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Special Relativity2 1.1 Introduction General relativity Spring 2021 Syksy R¨as¨anen CONTENTS 1 Contents 1 Special relativity2 1.1 Introduction................................2 1.2 Newtonian spacetime...........................3 1.2.1 Illustrating the structure of Newtonian spacetime.......3 1.2.2 Symmetries of Newtonian spacetime..............3 1.3 Minkowski space.............................5 1.3.1 The Minkowski metric......................5 1.3.2 Natural units...........................8 1.3.3 Sets of points at constant distance...............8 1.3.4 The line element.........................9 1.3.5 Poincar´etransformations.................... 11 1.3.6 Dot product and raising and lowering indices......... 14 1.3.7 Lorentz boosts and velocity................... 14 1.4 Dynamics in SR............................. 16 1.4.1 Generalising Newton's second law............... 16 1.5 Electrodynamics............................. 18 1.5.1 Maxwell equations........................ 18 1.5.2 The road in reverse........................ 20 Preface These are the lecture notes of the course General relativity at the University of Helsinki. They owe a heavy debt to the lecture notes of Hannu Kurki-Suonio, which were in turn influenced by Sean Carroll's lecture notes and his book Spacetime and Geometry. Unless otherwise noted, figures are by Hannu Kurki-Suonio. I thank Fernando Bracho Blok for LaTeXing parts of these lecture notes. Any errors are my responsibility; let me know when you catch them.1 General relativity is one of the two fundamental theories we currently have. The other is quantum field theory, in particular its application to the Standard Model of particle physics. Both of these branches grew from addressing inadequacies of Newtonian mechanics in the 20th century. Quantum physics was developed in bits and pieces in strong correspondence with observations as the classical description of matter was found to be inadequate. In contrast, observations played second fiddle to theoretical and mathematical arguments in the development of the theory of relativity. Special relativity was discovered by Albert Einstein in 1905 and the last bricks of general relativity were put in place by him and David Hilbert in 1915, after a years-long process involving also other collaborators. Also in contrast to the long development of quantum theory, the structure of special and general relativity was laid out almost in its entirety when they were first uncovered. Quantum field theory describes matter and its non-gravitational interactions, and general relativity covers spacetime and its interaction with matter, i.e. gravity. General relativity tells how spacetime behaves when it contains given kind of matter, but it is agnostic on what sort of matter exists. In the 1940s, quantum mechanics was 1Apparently, some early 20th century Arabic books included a disclaimer along the lines of \this book contains errors, but they can be easily picked up by an intelligent reader". That is a not unsuitable guideline also for these lecture notes. 1 SPECIAL RELATIVITY 2 unified with special relativity in Quantum Electrodynamics, the first quantum field theory. The unification of quantum physics with general relativity remains one of the greatest open questions in all of physics. It has been achieved to a limited extent in the theory of cosmic inflation, where linear perturbations of spacetime are quantised, and their quantum properties have been calculated and compared to observations of the cosmic microwave background and large scale structure with much success. We will discuss the intersection of quantum physics and general relativity only in passing when covering the Hawking radiation of black holes; otherwise we stick to classical (i.e. non-quantum) physics. The essence of general relativity is that gravity is an aspect of spacetime curva- ture, and that locally spacetime looks flat. We therefore start with a refresher on flat spacetime, i.e. special relativity formulated in terms of spacetime. 1 Special relativity 1.1 Introduction Familiarity with Newtonian physics and special relativity (SR) is assumed in this overview. The structure of SR is actually rather simple, the most difficult part is unlearning Newtonian ideas about space and time. Relativistic spacetime consists of events, corresponding to spacetime points that can be labelled with an integer number d of real numbers, called coordinates.2 There are infinitely many different ways to assign numbers to spacetime points, i.e. different coordinate systems. We label the coordinates xα, where α = 0; 1; : : : ; d − 1. We will mostly consider four-dimensional spacetime, d = 4, where we have one time direction and three spatial directions. (We will soon say what it means for a coordinate to be a time coordinate.) The time coordinate is assigned number α = 0, and also denoted x0 = ct, where c is the speed of light (we discuss the physical meaning of c below). We use Greek letters for spacetime indices and Latin indices for spatial indices, which run from 1 to d − 1. Just as the components of a vector in Newtonian physics are labelled xi, the coordinates xα are the components of a vector in spacetime (again, we will be more precise when we get to GR). Vectors in spacetime are sometimes called four-vectors for emphasis. We use the following notations for xα (and similarly for other vectors): xα = (x0; x1; x2; x3) = (x0; xi) = (x0; ~x) : (1.1) When talking about variables of functions, we will often drop the index and write f(x) instead of f(xα). The notation xα refers both to the set of components and to the specific component numbered α; as is the case for xi. Often we will even lazily conflate the set of components and the vector. We also denote four-vectors with an underline, x, and three-vectors with an arrow on top, ~x. In SR, we will mostly stick to Cartesian coordinates that are familiar from elementary treatments of Newtonian physics. We could also consider SR in general coordinates, but we would have to introduce half of the machinery of GR, so we leave that for the GR discussion. We begin with a brief overview of Newtonian mechanics in terms of symmetry, and then approach SR the same way. 2When we come to general relativity (GR), we will be more precise about the structure of spacetime and the meaning of coordinates. 1 SPECIAL RELATIVITY 3 1.2 Newtonian spacetime 1.2.1 Illustrating the structure of Newtonian spacetime A system of Newtonian mechanics consists of point particles moving in space. (Ex- tended objects can be understood as composites of point particles stuck together.) A system's state is fully determined when the position xi of each point particle is known as a function of time t: the central object is particle trajectory xi(t). The line drawn in spacetime (direct product of space and time) by a particle is called the worldline of the particle. The structure of Newtonian spacetime (maybe more appropriately called \space and time") is illustrated in terms of worldlines in figure 1. In Newtonian spacetime, the spatial slices of constant time are absolute, i.e. they are the same for all observers. In other words, Newtonian physics has absolute space. This also implies that simultaneity is absolute: all observers agree on whether two points lie on the same time slice. There are three kinds of directions in spacetime, illustrated in figure2. Spacelike directions are parallel to the slices of constant time, future timelike directions point up from the slices of constant time and past timelike directions point down from the slices of constant time. Particle trajectories always move up (i.e. are future timelike), but are otherwise arbitrary. In particular, the slope of the line, corresponding to the velocity, can have any value. Figure 1: Newtonian time slices. Figure 2: The three kinds of direc- Events A and B are simultaneous, as tions in Newtonian spacetime. are C and D. All depicted worldlines are future timelike. 1.2.2 Symmetries of Newtonian spacetime Newtonian spacetime is symmetric under the coordinate transformations t ! t0(t) = t + A (1.2) i 0i i j i i x ! x (t; ~x) = R jx + A + v t ; (1.3) i where A is a constant, R j is a constant rotation matrix (we discuss this in more detail in section 1.18), and Ai and vi are constant vectors. We have adopted the Einstein summation convention according to which any index that appears once upstairs and once downstairs is summed over, unless otherwise noted. This applies equally to Greek and Latin indices. Such an index is called a summation index or a 1 SPECIAL RELATIVITY 4 i j P i j dummy index. So R jx ≡ j=1:::3 R jx . The spacetime is also symmetric under time reversal, t ! −t, and spatial reflections (also called parity transformations), xi ! −xi (for any or all of the spatial directions). Symmetry under coordinate transformations means that the spacetime before and after the transformation describes the same physical situation. Objects that do not remain invariant are relative, while those that are invariant are called ab- solute. The transformations (1.2) and (1.3) of involve four separate symmetries. Invariance under constant time shifts means that the value of the time coordinate is not absolute. However, time intervals are absolute (and hence simultaneity is absolute), so we say that time is absolute. Invariance under constant translations means that there is no absolute position in space. However, spatial intervals are absolute, i.e. lengths are absolute. Invariance under spatial rotations means there is no absolute direction in space. However, relative angular positions of particles are absolute. Overall, we say that space is absolute as the symmetry transforma- tions translate points on a slice of constant time to points on the same slice. The time-dependent shift by vit is called the Galilei transformation. Invariance under it means that there is no absolute velocity.
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