In semiclassical gravity, Hawking’s thermal radiation implies monotonic entropy growth, violating unitarity; instead unitarity demands a Page curve with eventual entropy decrease. Recent work using quantum extremal surfaces and entanglement islands shows that including new interior island regions can produce a Page curve consistent with unitarity. Using an influence-functional (open-system) approach to Hawking radiation with gravitational replica wormholes, we find that integrating out gravity in the replica path integral (including connected wormhole saddles) gives a matter influence functional that fails to factorize across replicas. Equivalently, graviton exchange between replicas couples different emission times, imprinting a nonlocal bilocal memory kernel on the radiation entropy. The wormhole saddle yields an n-independent bilocal correction: the entropy acquires an additive two-point kernel term, yielding a nonlocal memory contribution. In the early (no-island) phase, the entropy grows as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S \simeq S_{\text {bulk}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>≃</mo> <msub> <mi>S</mi> <mtext>bulk</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S_{\text {bulk}} = S_{\text {bulk,local}} + S_{\text {memory}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mtext>bulk</mtext> </msub> <mo>=</mo> <msub> <mi>S</mi> <mtext>bulk,local</mtext> </msub> <mo>+</mo> <msub> <mi>S</mi> <mtext>memory</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S_{\text {memory}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mtext>memory</mtext> </msub> </math></EquationSource> </InlineEquation> is given by a two-time kernel integral. The Page curve is thus modified: the initial rise has an enhanced slope and eventual saturation due to memory effects, while at late times extremizing the full entropy (including <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_{\text {memory}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mtext>memory</mtext> </msub> </math></EquationSource> </InlineEquation>) yields a minimal island-phase entropy at the Bekenstein–Hawking value. Our results imply that these non-Markovian gravitational correlations – effectively a form of gravitational memory or soft hair – are essential for a consistent semiclassical Page curve. This reinterprets the island prescription as encoding delayed gravitational interactions, providing a concrete mechanism to restore unitarity in black hole evaporation.

Non-Markovian Memory Effects and Bilocal Entropy Corrections from Gravitational Replica Wormholes

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发表日期：

2026年5月30日

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Alberto Miró Morán

摘要

In semiclassical gravity, Hawking’s thermal radiation implies monotonic entropy growth, violating unitarity; instead unitarity demands a Page curve with eventual entropy decrease. Recent work using quantum extremal surfaces and entanglement islands shows that including new interior island regions can produce a Page curve consistent with unitarity. Using an influence-functional (open-system) approach to Hawking radiation with gravitational replica wormholes, we find that integrating out gravity in the replica path integral (including connected wormhole saddles) gives a matter influence functional that fails to factorize across replicas. Equivalently, graviton exchange between replicas couples different emission times, imprinting a nonlocal bilocal memory kernel on the radiation entropy. The wormhole saddle yields an n-independent bilocal correction: the entropy acquires an additive two-point kernel term, yielding a nonlocal memory contribution. In the early (no-island) phase, the entropy grows as $S \simeq S_{\text {bulk}}$ $S ≃ S_{bulk}$ , where $S_{\text {bulk}} = S_{\text {bulk,local}} + S_{\text {memory}}$ $S_{bulk} = S_{bulk,local} + S_{memory}$ and $S_{\text {memory}}$ $S_{memory}$ is given by a two-time kernel integral. The Page curve is thus modified: the initial rise has an enhanced slope and eventual saturation due to memory effects, while at late times extremizing the full entropy (including $S_{\text {memory}}$ $S_{memory}$ ) yields a minimal island-phase entropy at the Bekenstein–Hawking value. Our results imply that these non-Markovian gravitational correlations – effectively a form of gravitational memory or soft hair – are essential for a consistent semiclassical Page curve. This reinterprets the island prescription as encoding delayed gravitational interactions, providing a concrete mechanism to restore unitarity in black hole evaporation.