On holographic time-like entanglement entropy

TL;DR

The work addresses defining holographic time-like entanglement entropy in AdS/CFT without relying on analytic continuation. It introduces complex-valued weak extremal surfaces (CWES) as a principled way to select a unique complex area 𝒜(Γ) when mixing space-like and time-like segments, and then defines S_A via a dominance-based extremization over CWES. Through detailed analyses in AdS3/CFT2 (Poincaré and global AdS3) and BTZ black holes, the authors show that CWES reproduces known results, including the correct imaginary parts associated with timelike intervals, and extends to multi-interval and finite-temperature contexts. The framework offers a self-contained holographic mechanism to study timelike subregions and hints at applications to probing black hole interiors and more general spacetimes.

Abstract

In order to study the pseudo entropy of time-like subregions holographically, the previous smooth space-like extremal surface was recently generalized to mix space-like and time-like segments and the area becomes complex value. This paper finds that, if one tries to use such kind of piecewise smooth extremal surfaces to compute time-like entanglement entropy holographically, the complex area is not unique in general. We then generalize the original holographic proposal of space-like entanglement entropy to pick up a unique area from all allowed ``space-like+time-like'' piecewise smooth extremal surfaces for a time-like subregion. We will give some concrete examples to show the correctness of our proposal.

Figures (10)

• Figure 1: Penrose diagrams on Poincaré patch of AdS$_3$ spacetime and geodesics connecting $\partial A$. Left panel: The blue solid curves $A_1A_2$ and $B_1B_2$ stand for the space-like geodesics used in Ref. Doi:2022iyj and the red solid curve $A_2B_2$ stands for the time-like geodesic. The blue solid curves $A_1A_3$ and $B_1B_3$ and the red solid curve $A_3B_3$ stand for other possible geodesics. Right panel: Using two space-like geodesics to connect the endpoints of time-like interval $A$.
• Figure 2: The analog of multiple variables function for the local extremal points. When the maximal point is in the interior of definition domain, we require $\partial_xf=\partial_yf=0$, i.e. $\text{d} f=0$. However, it the maximal point locates at boundary of definition domain, we only needs $\partial_yf=0$ but $\text{d} f\neq0$ in general.
• Figure 3: There are three different possible configurations to connect the endpoints of time-like interval $A$ by a few of geodesics.
• Figure 4: Left: For a timelike interval in a Poincaré patch of AdS$_{3}$ spacetime, we can always use four null geodesics to connect its endpoints. Such piecewise null geodesics has infinitely many different choices but they all have zero area. Right: the CWES contains a connected branch $A_1A_2B_2B_1$ and a closed timelike branch $Q$.
• Figure 5: The global coordinate of AdS$_3$ spacetime. The green boundary stands for the null boundary of Poincaré patch. The blue and red dashed curves stand for the CWES of Poincaré patch. The right panel is the cut of $\phi=0$. The blue and red solid curves stand for the spacelike and timelike geodesics. Note that the global AdS$_3$ spacetime has topology $S^1\times R^2$, of which the time coordinate is periodic $\tau\sim\tau+2\pi$. Thus, the point $A_3$ and $B_3$ are same point and curve $A_2A_3B_3B_2A_2$ forms a closed circle.