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Introduction, Black hole information is one of the greatest puzzles of theoretical. physics from the 20th century that has persisted into the 21st. After Stephen Hawking discovered black hole evaporation in 1974. in 1976 he predicted that black hole formation and evaporation. would cause a pure quantum state to change into a mixed state. effectively losing information from the universe, In 1979 I questioned this conclusion as years later did many others. and in 2004 Hawking conceded that black hole evaporation does. not lose information However there are many other gravitational. theorists who have not accepted Hawking s concession. There do remain many puzzles about black hole information such. as how it gets out if it indeed does and whether there are. firewalls at the surfaces of old black holes that would immediately. destroy anything falling in,Black Hole Temperature and Entropy. In Planck units in which c G k 4 0 1 Hawking, showed that a black hole of surface gravity and event horizon. area A has temperature,and entropy, For a static uncharged nonrotating black hole of mass M in.

vacuum asymptotically flat spacetime Schwarzschild metric in. which the event horizon radius is rh 2M the surface gravity is. M rh2 1 4M and the event horizon area is,A 4 rh2 16 M 2 the temperature is. and the entropy is,Black Hole Evaporation Rates, D N Page Particle Emission Rates from a Black Hole Massless. Particles from an Uncharged Nonrotating Hole Phys Rev D 13. D N Page Particle Emission Rates from a Black Hole 2. Massless Particles from a Rotating Hole Phys Rev D 14 3260. D N Page Particle Emission Rates from a Black Hole 3. Charged Leptons from a Nonrotating Hole Phys Rev D 16. Photon and graviton emission from a Schwarzschild black hole. I dM dt M 2 0 000 037 474 M 2,I d S BH dt 8 M 0 000 941 82 M. I d S rad dt 0 001 398 4 M d S BH dt,I d S rad dt d S BH dt 1 4847. Hawking s Argument for Information Loss, S W Hawking Breakdown of Predictability in Gravitational.

Collapse Phys Rev D 14 2460 1976 used quantum field. theory in a classical dynamical black hole background to argue that. information was lost into the absolute event horizon and could not. get out so that when the black hole evaporated away the. information was lost from the universe resulting in the change. from an initial pure quantum state to a mixed state of thermal. Hawking radiation This is certainly what one would get from local. quantum field theory in a definite metric with an horizon out from. which signals cannot escape since they would have to travel faster. than the speed of light impossible in local quantum field theory. with the region behind the horizon collapsing into a spacetime. singularity One might have said the information is still inside the. black hole but if the black hole completely evaporates away after. it is gone the information would have completely disappeared from. the universe,My Objections to Hawking s Argument, D N Page Is Black Hole Evaporation Predictable Phys Rev. Lett 44 301 1980 pointed out that Hawking s proposal violated. CPT invariance and that Hawking s calculations have been made. in the semiclassical approximation of a fixed background metric. which breaks down long before the final stages of evaporation for. example by the stochastic recoil of the black hole to the quantum. fluctuations of the momentum of the Hawking radiation. I listed 8 possible alternatives and suggested that it seemed most. productive to pursue the most conservative possibility a unitary. S matrix I noted Hawking suggested that God not only plays. dice He sometimes throws the dice where they cannot be seen. But it may be that if God throws dice where they cannot be seen. they cannot affect us,Turning the Tide, For several years little attention was given to black hole. information Relativists such as William Unruh and Robert Wald. tended to support Hawking s position that information is lost and. particle physicists such as Edward Witten tended to support my. position that information is not lost Although he was not involved. in the earliest days of the debate to cover them in his 2008 book. Leonard Susskind s book Black Hole War My Battle with. Stephen Hawking to Make the World Safe for Quantum Mechanics. is one entertaining perspective of the debate I think it was mainly. the AdS CFT conjecture of Juan Maldacena that turned the tide. toward the view that information is not lost This conjecture was. that a bulk gravitational theory in asymptotically anti de Sitter. AdS spacetime is dual to a conformal field theory CFT on the. conformal boundary Since the CFT is manifestly unitary with no. loss of information so should be the bulk gravitational field. Black Hole Firewalls, Interest in black hole information has surged recently with. A Almheiri D Marolf J Polchinski and J Sully Black Holes. Complementarity or Firewalls JHEP 1302 2013 062, They give a provocative argument that suggests that an infalling. observer burns up at the horizon of a sufficiently old black hole. so that the horizon becomes what they called a firewall. Unitary evolution suggests that at late times the Hawking radiation. is maximally entangled with the remaining black hole and. neighborhood including the modes just outside the horizon. This further suggests that what is just outside cannot be. significantly entangled with what is just inside, But without this latter entanglement an observer falling into the.

black hole should be burned up by high energy radiation. Time Dependence of Hawking Radiation Entropy, One cannot externally observe entanglement across the horizon. However it should eventually be transferred to the radiation. Therefore we would like to know the retarded time dependence of. the von Neumann entropy of the Hawking radiation, A Strominger Five Problems in Quantum Gravity Nucl Phys. Proc Suppl 192 193 119 2009 arXiv 0906 1313 hep th has. emphasized this question and outlined five candidate answers. I bad question,I information destruction,I long lived remnant. I non local remnant,I maximal information return, I shall assume without proof maximal information return. Assumptions,I Unitary evolution no loss of information.

I Initial approximately pure state,e g SvN 0 S star 1057 S BH 0 1077. I Nearly maximal entanglement between hole and radiation. I Complete evaporation into just final Hawking radiation. I Nonrotating uncharged Schwarzschild black hole,I Initial black hole mass large M0 M. I Massless photons and gravitons other particles m 10 10 eV. I Therefore essentially just photons and gravitons emitted. Arguments for Nearly Maximal Entanglement, D N Page Average Entropy of a Subsystem Phys Rev Lett. 71 1291 1993 gr qc 9305007, There is less than one half unit of information on average in the. smaller subsystem of a total system in a random pure state. D N Page Information in Black Hole Radiation Phys Rev. Lett 71 3743 1993 hep th 9306083, If all the information going into gravitational collapse escapes.

gradually from the apparent black hole it would likely come at. initially such a slow rate or be so spread out that it could never. be found or excluded by a perturbative analysis, Y Sekino and L Susskind Fast Scramblers JHEP 0810 065. 2008 arXiv 0808 2096 hep th conjecture, I The most rapid scramblers take a time logarithmic in the. number of degrees of freedom, I Black holes are the fastest scramblers in nature. These conjectures support my results using an average over all. pure states of the total system of black hole plus radiation. von Neumann Entropies of the Radiation and Black Hole. Take the semiclassical entropies S rad t and S BH t to be. approximate upper bounds on the von Neumann entropies of the. corresponding subsystems with the same macroscopic parameters. Therefore the von Neumann entropy of the Hawking radiation. SvN t which assuming a pure initial state and unitarity is the. same as the von Neumann entropy of the black hole should not be. greater than either S rad t or S BH t,Take my 1993 results as suggestions for the. Conjectured Anorexic Triangle Hypothesis CATH, Entropy triangular inequalities are usually nearly saturated.

This leads to the assumption of nearly maximal entanglement. between hole and radiation so SvN t should be near the minimum. of S rad t and S BH t,Time of Maximum von Neumann Entropy. Since the semiclassical radiation entropy S rad t is monotonically. increasing with time and since the semiclassical black hole entropy. S BH t is monotonically decreasing with time the maximum von. Neumann entropy is at the crossover point at time, t tdecay 0 5381 tdecay 4786 M03 6 236 1066 M0 M 3 yr. 1 1 3 2 0 5381,at which time the mass of the black hole is. M 1 1 2 M0 0 7730 M0, and its semiclassical Bekenstein Hawking entropy 4 M 2 is. S BH 1 S BH 0 0 5975 S BH 0, Maximum von Neumann Entropy of the Hawking Radiation.

At the time t when S rad t S BH t the von Neumann entropy. of the radiation and of the black hole is maximized and has the. S SvN t S rad t S BH t 4 M02 0 5975 S BH 0,0 5975 4 M02 7 509M02 6 268 1076 M0 M 2. Note that this maximum of the von Neumann entropy is about. 19 5 greater than half the original semiclassical, Bekenstein Hawking entropy of the black hole The time t for the. maximum von Neumann entropy is about 0 8324 times the time. t1 2 1 2 3 2 tdecay 0 6464 tdecay 1 201 t for the black. hole to lose half its area and semiclassical Bekenstein Hawking. Time Dependence of the Entropy of the Hawking Radiation. The von Neumann entropy of the Hawking radiation SvN t from a. large nonrotating uncharged black hole is very nearly the. semiclassical radiation entropy S rad t for t t and is very nearly. the Bekenstein Hawking semiclassical black hole entropy S BH t. for t t or using the Heaviside step function x,SvN t 4 M02 1 1 t t. 4 M0 1 t t,4 1 4847 M02 1 1 4786M03 t,4 M0 1 t 4786M03. Black Hole Information and Firewalls Don N Page University of Alberta 2017 July 7 Introduction Black hole information is one of the greatest puzzles of theoretical physics from the 20th century that has persisted into the 21st After Stephen Hawking discovered black hole evaporation in 1974 in 1976 he predicted that black hole formation and evaporation would cause a pure quantum state to

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