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Black Holes Msc in Quantum Fields and Fundamental Forces Imperial College London Black Holes MSc in Quantum Fields and Fundamental Forces Imperial College London Fay Dowker Blackett Laboratory, Imperial College, London, SW7 2AZ, U.K. Contents 1 The Schwarzschild Black Hole3 1.1 Eddington-Finkelstein Coordinates7 1.2 Kruskal-Szekeres Coordinates 10 1.3 Penrose Diagrams 14 2 Charged & Rotating Black Holes 23 2.1 The Reissner-Nordstr¨omSolution 24 2.2 Rotating Black Holes 27 3 Killing Vectors & Killing Horizons 30 3.1 Symmetries & Killing Vectors 30 3.2 Conservation Laws 33 3.3 Hypersurfaces 37 3.4 Killing Horizons 38 3.5 Black Hole Uniqueness 40 3.6 Komar Integrals 42 4 Black Hole Thermodynamics 45 4.1 Overview 45 4.2 The First Law of Black Hole Mechanics 47 4.3 Working up to Hawking's Area Theorem 48 4.4 The Raychaudhouri Equation 51 4.5 Causal Structure 53 4.6 Hawking's Area Theorem 55 5 Hawking Radiation 56 5.1 Quantum Field Theory in Curved Spacetime 58 5.2 QFT in Rindler Space in 1 + 1 dimensions 62 5.3 QFT in the spacetime of a spherical collapsing star: Hawking Radiation 70 Recommended Reading: Popular/Historical: • K.S. Thorne, Black Holes and Time Warps, Picador, 1994. • W. Israel, Dark Stars: The Evolution of an Idea, in Three Hundred Years of Gravitation (S.W. Hawking & W. Israel, eds.), Cambridge University Press, 1987. Textbooks: • S.W. Hawking & G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, 1973. • R.M. Wald, General Relativity, Chicago University Press, 1984. • S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, 2004. Lecture Notes: • P. Townsend, Black Holes. Available online at arXiv:gr-qc/9707012. General: • S. W. Hawking & R. Penrose, The Nature of Space and Time, Princeton Uni- versity Press, 1996. This is a series of lectures by Hawking and Penrose concluded by a final debate. Hawking's part of the lectures is available online at arXiv:hep-th/9409195. { 2 { 1 The Schwarzschild Black Hole The Schwarzschild metric (1916) is a solution to the vacuum Einstein equations Rµν = 0. It is given by 2M 2M −1 ds2 = g dxµdxν = − 1 − dt2 + 1 − dr2 + r2dΩ2; (1.1) µν r r 2 2 2 2 2 where 0 < r < 1 is a radial coordinate and dΩ2 = dθ + sin θdφ is the round metric on the two-sphere. The line-element (1.1) is the unique spherically symmetric solution to the vac- uum Einstein equations. This result is known as Birkhoff's theorem and it has strong implications. For example, since (1.1) is independent of t, Birkhoff's theorem implies that any spherically symmetric vacuum solution must be time-independent (static). To study the physics of black holes, we will start with a simple model of spheri- cally symmetric gravitational collapse: a ball of pressure-free dust that collapses un- der its own gravity. Since the dust ball is spherically symmetric, Birkhoff's theorem tells us that outside of it (where the energy-momentum tensor vanishes), the metric must be Schwarzschild. Since the metric is continuous, it must be Schwarzschild on the surface, too. Let us follow the path of a massive particle on the surface of the dust ball. Assume it follows a radial geodesic, dθ = dφ = 0. Since the pull of gravity in the r-direction is the same for all particles at a given radius, the particle will remain at the surface throughout its trajectory. The action of a massive relativistic particle moving in a spacetime (M; g) along a trajectory xµ(τ) parametrised by τ is given by Z 1 Z dxµ dxν S = L dτ = g − m2 dτ: (1.2) 2m µν dτ dτ An implicit assumption in this form of the action is that the parameter τ is equal to the proper time measured along the particle's worldline (more on this in section 3.2). µ For the radial trajectory of the surface particle we have x (τ) = (t(τ); r(τ); θ0; φ0). Define R(t) ≡ r(τ(t)); (1.3) the radius of the particle's position as a function of coordinate-time t. We want to find the equation of motion for R(t). A simple way to get the first equation is from dτ 2 = −ds2, which implies " # 2M 2M −1 dR2 dt 2 1 = 1 − − 1 − : (1.4) R R dt dτ { 3 { The factor dt/dτ can be found from the Euler-Lagrange equations of the action: @L d @L − = 0: (1.5) @xµ dτ @(dxµ/dτ) The action of the dust particle in the Schwarzschild background takes the form ! 1 Z 2M dt 2 2M −1 dR2 S = − 1 − + 1 − − m2 dτ: (1.6) 2m R dτ R dτ Since it has no explicit dependence on the t-coordinate (it only depends on dt/dτ), the Euler-Lagrange equation for the t-coordinate d @L d 2M dt = −2 1 − = 0 (1.7) dτ @(dt/dτ) dτ R dτ reveals a constant of motion along the geodesic xµ(τ): 2M dt = 1 − = const: (1.8) R dτ The constant is equal to the energy per unit rest mass of the particle, as mea- sured by an inertial observer at r = 1. To see this, recall that the energy of a relativistic particle is given by the t-component of the momentum (co-)vector µ pµ = (−E; p) = mgµν dx /dτ. Then dt 2M dt E=m = −p =m = g = 1 − = > 0: (1.9) 0 00 dτ R dτ A particle starting at rest at infinity has = 1 at R = 1 and therefore = 1 ev- erywhere along its geodesic. Consequently, a particle starting at rest at some finite radius R = Rmax must have < 1. We will therefore assume 0 < < 1 from now on. Plugging (1.8) into (1.4) we obtain " # 2M 2M −1 2M −2 1 = 1 − − 1 − R_ 2 1 − 2; (1.10) R R R where we have defined R_ ≡ dR=dt. This gives the following equation for R_ : 2M 2 2M R_ 2 = −2 1 − − 1 + 2 : (1.11) R R We immediately see that R_ vanishes at 2 R = Rmax ≡ 2M=(1 − ) > 2M (1.12) { 4 { Figure 1. R_ 2 as a function of R for the massive surface particle. and at R = 2M, as illustrated in Figure1. The zero at Rmax simply corresponds _ to the initial condition that the particle starts off at rest at Rmax. However, R ap- proaches zero again as R ! 2M +: the particle appears to slow down as it approaches the Schwarzschild radius. How much time t2M does it take the particle to reach R = 2M from R = Rmax? Integrating (1.11), we find −1 − 1 Z t2M Z 2M 2 2M 2M 2 t2M = dt = − dR 1 − − 1 + = 1 ! (1.13) 0 Rmax R R To convince yourself that this integral is infinite, look at the contribution to the integral from the region near R = 2M. Let R = 2M + ρ with 0 < ρ 2M and Taylor expand in ρ: −1 − 1 Z 0 2M 2M 2 t = finite number − dρ 1 − − 1 + 2 2M 2M + ρ 2M + ρ ρ (1.14) Z 0 2M = finite number − dρ + O(ρ0) : ρ ρ This is infinite since the 1/ρ term in the integral diverges logarithmically at 0. Hence, as far as the coordinate time t is concerned, the particle never reaches the Schwarzschid radius. This result seems physically rather strange. The surface of a collapsing ball of dust mysteriously slows down as it approaches R = 2M and never actually reaches R = 2M. However, notice that while t may be a meaningful measure of time for { 5 { Figure 2.(dR/dτ)2 as a function of R for the massive surface particle. inertial observers at r = 1 (t is proportional to their proper time since the metric is flat at infinity), it has no physical meaning to the infalling particle itself (or to an observer falling along with it). This should not be surprising. After all, t is just a coordinate, and coordinates in general relativity have no physical meaning. So let us instead work with the proper time τ of the infalling particle. How does R change as a function of τ? Using (1.8) and (1.4) we find dR2 R = (1 − 2) max − 1 : (1.15) dτ R We see that nothing curious at all happens as R ! 2M +. The particle (and therefore the surface of the dust ball) passes smoothly through R = 2M. This is confirmed by the plot of (dR/dτ)2 against R in Figure2. Let us calculate the proper time τ2M that elapses for the particle between R = Rmax and R = 2M. Using (1.15), we find (exercise) Z τ2M Z 2M p dτ 2M 2 τ2M = dτ = dR = 1 − + arcsin ; (1.16) 2 3 0 Rmax dR (1 − ) 2 which is indeed finite. The strange behaviour near R = 2M has disappeared | a first indication that the singularity in the metric at R = 2M is due to a pathology in the time coordinate t. { 6 { The proper time taken to reach r = 0 is also finite: Z 0 dτ πM τ0 = dτ = : (1.17) 2 3 Rmax dR (1 − ) 2 This means that the entire dust star collapses to the single point r = 0 in a finite proper time, a first sign that there is a true singularity at r = 0 | a curvature singu- µνρσ larity | at which physical (coordinate-independent) quantities such as RµνρσR diverge. While the calculations above have given us some useful insights, we have been a bit careless. We worked out the geodesics using the Schwarzschild metric, and traced them through r = 2M, even though the metric (1.1) breaks down at r = 2M.
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