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Special Relativity
Special Relativity
A Wikibook
http://en.wikibooks.org/wiki/Special_relativity
Part 1: Introductory text Cover picture: Albert Einstein and Hendrik Lorentz photographed by Ehrenfest (1880-1933) in
front of his home in Leiden in 1921. Source: Museum Boerhaave, Leiden Table of Contents
Introduction.......................................................................................................................................... 4
Intended Audience........................................................................................................................... 8
What's so special?............................................................................................................................ 8
Common Pitfalls in Relativity......................................................................................................... 9
A Word about Wiki..........................................................................................................................9
The principle of relativity...................................................................................................................10
Frames of reference, events and transformations.......................................................................... 10
Special relativity............................................................................................................................ 12
The postulates of special relativity................................................................................................ 13
The spacetime interpretation of special relativity.............................................................................. 14
Spacetime.......................................................................................................................................21
The lightcone................................................................................................................................. 22
The Lorentz transformation equations................................................................................................24
More about the relativity of simultaneity and the Andromeda paradox.............................................26
The nature of length contraction....................................................................................................28
Evidence for length contraction, the field of an infinite straight current.......................................30
De Broglie waves...........................................................................................................................32
More about time dilation................................................................................................................33
The twin paradox........................................................................................................................... 34
Jim and Bill's view of the journey................................................................................................. 37
The Pole-barn paradox...................................................................................................................40
Addition of velocities......................................................................................................................... 42
Relativistic Dynamics.........................................................................................................................45
Momentum.....................................................................................................................................45
Force.............................................................................................................................................. 50
Energy............................................................................................................................................50
Nuclear Energy...................................................................................................................................52
Light propagation and the aether........................................................................................................ 55
The aether drag hypothesis............................................................................................................ 55
The Michelson-Morley experiment............................................................................................... 59
Measuring aether.......................................................................................................................60
The experiments........................................................................................................................ 61
The most famous failed experiment..........................................................................................62
Fallout....................................................................................................................................... 64
Mathematical analysis of the Michelson Morley Experiment....................................................... 65
Coherence length........................................................................................................................... 69
Lorentz-Fitzgerald Contraction Hypothesis...................................................................................70
External links................................................................................................................................. 70
Appendix 1......................................................................................................................................... 71
Mathematics of the Lorentz Transformation Equations........................................................... 71
1 Introduction
The Special Theory of Relativity is a physical theory that was developed at the end of the nineteenth century
and the beginning of the twentieth century. It replaced older theories such as Newtonian Physics and led to
early Quantum Theory and General Relativity.
Special Relativity begins by re-examining the basis of Newtonian Physics. In Special Relativity it is shown
that the Newtonian treatment of relative motion is incorrect and that the whole of physics must be rebuilt to
account for this problem.
The following example serves to introduce the importance of
relative motion when observing the world. Jim is standing on
the street corner looking at a nearby stationary dog. Bob rides
by on a bus. Jim and Bob both use various pieces scientific
equipment to measure the apparent velocity of the dog. From
everyday experience you should already be able to determine
the results. Bob, seeing the dog on the street move by,
determines that the dog is moving at the same speed as the
bus. Jim on the other hand, determines that the dog is not
moving at all.
The results obtained by Jim and Bob are different, but they make perfect sense. Jim and Bob are in different
frames of reference. It seems that velocity measurements depend greatly on the frame of reference from
which one takes the measurements. As we shall see, measurements of things we often take for granted, like
time and space, also depend on the frame of reference.
The question we now ask is, "Which frame of reference is better, Jim's or Bob's?" Some would immediately
say that performing measurements of distant objects from a moving bus is impractical, and anything so
serious must be done while standing still. Unfortunately it is often the case that we don't have such a
stationary frame of reference at our disposal.
When measuring the motion of distant planets the measurements must be performed on Earth, a moving
planet in itself. In fact the Earth is behaving much worse than a bus; it is rotating and falling through space in
an elliptical path! In such a case one may insist that all recorded data is transformed to the Sun's frame of
reference, thereby defining the Sun as stationary. Then it is easier to conceptualize the nature of our solar
system. But isn't the Sun also moving with respect to the other stars and the universe in
general?
Indeed one may consider many ways to orient a frame of reference in the universe. But the
question still remains, "Which is better?" This question bothered many scientists in the late
2
Jim, dog and bus
Maxwell 19th century when Maxwell's new theory of electromagnetism produced a number for the speed o