Lecture Notes for FY3452 Gravitation and Cosmology M. Kachelrieß M. Kachelrieß Institutt for fysikk NTNU, Trondheim Norway email: [email protected] Watch out for errors, most was written late in the evening. Corrections, feedback and any suggestions always welcome! Copyright © M. Kachelrieß 2010–2015, 2020. Last up-date November 9, 2020 Contents 1 Special relativity 8 1.1 Newtonian mechanics and gravity . 8 1.2 Minkowski space . 10 1.3 Relativistic mechanics . 14 1.A Appendix: Comments and examples on tensor and index notation . 16 2 Lagrangian mechanics and symmetries 21 2.1 Calculus of variations . 21 2.2 Hamilton’s principle and the Lagrange function . 22 2.3 Symmetries and conservation laws . 24 2.4 Free relativistic particle . 26 3 Basic differential geometry 28 3.1 Manifolds and tensor fields . 29 3.2 Tensor analysis . 31 3.2.1 Metric connection and covariant derivative . 32 3.2.2 Geodesics . 34 3.A Appendix: a bit more... 35 3.A.1 Affine connection and covariant derivative . 35 3.A.2 Riemannian normal coordinates . 36 4 Schwarzschild solution 38 4.1 Spacetime symmetries and Killing vectors . 38 4.2 Schwarzschild metric . 40 4.3 Gravitational redshift . 40 4.4 Orbits of massive particles . 41 4.5 Orbits of photons . 45 4.6 Post-Newtonian parameters . 47 4.A Appendix: General stationary isotropic metric . 48 5 Gravitational lensing 49 6 Black holes 53 6.1 Rindler spacetime and the Unruh effect . 53 6.2 Schwarzschild black holes . 57 6.3 Kerr black holes . 62 6.4 Black hole thermodynamics and Hawking radiation . 67 6.A Appendix: Conformal flatness for d = 2 ..................... 69 7 Classical field theory 70 7.1 Lagrange formalism . 70 3 Contents 7.2 Noether’s theorem and conservation laws . 71 7.3 Perfect fluid . 75 7.4 Klein-Gordon field . 76 7.5 Maxwell field . 78 8 Einstein’s field equation 84 8.1 Curvature and the Riemann tensor . 84 8.2 Integration, metric determinant g, and differential operators . 86 8.3 Einstein-Hilbert action . 88 8.4 Dynamical stress tensor . 91 8.4.1 Cosmological constant . 92 8.4.2 Equations of motion . 92 8.5 Alternative theories . 93 9 Linearized gravity and gravitational waves 96 9.1 Linearized gravity . 96 9.1.1 Metric perturbations as a tensor field . 96 9.1.2 Linearized Einstein equation in vacuum . 97 9.1.3 Linearized Einstein equation with sources . 99 9.1.4 Polarizations states . 100 9.2 Stress pseudo-tensor for gravity . 103 9.3 Emission of gravitational waves . 106 9.4 Gravitational waves from binary systems . 109 9.4.1 Weak field limit . 109 9.4.2 Strong field limit and binary merger . 112 9.A Appendix: Projection operator . 114 10 Cosmological models for an homogeneous, isotropic universe 117 10.1 Friedmann-Robertson-Walker metric for an homogeneous, isotropic universe . 117 10.2 Geometry of the Friedmann-Robertson-Walker metric . 119 10.3 Friedmann equations . 124 10.4 Scale-dependence of different energy forms . 126 10.5 Cosmological models with one energy component . 127 10.6 The ΛCDM model . 128 10.7 Determining Λ and the curvature R0 from ρm;0, H0; q0 ............. 129 10.8 Particle horizons . 131 11 Cosmic relics 132 11.1 Time-line of important dates in the early universe . 132 11.2 Equilibrium statistical physics in a nut-shell . 134 11.3 Big Bang Nucleosynthesis . 137 11.3.1 Equilibrium distributions . 137 11.3.2 Proton-neutron ratio . 138 11.3.3 Estimate of helium abundance . 140 11.3.4 Results from detailed calculations . 140 11.4 Dark matter . 141 11.4.1 Freeze-out of thermal relic particles . 141 4 Contents 11.4.2 Hot dark matter . 143 11.4.3 Cold dark matter . 144 12 Inflation and structure formation 146 12.1 Inflation . 146 12.1.1 Scalar fields in the expanding universe . 147 12.1.2 Generation of perturbations . 150 12.1.3 Models for inflation . 152 12.2 Structure formation . 152 12.2.1 Overview and data . 152 12.2.2 Jeans mass of baryons . 153 12.2.3 Damping scales . 154 12.2.4 Growth of perturbations in an expanding Universe: . 155 12.2.5 Recipes for structure formation . 156 12.2.6 Results . 157 5 Preface These notes summarise the lectures for FY3452 Gravitation and Cosmology I gave in 2009 and 2010. Asked to which of the three more advanced topics black holes, gravitational waves and cosmology more time should be devoted, students in 2009 voted for cosmology, while in 2010 black holes and gravitational waves were their favourites. As a result, the notes contain probably slightly more material than manageable in an one semester course. For 2020, we will have to make a similar decision, and there will be a vote in the first week of the lecturing period. I’m updating the notes throughout the semester. Compared to the last (2015) version, the order of topics is changed, some sections are streamlined to get space for new stuff, some like the one about Noether’s theorem improved, and conventions will be unified. At the moment, chapters 1–2, 4, 6–7, and partly 8 are updated. There are various differing sign conventions in general relativity possible – all ofthemare in use. One can define these choices as follows αβ η = S1 × [−1; +1; +1; +1]; (0.1a) α × α − α α κ − α κ R βρσ = S2 [@ρΓ βσ @σΓ βρ + Γ κρΓ βσ Γ κσΓ βρ]; (0.1b) Gαβ = S3 × 8πG Tαβ; (0.1c) × ρ Rαβ = S2S3 R αρβ: (0.1d) We choose these three signs as Si = {−; +; +g. Conventions of other authors are summarised in the following table: HEL dI,R MTW, H W [S1] -- + + [S2] + + + - [S3] -- + - Some useful books: H: J. B. Hartle. Gravity: An Introduction to Einstein’s General Relativity (Benjamin Cummings) HEL: Hobson, M.P., Efstathiou, G.P., Lasenby, A.N.: General relativity: an introduction for physicists. Cambridge University Press 2006. [On a somewhat higher level than Hartle.] • Robert M. Wald: General Relativity. University of Chicago Press 1986. [Uses a modern mathematical language] • Landau, Lev D.; Lifshitz, Evgenij M.: Course of theoretical physics 2 - The classical theory of fields. Pergamon Press Oxford, 1975. MTW: Misner, Charles W.; Thorne, Kip S.; Wheeler, John A.: Gravitation. Freeman New York, 1998. [Entertaining and nice description of differential geometry - but lengthy.] 6 Contents • Schutz, Bernard F.: A first course in general relativity. Cambridge Univ. Press, 2004. • Stephani, Hans: Relativity: an introduction to special and general relativity. Cambridge Univ. Press, 2004. W: Weinberg, Steven: Gravitation and cosmology. Wiley New York, 1972. [A classics. Many applications; outdated concerning cosmology.] • Weyl, Hermann: Raum, Zeit, Materie. Springer Berlin, 1918 (Space, Time, Matter, Dover New York, 1952). [The classics.] Finally: If you find typos (if not, you havn’t read carefully enough) in the part whichis already updated, conceptional errors or have suggestions, send me an email! 7 1 Special relativity 1.1 Newtonian mechanics and gravity Inertial frames and the principle of relativity Newton presented his mechanics in an ax- iomatic form. His Lex Prima (or the Galilean law of inertia) states: Each force-less mass point stays at rest or moves on a straight line at constant speed. Distinguishing between straight and curved lines requires an affine structure of space, while measuring velocities relies ona metric structure that allows one to measure distances. In addition, we have to be able to compare time measurements made at different space points. Thus, in order to apply Newton’s first law, we have to add some assumptions on space and time. Implicitly, Newton assumed an Euclidean structure for space, and thus the distance between two points P1 = (x1; y1; z1) and P2 = (x2; y2; z2) in a Cartesian coordinate system is 2 − 2 − 2 − 2 ∆l12 = (x1 x2) + (y1 y2) + (z1 z2) (1.1) or, for infinitesimal distances, dl2 = dx2 + dy2 + dz2 : (1.2) Moreover, he assumed the existence of an absolute time t on which all observers can agree. In a Cartesian inertial coordinate system, Newton’s lex prima becomes then d2x d2y d2z = = = 0 : (1.3) dt2 dt2 dt2 Most often, we call such a coordinate system just an inertial frame. Newton’s first law is not just a trivial consequence of its second one, but may be seen as a practical definition of those reference frames for which his following laws are valid. Which are the transformations which connect these inertial frames or, in other words, which are the symmetries of empty space and time? We know that translations a and rotations R are symmetries of Euclidean space: This means that using two different Cartesian coordinate systems, say a primed and an unprimed one, to label the points P1 and P2, their distance defined by Eq. (1.3) remains invariant, cf. with Fig. 1.1. The condition that the normof 0 distance vector l12 is invariant, l12 = l12, implys l0T l0 = lT RT Rl = lT l (1.4) or RT R = 1. Thus rotations acting on a three-vector x are represented by orthogonal matrices, R 2 O(3). All frames connected by x0 = Rx + a to an inertial frame are inertial frames too. In addition, there may be transformations which connect inertial frames which move with a given relative velocity. In order to determine them, we consider two frames with relative velocity v along the x direction: The most general linear1 transformation between these two 1A non-linear transformation would destroy translation invarince, cf. with Ex.xx 8 1.1 Newtonian mechanics and gravity y y0 P b x0 x Figure 1.1: The point P is invariant, with the coordinates (x; y) and (x0; y0) in the two coor- dinate systems.