Time Dependence of Hawking Radiation Entropy
By D. Page. 2013
366 citations recorded by the source corpus.
If a black hole starts in a pure quantum state and evaporates completely by a unitary process, the von Neumann entropy of the Hawking radiation initially increases and then decreases back to zero when the black hole has disappeared. Here numerical results are given for an approximation to the time dependence of the radiation entropy under an assumption of fast scrambling, for large nonrotating black holes that emit essentially only photons and gravitons. The maximum of the von Neumann entropy then occurs after about 53.81% of the evaporation time, when the black hole has lost about 40.25% of its original Bekenstein-Hawking (BH) entropy (an upper bound for its von Neumann entropy) and then has a BH entropy that equals the entropy in the radiation, which is about 59.75% of the original BH entropy 4πM02, or about 7.509M02 ≈ 6.268 × 1076(M0/M⊙)2, using my 1976 calculations that the photon and graviton emission process into empty space gives about 1.4847 times the BH entropy loss of the black hole. Results are also given for black holes in initially impure states. If the black hole starts in a maximally mixed state, the von Neumann entropy of the Hawking radiation increases from zero up to a maximum of about 119.51% of the original BH entropy, or about 15.018M02 ≈ 1.254 × 1077(M0/M⊙)2, and then decreases back down to 4πM02 = 1.049 × 1077(M0/M⊙)2.
Subfields: Astronomy and Astrophysics, Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics
View Paper: https://www.semanticscholar.org/paper/e37fcb725d26bcc244aa48402dc36ce16b2e0728
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Citations in the corpus
- Jerusalem Lectures on Black Holes and Quantum Information (2014, methodology): “By being more careful about details of the evaporation, such as greybody factors and the number and helicities of the available massless particles, one can work out more of the quantitative details about exactly when the curve turns over and at what value of the entropy [41].”
- Islands in Schwarzschild black holes (2020, neutral): “The time evolution of the entanglement entropy is described by the so-called Page curve [3, 4].”
- Islands in asymptotically flat 2D gravity (2020, background): “5, has the form anticipated by Page [49,50]: It grows linearly until an O(1) fraction of the black”
- Information flow in black hole evaporation (2019, background): “he bath, but then there is a sharp transition to a phase where it decreases again. This rise and fall of the black hole entropy are characteristic of the behaviour exhibited by the classic Page curve [33,34]. This novel transition occurs in this holographic model (and in the model described by [1]) ”
- Lessons from the information paradox (2020, result): “Simple estimates — obtained by comparing the entropy of Hawking radiation with the original Bekenstein-Hawking entropy — suggest that S ′ S is O (1) [226], which implies that e ′ e.”
- Diving into traversable wormholes (2017, background): “Another issue we can discuss is the famous Page curve (see figure 1 in [41]).”
- Entanglement wedge reconstruction and the information paradox (2019, background): “occurs when the horizon area Ahor > Ahor/2, despite commonly being called the halfway point of the evaporation [6].”
- Black Hole Remnants and the Information Loss Paradox (2014, background): “Note that due to the gray-body factor, the turning point is not exactly at f = 1/2 [18].”
- The Page curve of Hawking radiation from semiclassical geometry (2019, background): “In particular, unitarity implies that the von Neumann entropy of the Hawking radiation should initially rise but then fall back down, following the so called “Page curve” [3,4].”