# Einstein's Special Theory of Relativity

### WORK IN PROGRESS!! - LAST UPDATED 2 SEP 2001

In 1905, Albert Einstein1 published three very important papers which helped to revolutionize Physics. One paper related to Brownian Motion. Another contained an explanation of the 'Photoelectric Effect' which contributed to the question of the nature of light (wave or particle? See Wave-Particle Duality) and to the development of Quantum Theory. The third of these papers, titled 'On the Electrodynamics of Moving Bodies' contained what has come to be known as the Special Theory of Relativity. This is Einstein's most famous, and most often misunderstood theory.

### Why Relativity was Necessary

There were certain major problems in classical Physics which Einstein addressed with his Special Theory. To understand the weight of the theory, it will be good to see first what deficiencies in Physics made it necessary. One problem related to the idea of Absolute Motion. Another was a flaw in Maxwell's Equations for electricity and magnetism. Perhaps the most subtle and bizarre problem arose from looking at these two ideas together. Let's see what these problems were:

#### Absolute Space and Time

Isaac Newton2, the undisputed god of Classical Physics, spelled out in his Principia Mathematica what everyone believed about the space, time and motion.

Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external. . . Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. . . Absolute motion is the translation of a body from one absolute place into another. . . Real, absolute rest is the continuance of a body in the same part of immovable space. . . As the order of the parts of time is immutable, so also is the order of the parts of space. . .

- Isaac Newton, Principia Mathematica, 1686, General Scholium

This seems reasonable, if not exactly useful. We may never find an object which is at Absolute Rest, against which Absolute Motion could be measured, but common sense suggests that such an object could exist. Around the turn of the Twentieth Century, certain attempts were made to measure the Absolute Motion of the Earth through space. One of these attempts, the Michelson-Morley experiment, is renowned as one of the greatest failures in the history of Physics.

Michelson and Morley's idea was this - It was well known3 that light consists of waves which travel at some high velocity through space. Any wave requires a medium through which to travel - for example, sound waves travel through air, so where there is no air, like in space, there is no sound. The medium through which light travels - call it Ether - must completely fill the vast reaches of interstellar space, and that is why we can see stars. Presumably, this Ether is stationary, and the Earth moves through it. It would seem that if we measure the speed of light which we're moving along with, we should come up with a slower speed than when we measure the speed of light moving in a direction across or opposite our direction of travel. Michelson and Morley constructed an Interferometer - a very clever apparatus with could compare speeds of light in different directions - and they failed, utterly and repeadedly, to find any difference in speeds, no matter which way they pointed it.

To summarize, Newton's world-view suggested that anything which moves, such as light, or the Earth, has some Absolute Speed with respect to Absolute Space. Just when it seemed that we could finally measure Absolute Speeds, we found ourselves unable to do so, and at a loss to explain why. Light, it seemed, did not behave the way Classical Physics predicted; it did not have a Newtonian Absolute Motion.

#### Maxwell and Electromotive Force

Meanwhile, James Clark Maxwell4, a Nineteenth Century Scottish Physicist, had written down four brilliant equations which described all known phenomena involving electricity and magnetism. Einstein studied these equations in school and noticed something about them that wasn't quite aesthetically satisfying. The problem is easily seen with a simple scenario.

Imagine you are holding a bar magnet in your right hand and a metal coat hanger in your left hand. If you push the end of the bar magnet through the loop formed by the coat hanger, this will produce an electrical current in the hanger. It doesn't matter whether you hold your left hand stationary and move your right hand, or whether you hold your right hand stationary and move your left hand, the effect will be the same. You could even move both hands simultaneously.

The problem here is that when you push the magnet through the coat hanger, Maxwell's equations offered a completely different explanation than it offered in the case where you move the wire over the magnet. When the magnet is moving, the current is induced by an electrical field which arises around any moving magnet. When the wire is moving however, Maxwell couldn't appeal to a moving magnet, so he had to postulate a mysterious 'Electromotive Force' which arises any time a charged particle (like an electron in the wire) moves across a magnetic field.

To say that an electrical field is acting when one hand moves, but that an Electromotive force is active when the other hand moves, seemed silly to Einstein. Clearly the same thing is happening in both cases. What if we didn't even know which was moving? We should still have laws that can account for the current by looking at the relative motions of the magnet and the coat-hanger.

So, this second problem seems unrelated to the first one. While that had to do with Absolute Motion and the speed of light, here is a problem relating to how electromagnetic phenomena are described. However, the speed of light and electromagnetism are very closely linked together.

#### Maxwell and the Speed of Light

Maxwell's equations said, among other things, that a changing electrical field will produce a magnetic field, and vice versa: that a changing magnetic field will produce an electrical field. This effect feeds upon itself, with an electrical impulse producing a magnetic impulse, which produces an electrical impulse... and this series of impulses propagates, wave-like, through space. By analyzing the equations, clever Physicists were able to determine the velocity with which these waves of electromagnetic radiation propagate. The answer they arrived at was, in a stunning coincidence, just exactly the speed of light! So light is nothing more than electromagnetic radiation, described by Maxwell's equations.

Now this was strange. Most moving things move at speeds that could be different. If you are driving down the road, and you apply the brakes on your car, then your speed can change without any unduly odd effect on the rest of the world. The speed of light, however, is a consequence of the laws of physics (Maxwell's equations, anyway). This speed could not be different unless those laws were different. For the speed of light to seem different to different observers, as common sense tells us it would, the laws of physics would have to be different for different observers! In order to preserve the same laws fo physics for different observers, we have to accept the counter-intuitive notion that the speed of light is the same, no matter who is measuring it, no matter what direction they are moving relative to the light. This is what was confirmed my Michelson and Morley's failed experiment.

### Einstein's Assumptions

#### The Constancy of the Speed of Light

Mindful of Michelson and Morley's experiment, Einstein faced the fact, unreasonable as it may seem, that the speed of light seems to come out the same no matter how the person measuring it is moving. Since this fact flies in the face of our common-sense ideas about Space and Time, Einstein suspended his belief in common-sense and explored what Space and Time would have to be like to make possible the observed behaviour of light.

Specifically, he said that we could use the constancy of the speed of light as a way to synchronize clocks. Thus, our idea of time is determined by light's velocity, instead of the other way around.

#### The Principle of Relativity

Meanwhile, Einstein wanted to avoid Maxwell's problem with relative motions, so he adopted the Principle of Relativity as his second assumption, along with the contancy of light speed. Loosely put, the Principle states that if two unpowered space ships drift past each other, and each says that they are stationary while the other moves, then there's no way to tell who's right. More formally put, there is no possible experiment which can establish such a thing as absolute rest or absolute motion. (In Special Relativity, 'motion' is taken to always mean uniform, straight-line motion. In General Relativity, Einstein took accelerated motion into account as well.)

The problem in Maxwell's physics, with the magnet and the coat hanger in relative motion, is only a violation of this Principle at an aesthetic level. Maxwell recognized that the experiment comes out the same, whether the magnet or the wire moves; either way a current is generated. The problem is just that Maxwell gives different names to the effect when viewed from different frames of reference, which is somewhat inelegant.

### Consequences

Starting from Einstein's two assumptions, it is a not too difficult calculus problem to say how Space and Time are affected. The results obtained are somewhat surprising.

#### Simultaneity

The first common-sense notion to go out the window is that of simultaneity. We say that two events are simultaneous if they happen at the same time. It turns out under Relativity that simultaneity is in they eye of the beholder. Events which are simultaneous for one observer need not be for an observer moving relatively to the first one. Clocks which are synchronized on a moving train will not appear synchronized when viewed from the station platform.

There are limits to this effect. There is no frame of reference from which the Magna Carta and the Treaty of Versailles would appear simultaneous. For the order of two events to be in question, they must be separated in such a way that no communication could possibly travel from one event to the other, even at the speed of light.

#### Length Contraction and Time Dilation

Two of the most celebrated and confusing consequences of relativity are called Length Contraction and Time Dilation. These mysterious phenomena are most easily understood as consequences of the relativity of simultaneity.

Length Contraction is the name for the fact that measurements of length come out differently for observers in relative motion. The passengers on a train might believe that their train is 1 kilometer long, and that the station platform whizzing by is 1/2 kilometer long. Meanwhile, observers on the platform will measure the train to be 1/2 kilometer long and the platform to be 1 kilometer long. (This effect will occur if the relative speed with which the train moves by the platform is around 86% of the speed of light.)

The connection between Length Contraction and simultaneity is clear if we consider how these measurements of length are made. The observer on the platform, wishing to measure the train, will reason as follows: 'At a certain time, the front end of the train passed a certain point on the platform. At the same time, the rear end of the train passed a point on the platform 1/2 kilometer from the first point. Therefore, the length of the train is 1/2 kilometer.' An observer on the train will disagree about the simultaneity of these two measurements. He will say that the rear of the train was measured later than the front of the train, causing the length measurement to come out too short.

Time Dilation is a related effect in which Time is revealed to be as susceptible as Space to relativistic strangeness. Basically, sticking with the train example, the observers on the platform will say that the clocks they see on the train all run slow. Meanwhile, the observers on the train will say that the clocks on the platform run slow. Each is correct, from their own frame of reference.

#### Composition of Velocities

Under relativity, the speed of light is the cosmic speed limit; nothing can go faster than light. Suppose we were to try with the following ruse. First, board a train which moves at a high speed relative to the ground, say, 9/10 the speed of light. Now, standing on the train, walk forward along its length at a high speed, say, 1/2 the speed of light. Common sense would say that your speed relative to the ground should be the sum of these two speeds, or 1.4 times the speed of light.

Einstein shows, however, that this naive approach is no good. When adding velocities, we must keep our strange assumptions in mind. Using the formula which he derives5 , we add 9/10 the speed of light to 1/2 the speed of light and come up with a total of 28/29 of the speed of light - close, but still under the limit.

#### E=mc2

E=mc2 is probably the most well known equation in all of Physics.

11879-195521642-17273At least it was believed to be well known... see Wave-Particle Duality41831-1879 Interestingly, Maxwell died in the year that Einstein was born. This sort of thing happens all the time.5Using V1 and V2 to represent the two velocities and C to represent the speed of light, the sum of V1 and V2 is equal to (V1 + V2)/(1 + V1V2/C2)