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Hawking Radiation and Black Hole Evaporation Wolfram Schmidt www.astro.uni-wuerzburg.de/∼schmidt ART Vorlesung, 9 February 2005 – p.1/21 Overview Classical black hole mechanics The quantum vacuum in Minkowski spacetime The Unruh effect Hawking radiation Black hole entropy ART Vorlesung, 9 February 2005 – p.2/21 The Zeroth Law ✦ Zeroth Law of Black Hole Mechanics The surface gravity κ is constant at the horizon of a stationary black hole ✦ Zeroth Law of Thermodynamics The temperature T is constant throughout a system in thermal equilibrium ART Vorlesung, 9 February 2005 – p.3/21 The First Law ✦ First Law of Black Hole Mechanics If mass-energy δM carrying angular momentum δJ and zero electric charge falls into a black hole, the change of surface area of the horizon δA is given by κ δA + ΩδJ δM = 8π ✦ First Law of Thermodynamics The change of internal energy in a quasistatic process is the sum of the added heat and mechanical work done on the system: δE = T δS + P δV ART Vorlesung, 9 February 2005 – p.4/21 The Second Law ✦ Second Law of Black Hole Mechanics (Area theorem by H AWKING, 1971): The surface area of the horizon can never decrease, δA ≥ 0 ✦ Second Law of Thermodynamics The total entropy of an isolated system can never decrease: δS ≥ 0 ART Vorlesung, 9 February 2005 – p.5/21 The Area Theorem The total area of black hole horizons is monotonically increasing with time ART Vorlesung, 9 February 2005 – p.6/21 Black Hole Thermodynamics? Analogy to thermodynamics ☛ Surface gravity κ is a measure of temperature T ☛ Horizon area A is proportional to the entropy S Refutations (from the purely classical point of view): ✘ The temperature of a black hole vanishes! ✘ Entropy is dimensionless, the horizon area is a length squared! ✘ The area is separately non-decreasing, whereas only the total thermodynamical entropy is non-decreasing! ✘ Numerically, the black hole entropy is vastly larger than the entropy of a star from which the hole could have formed! ART Vorlesung, 9 February 2005 – p.7/21 The Quantum Vacuum The definition of a particle (quantum) depends on the frame of reference If the frames of two observers differ only by a Lorentz transformation, then they will agree about the particle content If they have relative acceleration, then they will measure different particle numbers! The vacuum in Minkowski spacetime appears to be a thermal state when viewed by an accelerating observer (DAVIES, 1975, and U NRUH, 1976) ART Vorlesung, 9 February 2005 – p.8/21 Minkowski Spacetime Geometric units: c = 1, G = 1, kB = 1 ds2 = −dt2 + dr2 + r2 dΩ ✦ r → ∞ ➢ spacelike infinity I 0 ✦ t → +∞ ➢ future timelike infinity I + ✦ t → −∞ ➢ past timelike infinity I − ds2 = −dudv + 41 (u − v)2 dΩ ✦ u = t − r → ∞ ➢ future null infinity I + ✦ v = t + r → ∞ ➢ past null infinity I − ART Vorlesung, 9 February 2005 – p.9/21 Conformal Minkowski Spacetime Compactify the manifold to include the boundaries I − , I − , I 0 , I + and I + ART Vorlesung, 9 February 2005 – p.10/21 QFT in a Nutshell Massless scalar quantum field φ in 1+1 Minkowski spacetime Field equation ∂µ ∂ µ φ = −φ,tt + φ,rr = 0 √ ± −iω(t∓r) ➯ plane-wave solutions fω = e / 2ω Wave-packets following outgoing null rays onto I + : Z + + † + ∗ φ = dω [a+ f + (a ω ω ω ) (fω ) ] † + ✦ (a+ ω ) and aω are, respectively, creation and annihilation operators ✦ Number operator for outgoing particles of frequency ω is + † + N+ = (a ω ) aω inert ART Vorlesung, 9 February 2005 – p.11/21 Inertial vs. Accelerated Observer Basis modes in the inertial frame √ √ −iωu −iωv + − / 2ω , fω = e / 2ω fω = e For a Rindler observer moving with const. acceleration a, the null coordinates transform to u0 and v 0 The transformation from the inertial to 0the0 accelerated frame is given by ds2 = −dudv = −ea(v −u ) du0 dv 0 The Rindler observer measures different time t0 = 21 (u0 + v 0 ) and length r0 = 21 (v 0 − u0 ) and has different 0 √ 0 √ −iωu + −iωv − / 2ω , gω = e basis modes gω = e / 2ω Z + † + ∗ + + (b g φ = dω [b+ ω ) (gω ) ] ω ω ART Vorlesung, 9 February 2005 – p.12/21 The Unruh Effect − )i = 0 vacuum state |0(I − )i, ∀ω : a− |0(I ω ☞ There are no in/outcoming particles seen by an inertial observer Number operator for outgoing modes in the accelerated + ) † b+ frame is N+ = (b ω ω Rindler For the vacuum state |0(I − )i, the Rindler observer finds h0(I − − )|N+ |0(I )i Rindler = 1 e2πω/a − 1 ☞ The Rindler observer beholds a flux of thermal radiation propagating toward I + ☞ The temperature of the vacuum is proportional to acceleration: T = ~a/2π ART Vorlesung, 9 February 2005 – p.13/21 The Collapsing Star Spacetime A black hole is a region in spacetime which is not in the past of of I + ART Vorlesung, 9 February 2005 – p.14/21 QFT in Curved Spacetime Use a prescribed background geometry with metric gµν : g µν ∂µ ∂ν φ = 0 H AWKING (1973) studied a scalar field φ propagating in the background spacetime around a star collapsing to a black hole ✦ Towards I − , the spacetime is asymptotically Minkowskian ✦ Positive frequency modes are partially converted into negative frequency modes near the horizon (pair creation) ✦ Whereas positive frequency modes scatter off the horizon, negative frequency modes are absorbed ✦ Assuming the state |0(I − )i in the remote past (the Boulware state), H AWKING found that in the future of the collapse, there is a flux of thermal particles toward I + ART Vorlesung, 9 February 2005 – p.15/21 Pair Creation Near the Event Horizon Virtual pair creation close to the horizon can result in a real particle escaping to I + which carries away a fraction of the black hole mass ART Vorlesung, 9 February 2005 – p.16/21 Hawking Radiation A black hole produces thermal radiation with T = ~ 2π κ The temperature is proportional to the surface gravity κ = 12 f 0 (RH ) = ~/8πM , where f (r) = 1 − 2M/r The wavelength of Hawking radiation is λH = 8π 2 RH T vanishes in the classical limit ~ → 0 a = κ/f 1/2 (r) is the local acceleration of a stationary observer Hawking radiation is a consequence of the state near the horizon being the vacuum as viewed by free-fall observers ART Vorlesung, 9 February 2005 – p.17/21 Black Hole Evaporation Negative frequency (energy) particle flux into H+ diminishes the black hole mass The the rate of positive energy emission to I + is ∝ M −2 Black hole life time is of the order M 3 ✦ (M/MP )3 TP ∼ 10−28 M 3 [cgs] ✦ A solar mass black hole (M ∼ 1033 g) emits thermal radiation of roughly 10−11 eV and lives about 1054 times the age of the universe ✦ Only primordial mini black holes could be observed if they were around ART Vorlesung, 9 February 2005 – p.18/21 Bekenstein-Hawking Entropy There is entropy associated with a black hole horizon: SBH AH = 4~ The unit of horizon area is L2P ∼ 10−66 cm2 For a stellar mass black hole, SBH ∼ 1078 which is about 1021 times the entropy of a collapsing star from which the hole could have formed Generalised second law: The sum of BH entropy and the entropy outside black holes can never decrease δ(Soutside + SBH ) ≥ 0 ART Vorlesung, 9 February 2005 – p.19/21 The Meaning of Black Hole Entropy H AWKING (until last year): extra quantum uncertainty ✦ SBH corresponds to the maximum information lost in the BH ✦ Information about quantum states can not be restored P ENROSE (The Emperor’s New Mind): complementary quantum uncertainty ✦ Loss of information corresponds to shrinking of phase space volume as trajectories converge towards singularities ✦ This is counter-balanced by quantum measurements in which information and, thus, phase space volume is regained M- THEORY: holographic principle and such ✦ Information is stored in the quantised horizon and eventually will be rejuvenated by BH evaporation ART Vorlesung, 9 February 2005 – p.20/21 More to Read... S. H AWKING, Quantum Mechanics of Black Holes, Scientific American, January 1977 S. H AWKING and R. P ENROSE, The Nature of Space and Time, Princeton University Press, 1996 T. JACOBSON, Introductory Lectures on Black Hole Thermodynamics, www.glue.umd.edu/∼tajac/BHTlectures/lectures.ps J. T RASCHEN, An Introduction to Black Hole Evaporation, de.arxiv.org/abs/gr-qc/?0010055 The Holographic Principle and M-theory www.damtp.cam.ac.uk/user/gr/public/holo/ ART Vorlesung, 9 February 2005 – p.21/21