Concerns about the replica wormhole derivation of the island conjecture

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 ${\rm tr}(ρ_\textsf{R}^2)$, 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 $ρ_\textsf{R}=ρ_\textsf{R}(σ)$ with $σ$ the set of variables used in the average, the ensemble average is only an accurate estimate of the entropy after $σ$ has been measured. At the formation of each Hawking pair, there is a distribution over $σ$ that affects the entropy. Without a real time determination of $σ$, 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.

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