Special Relativity/Introduction

Introduction
The Special Theory of Relativity is a theory of classical physics that was developed at the end of the nineteenth century and the beginning of the twentieth century. It changed our understanding of older physical theories such as Newtonian Physics and led to early Quantum Theory and later the Theory of General Relativity. Special Relativity is one of the foundation blocks of physics.

This book will introduce the reader to, perhaps, the most profound discovery of the twentieth century and the modern world: the universe has at least four dimensions.

Historical Development
Special Relativity is not a theory about light, it is a theory about space and time, but it was the strange behaviour of light that first alerted scientists to the possibility that the universe had an unexpected geometry. The short history of Special Relativity given here will start with light but will end with the discovery that the behaviour of light is related to the geometry of the universe.

In the nineteenth century it was widely accepted that light travelled as waves in a substance called the “aether”. Light was thought to travel in this aether in a similar way to how other types of waves travel in material substances; for example, sound waves travel in air (and other substances). Light would travel to our eyes as waves through the aether like sound travels to our ears as waves in the air.

The nature of the aether was unknown but a possible link between the aether and electrical and magnetic fields became apparent during the first half of the nineteenth century. Faraday demonstrated that the polarisation of light was affected by magnetic fields and Weber showed that electrical effects could be transmitted across non-conducting materials so there was a strong suggestion that light might be some sort of electromagnetic effect.



In 1865 the Scottish physicist James Clerk Maxwell drew together the various experiments on electricity and magnetism into an electromagnetic theory of light based on the idea of an aether. He identified that electricity and magnetism, previously considered to be separate forces, were two aspects of one force. He was able to calculate the speed at which electromagnetic waves moved as a simple ratio between the strength of the electric field and the strength of the magnetic field. From this, one of his key observations was that electromagnetic effects seemed to propagate at nearly light speed. He wrote of the velocity of electrical interactions that:

“This velocity is so nearly that of light that it seems we have strong reason to conclude that light itself (including radiant heat and other radiations, if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.”

Maxwell's theory explained radio, heat radiation, light and many other phenomena as electromagnetic waves travelling in an aether. The velocity of these waves depended upon the properties of the aether itself. Someone who was stationary within the aether would measure the speed of light to be constant as a result of the constant properties of the aether. A light ray going from one stationary observer to another in the aether would take the same amount of time to make the journey no matter which stationary observer measured it. However, although stationary observers would all observe the same velocity for light, moving observers would measure the velocity of light as the sum of their velocity relative to the aether and the velocity of light in the aether.

If space were indeed full of an aether then the motion of objects through this aether should be detectable by measuring the velocity of light rays. In practice it is difficult to measure the velocity of light with sufficient precision. Maxwell suggested that an instrument called an "interferometer" would provide the required accuracy. He proposed that if an interferometer were moved through the aether the addition of the velocity of the equipment to the velocity of the light in the aether would cause a distinctive interference pattern. Maxwell's idea was submitted as a letter to Nature in 1879 (posthumously).

Albert Michelson read Maxwell's paper and in 1887 Michelson and Morley performed an 'interferometer' experiment to test whether the observed velocity of light is indeed the sum of the speed of light in the aether and the velocity of the observer. Michelson and Morley discovered that the measured velocity of light did not change with the velocity of the observer. To everyone's surprise the experiment showed that the speed of light was independent of the speed of the destination or source of the light in the proposed aether. How might this "null result" of the interferometer experiment be explained? How could the speed of light in a vacuum be constant for all observers no matter how they are moving themselves? It was possible that Maxwell's theory was correct but the theory about the way that velocities add together (known as Galilean Relativity) was wrong. Alternatively it was possible that Maxwell's theory was wrong and Galilean Relativity was correct. However, the most popular interpretation at the time was that both Maxwell and Galileo were correct and something was happening to the measuring equipment. Perhaps the instrument was being squeezed in some way by the aether or some other material effect was occurring.

Various physicists attempted to explain the Michelson and Morley experiment. George Fitzgerald (1889) and Hendrik Lorentz (1895) suggested that objects tend to contract along the direction of motion relative to the aether and Joseph Larmor (1897) and Hendrik Lorentz (1899) proposed that moving objects are contracted and that moving clocks run slow as a result of motion in the aether. Fitzgerald, Larmor and Lorentz's contributions to the analysis of light propagation are of huge importance because they produced the "Lorentz Transformation Equations". The Lorentz Transformation Equations were developed to describe how physical effects would need to change the length of the interferometer arms and the rate of clocks to account for the lack of change in interference fringes in the interferometer experiment. It took the rebellious streak in Einstein to realise that the equations could also be applied to changes in space and time itself.

By the late nineteenth century it was becoming clear that aether theories of light propagation were problematic. Any aether would have properties such as being massless, incompressible, entirely transparent, continuous, devoid of viscosity and nearly infinitely rigid. In 1905 Albert Einstein realised that Maxwell's equations did not require an aether. On the basis of Maxwell's equations he showed that the Lorentz Transformation was sufficient to explain that length contraction occurs and clocks appear to go slow provided that the old Galilean concept of how velocities add together was abandoned. Einstein's remarkable achievement was to be the first physicist to propose that Galilean relativity might only be an approximation to reality. He came to this conclusion by being guided by the Lorentz Transformation Equations themselves and noticing that these equations only contain relationships between space and time without any references to the properties of an aether.

In 1905 Einstein was on the edge of the idea that made relativity special. It remained for the mathematician Hermann Minkowski to provide the full explanation of why an aether was entirely superfluous. He announced the modern form of Special Relativity theory in an address delivered at the 80th Assembly of German Natural Scientists and Physicians on September 21, 1908. The consequences of the new theory were radical, as Minkowski put it:

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

What Minkowski had spotted was that Einstein's theory was actually related to the theories in differential geometry that had been developed by mathematicians during the nineteenth century. Initially Minkowski's discovery was unpopular with many physicists including Poincaré, Lorentz and even Einstein. Physicists had become used to a thoroughly materialist approach to nature in which lumps of matter were thought to bounce off each other and the only events of any importance were those occurring at some universal, instantaneous, present moment. The possibility that the geometry of the world might include time as well as space was an alien idea. The possibility that phenomena such as length contraction could be due to the physical effects of spacetime geometry rather than the increase or decrease of forces between objects was as unexpected for physicists in 1908 as it is for the modern high school student. Einstein rapidly assimilated these new ideas and went on to develop General Relativity as a theory based on differential geometry but many of the earlier generation of physicists were unable to accept the new way of looking at the world.

The adoption of differential geometry as one of the foundations of relativity theory has been traced by Walter (1999). Walter's study shows that by the 1920's modern differential geometry had become the principal theoretical approach to relativity, replacing Einstein's original electrodynamic approach.

It has become popular to credit Henri Poincaré with the discovery of the theory of Special Relativity, but Poincaré got many of the right answers for some of the wrong reasons. He even came up with a version of $$E=mc^2$$. In 1904 Poincaré had gone as far as to enunciate the "principle of relativity" in which "The laws of physical phenomena must be the same, whether for a fixed observer, as also for one dragged in a motion of uniform translation, so that we do not and cannot have any means to discern whether or not we are dragged in a such motion." Furthermore, in 1905 Poincaré coined the term "Lorentz Transformation" for the equation that explained the null result of the Michelson Morley experiment. Although Poincaré derived equations to explain the null result of the Michelson Morley experiment, his assumptions were still based upon an aether. It remained for Einstein to show that an aether was unnecessary.

It is also popular to claim that Special Relativity and aether theories such as those due to Poincaré and Lorentz are equivalent and only separated by Occam's Razor. This is not strictly true. Occam's Razor is used to separate a complex theory from a simple theory, the two theories being different. In the case of Poincare's and Lorentz's aether theories both contain the Lorentz Transformation which is already sufficient to explain the Michelson and Morley Experiment, length contraction, time dilation etc. without an aether. The aether theorists simply failed to notice that this is a possibility because they rejected spacetime as a concept for reasons of philosophy or prejudice. In Poincaré's case he rejected spacetime because of philosophical objections to the idea of spatial or temporal extension (see note 1).

It is curious that Einstein actually returned to thinking based on an aether for philosophical reasons similar to those that haunted Poincaré (See Granek 2001). The geometrical form of Special Relativity as formalised by Minkowski does not forbid action at a distance and this was considered to be dubious philosophically. This led Einstein, in 1920, to reintroduce some of Poincaré's ideas into the theory of General Relativity. Whether an aether of the type proposed by Einstein is truly required for physical theory is still an active question in physics. However, such an aether leaves the spacetime of Special Relativity almost intact and is a complex merger of the material and geometrical that would be unrecognised by 19th century theorists.

Intended Audience
This book presents special relativity (SR) from first principles and logically arrives at the conclusions. There will be simple diagrams and some thought experiments. Although the final form of the theory came to use Minkowski spaces and metric tensors, it is possible to discuss SR using nothing more than high school algebra. That is the method used here in the first half of the book. That being said, the subject is open to a wide range of readers. All that is really required is a genuine interest.

For a more mathematically sophisticated treatment of the subject, please refer to the Advanced Text in Wikibooks.

The book is designed to confront the way students fail to understand the relativity of simultaneity. This problem is well documented and described in depth in: Student understanding of time in special relativity: simultaneity and reference frames by Scherr et al.

What's so special?
The special theory was suggested in 1905 in Einstein's article "On the Electrodynamics of Moving Bodies", and is so called because it applies in the absence of non-uniform gravitational fields.

In search of a more complete theory, Einstein developed the general theory of relativity published in 1915. General relativity (GR), a more mathematically demanding subject, describes physics in the presence of gravitational fields.

The conceptual difference between the two is the model of spacetime used. Special relativity makes use of a Euclidean-like (flat) spacetime. GR lives in a spacetime that is generally not flat but curved, and it is this curvature which represents gravity. The domain of applicability for SR is not so limited, however. Spacetime can often be approximated as flat, and there are techniques to deal with accelerating special relativistic objects.

Common Pitfalls in Relativity
Here is a collection of common misunderstandings and misconceptions about SR. If you are unfamiliar with SR then you can safely skip this section and come back to it later. If you are an instructor, perhaps this can help you divert some problems before they start by bringing up these points during your presentation when appropriate.

Beginners often believe that special relativity is only about objects that are moving at high velocities. Strictly speaking, this is a mistake. Special relativity applies at all velocities but at low velocity the predictions of special relativity are almost identical to those of the Newtonian empirical formulae. As an object increases its velocity the predictions of relativity gradually diverge from Newtonian Mechanics.

There is sometimes a problem differentiating between the two different concepts "relativity of simultaneity" and "signal latency/delay." This book text differs from some other presentations because it deals with the geometry of spacetime directly and avoids the treatment of delays due to light propagation. This approach is taken because students would not be taught Euclid's geometry using continuous references to the equipment and methods used to measure lengths and angles. Continuous reference to the measurement process obscures the underlying geometrical theory whether the geometry is three dimensional or four dimensional.

If students do not grasp that, from the outset, modern Special Relativity proposes that the universe is four dimensional, then, like Poincaré, they will consider that the constancy of the speed of light is just an event awaiting a mechanical explanation and waste their time by pondering the sorts of mechanical or electrical effects that could adjust the velocity of light to be compatible with observation.

A Word about Wiki
This is a wikibook. That means it has great potential for improvement and enhancement. The improvement can be in the form of refined language, clear mathematics, simple diagrams, and better practice problems and answers. The enhancement can be in the form of artwork, historical context of SR, anything. Feel free to improve and enhance Special Relativity and other wikibooks as you see necessary.