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PHY483F/1483F Relativity Theory I (2020-21) Department of Physics University of Toronto

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PHY483F/1483F Relativity Theory I (2020-21) Department of Physics University of Toronto Instructor: Prof. A.W. Peet Sources:- • M.P. Hobson, G.P. Efsthathiou, and A.N. Lasenby, \General relativity: an introduction for physicists" (Cambridge University Press, 2005) [recommended textbook]; • Sean Carroll, \Spacetime and geometry: an introduction to general relativity" (Addison- Wesley, 2004); • Ray d'Inverno, \Introducing Einstein's relativity" (Oxford University Press, 1992); • Jim Hartle, \Gravity: an introduction to Einstein's general relativity" (Pearson, 2003); • Bob Wald, \General relativity" (University of Chicago Press, 1984); • Tom´asOrt´ın,\Gravity and strings" (Cambridge University Press, 2004); • Noel Doughty, \Lagrangian Interaction" (Westview Press, 1990); • my personal notes over three decades. Version: Monday 16th November, 2020 @ 10:55 Licence: Creative Commons Attribution-NonCommercial-NoDerivs Canada 2.5 Contents 1 R10Sep iii 1.1 Invitation to General Relativity . iii 1.2 Course website . .1 2 M14Sep 2 2.1 Galilean relativity, 3-vectors in Euclidean space, and index notation . .2 3 R17Sep 8 3.1 Special relativity and 4-vectors in Minkowski spacetime . .8 3.2 Partial derivative 4-vector . 12 4 M21Sep 14 4.1 Relativistic particle: position, momentum, acceleration 4-vectors . 14 4.2 Electromagnetism: 4-vector potential and field strength tensor . 18 5 R24Sep 21 5.1 Constant relativistic acceleration and the twin paradox . 21 5.2 The Equivalence Principle . 24 5.3 Spacetime as a curved Riemannian manifold . 25 6 M28Sep 27 6.1 Basis vectors in curved spacetime . 27 6.2 Tensors in curved spacetime . 29 6.3 Rules for tensor index gymnastics . 30 7 R01Oct 32 7.1 Building a covariant derivative . 32 7.2 How basis vectors change: the role of the affine connection . 32 7.3 The covariant derivative and parallel transport . 36 8 M05Oct 38 8.1 The geodesic equations for test particle motion in curved spacetime . 38 8.2 Example computation for affine connection and geodesic equations . 40 9 R08Oct 44 9.1 Spacetime curvature . 44 9.2 The Riemann tensor . 44 9.3 Example computations for Riemann . 46 10 R15Oct 50 10.1 Geodesic deviation . 50 10.2 Tidal forces and taking the Newtonian limit for Christoffels . 51 11 M19Oct 56 11.1 Newtonian limit for Riemann . 56 11.2 Riemann normal coordinates and the Bianchi identity . 58 i 11.3 The information in Riemann . 60 12 R22Oct 62 12.1 Lie derivatives . 62 12.2 Killing vectors and tensors . 64 13 M26Oct 67 13.1 Maximally symmetric spacetimes . 67 13.2 Einstein's equations . 70 14 R29Oct 73 14.1 Birkhoff's theorem and the Schwarzschild black hole . 73 15 M02Nov 79 15.1 TOV equation for a star . 79 15.2 Geodesics of Schwarzschild . 81 16 R05Nov 85 16.1 Causal structure of Schwarzschild . 85 17 M16Nov 91 17.1 Charged black holes . 91 17.2 Rotating black holes . 94 18 R19Nov 96 18.1 The Kerr solution . 96 18.2 The Penrose process . 98 19 M23Nov 101 19.1 Gravitational redshift . 101 19.2 Planetary perihelion precession . 103 20 R26Nov 106 20.1 Bending of light . 106 20.2 Radar echoes . 109 21 M30Nov 111 21.1 Geodesic precession of gyroscopes . 111 21.2 Accretion disks . 114 22 R03Dec 116 22.1 Finding the wave equation for metric perturbations . 116 22.2 Solving the linearized Einstein equations . 118 23 M07Dec 122 23.1 Gravitational plane waves . 122 23.2 Energy loss from gravitational radiation . 125 ii 1 R10Sep 1.1 Invitation to General Relativity From a particle physics perspective, the gravitational force is the weakest of the four known forces. So why does gravity dominate the dynamics of the universe? A simple first answer is that there is a lot of matter in the universe that gravitates. Even though the gravitational attraction between any two subatomic particles is weak, if you get enough of them together you can eventually make a black hole! A slightly more sophisticated answer focuses on the range of the gravitational force and what sources it. The only two long-range forces we know of in Nature are gravity and electromagnetism. By contrast, the strong nuclear force binding atomic nuclei and the weak nuclear force responsible for the fusion reaction powering our Sun are very short-range. Electromagnetic fields are sourced by charges and currents, but the universe is electrically neutral on average, so electromagnetism does not dominate its evolution. Gravity, on the other hand, is sourced by energy-momentum. Since everything has energy-momentum, even the graviton, you can never get away from gravity. Newton wowed the world a third of a millennium ago with his Law of Universal Gravita- tion, which explained both celestial and terrestrial observations. Our focus in this course is on explaining Einstein's famous General Theory of Relativity (GR), which is a gener- alization of both Newtonian gravity and Special Relativity proven useful for describing the dynamics of the cosmos. By the end of term, you will be familiar with Albert Einstein's famous equation for the gravitational field gµν(x) 1 R − g R + Λg = −8πG T ; (1.1) µν 2 µν µν N µν where Rµν and R involve (first and) second derivatives of gµν and Tµν describes the energy- momentum tensor of all non-gravitational fields which are collectively known as matter fields. You will also understand how GR gives back Newton's theory of gravity in the limit where speeds are small and spacetime is weakly curved. You can think of Einstein's GR as Gravity 2.0, built on the foundation of Gravity 1.0 established by Newton { an upgrade. The name for this course is \Relativity Theory 1". Another name by which it is com- monly known is \GR 1", which stands for \General Relativity 1". The main thing we learn how to do in this course is how to write the dynamical equations of physics in the language of tensor analysis. Tensor analysis always sounds scary when you start, but it is not much more complicated conceptually than vector analysis, something you have been doing for years. We will show how to take your vector analysis knowledge from flat space and generalize it to spacetime. We begin with flat spacetime, which is pertinent to Special Relativity, and then we build on that knowledge to figure out how to write dynamical equations of physics even in curved spacetime. Einstein taught us that the speed of light is constant and is the same in all inertial frames of reference. We will therefore adopt the relativistic convention that c = 1 throughout the course. This implies that time is measured in metres, and mass is measured in units of energy, e.g. me=511 keV. We will keep all other physical constants explicit, such as Planck's constant ~ characterizing the strength of quantum effects and the Newton constant GN characterizing the strength of gravity. If you feel queasy about missing factors of c in any equation, they can always be easily restored by using dimensional analysis. iii 1.2 Course website Please have a careful read of the course website at https://ap.io/483f/. It contains lots of vital and useful information for all students taking PHY483F/PHY1483F, including the syllabus, online lecture notes, and how to contact me. Almost everything you need to know about the course is contained in the pages listed, and in all the clickable links in those pages. The remaining tiny fraction of information that needs to be hidden behind a UofT firewall for class members only can be found on Quercus, in the Announcements (from the Prof.) and Modules (from the TA). Remarks in my notes intended for more advanced/interested students are indicated in blue. 1 2 M14Sep 2.1 Galilean relativity, 3-vectors in Euclidean space, and index notation Before we review some aspects of Special Relativity and introduce some new ones, let us begin by reminding ourselves of the non-relativistic version of relativity, also known as Galilean relativity. When we want to transform from one inertial frame of reference to another moving at relative velocity v, there are three things we must think about: (a) how time intervals relate, (b) how spatial position intervals relate, (c) how velocities relate. In Galilean relativity, all clocks are synchronized, dt0 = dt ; (2.1) displacements are related via dx0 = dx − vdt ; (2.2) and velocities u = dx=dt compose by simple addition, u0 = u − v ; (2.3) where v is the relative velocity between the unprimed and primed frames of reference. Ein- stein upgraded these formulæ when he invented Special Relativity, and you have seen the results before: they are known as Lorentz Transformations. We will get to them soon enough { and we will show you how simple they can look when written in terms of rapidity rather than velocity. But for now, let us inspect how 3-vectors work more closely, in some detail. This will serve as a pattern for the relativistic case. Lots of things of interest in physics are vectors, which are in essence things that point.I like to say that a vector has a `leg' that sticks out, telling you where it points. Mathematically, the vector components are what you get when you resolve the vector along an orthonormal basis. In Special and General Relativity, we will need to be scrupulously careful to distinguish where we put our indices (up/down and left/right). For arbitrary vectors v, we write the index telling you which component is which with an upstairs index: v1; v2; : : : ; vd, where i = 1; : : : ; d and d is the spatial dimension. Note that the upstairs index i used here is not a power; instead, it specifies which component vi is being discussed: the ith one.
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