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The Physics of Black Holes The Physics of Black Holes Trial Lecture Nikolai Fomin University of Bergen 25/08/2020 1/43 The Physics of Black Holes Plan of the Lecture 1 Introduction 2 Formation and Classification of Black Holes 3 Black Hole Mechanics 4 Experimental Results 5 Hawking Radiation 6 Conclusion 7 Backup 2/43 The Physics of Black Holes Introduction Enigmatic objects Wide spread in media and fiction Naive definition Black holes are objects with such strong gravitational pull that not even light can escape Image source: European Southern Observatory 3/43 The Physics of Black Holes Historical Developments 1915 Einstien Field Equations (EFE) Gµν + Λgµν = κTµν 1916 Schwarzschild metric { solution of EFE 2GM Schwarzschild radius rs = c2 1958 Finkelstein and Misner { identified rs with the event horizon of a black hole Images source: wikimedia 4/43 The Physics of Black Holes Historical Developments 1963 Kerr { EFE solution for rotating black holes 1965 Newman { rotating, charged black hole 1967 Israel et al. { no-hair theorem 1970s Hawking et al. { laws of black hole mechanics Image source: 1974 Hawikng { Hawking radiation The royal society No-hair theorem An isolated black hole is fully described by it's mass, angular momentum and charge Image source: University of Pittsburgh 5/43 The Physics of Black Holes What Is a Black Hole? Classical Definition (Hawking, S. & Ellis, G.) A region of the spacetime that is not contained in the causal past of the infinite future. Divide universe in two mutually exclusive regions Exterior causally connected to ”infinitely far away" Interior is a region such that anything that enters it remains there The boundary is defined to be it's event horizon 6/43 The Physics of Black Holes What Is a Black Hole? The definition of black holes and event horizons is global No local experiment can map out the event horizon Whether a ray can escape to infinity starting today depends on the position of the event horizon next week What we know There are objects in the universe that approximately behave like the idealized black holes for sufficiently long time 7/43 The Physics of Black Holes Black Holes Formation An active star with radius R and mass M Equilibrium of kinetic and gravitational energy When the star runs out of fuel { gravitational collapse Possible equilibrium states Image source: ASPIRE 8/43 The Physics of Black Holes Chandrasekhar limit Gravitational energy Electron degeneracy pressure GM2 α Egrav ∼ − R = − R , 3 M ∼ nmpR =) Kinetic energy M n = ne ∼ m R3 E ∼ nR3hEi p kin For non-relativistic electrons −1=3 ~ 2 ne ∼ hpe i hpe i hEi ∼ me 2 Pauli exclusion principle becomes ~ M 5=3 1 β Ekin ∼ m ( m ) R2 = R2 relevant e p 9/43 The Physics of Black Holes Chandrasekhar limit α β E = Egrav + Ekin ∼ − R + R2 has a minimum for non-zero R Relativistic electrons 1=3 hEi ∼ hpeic ∼ ~cne E ∼ c( M )4=3 1 = γ kin ~ mp R R α γ E ∼ − R + R 1 ~c 3=2 =) Mcritical ∼ 2 ( ) mp G Mcritical ≈ 1:4M White dwarf Image source: P.K. Townsend 10/43 The Physics of Black Holes Tolman{Oppenheimer{Volkoff limit M > Mcritical Electron capture Collapse to nuclear matter density e− + p+ ! n + ν e Neutron degeneracy pressure ∆m ∼ 3me Tolman{Oppenheimer{Volkoff limit Electrons in white dwarfs are MTOV ≈ 2:3M non-relativistic 2 Neutron star EF / mec Heavier stars collapse into black holes 11/43 The Physics of Black Holes Black Holes Classification Intermediate-mass black holes Primordial black holes 2 5 Mass ranges 10 − 10 M Formed in the early universe Too massive to come from a Can have masses below TOV limit gravitational collapse of a star "Normal" or stellar black holes Supermassive black holes Produced in the gravitational Mass ranges 105 − 109M collapse of a star Centers of galaxies Typical mass ranges 2 [3; 100]M Sagittarius A∗ 12/43 The Physics of Black Holes Black Hole Mechanics Variables typically used to Surface gravity κ describe black holes Formal definition Horizon area A Magnitude of the gradient of the norm of the Mass M horizon generating Killing field χα evaluated at Energy E the horizon. Angular velocity Ω Equivalently Angular momentum J κ is the force per unit mass that must be Electric surface potential Φ applied at infinity in order to hold a particle on Electric charge Q its path. 13/43 The Physics of Black Holes Surface Gravity Massless inextensible string Raise a unit mass particle by the distance dl at radius r dEr = gr dl Distant observer pulling the string dE1 = g1dl Gravitational redshift Image source: wikimedia 14/43 The Physics of Black Holes Surface Gravity gr = Er = (1 − 2GM )−1=2 g1 E1 r Surface gravity for Schwarzschild Schwarzschild black hole black holes g = GM (1 − 2GM )−1=2 r r 2 r κ = 1 GM 4GM =) g1 = 2 r For rotating Kerr black holes As r ! rs = 2GM, 1 2 κ = 4GM − MGΩ gr ! 1, g1 ! κ 15/43 The Physics of Black Holes Mechanics of Black Holes Zeroth law of thermodynamics If two systems are both in thermal equilibrium with a third system then 1970s { Bardeen, Carter, Hawking they are in thermal equilibrium with each other. Classical thermodynamics analogy κ local ! global Black hole equivalent Surface gravity κ is constant on the κ $ classical temperature horizon of a stationary black hole. Dominant energy condition is assumed. 16/43 The Physics of Black Holes Mechanics of Black Holes First law of thermodynamics Conservation of energy Angular velocity Ω dE = TdS − pdV + µdN Electric potential of the horizon Φ Black hole equivalent First term { "entropy" If a black hole with mass M, charge Q Other terms { changes due to and angular momentum J is perturbed rotation and electromagnetism κ dM = 8π dA + ΩdJ + ΦdQ 17/43 The Physics of Black Holes Mechanics of Black Holes Second law of thermodynamics Entropy of a closed system never decreases Hawking's area theorem dS ≥ 0 Black hole cannot split in two Black hole equivalent M3 ! M1 + M2, M3 ≥ M1 + M2 p 2 2 The area of the future event horizon is M3 ≤ M1 + M2 ≤ M1 + M2 a non-decreasing function of time. dA ≥ 0 18/43 The Physics of Black Holes Mechanics of Black Holes Third law of thermodynamics It is impossible for any process, no Weaker version of the Third Law than the matter how idealized, to reduce the Planck version entropy of a system to its absolute-zero 1 Simplest black hole { κ ∼ 4M value in a finite number of operations. Adding charge Q =) electrostatic Black hole equivalent repulsion Surface gravity κ cannot be reduced to Similar for angular momentum J zero in a finite number of operations. 19/43 The Physics of Black Holes Gravitational Waves Gravitation { manifestation of the curvature of the spacetime Gravitational waves Propagate with the speed of light Typical sources { binaries Hulse - Taylor pulsar 1913+16 20/43 The Physics of Black Holes Gravitational Waves Laser interferometric detectors LIGO, VIRGO, GEO 600, TAMA 300 2015 LIGO { first direct observation Observation of Gravitational Waves from a Binary Black Hole Merger LISA, DECIGO { new generation of space-based detectors Image source: wikimedia 21/43 The Physics of Black Holes Gravitational Waves Images sourse: LIGO 22/43 The Physics of Black Holes Photon Orbit Black holes invisible to radio telescopes Spacetime around black holes curved Photon orbit Lower bound for a stable orbit 2019 Event Horizon Telescope { supermassive black hole in the Messier 87 galaxy Image source: Event Horizon Telescope 23/43 The Physics of Black Holes Accretion Disks Accumulation of matter around a central body Orbiting in inward spirals Gravitational potential energy converted into heat Black holes { hot enough to emit x-rays Hubble Space Telescope 5 Intermediate-mass black hole M ∼ 5 × 10 M Messier 87 black-hole-powered jet Images source: NASA 24/43 The Physics of Black Holes Black Hole Entropy So far T = 0 dStot dSBH dSmatter Second law of thermodynamics dt = dt + dt ≥ 0 Throw a box of gas in the black Do we also get the TdSBH = dE? hole 1974 Hawking confirmed the What happens to the entropy? conjecture A SBH = 2 1973 Bekenstein { black hole 4Lp entropy postulate 25/43 The Physics of Black Holes Black Hole Entropy Box with gas { linear size L, mass ρ ∼ L if δr ∼ L2=GM m and temperature T The energy is red shifted q 2GM mL Black hole { mass M, horizon ∆M ∼ m 1 − 2GM+δr ∼ GM radius rs = 2GM Assume L ∼ ~=T ∆S ∼ −m=T ∆S ∼ −m=T ∼ −mL=~ ∼ Schwarzschild black hole proper −GM∆M= ∼ − ∆A ~ ~G distance Generalizes to Kerr black holes p ρ ∼ GMδr and more general systems 26/43 The Physics of Black Holes Hawking Radiation Inside the horizon components of In 1974 Hawking { black holes the metric change signs emit blackbody radiation Ingoing virtual particle with Virtual particle-antiparticle pairs negative energy relative to external Energy conservation { ±E observer can have positive energy Exist for t ∼ ~=E relative to an observer inside the Energy frame dependent horizon Black hole evaporation 27/43 The Physics of Black Holes Hawking Radiation Black hole radiating with Proper time to reach hoziron p characteristic temperature ∼ GMδr kT ∼ ~=GM jEj ∼ p ~ GMδr ∆S = ∆M=T ∼ GM∆M=~ Redshifted at infinity q =) ∆S ∼ ∆A 2GM ~G E1 ∼ p ~ 1 − GMδr 2GM+δr Precise calculations give ~ E1 ∼ GM kTHawking = ~κ/2π 28/43 The Physics of Black Holes Hawking Radiation - Experiment −8 THawking ∼ 6 × 10 M =M K No chance to observe In TeV-scale gravity models could be studied at LHC Analogs in condensed matter systems Acoustic black holes, Bose-Einstein condensates, slow light, superfluid quasiparticles Observation of Stimulated Hawking Radiation in an Optical Analogue 29/43 The Physics of Black Holes Entanglement For entangled states only the Consider two separated electrons with
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