Entanglement Islands and Infrared Anomalies in Schwarzschild Black Hole

Abstract

In this paper, island formation for entangling regions of finite size in the asymptotically flat eternal Schwarzschild black hole is considered. We check the complementarity property of entanglement entropy which was implicitly assumed in previous studies for semi-infinite regions. This check reveals the emergence of infrared anomalies after regularization of a Cauchy surface. A naive infrared regularization based on ``mirror symmetry'' is considered and its failure is shown. We introduce an improved regularization that gives a correct limit agreed with the semi-infinite results from previous studies. As the time evolution goes, the endpoints of a finite region compatible with the improved regularization become separated by a timelike interval. We call this phenomenon the ``Cauchy surface breaking''. Shortly before the Cauchy surface breaking, finite size configurations generate asymmetric entanglement islands in contrast to the semi-infinite case. Depending on the size of the finite regions, qualitatively new behaviour arises, such as discontinuous evolution of the entanglement entropy and the absence of island formation. Finally, we show that the island prescription does not help us to solve the information paradox for certain finite size regions.Comment: v1: 55 pages, 19 figures; v2: 57 pages, 19 figures, references added, Sec. 5 presentation improve