Abstract
We comment on the role of the graviton mass in recent calculations of the Page curve using holographic ideas. All reliable calculations of the Page curve in more than 2+1 spacetime dimensions have been performed in systems with massive gravitons. A crucial ingredient in these calculations is the formation of islands, regions that contribute to the entropy of degrees of freedom located elsewhere. While most often simply ignored, it is indeed true that mass of the graviton does not appear to significantly affect the calculations that appeared in the literature. We use the freedom to change the graviton mass to give an extremely simple model of analytically tractable island formation in general dimensions. We do however note that if one attempts to take the limit of zero graviton mass, any contribution from the islands disappears. This raises the question to what extent entanglement islands can play a role in standard massless gravity.
Highlights
We comment on the role of the graviton mass in recent calculations of the Page curve using holographic ideas
The description 1) is the full holographic dual, a BCFT living on the part of the AdSd+1 boundary that has not been removed by the brane, with boundary conditions imposed at the location of the intersection of brane and boundary
Any boundary conditions that allow energy to be transferred from the AdSd brane to the bulk of the BCFT will lead to energy non-conservation on the defect and an anomalous dimension for the corresponding stress tensor and so a mass for the bulk graviton
Summary
As we reviewed in the introduction, massive gravitons are unavoidable anytime we impose transparent boundary conditions on AdS gravity. We study a d-dimensional field theory living on Minkowski space and pick an otherwise undistinguished plane to call it the defect and apply the rules of BCFTs to it In this case the BOPE is a Taylor series expansion and so the lowest operator appearing in the BOPE of the dimension ∆ = d stress tensor has dimension ∆ = d. One point we wish to make here is that if we do calculations with a graviton whose mass is of order 1/l in any case (that is, a stress tensor whose BOPE starts with an operator of dimension ∆ that is not infinitessimally close to d − 1), one might as well go all the way to the theory at λ = 0. The bulk calculations are analytically tractable and do not require numerical GR
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