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Special Relativity Special Relativity Wikibooks.org March 12, 2013 On the 28th of April 2012 the contents of the English as well as German Wikibooks and Wikipedia projects were licensed under Creative Commons Attribution-ShareAlike 3.0 Unported license. An URI to this license is given in the list of figures on page 155. If this document is a derived work from the contents of one of these projects and the content was still licensed by the project under this license at the time of derivation this document has to be licensed under the same, a similar or a compatible license, as stated in section 4b of the license. The list of contributors is included in chapter Contributors on page 151. The licenses GPL, LGPL and GFDL are included in chapter Licenses on page 159, since this book and/or parts of it may or may not be licensed under one or more of these licenses, and thus require inclusion of these licenses. The licenses of the figures are given in the list of figures on page 155. 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Contents 1 Introduction 3 1.1 Introduction . 3 1.2 Historical Development . 4 1.3 Intended Audience . 11 1.4 What's so special? . 11 1.5 Common Pitfalls in Relativity . 11 1.6 A Word about Wiki . 12 2 The principle of relativity 13 2.1 The principle of relativity . 13 2.2 Inertial reference frames . 21 3 Spacetime 25 3.1 The modern approach to relativity . 25 3.2 The lightcone . 36 3.3 The Lorentz transformation equations . 38 3.4 More about the relativity of simultaneity . 39 3.5 The nature of length contraction . 44 3.6 More about time dilation . 46 3.7 The twin paradox . 47 3.8 The Pole-barn paradox . 54 3.9 The transverse doppler effect . 63 3.10 Relativistic transformation of angles . 63 3.11 Addition of velocities . 64 3.12 Introduction . 65 3.13 Momentum . 66 3.14 Force . 70 3.15 Energy . 71 3.16 Nuclear Energy . 75 4 aether 77 4.1 Introduction . 77 4.2 The aether drag hypothesis . 78 4.3 The Michelson-Morley experiment . 83 4.4 Mathematical analysis of the Michelson Morley Experiment . 91 4.5 Lorentz-Fitzgerald Contraction Hypothesis . 95 4.6 External links . 96 5 mathematical approach 97 5.1 Vectors . 97 III Contents 5.2 Matrices . 103 5.3 Indicial Notation . 108 5.4 Analysis of curved surfaces and transformations . 111 6 Waves 115 6.1 Applications of Special Relativity . 115 6.2 External Links . 121 7 Mathematical Transformations 123 7.1 The Lorentz transformation . 123 7.2 Hyperbolic geometry . 138 7.3 Mathematics of the Lorentz Transformation Equations . 145 8 Contributors 151 List of Figures 155 9 Licenses 159 9.1 GNU GENERAL PUBLIC LICENSE . 159 9.2 GNU Free Documentation License . 160 9.3 GNU Lesser General Public License . 161 1 1 Introduction 1.1 Introduction The Special Theory of Relativity was the result of developments in physics at the end of the nineteenth century and the beginning of the twentieth century. It changed our understanding of older physical theories such as Newtonian Physics and led to early Quantum Theory and General Relativity. Special Relativity does not just apply to fast moving objects, it affects the everyday world directly through "relativistic" effects such as magnetism and the relativistic inertia that underlies kinetic energy and hence the whole of dynamics. Special Relativity is now one of the foundation blocks of physics. It is in no sense a provisional theory and is largely compatible with quantum theory; it not only led to the idea of matter waves but is the origin of quantum 'spin' and underlies the existence of the antiparticles. Special Relativity is a theory of exceptional elegance, Einstein crafted the theory from simple postulates about the constancy of physical laws and of the speed of light and his work has been refined further so that the laws of physics themselves and even the constancy of the speed of light are now understood in terms of the most basic symmetries in space and time. Further Reading Feynman Lectures on Physics. Symmetry in Physical Laws. (World Student) Vol 1. Ch 52. Gross, D.J. The role of symmetry in fundamental physics. PNAS December 10, 1996 vol. 93 no. 25 14256-14259 http://www.pnas.org/content/93/25/14256.full 3 Introduction 1.2 Historical Development Figure 1 James Clerk Maxwell In the nineteenth century it was widely believed that light was propagated in a medium called the "aether". In 1865 James Clerk Maxwell produced a theory of electromagnetic waves that seemed to be based on this aether concept. According to his theory the velocity of electromagnetic waves such as light would depend on two constant factors, the permittivity and permeability constants, which were properties of the aether. Anyone who was stationary within the aether would measure the speed of light to be constant as a result of these constant properties of the aether. A light ray going from one stationary point to another in the aether would take the same amount of time to make the journey no matter who observed 4 Historical Development it and observers would measure the velocity of any light that reached them as the sum of their velocity relative to the aether and the velocity of light in the aether. If space were indeed full of an aether then objects would move through this aether and this motion should be detectable. Maxwell proposed that the state of motion of an observer relative to an aether might be tested experimentally by reflecting beams of light at right angles to each other in an interferometer. As the interferometer was moved through the aether the addition of the velocity of the equipment to the velocity of the light in the aether would cause a distinctive interference pattern. Maxwell's idea was submitted as a letter to Nature in 1879 (posthumously). Albert Michelson read Maxwell's paper and in 1887 Michelson and Morley performed an 'interferometer' experiment to test whether the observed velocity of light is indeed the sum of the speed of light in the aether and the velocity of the observer. To everyone's surprise the experiment showed that the speed of light was independent of the speed of the destination or source of the light in the proposed aether. 5 Introduction Figure 2 Albert Abraham Michelson How might this "null result" of the interferometer experiment be explained? How could the speed of light in a vacuum be constant for all observers no matter how they are moving themselves? It was possible that Maxwell's theory was correct but the theory about the way that velocities add together (known as Galilean Relativity) was wrong. Alternatively it was possible that Maxwell's theory was wrong and Galilean Relativity was correct. However, the most popular interpretation at the time was that both Maxwell and Galileo were correct and something was happening to the measuring equipment. Perhaps the instrument was being squeezed in some way by the aether or some other material effect was occurring. 6 Historical Development Various physicists attempted to explain the Michelson and Morley experiment. George Fitzgerald (1889) and Hendrik Lorentz (1895) suggested that objects tend to contract along the direction of motion relative to the aether and Joseph Larmor (1897) and Hendrik Lorentz (1899) proposed that moving objects are contracted and that moving clocks run slow as a result of motion in the aether. Fitzgerald, Larmor and Lorentz's contributions to the analysis of light propagation are of huge importance because they produced the "Lorentz Transformation Equations". The Lorentz Transformation Equations were developed to describe how physical effects would need to change the length of the interferometer arms and the rate of clocks to account for the lack of change in interference fringes in the interferometer experiment. It took the rebellious streak in Einstein to realise that the equations could also be applied to changes in space and time itself. Figure 3 Albert Einstein 7 Introduction By the late nineteenth century it was becoming clear that aether theories of light propagation were problematic. Any aether would have properties such as being massless, incompressible, entirely transparent, continuous, devoid of viscosity and nearly infinitely rigid. In 1905 Albert Einstein realised that Maxwell's equations did not require an aether. On the basis of Maxwell's equations he showed that the Lorentz Transformation was sufficient to explain that length contraction occurs and clocks appear to go slow provided that the old Galilean concept of how velocities add together was abandoned. Einstein's remarkable achievement was to be the first physicist to propose that Galilean relativity might only be an approximation to reality.
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