# Special Relativity

In spite of the success of Newton's three laws of motion for two centuries some problems began to arise. These problems were the absence of an aether electromagnetic waves were though to travel on, a precession in the orbit of Mercury not accounted by perturbations from the other planets, and the inability of classical physics to account for the distribution of light frequencies by a non-reflecting hot body, called a black body. The last of these loose threads in the physical world view was pulled by Max Planck in 1895. The resolution of this black body problem was solved by assuming that radiation occurred in discrete energy packets. This lead to the concept of the quanta, or that energy states on a small scale occurred not in a continuum, but rather in discrete energy levels. Of course this lead further to the theory of quantum mechanics. The first of these threads lead to the prospect that electromagnetic radiation did not exist on a fixed reference frame. This stimulated ideas by Lorentz and Einstein to develop the special theory of relativity. The last problem was solved by Einstein's general theory of relativity. By 1915 the world view of physicists changed dramatically. While the story of quantum mechanics is fascinating in its own right, we will concentrate on the development of special relativity.

James Maxwell developed the equations which unified the electric field and the magnetic field into a single set of equations that described the electromagnetic field [6.1] in 1865. It was known previously, mostly by the work of Faraday, that a current could induce a magnetic field and that a changing electric field could produce a changing magnetic field, and the converse as well. Maxwell demonstrated how a changing current, known to be associated with a changing magnetic field, could produce a changing electric field. This changing electric field was a "displacement current" added to the Faraday equation. This unification of the electric and magnetic field resulted in the understanding of how a bundle of oscillating electric and magnetic field could propagate though space as a wave. It was further demonstrated that light is a form of electromagnetic wave. This electromagnetic field as a wave propagates through space at a speed c ~ 300,000 km/sec. Further, the wave equations demand that this should be the case universally.

In spite of this physicists started asking the wrong questions. They assumed that electromagnetic waves were similar to water waves, and so there had to be a medium these wave propagated within. This gave a fixed frame to the universe for electromagnetic wave propagation. Yet it should have been apparent to any inquiring physicist in the late 19th century that there was a problem here. Newton's laws give a hint of this. The first law tells us that a body has inertia, and so the proper reference frame for observing physics is inertial, or not accelerating. The second law tells us that a body accelerates in direct proportion to the force applied, where the proportionality factor is that body's mass. Again the second law is only properly applied or observed from an inertial reference frame. The third law says that the force one body exerts on another is equal in magnitude to a force exerted on the first by the second in the opposite direction. This law of motion indicates that no matter the direction with which this is arranged the result is the same. Also by the first law this obtains no matter how fast these two bodies are travelling with respect to an inertial reference frame of observation. Further, this happens everywhere in space. The third law tells us that space is the same as measured by any inertial observer travelling at any velocity, that space is isotropic, for it does not depend upon any direction, and finally that space is homogenous so that the dynamical principles are independent of where in space one observes motion. This is referred to as the Galilean relativity principle. The imposition of some aether amounts to a violation of this relativity principle. The imposition of this aether amounts to the imposition of a preferred inertial reference frame with zero velocity. This is in fact an implicit violation of Newton's first and third laws of motion, at least with respect to the motion of an electromagnetic wave.

This apparent contradiction between Newtonian mechanics and the aether theory failed to grab the attention of physicists. However, this preferred frame would have measurable consequences. If an observer were travelling at 10% the speed of light with respect to the aether frame light propagating in the same direction would be seen to be propagating at 90% the speed of light. Similarly light travelling in the opposite direction would be seen to propagate at 110% the speed of light. Just as a swimmer crossing Fig. 6.1. Schematic of the Michelson-Morley interferometer.

a river is carried down stream by the current, light travelling in a direction perpendicular to this reference frame would be diverted. The consequences of this would be measurable. Since the Earth executes a near circular motion around the sun these effects would be seen at different times of the year as the Earth travels through the aether in different directions and different velocities. Of course the effect might be subtle, for the 29.5 km/sec orbital speed of the Earth around the sun is ~ 10~4 that of light speed. Yet by the end of the 19th century optical technology was up to the job.

Albert Michelson and Edward Morley designed just the device to do this in 1887. They floated a table in a vat containing liquid mercury, where upon the table were a light source, a half mirror beam splitter, two full mirrors and a scope. The half mirror split the light so that it would travel in two directions perpendicular to each other. Only a very narrow range of light frequencies went through this interferometer. This light would then reflect off the two mirrors and recombine at the scope. If the two light beams travelled different distances the recombined beam exhibits interference fringes where the two waves combine and cancel. Further, if as the Earth orbits the sun the speed of this light would be seen to change as the Earth travels at different directions and velocity with respect to the aether. This would mean the interference fringes would be seen to change over a year period. Michelson and Morley found no such change. The effect of the aether was absent. This cast doubt upon whether the aether existed or whether there were competing effects that masked the appearance of the aether. Michelson won the Nobel prize for this result in 1907.

George FitzGerald and Hendrik Lorentz, in 1904 and 1905 respectively, accounted for this loss of interference change by calculating how much the apparatus would have to shrink in the direction along its motion with respect to the aether. This is the famous Lorentz-FitzGerald contraction formula L' = L\J 1 — (v/c)2 [6.2]. This lead to Lorentz publishing his Lorentz transformations which removed the effect of the aether. These transformations meant that lengths along the direction of motion are reduced and a clock on that moving frame is seen to slow down with respect to an observer fixed to the aether. This result is based on largely phenomenological arguments, or arguments concerning a measurement outcome within some established physical theory. The Lorentz transformations are an end result of Einstein's special theory of relativity, yet Einstein's approach was to restructure the meaning of space and time by framing it on the invariance of the speed of light.

Albert Einstein at the age of 15 read a popularization of electricity and magnetism. He learned the essential nature of electromagnetism: An electromagnetic wave consists of oscillating electric and magnetic fields that propagates at a predicted speed c ~ 3 x 108 m/sec. He then imagined what would happen if somebody were travelling along side the wave. This observer would see the electric and magnetic field oscillate, but remain at rest with respect to that observer's frame. However, Einstein realized that this observation would violate what Maxwell told us: Electromagnetic waves travel at the speed of light. Einstein saw a contradiction in the standard concepts of physics as a teenager, at a time when the physics community was beginning to struggle with increasingly irreconcilable problems.

Einstein realized the crux of the problem in 1905. His approach to the problem was to look at the motion of charged particles in spacetime. In particular a charged particle with an electric field will have a magnetic field circulating around it as measured by an observer in an inertial reference frame moving with some velocity relative to that particle. Einstein simply added the requirement that the speed of light must remain the same relative to all reference frames. His analysis found the Lorentz transformations, but with the concept that space and time are inter-changable. Just as a rotation of an x — y axis about its origin to some new x' — y' axis inter-changes the meaning of the two coordinates, a spatial direction and time can be similarly inter-changed. However, this "rotation" is a pseudo-rotation that is hyperbolic instead of elliptical (circular).

To illustrate the nature of special relativity we will consider the transformation of vectors in space plus time, or spacetime. A vector is an array of components for the values of a vector at some point. So for some arbitrarily given origin with ct = 0, x = 0, y = 0, z = 0 a vector from this origin to some other point ct, x, y, z is written as