15 The Speeial Theory of Relativity 15-1 The principle of relativity For over 200 years the equations of motion enunciated by Newton were be- 15-1 The principle of relativity lieved to describe nature correctly,and the first time that an error in these laws was discovered,the way to correct it was also discovered.Both the error and its 15-2 The Lorentz transformation correction were discovered by Einstein in 1905. 15-3 The Michelson-Morley Newton's Second Law,which we have expressed by the equation experiment F=d(mu)/dt, 15-4 Transformation of time 15-5 The Lorentz contraction was stated with the tacit assumption that m is a constant,but we now know that this is not true,and that the mass of a body increases with velocity.In Einstein's 15-6 Simultaneity corrected formula m has the value 15-7 Four-vectors mo (15.1) 15-8 Relativistic dynamics √1-02/c2 15-9 Equivalence of mass and energy where the "rest mass"mo represents the mass of a body that is not moving and c is the speed of light,which is about3X105km·sec-1 or about186,000 mi·sec-l. For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity-it just changes Newton's laws by introducing a correction factor to the mass.From the formula itself it is easy to see that this mass increase is very small in ordinary circumstances.If the velocity is even as great as that of a satellite,which goes around the earth at 5 mi/sec, then v/c 5/186,000:putting this value into the formula shows that the cor- rection to the mass is only one part in two to three billion,which is nearly impossible to observe.Actually,the correctness of the formula has been amply confirmed by the observation of many kinds of particles,moving at speeds ranging up to practi- cally the speed of light.However,because the effect is ordinarily so small,it seems remarkable that it was discovered theoretically before it was discovered experimentally.Empirically,at a sufficiently high velocity,the effect is very large, but it was not discovered that way.Therefore it is interesting to see how a law that involved so delicate a modification (at the time when it was first discovered) was brought to light by a combination of experiments and physical reasoning. Contributions to the discovery were made by a number of people,the final result of whose work was Einstein's discovery. There are really two Einstein theories of relativity.This chapter is concerned with the Special Theory of Relativity,which dates from 1905.In 1915 Einstein published an additional theory,called the General Theory of Relativity.This latter theory deals with the extension of the Special Theory to the case of the law of gravitation;we shall not discuss the General Theory here. The principle of relativity was first stated by Newton,in one of his corollaries to the laws of motion:"The motions of bodies included in a given space are the same among themselves,whether that space is at rest or moves uniformly forward in a straight line."This means,for example,that if a space ship is drifting along at a uniform speed,all experiments performed in the space ship and all the phenom- ena in the space ship will appear the same as if the ship were not moving,pro- vided,of course,that one does not look outside.That is the meaning of the princi- ple of relativity.This is a simple enough idea,and the only question is whether it is true that in all experiments performed inside a moving system the laws of physics 15-1