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      Time Dependence of Hawking Radiation Entropy

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          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.

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          Quantum source of entropy for black holes

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            The Stretched Horizon and Black Hole Complementarity

            Three postulates asserting the validity of conventional quantum theory, semi-classical general relativity and the statistical basis for thermodynamics are introduced as a foundation for the study of black hole evolution. We explain how these postulates may be implemented in a ``stretched horizon'' or membrane description of the black hole, appropriate to a distant observer. The technical analysis is illustrated in the simplified context of 1+1 dimensional dilaton gravity. Our postulates imply that the dissipative properties of the stretched horizon arise from a course graining of microphysical degrees of freedom that the horizon must possess. A principle of black hole complementarity is advocated. The overall viewpoint is similar to that pioneered by 't~Hooft but the detailed implementation is different.
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              Average Entropy of a Subsystem

                (2010)
              If a quantum system of Hilbert space dimension \(mn\) is in a random pure state, the average entropy of a subsystem of dimension \(m\leq n\) is conjectured to be \(S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}\) and is shown to be \(\simeq \ln m - \frac{m}{2n}\) for \(1\ll m\leq n\). Thus there is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state.
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                Author and article information

                Journal
                21 January 2013
                2013-08-09
                Article
                10.1088/1475-7516/2013/09/028
                1301.4995

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                50 pages, LaTeX; results added for black holes initially in impure states
                hep-th gr-qc

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