Time Dependence of Hawking Radiation Entropy
Time Dependence of Hawking Radiation Entropy
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DOI：
10.1088/1475-7516/2013/09/028
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发表时间：
2013-01
影响因子：
6.4
通讯作者：
D. Page
中科院分区：
文献类型：
--
作者：
D. Page
如果一个黑洞从一个纯量子态开始，并通过一个幺正过程完全蒸发，霍金辐射的冯诺依曼熵最初会增加，然后当黑洞消失时会减少到零。在这里给出的数值结果的近似的时间依赖性的辐射熵的假设下快速混乱，大的非旋转黑洞，基本上只发射光子和引力子。冯诺依曼熵的最大值出现在蒸发时间的约53.81%之后，此时黑洞已经失去了其原始贝肯斯坦-霍金（BH）熵的约40.25（其冯诺依曼熵的上限），然后具有等于辐射中的熵的BH熵，其约为原始BH熵4πM02的59.75%，或约7.509M02 <$6.268 × 1076（M0/M <$）2，使用我1976年的计算，光子和引力子发射到真空中的过程给出了黑洞BH熵损失的约1.4847倍。结果也给出了黑洞在初始不纯状态。如果黑洞从最大混合态开始，霍金辐射的冯诺依曼熵从零增加到最大值，约为原始BH熵的119.51%，或约为15.018 M02 <$1.254 × 1077（M0/M <$）2，然后下降到4πM02 = 1.049 × 1077（M0/M <$）2。
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.