## PHYS102 Study Guide

### 8a. identify the postulates upon which the Special Theory of Relativity is based

1. What is an inertial reference frame?
2. According to the Special Theory of Relativity, how do physical laws vary when observed from different inertial reference frames?
3. 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 the 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 would look exactly the same in all inertial frames.

The second postulate of the Special Theory of Relativity is that the speed of light in a vacuum is a constant, and is approximately 3 × 108 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.

Introduction to Relativity gives the historical context for the Special Theory and introduces its two postulates.

### 8b. solve problems involving time dilation and length contraction

1. What is the Lorentz factor, and how does it depend on the velocity of a traveling object?
2. How does time flow differently in reference frames traveling at speeds near the speed of light?
3. 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 that 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 transformations of positions measured in different inertial frames are called Lorentz transformations, which 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 can be seen algebraically that this factor is always greater than 1.

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 $\gamma$ 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.

Work through this solved example, which illustrates the consistency of the time dilation and length contraction expressions in the Special Theory of Relativity.

### 8c. explain the principle of equivalence as introduced in the General Theory of Relativity

1. The Special Theory of Relativity deals only with inertial frames of reference when the reference frames move at a constant velocity relative to one another. How does the General Theory include non-inertial frames of reference by establishing the principle of equivalence?

Non-inertial frames of reference, by definition, are reference frames that move under acceleration as observed from an inertial frame of reference. The principle of equivalence states that accelerated motion is indistinguishable from motion under the influence of gravity. This means that the acceleration of an object is equivalent to the gravitational pull on it from another object of the appropriate mass. In order for an observer in a non-inertial frame to provide a picture of an event consistent with that of an observer in an inertial frame, gravity has to be included. This could be expressed by comparing the inertial mass and gravitational mass. Newton's Second Law states that $F = ma$, where $m$ is inertial mass; it indicates how much the velocity of an object changes under the influence of a force $F$. If $F$ is the force of gravity exerted by the Earth on the object, then $a$ must be $g$, gravitational acceleration, so that $F_\mathrm{gravity}=mg$. This also means that $m$ must be gravitational mass, which is the gravitational attraction between the object and the Earth. The principle of equivalence postulates that the two masses are exactly the same.

Read more about the principle of equivalence in General Relativity and Black Holes.

### 8d. compare and contrast the special and general theories of relativity

1. Which postulate is part of both the special and general theories of relativity?
2. What is the primary difference between the special and general theories?

The common basis of both the special and general theories of relativity is the relativity principle. It states that all physical laws are the same in all reference frames. For the Special Theory of Relativity, this means all inertial reference frames. The General Theory of Relativity includes both inertial and non-inertial reference frames.

The primary difference between the two theories is that the Special Theory of Relativity is limited in scope: it applies only to events observed from inertial frames of reference. The General Theory of Relativity includes both inertial and non-inertial frames. Most importantly, it also incorporates gravity. The general theory is mainly a theory of gravity, while the special theory primarily describes events in different inertial frames of reference.

To compare and contrast the two theories, read Special Relativity and The General Theory of Relativity.

### 8e. list the most significant consequences of Einstein's special and general theories of relativity

1. What are three major consequences of the Special Theory of Relativity? Consider our understanding of space, time, and energy.
2. What are two major consequences of the General Theory of Relativity? Consider the effect gravity has on light.

The consequences of special relativity include:

• Measurements of length and time are relative and depend on the observer who measures them. Time flows more slowly if measured in a reference frame moving at a speed close to the speed of light, which is known as time dilation. Lengths appear shorter when measured at a speed close to the speed of light, which is known as length contraction.
• No massive object can move at a speed greater than the speed of light in a vacuum relative to any observer. No information can propagate at a speed greater than the speed of light in a vacuum.
• All massive objects possess energy proportional to their mass at rest ("rest mass"), $E=mc^2$. This energy can convert to other forms of energy. In nuclear reactions, a particle can become a particle of a different mass, with a corresponding release or intake of energy.

See the list of the consequences of the theory in Special Relativity.

The consequences of general relativity include:

• Light can be deflected by gravity in a phenomenon known as gravitational lensing. In the General Theory of Relativity, the presence of a massive body is equivalent to a curvature in space-time. Since light travels along the shortest path between two points, it will not take a straight line to get from one point to another on a curved surface, since that would not be the shortest path. Instead, it would move along a curve.
• Time flows more slowly in a strong gravitational field. While this sounds similar to the time dilation arising as a consequence of special relativity, gravitational time dilation is a distinct effect.

The experimental confirmation of these consequences, as well as some others, is found in General Relativity and Black Holes and The General Theory of Relativity.

### 8f. explain the results of the Michelson-Morley experiment using the Special Theory of Relativity

1. What was the goal of the Michelson-Morley experiment?
2. Did the experiment produce the expected result?
3. 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 since light was a wave, it would require a medium in order to propagate, much like sound requires air. The setup of the experiment involved producing an interference pattern between two beams of light, one parallel and another perpendicular to the surface of the Earth. The interference pattern would depend on the orientation of the interferometer, the time of the day, and the time of the year; the changes in the pattern would yield measurements of the speed of the Earth relative to the ether. However, no such changes were ever detected. 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.

In 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.

Read a short summary of the Michelson-Morley experiment in Relativity.

### Unit 8 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.

Try to think of the reason why each term is included.

• Frame of reference (inertial and non-inertial)
• General Theory of Relativity
• Gravitational Lensing
• Length contraction
• Lorentz factor
• Mass, inertial and gravitational
• Principle of Equivalence
• Principle of Relativity
• Rest energy
• Rest mass
• Special Theory of Relativity
• Time Dilation