Small Schwarzschild de Sitter black holes, quantum extremal surfaces and islands

Abstract

We study 4-dimensional Schwarzschild de Sitter black holes in the regime where the black hole mass is small compared with the de Sitter scale. Then the de Sitter temperature is very low compared with that of the black hole and we study the black hole, approximating the ambient de Sitter space as a frozen classical background. We consider distant observers in the static diamond, far from the black hole but within the cosmological horizon. Using 2-dimensional tools, we find that the entanglement entropy of radiation exhibits linear growth in time, indicative of the information paradox for the black hole. Self-consistently including an appropriate island emerging at late times near the black hole horizon leads to a reasonable Page curve. There are close parallels with flat space Schwarzschild black holes in the regime we consider.

Small Schwarzschild de Sitter black holes, the future boundary and islands

“…This paper continues the study in [92] of "small" Schwarzschild de Sitter black holes, with the black hole mass m small compared with the de Sitter scale l, but large enough that a quasi-static approximation to the geometry is valid. The de Sitter temperature is very low compared with that of the black hole so the ambient de Sitter space is approximated as a frozen classical background.…”

“…In [92], we considered the radiation region to be in the static diamond bounded by the black hole and cosmological horizons in the Schwarzschild de Sitter background: this static patch is parametrized by certain Kruskal coordinates [136] in the vicinity of the black hole horizon. For our present purposes, we would like to analytically extend the Kruskal JHEP05(2024)016 coordinates (U D , V D ) defined in the static patch in the vicinity of the cosmological horizon and (U S , V S ) near the black hole horizon to a new set of Kruskal coordinates (U ′ D , V ′ D ) lying within the future universe, near the future boundary, and (U ′ S , V ′ S ) in the interior of black hole (inside the horizon) respectively.…”

“…We imagine that the black hole has formed from initial matter in a pure state which is a reasonable approximation since the de Sitter temperature is very low (more generally, the bulk matter CFT is in a thermal state at the de Sitter temperature). In [92], we focussed on one black hole coordinate patch in the Penrose diagram (roughly a line of alternating Schwarzschild and de Sitter patches, see figure 1) and considered observers in the static diamond patches far outside the black hole but within the cosmological horizons. While the entanglement entropy of the radiation region exhibits unbounded growth, reflecting the information paradox for the black hole (which has finite entropy), including appropriate island contributions recovers finiteness of entanglement, and thereby expectations on the Page curve.…”

“…Section 3 discusses the entanglement entropy without islands (details in appendix B), while section 4 discusses the island calculation (details in appendix C). Appendix A is a brief review of the analysis in [92] for radiation entropy with islands from the point of view of the static diamond observers. Appendix D.1-D.2 discuss inconsistencies in other potential island solutions, while appendix E discusses timelike separated quantum extremal surface solutions for future boundary observers.…”

An outsider’s perspective on information recovery in de Sitter space

“…The reason why it is even possible to apply the derived expressions for the entropy to de Sitter space is that they are agnostic about the precise background under consideration. While originally studied in the context of holography in Anti-de Sitter space, JHEP01(2023)129 by now many works have applied this so-called island formula to other backgrounds, such as a plethora of black hole solutions [16][17][18][19][20][21][22][23][24][25] and cosmological spacetimes [14,15,[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. As long as it is possible to unambiguously define a subsystem entangled with a gravitating region, the island formula applies.…”