Unit 8: Special Relativity
The physical descriptions we have studied to this point were based on a notion of absolute space and time. A model for this point of view was that space is filled everywhere by a continuous medium called the ether. Light and other forms of electromagnetic radiation were waves in this ether, analogous to sound waves in air. All other phenomena were to be understood as various manifestations of Maxwell's electromagnetism, which was originally based on a mechanical model of ether. It seemed reasonable that the 19th Century "theory of everything" could be tied down by measuring the "elastic" properties of the ether.
Toward the end of the 1800s, however, this model became associated with more and more hastily patched cracks. The detailed history of the gradual realization that ether models were not quite right is complex and technical. However, there is one rather clear indication of trouble. In 1887, Albert Michelson and Edmund Morley of the Case Institute (now Case Western University) performed an experiment using an optical interferometer in which they compared the speed of light in two beams traveling at right angles to each other. If the speed of light relative to the ether was always the same, the measured speed of light would be larger or smaller depending on the direction the experiment was traveling through the ether. The motion of the Michelson-Morley experiment was provided by the rotation of the Earth on its axis and the orbital motion of the Earth around the Sun, as well as the absolute velocity (if any) of the Sun relative to the ether.
They expected to see both diurnal changes and yearly changes in the relative velocities of light in the two paths. True, the changes expected by classical ether theory were small (on the order of 0.01% of the velocity of light), but the Michelson-Morley interferometer was able to detect velocity changes about 6-7 times smaller. To the surprise of all, there were no changes whatsoever observed. This experiment was widely repeated, using constantly improving equipment – a new version of the experiment carried out in 2002 established that the velocity of light is constant to better than 1 part in 1,015 – one of the most precise physical measurements ever accomplished.
The explanation of the Michelson-Morley null result was length contraction, as developed by Hendrik Lorentz and George Francis FitzGerald. Length contraction explained the Michelson-Morley result, the idea being that matter is held together by electromagnetic forces (true), and so the actual size of objects will change with motion through the ether (false). In the end, it was Albert Einstein's formulation of the theory of Special Relativity that gave us a consistent explanation of all such phenomena. His primary postulate was to accept that the speed of light and the laws of physics are constant in all reference frames – including reference frames that are in motion. Oddly, despite the fact that Einstein's theory completely explained the Michelson-Morley result, he took no motivation for his theory from that experiment.
Completing this unit should take you approximately 10 hours.
Upon successful completion of this unit, you will be able to:
- identify the postulates that the Special Theory of Relativity is based on;
- solve problems involving time dilation and length contraction;
- state the law of velocity addition in special relativity;
- explain the results of the Michelson-Morley experiment using the Special Theory of Relativity;
- explain how the special theory of relativity relates to mass and energy; and
- calculate rest energy from the mass of an object.
8.1: Introduction to Relativity
Einstein stated that his motivation for developing the special theory of relativity was Maxwell's theory of electromagnetism – the same theory we learned about as the foundation of electromagnetic waves. Einstein's theory made it possible for physicists to accept the reality of electric and magnetic fields in their own right, whereas the ether theory interpreted those fields as "deformations" of the ether as a medium.
To follow Einstein's thinking, recall that Maxwell had predicted the speed of light to have a universal value called c, and that was done without reference to any material objects – the calculation worked entirely in empty space. Einstein asked what this apparent universality of c meant for material objects that are in motion while they send and receive light pulses. Essentially, Einstein elevated the constancy of c (in vacuum) to the level of a law of nature.
It turns out that the speed of light can only be constant for all observers if we also admit that c plays a special role in the mechanics of material objects. This means that c was discovered to be a universal speed limit not just for light but for all objects that show any kind of motion relative to some observer!
Watch this video, which provides an overview over the often counter-intuitive conclusions that must be drawn from this fact.
8.2: Simultaneity, Time Dilation, and Length Contraction
The revolutionary aspect of Einstein's theory is that it translated a fact from electrodynamics to the wider world of physics by re-formulating the foundations of Newton's classical mechanics itself. The biggest contradiction to Newton's worldview concerns the nature of time and space themselves.
The nature of space is changed in Einstein's theory because the length of objects now becomes dependent on their velocity relative to the observer.
Read this text to learn more about this phenomenon.
Because Einstein identified the speed of light as a universal speed limit, it can now no longer be strictly true that the ground speed of a person running forward on a fast-moving airplane will just be the sum of the plane's ground speed and the person's speed relative to the plane. This velocity-addition formula, which goes back all the way to Galileo, would in principle allow ground speeds larger than c to be achieved by adding speeds that are individually smaller than c.
As this text discusses, Einstein's Law for Velocity addition is more complicated, but it becomes approximately the same as Galileo's old law when all the speeds involved are low compared to the speed of light.
8.3: Relativistic Momentum and Energy
From the point of view of a 19th century physicist, perhaps the most consequential achievement of Newton's classical mechanics was the idea that conservation laws exist. In particular, the quantities momentum and energy would never even have received their own name if it were not for the fact that they obey their own conservation laws: whenever objects interact with each other, they can exchange momentum and energy – but they can never create or destroy either of those quantities.
Conservation laws help make the world predictable even when we do not know all the details of all the objects we are tracking. This is why it is important to know how the concept of momentum needs to be modified so that the law of momentum conservation survives Einstein's rewriting of mechanics. Read this text for additional explanation.
Without the concept of energy, the history of physics would look very different. Entire branches of physics are based on the study of how different forms of energy are transformed into each other. A theory that violates the conservation of energy would not be acceptable to most physicists, and therefore we now discuss how energy is defined in the special theory of relativity.
In this section, we find what is perhaps the most famous formula in all of physics:
.Read this text, which makes the important distinction between rest energy, rest mass, and the corresponding quantities measured by an observer in relative motion. The subscript on
indicates that it refers to the rest energy of an object, and likewise m stands for the rest mass.The reason this equation became famous is that it explains how atomic bombs create huge amounts of energy while reducing the mass contained in them by a tiny amount in a nuclear reaction. The same formula also explains why the Sun is continuously losing mass as it produces energy in nuclear reactions at its core.
In conclusion, the conservation laws of energy and momentum are still valid in Einstein's Special Theory of Relativity. Historically, this is not so surprising because kinetic energy was in fact the first thing on Einstein's mind when he began thinking about the implications of Maxwell's electromagnetism for the motion of material objects.
The title of Einstein's 1905 paper that started the revolution in our understanding of time and space is "On the Electrodynamics of Moving Bodies". There is no better way to end a course on electromagnetism than to come to the realization that we have not really reached the end of an exploration, but rather the beginning of a new era in science.