PHYS102: Introduction to Electromagnetism

8a. Identify the postulates that form the basis for the Special Theory of Relativity

• What is an inertial reference frame?
• According to the Special Theory of Relativity, how do physical laws vary when observed from different inertial reference frames?
• According to the Special Theory of Relativity, what is the difference between the speed of light and other materials, such as physical objects or mechanical waves?

The Special Theory of Relativity describes how observations of events change when conducted in different inertial reference frames. Inertial reference frames all move with constant velocities relative to one another; there is no preferred inertial frame.

The first postulate of the Special Theory of Relativity is that all physical laws are the same in all inertial frames of reference. In other words, if you try to perform an experiment to determine whether your reference frame is in motion, you would not be successful. All experiments will look exactly the same in all inertial frames.

The second postulate of the Special Theory of Relativity states that the speed of light in a vacuum is a constant, and is approximately $3\times 10^{8}$ m/s. This means that measurements of the speed of light performed by observers in inertial reference frames traveling at different velocities will all yield the same result. The speed of light in a vacuum is also independent of the source.

Review the origins of Einstein's ideas in Einstein's Postulates.

8b. Solve problems involving time dilation and length contraction

• What is the Lorentz factor, and how does it depend on the velocity of a traveling object?
• How does time flow differently in reference frames traveling at speeds near the speed of light?
• How are the measurements of length different in reference frames traveling at speeds near the speed of light?

One consequence of the Special Theory of Relativity (which follows directly from its two postulates) is that measurements of time and length are not the same in different inertial reference frames that move at different relative velocities. When you measure the length of a stick when it is on the ground and again when it is on a moving train, the result will be the same. Also, when you measure the duration of an event occurring on the ground and again on a moving train, you expect the results to be the same. However, according to the Special Theory of Relativity, this is not the case. The difference between the two results is not significant in real life, since a train moves very slowly. However, when that train moves at a speed close to the speed of light, the difference becomes significant.

The changes in lengths and time intervals measured in different inertial frames involve a factor $\gamma$ that depends on the speed $v$ of the relative motion of the frames.

This is called the Lorentz factor: $\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$. It turns out that no object can move at speeds larger than the speed of light, $c$. This means that $\gamma$ is larger than 1 (or equal to 1, for an object at rest).

The time interval t measured by an observer in a frame moving at speed $v$ will be measured as $t'=\gamma t=\frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$. Since $\gamma$ is greater than 1, $t'$ is greater than $t$. Time slows down when measured in a moving reference frame; this phenomenon is known as time dilation.

The length L measured by an observer in a frame moving at speed v will be measured as

$L'=\frac{L}{\gamma}=L\sqrt{1-\frac{v^{2}}{c^{2}}}$. Since is greater than 1, $L'$ is less than $L$. Lengths decrease when measured in a moving reference frame; this phenomenon is known as length contraction.

Review time dilation and length contraction in Simultaneity and Time Dilation and Length Contraction.

8c. State the law of velocity addition in special relativity

• What is the maximum speed any object can have?
• How does the law of velocity addition get modified when objects move at high speeds? Does the end result get larger or smaller than expected from the classical laws of Galileo and Newton?
• Can the relativistic law of velocity addition be applied to objects that are moving in opposite directions?

Imagine you are running on a moving train, from the back to the front of the compartment. To someone on the ground, it looks as if you are running faster than the train itself because the velocity of the train adds to the velocity with which you are running. But Einstein realized that this cannot actually be completely accurate, because in the hypothetical case that all the velocities are very large they could add up to more than the speed of light. The speed of light, $c$, is 300,000 km/s, which is the "universal speed limit" for all physical objects. The law of velocity addition as just described must break down when speeds approach $c$.

To write down the formula that fixes this issue, take the example of a person running on a moving train. Call the velocity of the train $v$, and the velocity of the running person relative to the train $u'$. Then the velocity of the person relative to the ground is

$u=(v+u')/(1+vu'/c^{2})$

The term "velocity addition" does not quite describe what you have to do here, because after adding the velocities, you have to divide the total by the extra factor ($1+vu'/c^{2}$). The effect is that the result ends up being less than the sum $v+u'$, just as Einstein knew it should, so that the combined velocity $u$ can never exceed the speed of light.

Velocities are vectors, so they have a direction, too. The above formula works for any two velocities that point either in the same or in opposite directions. If the velocities point in opposite directions, you have to call one direction positive and the other direction negative before combining them. But the formula does not work if the velocities are not aligned with each other, as in a boat crossing a river from north to south while the river flows from east to west.

Review Einstein's Law for Velocity Addition with examples in Relativistic Addition of Velocities.

8d. Explain the results of the Michelson-Morley experiment using the Special Theory of Relativity

• What was the goal of the Michelson-Morley experiment?
• Did the experiment produce the expected result?
• What postulate of special relativity explains the results of the Michelson-Morley experiment?

The goal of the Michelson-Morley experiment was to measure the speed of Earth relative to the ether, the hypothetical medium where light propagated. At the time, scientists assumed that light would require a medium to propagate because it is a wave, much like sound requires air.

The setup for their experiment involved producing an interference pattern between two beams of light, traveling back and forth at right angles to each other. The interference pattern would depend on the orientation of the interferometer, the time of the day, and the time of year; the changes in the pattern would yield measurements of the speed of the Earth relative to the ether. However, they never detected these types of changes. None of the proposed explanations were able to reconcile this result (or lack of result) with what was known about electricity, magnetism, and wave propagation.

Within the framework of the Special Theory of Relativity, the results of the Michelson-Morley experiment make sense. One of the postulates of the theory is that the speed of light is constant in all reference frames moving at any speed. Since light always propagates at the same speed, there is no need for a medium of propagation relative to which the speed of light should be measured. Thus, there is no need for ether. The results of the Michelson-Morley experiment indicate that ether does not exist. They also confirm that the speed of light is the same in all inertial frames of reference.

Review this brief description in Einstein's Postulates.

8e. Explain how the special theory of relativity relates to mass and energy

• What happens to the energy of an object if its speed approaches the speed of light?
• How is the energy of an object related to its mass?

An object's kinetic energy is the most directly observable form of energy because it is related to the speed at which the object is moving. But Einstein knew that no object can go faster than the speed of light, so it must become impossible to increase the energy $E$ of a fast-moving object when it is already going at this maximum speed. The formula that contains this insight is relativistic energy:

$E=\gamma \ mc^{2}$.

$m$ is the rest mass of the object (it is the same mass that appears in Newton's Second Law, so the qualifier "rest" is not really needed). $\gamma$ is the Lorentz factor, which contains the speed: $\gamma =\frac{1}{\sqrt{1-v^{2}/c^{2}}}$.

When the speed approaches the speed of light $c$, the Lorentz factor and the energy $E$ blow up to infinity. You cannot get larger than infinity, and this explains why the object cannot get any faster when it reaches this limit.

Review Relativistic Energy to practice applications of these relations.

8f. Calculate the rest energy from the mass of an object

• What happens to the energy of an object if its speed is zero?
• If a star emits energy in the form of light, how does this affect the mass of the star?

For slow-moving objects, the formula $E=\gamma \ mc^{2}$ agrees with the kinetic-energy formula known from Newton's classical physics, except for a constant term that can be considered just part of the potential energy of the object. This constant term is obtained by setting $\gamma =1$, corresponding to zero speed, in the formula for the energy:

$E=mc^{2}$.

This is also called the rest energy of the object. Therefore, Einstein's special relativity predicts that the rest energy and mass of an object are proportional to each other.

Around the same time that Einstein discovered the energy-mass relation above, other physicists also discovered the phenomenon of radioactive decay, where the nucleus of an atom can change its mass while emitting new forms of radiation. The energy $\Delta E$ contained in that radiation is related to the change in mass $\Delta m$ of the nucleus, and Einstein's formula predicts:

$\Delta E=(\Delta m)c^{2}$

See example 28.7 in Relativistic Energy.

Unit 8 Vocabulary

You should be familiar with these terms to complete the final exam.

• inertial frame of reference
• length contraction
• Lorentz factor
• Michelson-Morley experiment
• principle of relativity
• relativistic energy
• rest energy
• rest mass
• Special Theory of Relativity
• speed of light
• time dilation
• velocity addition law