de Sitter extremal surfaces, time contours, complexifications and pseudo-entropies

Abstract

We study no-boundary de Sitter extremal surfaces and their pseudo-entropy areas for generic subregions at the future boundary, building on previous work. For large subregions, timelike+Euclidean extremal surfaces exist with transparent geometric interpretations, as do complex ones. The situation for small subregions is analogous to Poincare $dS$ and only complex extremal surfaces exist. In general, the extremal surface area integrals are defined via time contours in the complex time plane. We find multiple extremal surfaces with indistinguishable areas whose time contours can be deformed into each other in the complex time plane without obstruction, which are equivalent for these purposes. This also suggests equivalences between complex $dS$ replica geometries. We discuss $dS_3$ as a simple example at length. This suggests a picture for multiple subregions and entropy inequalities in de Sitter, as encoding $AdS$ ones via analytic continuation. We also discuss mapping future boundary subregions and those on constant time slices in the static patch via lightrays.

Figures (3)

• Figure 1: $dS_3$ no-boundary extremal surfaces on an $S^2$ equator slice (right side; left is the "top view" from ${\mathcal{I}}^+$): the red IR extremal surface for maximal ${\mathcal{I}}^+$ subregion $[-{\pi\over 2}, {\pi\over 2}]$ ($S^{1}$ hemisphere), the tilted violet curve for non-maximal subregion $[-\theta_\infty, \theta_\infty]$, the blue limiting timelike+Euclidean surface for $[-{\pi\over 4}, {\pi\over 4}]$, and the green dashed complex extremal surface for smaller subregions.
• Figure 2: Time contours in the complex time $r$-plane for timelike+Euclidean (red or blue) and complex $dS_3$ extremal surfaces (IR = red, large subregion = blue, small subregion = green, with turning points on the imaginary-$r$ axis), and deformations thereof (skirting the potential pole at $r=l$).
• Figure 3: Lorentzian $dS$ and lightrays in the $(t,r)$-plane Penrose diagram (left) from observers defining the $r=r_0$ cutoff slice on the $t=const$ midslice in the static patch to the hemisphere subregion on the $t=0$ vertical slice at the cutoff future boundary at $r=R_c$. On the right, the horizontal circles represent the equatorial plane spheres, with sizes growing with $r$ from the midslice to ${\mathcal{I}}^+$. A small subregion (green) inflates to a large one at ${\mathcal{I}}^+$ along the lightrays (red). The blue shadow is the left-right extremal surface.