The Semiclassical Hawking Curve

Leading semiclassical pair creation makes radiation entropy grow because each outgoing mode is entangled with a new interior partner and is approximately independent of earlier radiation. Under those assumptions, adding a mode adds positive entropy. The curve cannot turn downward until the state, factorization, independence, or entropy prescription changes by an amount comparable to the pair entanglement.

Required background. Quantum Information supplies reduced states, entropy additivity, trace distance, and strong subadditivity used in the derivation.

Helpful background. Semiclassical Gravity supplies the slowly evaporating Schwarzschild background and backreaction regime; Quantum Fields in Curved Spacetime supplies Bogoliubov pair creation and the Hawking state; Black-Hole Information supplies the unitarity tension being diagnosed.

One regulated Hawking pair

For a free bosonic mode of positive Killing frequency $\omega$, the leading near-horizon approximation has the two-mode squeezed form

$$\lvert\chi_\omega\rangle_{bc} =\sqrt{1-e^{-\beta\omega}} \sum_{n=0}^{\infty}e^{-n\beta\omega/2} \lvert n\rangle_b\lvert n\rangle_c,$$

where $b$ is exterior, $c$ is its interior partner, and $\beta=T_{\mathrm H}^{-1}$. This is the pair before propagation through the exterior greybody barrier; it is a narrow-frequency packet approximation, not an exact decomposition of every localized detector mode. For a four-dimensional Schwarzschild black hole, $T_{\mathrm H}=1/(8\pi GM)$ in the global units. A finite set of packets still contains infinite oscillator occupation spaces. The displayed sum is convergent for $\beta\omega\gt0$; finite-dimensional entropy-continuity arguments additionally require an occupation cutoff and renormalization of the truncated state, or an energy-constrained bound. Fermionic modes instead have occupations zero and one.

Tracing over $c$ gives a thermal geometric distribution,

$$\rho_b=(1-e^{-\beta\omega}) \sum_{n=0}^{\infty}e^{-n\beta\omega}\lvert n\rangle\langle n\rvert, \qquad \bar n_\omega=\frac{1}{e^{\beta\omega}-1}.$$

Its entropy is

$$s_\omega=(\bar n_\omega+1)\ln(\bar n_\omega+1) -\bar n_\omega\ln\bar n_\omega\gt0.$$

The complete pair is pure. The positive exterior entropy records entanglement with $c$, not fundamental loss at one emission step.

Deriving monotonic growth

Let $R_{k-1}$ contain all previously emitted wave packets and let $b_kc_k$ be the next pair. The leading “nice-slice” approximation assumes

$$\rho_{R_{k-1}b_kc_k} \simeq\rho_{R_{k-1}}\otimes \lvert\chi_k\rangle\langle\chi_k\rvert_{b_kc_k}.$$

After adding $b_k$ to the radiation and tracing over $c_k$,

$$\rho_{R_k}\simeq\rho_{R_{k-1}}\otimes\rho_{b_k}, \qquad S(R_k)\simeq S(R_{k-1})+s_k.$$

Therefore $\Delta S_R\simeq s_k\gt0$. Greybody factors alter which wave packets escape and their occupation probabilities, but they do not produce the order-one early–late correlations needed for a descending fine-grained branch within this independent-pair approximation. Hawking’s endpoint interpretation promoted the accumulated exterior mixed state to a complete pure-to-mixed evolution Hawking 1976, §§ II–IV.

The first omitted effects are correlations between successive emissions, deviations of the pair from the local vacuum, backreaction beyond adiabatic evolution, and quantum-gravitational corrections to factorization or the entropy observable. A Page turnover requires one or more of these effects to invalidate the additive step.

Why small independent corrections are insufficient

The relevant smallness concerns the joint pair state, not the outgoing marginal alone. A transparent sufficient version uses three independent regulated factors $R,b,c$ in one state. Suppose

$$S(bc)\leq\delta_{\mathrm{pair}}, \qquad S(c)\geq\ln2-\delta_{\mathrm{partner}}.$$

Weak monotonicity, which follows from strong subadditivity, gives $S(Rb)+S(bc)\geq S(R)+S(c)$. Hence

$$S(Rb)-S(R) \geq\ln2-\delta_{\mathrm{pair}}-\delta_{\mathrm{partner}}.$$

No separate independence assumption between $b$ and $R$ was needed: a nearly pure entangled pair already restricts those correlations. Mathur derives entropy estimates from his specific small-amplitude correction convention and obtains $\Delta S_R\gt\ln2-2\epsilon$ Mathur 2009, § 5, Eqs. (53), (56), and (61)–(64). An arbitrary trace-distance parameter cannot simply be substituted for that $\epsilon$; converting it to entropy errors requires a continuity bound and its dimensions.

If the displayed entropy-error sum stays below $\ln2$ at every step, each increment remains positive, regardless of how many steps are taken. This conditional result leaves room for a change in the pair structure, nonlocal encoding, or the gravitational entropy prescription.

The distinction is qualitative as well as quantitative. Adding small phases to each pair may make amplitudes microstate-dependent while leaving the reduced eigenvalues and entropy almost unchanged. Information recovery needs correlations that alter the entanglement pattern, not merely an exponentially long list of individually tiny labels under assumptions that keep the entropy bound intact.

Checks and failure regimes

In the low-occupation limit $\beta\omega\to\infty$, $\bar n_\omega\to0$ and $s_\omega\to0$, so an unexcited mode contributes no entropy. For a maximally entangled qubit pair, $s_k=\ln2$ and $k$ independent pairs give $S(R_k)=k\ln2$, verifying additivity and normalization. If $b_k$ instead purifies a qubit already in $R_{k-1}$, the radiation entropy can decrease; that state explicitly violates pair independence.

Near the endpoint, curvatures and backreaction may invalidate the slowly varying semiclassical geometry. The derivation cannot be extrapolated through that failure and then cited as an exact quantum-gravity prediction. Conversely, an endpoint correction occurring only after an extensive entropy has accumulated must explain how it transfers the correlations without violating its own energy, causal, and Hilbert-space limits.

Relation to the Page curve

The Hawking curve and unitary Page curve condition on different dynamics. The first uses local vacuum pair creation and independent interior partners; the second uses global purity plus typical mixing of radiation and the remaining black hole. Their disagreement is the information problem, not an algebraic error. The entropy review makes these assumptions and the resulting conflict explicit Almheiri et al. 2021, §§ 2–5.

Generic entanglement inequalities belong to Quantum Information, the Bogoliubov calculation to Quantum Fields in Curved Spacetime, and the backreacting lifetime to Semiclassical Gravity. Explicit string or holographic mechanisms are outside this derivation.

Exercise

After the Page time, suppose the new qubit pair obeys $S(b_kc_k)\leq0.05$ nats and $S(c_k)\geq\ln2-0.05$ nats, with the independent factors assumed above. Can the bound permit a descending entropy step? Would knowing only that $\rho_{b_k}=I/2$ suffice?

Solution

The bound gives $\Delta S_R\geq\ln2-0.10\approx0.593$ nats, so no descending step is possible under these hypotheses. The marginal condition alone would not suffice: if $b_k$ is Bell-entangled with a qubit in $R$, its marginal is exactly $I/2$, yet adding it lowers the radiation entropy by $\ln2$. In that counterexample $b_kc_k$ is not a nearly pure entangled pair. Recovery therefore requires a change in the joint entanglement assumptions, subsystem independence, or applicable entropy prescription.

References

• Almheiri, Ahmed, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. 2021. “The Entropy of Hawking Radiation.” Reviews of Modern Physics 93: 035002. DOI.
• Hawking, Stephen W. 1976. “Breakdown of Predictability in Gravitational Collapse.” Physical Review D 14: 2460–2473. DOI.
• Mathur, Samir D. 2009. “The Information Paradox: A Pedagogical Introduction.” Classical and Quantum Gravity 26: 224001. Open text.