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Relativity Theory, We Often Use the Convention That the Greek6 Indices Run from 0 to 3, Whereas the Latin Indices Take the Values 1, 2, 3

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Relativity Theory, We Often Use the Convention That the Greek6 Indices Run from 0 to 3, Whereas the Latin Indices Take the Values 1, 2, 3 Relativity Theory Jouko Mickelsson with Tommy Ohlsson and H˚akan Snellman Mathematical Physics, KTH Physics Royal Institute of Technology Stockholm 2005 Typeset in LATEX Written by Jouko Mickelsson, 1996. Revised by Tommy Ohlsson, 1998. Revised and extended by Jouko Mickelsson, Tommy Ohlsson, and H˚akan Snellman, 1999. Revised by Jouko Mickelsson, Tommy Ohlsson, and H˚akan Snellman, 2000. Revised by Tommy Ohlsson, 2001. Revised by Tommy Ohlsson, 2003. Revised by Mattias Blennow, 2005. Solutions to the problems are written by Tommy Ohlsson and H˚akan Snellman, 1999. Updated by Tommy Ohlsson, 2000. Updated by Tommy Ohlsson, 2001. Updated by Tommy Ohlsson, 2003. Updated by Mattias Blennow, 2005. c Mathematical Physics, KTH Physics, KTH, 2005 Printed in Sweden by US–AB, Stockholm, 2005. Contents Contents i 1 Special Relativity 1 1.1 Geometry of the Minkowski Space . 2 1.2 LorentzTransformations. 4 1.3 Physical Interpretations . 5 1.3.1 LorentzContraction ............................ 7 1.3.2 TimeDilation................................ 7 1.3.3 Relativistic Addition of Velocities . 7 1.3.4 The Michelson–Morley Experiment . 8 1.3.5 The Relativistic Doppler Effect . 10 1.4 The Proper Time and the Twin Paradox . 11 1.5 Transformations of Velocities and Accelerations . 12 1.6 Energy, Momentum, and Mass in Relativity Theory . 13 1.7 The Spinorial Representation of Lorentz Transformations . 16 1.8 Lorentz Invariance of Maxwell’s Equations . 17 1.8.1 Physical Consequences of Lorentz Transformations . 20 1.8.2 TheLorentzForce ............................. 22 1.8.3 The Energy-Momentum Tensor . 24 1.9 Problems ...................................... 27 2 Some Differential Geometry 39 2.1 Manifolds ...................................... 39 2.2 Vector Fields and Tangent Vectors . 42 2.2.1 TensorFields ................................ 45 2.3 Geodesics ...................................... 46 2.3.1 Affine Connection and Christoffel Symbols . 46 2.3.2 ParallelTransport ............................. 48 2.4 TorsionandCurvature............................... 50 2.5 Metric and Pseudo-Metric . 53 2.6 Problems ...................................... 59 3 General Relativity 63 3.1 The Einstein Field Equations . 63 3.2 TheNewtonianLimit ............................... 66 3.3 The Schwarzschild Metric . 68 i 3.4 Experimental Tests of General Relativity . 70 3.4.1 The Gravitational Redshift . 71 3.4.2 The Perihelion Precession of Mercury . 72 3.4.3 TheBendingofLight ........................... 74 3.4.4 Radar(Laser)EchoDelay. 74 3.4.5 Black Holes, Binary Star Systems, and Star Evolution . 75 3.5 CosmologicalModels................................ 75 3.5.1 The Large Scale Structure of the Universe . 75 3.5.2 The Robertson–Walker Metric . 76 3.6 Problems ...................................... 79 4 Solutions to Problems 85 4.1 Solutions to Problems in Chapter 1 . 85 4.2 Solutions to Problems in Chapter 2 . 121 4.3 Solutions to Problems in Chapter 3 . 131 Useful Formulas in Relativity Theory 147 Hyperbolic Functions . 147 The Electromagnetic Field . 147 Metric, Connection, Curvature, and Torsion . 147 GeneralRelativity ....................................148 ii Chapter 1 Special Relativity Relativity theory has at least two independent roots. One is the independence of the state of motion of the observer for the description of physics, which goes at least back to Galilei, and an other the long cherished dream of physicists, since the days of Archimedes, of getting rid of motion by introducing geometry. The first line of thought says that the physics should not depend on the observers frame of reference, whether it is in a state of motion or not. Galilei seems to have realized this, and suggested that an experiment of free fall performed on a boat moving with constant velocity or at rest, should give the same result. In other words: free fall experiments cannot tell whether we are in a coordinate system moving with constant speed v or in a coordinate system at rest. This leads, (together with the invariance under translation and rotations), to the so called Galilei invariance of the equations of motion of Newton. In one space dimension the Galilei transformation is: x x′ = x vt, (1.1) → − t t′ = t, (1.2) → where v is the constant velocity of the observer. In Newton’s equation F = ma for a body of mass m acted upon by a force F , this invariance is immediately obvious, since the acceleration a is the second derivative of position with respect to time, i.e., a =x ¨(t), and the velocity v is independent of time. The second root, the geometrization of motion, was discussed already by the presocratic philosophers, and was especially promoted by Archimedes. This geometrization is not the same as that discussed later by Kepler and Galilei, who thought of motion in terms of orbits with geometrical shape: ellipses, parabolas and hyperbolas. These conical sections are geometrical, but the particles still move along these curves in space. The idea of the fourth dimension became prominent during the 19th century, especially by the German scientist Gustav Fechner, who wrote about the fourth dimension although with a slightly different aim than ours. Later the English science fiction written H. G. Wells introduced time as the fourth dimension in his novel “The time machine” from 1895. Even with Einstein’s relativity paper in 1905, the four-dimensional formulation was not part of his ideas. His discussion is focused on the relation between electricity and magnetism, and describes there mainly the notion of relative time and the concept of simultaneity. 1 2 CHAPTER 1. SPECIAL RELATIVITY Soon after Einstein’s formulation of the theory of relativity, his former teacher Hermann Minkowski at ETH in Z¨urich realized that the (special) theory of relativity could be formu- lated very elegantly in a four-dimensional space-time continuum: the Minkowski space. He published his formulation in 1908, just a short time before he untimely died. To follow Minkowski, we should first transform time to have the same dimension as the three other space coordinates. Fortunately this can readily be done, since the speed of light, c, by Einstein’s postulate is a universal constant of nature, the same for all observers related via Lorentz transformations to each other. (The Lorentz transformations leave Maxwell’s equations invariant.) The fourth coordinate is therefore to be chosen as x0 = ct. The motion of an object with constant velocity will then be represented by a straight line in this space, and all theorems about straight lines in four dimensions will have something to say about motions of a free particle. In a similar vein, curved lines will represent particles acted upon by forces, i e. accelerating or retarding particles. Thus motions can be described by geometry, and the old dream of Archimedes is, in a sense, fulfilled. In the four dimensions of Minkowski space the particles are not described by orbits, but by world-lines. These represent the space-time history of the particles. If a particle moves regularly in a Keplerian circular orbit in tree-dimensional space, its total motion, or world-line, will be a (hyper) spiral in Minkowski space, etc. The two roots are now possible to connect to each other, as we think of all transforma- tions that leave the geometrical forms invariant, or of all transformations that are able to superimpose one (four-dimensional) geometrical object onto another one. Below, we start out by studying the mathematical tools relevant to the description of special relativity in Minkowski space. 1.1 Geometry of the Minkowski Space Let M = R4 be the four dimensional real vector space consisting of all 4-tuples x = (x0,x1,x2,x3) of real numbers. Addition of vectors is defined as usual as x + y = (x0 + y0,x1 +y1,x2 +y2,x3 +y3) and multiplication by real scalars λ as λx = (λx0,λx1,λx2,λx3). Normally, in Euclidean geometry, the length of a vector x M is defined by the formula ∈ x 2 = (x0)2 + (x1)2 + (x2)2 + (x3)2. (1.3) | | The Euclidean inner product is x,y = xiyi, where i = 0, 1, 2, 3. From now on, we shall use the Einstein summation convention:h i Sum over repeated indices (on the same side of an equation). Minkowskian geometry differs from Euclidean geometry: The ‘length’ of a vector x M is defined by the formula ∈ x2 = (x0)2 (x1)2 (x2)2 (x3)2. (1.4) − − − Note that the right-hand side is indefinite; it can be either positive, zero, or negative. For this reason, we shall speak only about the length squared x2 and we are not taking the square root of this expression (which could be imaginary). We introduce the Minkowskian inner product as x y = x0y0 x1y1 x2y2 x3y3. (1.5) · − − − The Minkowskian inner product is obviously commutative, i.e., x y = y x. We shall also introduce the metric tensor η = (η ), where η = 0 if µ = ν,· η =· 1, and η = 1 µν µν 6 00 ii − 1.1. GEOMETRY OF THE MINKOWSKI SPACE 3 (no summation here!) when i = 1, 2, 3. This metric tensor is usually called the Minkowski 1 µν µν 1 metric. The inverse of the Minkowski metric is given by η− = (η ), where η = ηµν . The Minkowski metric and its inverse fulfill the relation λν ν ν ηµλη = ηµ = δµ, (1.6) ν ν ν where δµ is the Kronecker delta, δµ = 1 if µ = ν and δµ = 0 if µ = ν. In relativity theory, we often use the convention that the Greek6 indices run from 0 to 3, whereas the Latin indices take the values 1, 2, 3. With this convention, we can write µ ν µ ν x y = x ηµν y = ηµν x y . · Another often used notational convention: We lower and raise the vector and tensor indices according to the rules ν ν λν λω xµ = ηµν x , Aµ = ηµλA , Aµν = ηµληνωA , µ µν µ µλ µν µλ νω x = η xν , A ν = η Aλν , A = η η Aλω.
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