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Entropy transition in a specified two-candidate model

Physical scope

A conditional prediction for a boundary region whose prescribed candidate class consists of exactly two quantum-stationary surfaces with finite, continuously differentiable generalized-entropy branches on an open boundary time interval. The cusp is a property of the adopted minimum rule, not a deduction about an exact finite-cutoff microscopic entropy or about which saddle contributes in a different gravitational path integral.

Statement

Let $I \subset R$ be an open interval of boundary time, measured in seconds, and let $t_{0} \in I$ . Suppose the prescribed candidate class for the adopted entropy dictionary consists, near $t_{0}$ , of exactly two admissible surfaces $X_{1} (t)$ and $X_{2} (t)$ . Let their finite generalized entropies be $S_{1}, S_{2} \in C^{1} (I, R)$ , with values in $J K^{- 1}$ ; here $C^{1}$ means continuously differentiable. Assume $S_{1} (t_{0}) = S_{2} (t_{0})$ , that $S_{1} (t) \neq S_{2} (t)$ for all sufficiently close $t \neq t_{0}$ , and that $S_{1}^{'} (t_{0}) \neq S_{2}^{'} (t_{0})$ . Then the adopted semiclassical boundary-entropy prediction $S_{\partial}^{m o d e l} (t) = \min {S_{1} (t), S_{2} (t)}$ is continuous at $t_{0}$ and is not differentiable there. This conclusion concerns the selected two-candidate minimum. The hypotheses do not identify it with an exact finite-cutoff microscopic entropy, which need not have this cusp.

Statement references

Restricted semiclassical entropy dictionary

Ahmed Almheiri et al., The entropy of Hawking radiation

Facts & Assumptions

Given: The open time interval, the two prescribed admissible candidates, their finite $C^{1}$ generalized-entropy functions, the isolated crossing, and the unequal derivatives specified in the Statement.

[F1]

Restricted semiclassical entropy dictionary: for a specified candidate class with finite values and an attained minimum, the adopted boundary-entropy prediction is the minimum of the scheme-fixed generalized entropies; the postulate does not identify this prediction with the exact finite-cutoff entropy.

Proof

1.1givenF1algebra

For real numbers $x, y$ , $\min {x, y} = (x + y - ∣ x - y ∣) / 2$ . Therefore $S_{\partial}^{m o d e l} (t) = (S_{1} (t) + S_{2} (t) - ∣ S_{1} (t) - S_{2} (t) ∣) / 2$ . Since $S_{1}$ and $S_{2}$ are continuous, and absolute value is continuous, $S_{\partial}^{m o d e l}$ is continuous at $t_{0}$ . By [F1], this minimum is the model's semiclassical prediction.

1.2givenalgebra

Put $D (t) = S_{1} (t) - S_{2} (t)$ and $m = D^{'} (t_{0}) \neq 0$ . Since $D (t_{0}) = 0$ , the definition of derivative gives $D (t_{0} + h) = m h + r (h)$ with $r (h) / h \to 0$ as $h \to 0$ . Choose $δ > 0$ small enough that $∣ r (h) ∣ < ∣ m ∣ ∣ h ∣ / 2$ whenever $0 < ∣ h ∣ < δ$ . Then $D (t_{0} + h)$ has the sign of $m h$ on each side of $t_{0}$ . Hence the smaller branch on the left is different from the smaller branch on the right.

2.1givenstep 1.2algebra

On each side the minimum equals the selected $C^{1}$ branch from step 1.2. Its left derivative at $t_{0}$ is one of $S_{1}^{'} (t_{0}), S_{2}^{'} (t_{0})$ and its right derivative is the other. These numbers are unequal by hypothesis, so the two-sided derivative of $S_{\partial}^{m o d e l}$ does not exist.

3.1givenF1step 2.1algebra∎

Supplement the same model data with a regulated boundary code space $C^{2}$ and constant boundary-state family $ρ_{\partial} (t) = I_{2} / 2$ for $t \in I$ . This is a density operator supported on that code space. Its eigenvalues are $1 / 2, 1 / 2$ , so $S_{e x a c t} (t) = - k_{B} tr (ρ_{\partial} (t) \log ρ_{\partial} (t)) = k_{B} \log 2$ is constant and smooth. The postulate does not impose $S_{e x a c t} = S_{\partial}^{m o d e l}$ , so this assignment is compatible with the two-branch premises and the cusp established in step 2.1. Thus those premises do not force a cusp in exact finite-cutoff entropy.

Remarks

For the symmetric branches $S_{1} = S_{0} + a (t - t_{0})$ and $S_{2} = S_{0} - a (t - t_{0})$ , with $a > 0$ measured in $J K^{- 1} s^{- 1}$ , the plot below shows the crossing and the lower envelope. It is illustrative only; the proof above establishes the result for all functions satisfying the stated hypotheses.

Depends on

Restricted semiclassical entropy dictionary

Dependency tree · 2 levels

Sources

Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, The entropy of Hawking radiation, §§6 and 8