Special Relativity/Introduction
Introduction
The Special Theory of Relativity is a physical theory that was developed at the end of the nineteenth century and the beginning of the twentieth century. It entirely replaced older 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. In Special Relativity it is shown that the Newtonian treatment of relative motion is incorrect and that the whole of physics must be rebuilt to account for this problem.
The following example serves to introduce the importance of relative motion when observing the world. Jim is standing on the street corner looking at a nearby stationary dog. Bob rides by on a bus. Jim and Bob both use various pieces of scientific equipment to measure the apparent velocity of the dog. From everyday experience you should already be able to determine the results. Bob, seeing the dog on the street move by, determines that the dog is moving at the same speed as the bus. Jim on the other hand, determines that the dog is not moving at all.
The results obtained by Jim and Bob are different, but they make perfect sense. Jim and Bob are in different frames of reference. It seems that velocity measurements depend greatly on the frame of reference from which one takes the measurements. As we shall see, measurements of things we often take for granted, like time and space, also depend on the frame of reference.
The question we now ask is, "Which frame of reference is better, Jim's or Bob's?" Some would immediately say that performing measurements of distant objects from a moving bus is impractical, and anything so serious must be done while standing still. Unfortunately it is often the case that we don't have such a stationary frame of reference at our disposal.
When measuring the motion of distant planets the measurements must be performed on Earth, a moving planet in itself. In fact the Earth is behaving much worse than a bus; it is rotating and falling through space in an elliptical path! In such a case one may insist that all recorded data is transformed to the Sun's frame of reference, thereby defining the Sun as stationary. Then it is easier to conceptualize the nature of our solar system. But isn't the Sun also moving with respect to the other stars and the universe in general?
Indeed one may consider many ways to orient a frame of reference in the universe. But the question still remains, "Which is better?" This question bothered many scientists in the late 19th century when Maxwell's new theory of electromagnetism produced a number for the speed of electromagnetic wave propagation in vacuum (speed of light) but with no indication of the frame of reference. Some postulated that the speed would be measured with respect to "the one true frame." That is, that frame where the cosmic aether, a mysterious material permeating all of space through which light waves move, is at rest.
After Michelson's and Morley's famous experiment showed no indication that such a thing as a cosmic aether existed, and that the speed of light seemed to be the same in all available frames of reference, it was suggested that there is no true frame. That is, all reference frames are equally true and valid from the perspective of physics. In other words neither Jim's nor Bob's frame is closer to the natural frame than the other, because such a frame doesn't exist.
Special Relativity is built on this premise. As a result, the universe suddenly became much more bizarre than previously suspected. Clocks slowed down, twins were no longer the same age, traincars shrunk as they went by, and two people's perceptions of "right now" no longer seemed to correlate. For many people these developments were stranger facts than fiction!
This book will show you how the simple assumptions of Special Relativity imply these strange effects exist, and how to calculate the magnitude of such effects so as to prepare for them in the real world. It also attempts to explain the huge conceptual breakthrough that occurred in scientific thought a century ago.
Students should be aware that Special and General Relativity can be slightly different theories, the mathematical results of Special Relativity are contained within General Relativity but the two theories can differ in the interpretation of spacetime. No attempt is made in the following text to make the presentation of Special Relativity conform to any particular interpretation of General Relativity.
Historical Development
In the nineteenth century the idea that light was propagated in a medium called the "aether" was prevalent. James Clerk Maxwell in 1865 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. Initially these constants were interpreted as properties of the aether. They would be the same for all observers so in Maxwell's paper there was an implicit idea of a universal, stationary aether. Observers would measure the velocity of light to be the sum of their velocity and the velocity of light in the aether.
In 1887 Michelson and Morley performed an experiment that showed that the speed of light was independent of the speed of the
destination or source of the light in the proposed aether. It seemed that Maxwell's theory was correct but the theory about the way that velocities add together (known as Galilean Relativity) was wrong.
Various physicists attempted to explain the Michelson and Morley experiment. George Fitzgerald in 1889 and Hendrik Lorentz in 1895 suggested that objects tend to contract along the direction of motion relative to the aether. In 1897 Joseph Larmor and in 1899 Hendrik Lorentz proposed that moving objects are contracted and that moving clocks run slow. Fitzgerald, Larmor and Lorentz's contributions to the analysis of light propagation are of huge importance because they produced the Lorentz Transformation which is the mathematical equation required to explain how Maxwell's Equations might take precedence over the addition of velocities specified by Galilean Relativity. If the aether caused lengths to contract and clocks to run slow then, because velocity is just a ratio of length to time, velocities would no longer need to add up in a simple fashion and the speed of light could be constant for all observers.
By the late nineteenth century it was becoming clear that aether theories of light propagation were problematical. 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. The equations work whether or not there is an aether. He proposed that the laws of physics are the same for all inertial frames of reference and that Maxwell's Equations were correct so that the speed of light is a constant for all observers rather than a constant applying to an "aether". On the basis of these simple assumptions he was able to derive the Lorentz Transformation. He showed that the Lorentz Transformation itself was sufficient to explain how length contraction occurs and clocks appear to go slow. Einstein's remarkable achievement was to be the first physicist to show some understanding of the geometrical implications of the Lorentz Transformation.
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 the universal instantaneous present moment. The idea 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 the new "physicalism" 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) and by the 1920's it had become the principle theoretical approach to relativity.
It has become popular to credit Henri Poincaré with the discovery of the theory of Special Relativity, but sadly Poincaré got many of the right answers for all the wrong reasons. He even came up with a version of
! 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." 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, a conceptual leap that thwarts many students even today.
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. The aether theorists simply fail to notice that this is a possibility because they reject 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.
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.
- Einstein, A. (1905). Zur Elektrodynamik bewegter Körper, in Annalen der Physik. 17:891-921. http://www.fourmilab.ch/etexts/einstein/specrel/www/
- Granek, G (2001). Einstein's ether: why did Einstein come back to the ether? Apeiron, Vol 8, 3. http://citeseer.ist.psu.edu/cache/papers/cs/32948/http:zSzzSzredshift.vif.comzSzJournalFileszSzV08NO3PDFzSzV08N3GRF.PDF/granek01einsteins.pdf
- S. Walter. The non-Euclidean style of Minkowskian relativity. Published in J. Gray (ed.), The Symbolic Universe, Oxford University Press, 1999, 91–127. http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf
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. Problems at the end of each section challenge the reader to apply what he or she has learned. 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 second half of the book or to one of the references at the end.
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.
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 only 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.
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." When simultaneous events in one frame are viewed as not simultaneous in another it is either because:
- They truly aren't simultaneous in the second frame due to relativistic effects, or,
- They just appear that way due to delay of light,
or both. They can occur together but the two effects are not the same thing. One can always factor out the light delay by calculating when the signal was transmitted using the speed of light and the distance to the object. Relativity isn't based solely on the finite speed of light, crazy stuff is really happening.
A Word about Wiki
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