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Part 3 Black Holes Part 3 Black Holes Harvey Reall Part 3 Black Holes March 13, 2015 ii H.S. Reall Contents Preface vii 1 Spherical stars 1 1.1 Cold stars . .1 1.2 Spherical symmetry . .2 1.3 Time-independence . .3 1.4 Static, spherically symmetric, spacetimes . .4 1.5 Tolman-Oppenheimer-Volkoff equations . .5 1.6 Outside the star: the Schwarzschild solution . .6 1.7 The interior solution . .7 1.8 Maximum mass of a cold star . .8 2 The Schwarzschild black hole 11 2.1 Birkhoff's theorem . 11 2.2 Gravitational redshift . 12 2.3 Geodesics of the Schwarzschild solution . 13 2.4 Eddington-Finkelstein coordinates . 14 2.5 Finkelstein diagram . 17 2.6 Gravitational collapse . 18 2.7 Black hole region . 19 2.8 Detecting black holes . 21 2.9 Orbits around a black hole . 22 2.10 White holes . 24 2.11 The Kruskal extension . 25 2.12 Einstein-Rosen bridge . 28 2.13 Extendibility . 29 2.14 Singularities . 29 3 The initial value problem 33 3.1 Predictability . 33 3.2 The initial value problem in GR . 35 iii CONTENTS 3.3 Asymptotically flat initial data . 38 3.4 Strong cosmic censorship . 38 4 The singularity theorem 41 4.1 Null hypersurfaces . 41 4.2 Geodesic deviation . 43 4.3 Geodesic congruences . 44 4.4 Null geodesic congruences . 45 4.5 Expansion, rotation and shear . 46 4.6 Expansion and shear of a null hypersurface . 47 4.7 Trapped surfaces . 48 4.8 Raychaudhuri's equation . 50 4.9 Energy conditions . 51 4.10 Conjugate points . 52 4.11 Causal structure . 54 4.12 Penrose singularity theorem . 58 5 Asymptotic flatness 61 5.1 Conformal compactification . 61 5.2 Asymptotic flatness . 66 5.3 Definition of a black hole . 70 5.4 Weak cosmic censorship . 72 5.5 Apparent horizon . 75 6 Charged black holes 77 6.1 The Reissner-Nordstrom solution . 77 6.2 Eddington-Finkelstein coordinates . 78 6.3 Kruskal-like coordinates . 79 6.4 Cauchy horizons . 82 6.5 Extreme RN . 84 6.6 Majumdar-Papapetrou solutions . 85 7 Rotating black holes 87 7.1 Uniqueness theorems . 87 7.2 The Kerr-Newman solution . 89 7.3 The Kerr solution . 90 7.4 Maximal analytic extension . 91 7.5 The ergosphere and Penrose process . 92 Part 3 Black Holes March 13, 2015 iv H.S. Reall CONTENTS 8 Mass, charge and angular momentum 95 8.1 Charges in curved spacetime . 95 8.2 Komar integrals . 97 8.3 Hamiltonian formulation of GR . 99 8.4 ADM energy . 102 9 Black hole mechanics 105 9.1 Killling horizons and surface gravity . 105 9.2 Interpretation of surface gravity . 107 9.3 Zeroth law of black holes mechanics . 108 9.4 First law of black hole mechanics . 109 9.5 Second law of black hole mechanics . 113 10 Quantum field theory in curved spacetime 115 10.1 Introduction . 115 10.2 Quantization of the free scalar field . 116 10.3 Bogoliubov transformations . 119 10.4 Particle production in a non-stationary spacetime . 120 10.5 Rindler spacetime . 121 10.6 Wave equation in Schwarzschild spacetime . 126 10.7 Hawking radiation . 129 10.8 Black hole thermodynamics . 136 10.9 Black hole evaporation . 137 Part 3 Black Holes March 13, 2015 v H.S. Reall CONTENTS Part 3 Black Holes March 13, 2015 vi H.S. Reall Preface These are lecture notes for the course on Black Holes in Part III of the Cambridge Mathematical Tripos. Acknowledgment I am grateful to Andrius Stikonasˇ for producing some of the figures. Conventions We will use units such that the speed of light is c = 1 and Newton's constant is G = 1. This implies that length, time and mass have the same units. The metric signature is (− + ++) The cosmological constant is so small that is is important only on the largest length scales, i.e., in cosmology. We will assume Λ = 0 in this course. We will use abstract index notation. Greek indices µ, ν; : : : refer to tensor components with respect to some basis. Such indices take values from 0 to 3. An equation written with such indices is valid only in a particular basis. Spacetime coordinates are denoted xµ. Abstract indices are Latin indices a; b; c : : :. These are used to denote tensor equations, i.e., equations valid in any basis. Any object carrying abstract indices must be a tensor of the type indicates by its indices e.g. a X b is a tensor of type (1; 1). Any equation written with abstract indices can be written out in a basis by replacing Latin indices with Greek ones (a ! µ, b ! ν etc). Conversely, if an equation written with Greek indices is valid in any basis then Greek indices can be replaced with Latin ones. µ 1 µσ For example: Γνρ = 2 g (gσν,ρ + gσρ,ν − gνρ,σ) is valid only in a coordinate ab basis. Hence we cannot write it using abstract indices. But R = g Rab is a tensor equation so we can use abstract indices. Riemann tensor: R(X; Y )Z = rX Y − rY X − r[X;Y ]Z. vii CHAPTER 0. PREFACE Bibliography 1. N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, 1982. 2. Spacetime and Geometry, S.M. Carroll, Addison Wesley, 2004. 3. V.P. Frolov and I.D. Novikov, Black holes physics, Kluwer, 1998. 4. S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, 1973. 5. R.M. Wald, General relativity, University of Chicago Press, 1984. 6. R.M. Wald, Quantum field theory in curved spacetime and black hole ther- modynamics, University of Chicago Press, 1994. Most of this course concerns classical aspects of black hole physics. The books that I found most useful in preparing this part of the course are Wald's GR book, and Hawking and Ellis. The final chapter of this course concerns quantum field theory in curved spacetime. Here I mainly used Birrell and Davies, and Wald's second book. The latter also contains a nice discussion of the law of black hole mechanics. Part 3 Black Holes March 13, 2015 viii H.S. Reall Chapter 1 Spherical stars 1.1 Cold stars To understand the astrophysical significance of black holes we must discuss stars. In particular, how do stars end their lives? A normal star like our Sun is supported against contracting under its own gravity by pressure generated by nuclear reactions in its core. However, eventually the star will use up its nuclear "fuel". If the gravitational self-attraction is to be balanced then some new source of pressure is required. If this balance is to last forever then this new source of pressure must be non-thermal because the star will eventually cool. A non-thermal source of pressure arises quantum mechanically from the Pauli principle, which makes a gas of cold fermions resist compression (this is called degeneracy pressure). A white dwarf is a star in which gravity is balanced by electron degeneracy pressure. The Sun will end its life as a white dwarf. White dwarfs are very dense compared to normal stars e.g. a white dwarf with the same mass as the Sun would have a radius around a hundredth of that of the Sun. Using Newtonian gravity one can show that a white dwarf cannot have a mass greater than the Chandrasekhar limit 1:4M where M is the mass of the Sun. A star more massive than this cannot end its life as a white dwarf (unless it somehow sheds some mass e.g. in a supernova). Once the density of matter approaches nuclear density, the degeneracy pressure of neutrons becomes important (at such high density, inverse beta decay converts protons into neutrons). A neutron star is supported by the degeneracy pressure of neutrons. These stars are tiny: a solar mass neutron star would have a radius of around 10km (the radius of the Sun is 7×105km). Recall that validity of Newtonian gravity requires jΦj 1 where Φ is the Newtonian gravitational potential. At the surface of a such a neutron star one has jΦj ∼ 0:1 and so a Newtonian description 1 CHAPTER 1. SPHERICAL STARS is inadequate: one has to use GR. In this chapter we will see that GR predicts that there is a maximum mass for neutron stars. Remarkably, this is independent of the properties of matter at extremely high density and so it holds for any cold star. As we will explain, detailed calculations reveal the maximum mass to be around 3M . Hence a hot star more massive than this cannot end its life as a cold star (unless it somehow sheds some mass e.g. in a supernova). Instead the star will undergo complete gravitational collapse to form a black hole. In the next few sections we will show that GR predicts a maximum mass for a cold star. We will make the simplifying assumption that the star is spherically symmetric. As we will see, the Schwarzschild solution is the unique spherically symmetric vacuum solution and hence describes the gravitational field outside any spherically symmetric star. The interior of the star can be modelled using a perfect fluid and so spacetime inside the star is determined by solving the Einstein equation with a perfect fluid source and matching onto the Schwarzschild solution outside the star. 1.2 Spherical symmetry We need to define what we mean by a spacetime being spherically symmetric. You are familiar with the idea that a round sphere is invariant under rotations, which form the group SO(3).
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