Foundations of Modern Cosmology

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Chapter 7: The Special Theory of Relativity

Chapter Summary

Einstein's theory of Special Relativity is based on two postulates:

  1. Relativity Principle: The laws of nature are the same in all inertial reference frames
  2. The speed of light in a vacuum is the same in all inertial frames

All of the apparently strange consequences of special relativity follow from these two simple statements. Through a series of thought experiments we demonstrate how the 2nd postulate of relativity leads to the conclusion that space and time intervals are relative, not invariant. We find that moving clocks run slow, and moving meter sticks are length contracted. In both cases the amount of the time dilation or length contraction is specified by the boost factor Gamma. The boost factor for an object moving with velocity v is given by the formula

  • Gamma = 1/(1-(v/c)2 )1/2

The boost factor begins at one for zero velocity, but increases very rapidly the closer the approach to c. This table shows some sample values:

Velocity v/c Gamma value
0 1
0.1 1.005
0.87 2
0.9 2.29
0.99 7.1
0.999 22.4

The second postulate means that no one can ever travel faster than light. The Galilean velocity addition formula, vtotal = v1 + v2, is replaced by the relativistic velocity addition formula

  • vtotal = (v1 + v2 ) / (1+v1v2/c2 )

There are other new special relativistic formulae for Doppler shift and for energy, the latter given by Einstein's famous equation E=mc2.

An important new concept is spacetime. A point in spacetime, with three spatial coordinates and one time coordinate, is called an event. A sequence of events makes up a worldline, a path through spacetime. You should understand what spacetime diagrams are and how to draw them. The invariant measure of the separation between two events in spacetime is the spacetime interval. You should understand how this is defined and what it means to be timelike separated, lightlike separated, and spacelike separated. How do these ideas relate to the idea of causality?

Points to Ponder

The study of relativity paradoxes investigates the properties of special relativity. Try to understand the concept each one illustrates. Special relativity tells us that many things that we used to regard as invariant are actually relative: for example, length, time interval, and simultaneity depend upon the relative motion of two observers. Some invariant quantities are the spacetime interval, the proper time, the proper length, and the rest mass.

Exercise 7.13 is a particularly intriguing paradox, the Train and the Tunnel Paradox. If you can figure this one out you are well on your way to understanding special relativity.

Feeling confused? Find out what the US Senate once had to say about Einstein's Theory of relativity!

Potential Pitfalls

The constancy of the speed of light is one of the biggest stumbling blocks. Every observer always sees light traveling at c regardless of the speed of its source.

Time dilation and length contraction are easy to confuse. Keep in mind that an object is longest in its own rest frame, while clocks at rest tick most rapidly, i.e. show the shortest elapsed time interval between two events. As seen in a frame in which they are moving , objects are contracted and clocks tick more slowly. In their own rest frame, all lengths and clocks seem normal no matter how fast another observer sees them to move.

Some people seem to find it offensive that superluminal travel is impossible for material objects. But if instead of "speed" we consider the "boost factor" (gamma), then the speed of light does correspond to infinity. So the statement becomes "Infinite boost factor is the maximum possible speed." This doesn't sound so implausible, does it? In the viewpoint of modern physics, the speed of light actually represents the speed at which the two long-range forces, gravity and electromagnetism, are propagated.

Questions & Answers

Questions and Answers related to Chapter 7.

Web World Some relativity questions and answers are available at NASA Relativity Page

For a slightly more advanced discussion, see the online lecture notes from Professor Fowler of UVa on special relativity.