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 .....................................
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