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 c M. Kachelrieß 2010–2012, 2020. Last up-date November 23, 2020 Contents 1 Special relativity 8 1.1 Newtonian mechanics and gravity . ..... 8 1.2 Minkowskispace .................................. 10 1.3 Relativisticmechanics . 14 1.A Appendix: Comments and examples on tensor and index notation . 16 2 Lagrangian mechanics and symmetries 21 2.1 Calculusofvariations . .. .. .. .. .. .. .. 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 Manifoldsandtensorfields . 29 3.2 Tensoranalysis .................................. 31 3.2.1 Metric connection and covariant derivative . ....... 32 3.2.2 Geodesics .................................. 34 3.A Appendix:abitmore... ............................ 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 Schwarzschildmetric .. .. .. .. .. .. .. .. 40 4.3 Gravitationalredshift . 40 4.4 Orbitsofmassiveparticles. ..... 41 4.5 Orbitsofphotons................................. 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 RindlerspacetimeandtheUnruheffect . 53 6.2 Schwarzschildblackholes . 57 6.3 Reissner-Nordstr¨omblackhole . ...... 62 6.4 Kerrblackholes .................................. 62 6.5 Black hole thermodynamics and Hawking radiation . ......... 68 6.A Appendix: Conformal flatness for d =2 ..................... 69 3 Contents 7 Classical field theory 70 7.1 Lagrangeformalism ............................... 70 7.2 Noether’s theorem and conservation laws . ........ 71 7.3 Perfectfluid..................................... 75 7.4 Klein-Gordonfield ................................ 76 7.5 Maxwellfield .................................... 78 8 Einstein’s field equation 84 8.1 CurvatureandtheRiemanntensor . 84 8.2 Integration, metric determinant g, and differential operators . 86 8.3 Einstein-Hilbertaction. ..... 88 8.4 Dynamicalstresstensor . 91 8.4.1 Cosmological constant . 92 8.4.2 Equationsofmotion ............................ 92 8.5 Alternativetheories ............................. 93 9 Linearized gravity and gravitational waves 96 9.1 Linearizedgravity ............................... 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 Stresspseudo-tensorforgravity . ....... 103 9.3 Emission of gravitational waves . 106 9.4 Gravitational waves from binary systems . ........ 109 9.4.1 Weakfieldlimit............................... 109 9.4.2 Strong field limit and binary merger . 112 9.A Appendix: Projectionoperator . 115 9.B Appendix: Derivation of the retarded Green function . ........... 116 10 Cosmological models for an homogeneous, isotropic universe 118 10.1 Friedmann-Robertson-Walker metric for an homogeneous, isotropic universe . 118 10.2 Geometry of the Friedmann-Robertson-Walker metric . ........... 120 10.3 Friedmannequations . 126 10.4 Scale-dependence of different energy forms . ......... 127 10.5 Cosmological models with one energy component . ......... 128 10.6TheΛCDMmodel ................................. 129 10.7 Determining Λ and the curvature R0 from ρm,0, H0,q0 ............. 131 10.8 Particlehorizons ............................... 132 11 Cosmic relics 134 11.1 Time-line of important dates in the early universe . ........... 134 11.2 Equilibrium statistical physics in a nut-shell . ............. 136 11.3 BigBangNucleosynthesis . 139 11.3.1 Equilibriumdistributions . 139 11.3.2 Proton-neutronratio . 140 11.3.3 Estimateofheliumabundance . 142 4 Contents 11.3.4 Results from detailed calculations . 142 11.4Darkmatter..................................... 143 11.4.1 Freeze-out of thermal relic particles . ....... 143 11.4.2 Hotdarkmatter .............................. 144 11.4.3 Colddarkmatter.............................. 145 12 Inflation and structure formation 148 12.1Inflation....................................... 148 12.1.1 Scalar fields in the expanding universe . 149 12.1.2 Generation of perturbations . 152 12.1.3 Modelsforinflation .. .. .. .. .. .. .. .. 154 12.2 Structureformation .. .. .. .. .. .. .. .. 154 12.2.1 Overviewanddata............................. 154 12.2.2 Jeansmassofbaryons . 155 12.2.3 Dampingscales............................... 156 12.2.4 Growth of perturbations in an expanding Universe: . ......... 157 12.2.5 Recipes for structure formation . 158 12.2.6 Results ................................... 159 5 Preface These notes summarise the lectures for FY3452 Gravitation and Cosmology I gave in 2009, 2010 and 2020. 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 and 2020 black holes and gravitational waves were their favourites. As a result, the notes contain probably more material than manageable in an one semester course. 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 (e.g. GW discovery), some like the one about Noether’s theorem improved, and conventions will be unified. At the moment, chapters 1–2, 4, 6–9 are updated. There are various differing sign conventions in general relativity possible – all of them are in use. One can define these choices as follows ηαβ = S [ 1, +1, +1, +1], (0.1a) 1 × − Rα = S [∂ Γα ∂ Γα +Γα Γκ Γα Γκ ], (0.1b) βρσ 2 × ρ βσ − σ βρ κρ βσ − κσ βρ G = S 8πG T , (0.1c) αβ 3 × αβ R = S S Rρ . (0.1d) αβ 2 3 × αρβ We choose these three signs as S = , +, + . Conventions of other authors are summarised i {− } 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.] Schutz, Bernard F.: A first course in general relativity. Cambridge Univ. Press, 2004. • 6 Contents 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 which is 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 on a 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 ∆l2 = (x x )2 + (y y )2 + (z z )2 (1.1) 12 1 − 2 1 − 2 1 − 2 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 norm of the ′ distance vector l12 is invariant, l12 = l12,