Special Relativity/Introduction

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Special Relativity

Contents

[edit] Introduction

The Special Theory of Relativity was developed at the end of the nineteenth century and the beginning of the twentieth century. It entirely replaced older physical theories such as Newtonian Physics and led to early Quantum Theory and General Relativity.

Special Relativity begins by re-examining the basis of Newtonian Physics and demonstrating that the Newtonian treatment of relative motion is inaccurate for the fast moving objects. As a result the whole of classical physics must be modified to account for this error.

Special Relativity does not just apply to fast moving objects, it affects the everyday world directly through "relativistic" effects such as magnetism and the relativistic inertia that underlies kinetic energy and hence the whole of dynamics.

Special Relativity is now one of the foundation blocks of physics. It is in no sense a provisional theory and is largely compatible with quantum theory; it not only led to the idea of matter waves but is the origin of 'spin' and underlies the existence of the antiparticles. Contrary to popular belief modern Special Relativity is not invalidated by effects such as quantum entanglement but rather provides the understanding of space and time through which these effects might be understood.

[edit] Historical Development

James Clerk Maxwell

In the nineteenth century the idea that light was propagated in a medium called the "aether" was prevalent. In 1865 James Clerk Maxwell produced a theory of electromagnetic waves that initially seemed to be based on this aether concept. The theory was highly successful but it predicted that the velocity of electromagnetic waves would depend on two constant factors: the permittivity and permeability constants. At first these constants were interpreted as properties of the aether. The constants would be the same for all observers so there was an implicit idea of a universal, stationary aether. Observers would measure the velocity of any light that reached them as the sum of their velocity relative to the aether and the velocity of light in the aether.

Maxwell proposed that the state of motion of an observer relative to an aether might be tested experimentally by reflecting beams of light at right angles to each other in an interferometer. His 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. 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.

Albert Abraham Michelson

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 physical 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.

Albert Einstein

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.

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.

Henri Poincaré

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 = mc2. 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.

  • G. F. FitzGerald (1889), The Ether and the Earth’s Atmosphere, Science 13, 390.
  • Larmor, J. (1897), On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with material media, Phil. Trans. Roy. Soc. 190: 205–300, doi:10.1098/rsta.1897.0020
  • H. A. L. Lorentz (1895), Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern, Brill, Leyden.

Note 1: The modern philosophical objection to the spacetime of Special Relativity is that it acts on bodies without being acted upon, however, in General Relativity spacetime is acted upon by its content.

[edit] 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 carefully designed to attack the failure of students 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.

[edit] 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 mainly applies in a special case: frames of reference that are not accelerating, or inertial frames. This is the same restriction that applies to Newton's Laws of Motion. We also don't consider the effect of gravitational fields in special relativity.

In search of a more complete theory, Einstein developed the general theory of relativity published in 1915. General relativity (GR) is a mathematically more demanding subject but has the advantage of describing all frames. This includes accelerating frames and gravitational fields.

The conceptual difference between the two is the model of spacetime used. Special relativity makes use of a Euclidian-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.

[edit] 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. 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 pondering the sorts of mechanical or electrical effects that could adjust the velocity of light to be compatible with observation.

[edit] 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.

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