Time Dependence of Hawking Radiation Entropy (1301.4995v3)

Abstract: 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 4 pi M_0^2, or about 7.509 M_0^2 \approx 6.268 x 10^{76}(M_0/M_sun)^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 15.018 M_0^2 \approx 1.254 x 10^{77}(M_0/M_sun)^2, and then decreases back down to 4 pi M_0^2 = 1.049 x 10^{77}(M_0/M_sun)^2.

• The paper demonstrates that Hawking radiation entropy increases then decreases, peaking at approximately 53.81% of a black hole's lifetime with 59.75% of its original entropy remaining.
• The paper finds that for black holes beginning in mixed states, the von Neumann entropy can exceed the initial Bekenstein-Hawking entropy, peaking at about 119.51%.
• The paper supports the argument for unitarity in black hole evaporation and challenges firewall proposals through detailed numerical analysis.

Time Dependence of Hawking Radiation Entropy: A Numerical Analysis

The paper "Time Dependence of Hawking Radiation Entropy" by Don N. Page tackles a fundamental question in quantum gravity concerning the time evolution of von Neumann entropy in the context of Hawking radiation from black holes. Through numerical analysis, the paper focuses on the dynamical aspects of a black hole's von Neumann entropy under the assumption of fast scrambling, specifically in large Schwarzschild black holes emitting predominantly photons and gravitons.

The research begins with a central premise: if black hole evaporation is a unitary process starting from a pure quantum state, the von Neumann entropy of the Hawking radiation should initially increase but eventually decrease back to zero as the black hole vanishes. This trajectory results from the entanglement between the black hole and its emitted radiation, which transfers over time to an entanglement among the radiation particles themselves.

Key Contributions

1. Numerical Insights into Entropy Evolution: The paper provides numerical results depicting the entropy evolution for large nonrotating black holes. The peak von Neumann entropy of the Hawking radiation is shown to occur after approximately 53.81% of the black hole's lifetime, at which point the black hole retains approximately 59.75% of its original Bekenstein-Hawking (BH) entropy. This finding corroborates the notion that the semiclassical approximation effectively governs the entropy dynamics until equilibrium is reached between the black hole and its radiation.
2. Entropy in Initially Mixed States: The author extends the analysis to scenarios where the black hole starts in a mixed quantum state. Here, the von Neumann entropy can actually exceed that of the initial BH entropy, peaking at about 119.51% of its initial value before falling back to the entropy associated with the complete black hole evaporation.
3. Theoretical Relevance: This paper emphasizes the need for a consistent holographic understanding and offers numerically backed evidence to challenge the firewall arguments postulated in the AMPS (Almheiri, Marolf, Polchinski, Sully) proposal.

Implications

The implications of this work are twofold. Practically, it provides a method to numerically approximate the Hawking radiation entropy curve, potentially guiding future observational strategies related to black hole evaporation. Theoretically, it strengthens the argument for unitarity in black hole evolution, supporting the view that information is conserved and challenging the concept of firewalls proposed to solve the information paradox.

Future Directions

In advancing this work, future research could explore validations from a broader range of black hole parameters, including rotation and charge to more comprehensively address Kerr-Newman black holes. Additionally, further investigation into quantum gravity theories that align with these numerical findings remains vital to bolster or refine the conclusions drawn here.

In conclusion, Don N. Page's investigation into the time dependence of Hawking radiation entropy provides relevant insights into the dynamics of black hole evaporation and contributes significantly to the discourse on quantum gravity's fundamental principles, particularly addressing the intricacies of information preservation in black holes.