Concerns about the replica wormhole derivation of the island conjecture

Abstract

In a close look at the replica wormhole derivation of the island conjecture, we note a discrepancy in the derived matrix eigenvalues in the limit where the Hawking radiation entropy goes to zero. The small corrections to the density matrix for the radiation infer that ${\rm tr}(\rho_\textsf{R}^2)$, equally a sum over the eigenvalues squared, must be small ($\ll1$) at all times. This is in contradiction with an ensemble dominated by a density matrix with an eigenvalue close to unity. Also, the radiation entropy has been approximated by the ensemble averaged entropy, as in a final state analysis. For $\rho_\textsf{R}=\rho_\textsf{R}(\sigma)$ with $\sigma$ the set of variables used in the average, the ensemble average is only an accurate estimate of the entropy after $\sigma$ has been measured. At the formation of each Hawking pair, there is a distribution over $\sigma$ that affects the entropy. Without a real time determination of $\sigma$, the radiation entropy follows Hawking's result during the evaporation. A question remains of how quickly such determinations can be assumed to occur. A complete argument for that the radiation entropy follows the Page curve should address this.