. On the field theory side, this gravity background corresponds to a sudden quench which leads to the thermalization of the strongly-coupled dual conformal field theory. We find the time-dependent extremal volume surface by numerically solving a partial differential equation with boundary condition given by the Hubeny-Rangamani-Takayanagi surface, and we use this solution to compute holographic subregion complexity as a function of time. Approximate analytical expressions valid at early and at late times are derived.

Figures (9)

• Figure 1: Kinds of space-like geodesics as function of $(J,E)$.
• Figure 2: Plots of the space-like geodesic (\ref{['geo1']}) in BTZ spacetime with $E=0$ and different values of the parameter $J$, with $r_{h}=1$. The blue curve represents $x_{+}(r)$, while the yellow one represents $x_{-}(r)$.
• Figure 3: Plots of the space-like geodesic (\ref{['geo1']}) in BTZ spacetime with different values of the parameters $(E,J)$, with $r_h=1$. The blue curve represents $x_{+}(r)$, while the yellow one represents $x_{-}(r)$.
• Figure 4: The plots show $r_{s}$ (solid line) and $r_{*}$ (dashed line) as a function of the boundary time $t$. Here $r_h=1$, and we set $l=5$ on the left and $l=12$ on the right.
• Figure 5: Time evolution of the geodesic for $l=8, r_h=1$. The black and red curves respectively denote branch 1 and 2 in the BTZ part; the blue curves denote the AdS part of the full geodesic.