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Introduction

Introduction

Albert Einstein's education in late-nineteenth-century Germany was neither modern nor liberal. He did well in the early grades,1 but in high school and college he began to get in trouble for what today's edspeak calls “critical thinking.” Indeed, there was much that deserved criticism in the state of physics at that time. There was a subtle contradiction between the theory of light as a wave and Galileo's principle that all motion is relative. As a teenager, Einstein began thinking about this on an intuitive basis, trying to imagine what a light beam would look like if you could ride along beside it on a motorcycle at the speed of light. Today we remember him most of all for his radical and far-reaching solution to this contradiction, his theory of relativity, but in his student years his insights were greeted with derision from his professors. One called him a “lazy dog.” Einstein's distaste for authority was typified by his decision as a teenager to renounce his German citizenship and become a stateless person, based purely on his opposition to the militarism and repressiveness of German society. He spent his most productive scientific years in Switzerland and Berlin, first as a patent clerk but later as a university professor. He was an outspoken pacifist and a stubborn opponent of World War I, shielded from retribution by his eventual acquisition of Swiss citizenship.

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Figure a: Albert Einstein.

As the epochal nature of his work became evident, some liberal Germans began to point to him as a model of the “new German,” but after the Nazi coup d'etat, staged public meetings began, at which Nazi scientists criticized the work of this ethnically Jewish (but spiritually nonconformist) giant of science. When Hitler was appointed chancellor, Einstein was on a stint as a visiting professor at Caltech, and he never returned to the Nazi state. World War II convinced Einstein to soften his strict pacifist stance, and he signed a secret letter to President Roosevelt urging research into the building of a nuclear bomb, a device that could not have been imagined without his theory of relativity. He later wrote, however, that when Hiroshima and Nagasaki were bombed, it made him wish he could burn off his own fingers for having signed the letter.

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Figure b:  The first nuclear explosion on our planet, Alamogordo, New Mexico, July 16, 1945.

Einstein has become a kind of scientific Santa Claus figure in popular culture, which is presumably why the public is always so titillated by his well-documented career as a skirt-chaser and unfaithful husband. Many are also surprised by his lifelong commitment to socialism. A favorite target of J. Edgar Hoover's paranoia, Einstein had his phone tapped, his garbage searched, and his mail illegally opened. A censored version of his 1800-page FBI file was obtained in 1983 under the Freedom of Information Act, and a more complete version was disclosed recently.2 It includes comments solicited from anti-Semitic and pro-Nazi informants, as well as statements, from sources who turned out to be mental patients, that Einstein had invented a death ray and a robot that could control the human mind. Even today, an FBI web page3 accuses him of working for or belonging to 34 “communist-front” organizations, apparently including the American Crusade Against Lynching. At the height of the McCarthy witch hunt, Einstein bravely denounced McCarthy, and publicly urged its targets to refuse to testify before the House Unamerican Activities Committee. Belying his other-worldly and absent-minded image, his political positions seem in retrospect not to have been at all clouded by naivete or the more fuzzy-minded variety of idealism. He worked against racism in the U.S. long before the civil rights movement got under way. In an era when many leftists were only too eager to apologize for Stalinism, he opposed it consistently.

This chapter is specifically about Einstein's theory of relativity, but Einstein also began a second, parallel revolution in physics known as the quantum theory, which stated, among other things, that certain processes in nature are inescapably random. Ironically, Einstein was an outspoken doubter of the new quantum ideas that were built on his foundations, being convinced that “the Old One [God] does not play dice with the universe,” but quantum and relativistic concepts are now thoroughly intertwined in physics.

The Principle of Relativity

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Figure c: Albert Michelson, in 1887, the year of the Michelson-Morley experiment., George FitzGerald, 1851-1901,and Hendrik Lorentz, 1853-1928.

By the time Einstein was born, it had already been two centuries since physicists had accepted Galileo's principle of inertia. One way of stating this principle is that experiments with material objects don't come out any different due the straight-line, constant-speed motion of the apparatus. For instance, if you toss a ball up in the air while riding in a jet plane, nothing unusual happens; the ball just falls back into your hand. Motion is relative. From your point of view, the jet is standing still while the farms and cities pass by underneath.

The teenage Einstein was suspicious because his professors said light waves obeyed an entirely different set of rules than material objects, and in particular that light did not obey the principle of inertia. They believed that light waves were a vibration of a mysterious substance called the ether, and that the speed of light should be interpreted as a speed relative to this ether. Thus although the cornerstone of the study of matter had for two centuries been the idea that motion is relative, the science of light seemed to contain a concept that a certain frame of reference was in an absolute state of rest with respect to the ether, and was therefore to be preferred over moving frames.

Experiments, however, failed to detect this mysterious ether. Apparently it surrounded everything, and even penetrated inside physical objects; if light was a wave vibrating through the ether, then apparently there was ether inside window glass or the human eye. It was also surprisingly difficult to get a grip on this ether. Light can also travel through a vacuum (as when sunlight comes to the earth through outer space), so ether, it seemed, was immune to vacuum pumps.

Einstein decided that none of this made sense. If the ether was impossible to detect or manipulate, one might as well say it didn't exist at all. If the ether doesn't exist, then what does it mean when our experiments show that light has a certain speed, 3×108 meters per second? What is this speed relative to? Could we, at least in theory, get on the motorcycle of Einstein's teenage daydreams, and travel alongside a beam of light? In this frame of reference, the beam's speed would be zero, but all experiments seemed to show that the speed of light always came out the same, 3×108 m/s. Einstein decided that the speed of light was dictated by the laws of physics, so it must be the same in all frames of reference. This put both light and matter on the same footing: both obeyed laws of physics that were the same in all frames of reference.

The principle of relativity

The results of experiments don't change different due to the straight-line, constant-speed motion of the apparatus. This includes both light and matter. This is almost the same as Galileo's principle of inertia, except that we explicitly state that it applies to light as well.

This is hard to swallow. If a dog is running away from me at 5 m/s relative to the sidewalk, and I run after it at 3 m/s, the dog's velocity in my frame of reference is 2 m/s. According to everything we have learned about motion, the dog must have different speeds in the two frames: 5 m/s in the sidewalk's frame and 2 m/s in mine. How, then, can a beam of light have the same speed as seen by someone who is chasing the beam?

In fact the strange constancy of the speed of light had already shown up in the now-famous Michelson-Morley experiment of 1887. Michelson and Morley set up a clever apparatus to measure any difference in the speed of light beams traveling east-west and north-south. The motion of the earth around the sun at 110,000 km/hour (about 0.01% of the speed of light) is to our west during the day. Michelson and Morley believed in the ether hypothesis, so they expected that the speed of light would be a fixed value relative to the ether. As the earth moved through the ether, they thought they would observe an effect on the velocity of light along an east-west line. For instance, if they released a beam of light in a westward direction during the day, they expected that it would move away from them at less than the normal speed because the earth was chasing it through the ether. They were surprised when they found that the expected 0.01% change in the speed of light did not occur.

Although the Michelson-Morley experiment was nearly two decades in the past by the time Einstein published his first paper on relativity in 1905, he probably did not even know of the experiment until after submitting the paper.4 At this time he was still working at the Swiss patent office, and was isolated from the mainstream of physics.

How did Einstein explain this strange refusal of light waves to obey the usual rules of addition and subtraction of velocities due to relative motion? He had the originality and bravery to suggest a radical solution. He decided that space and time must be stretched and compressed as seen by observers in different frames of reference. Since velocity equals distance divided by time, an appropriate distortion of time and space could cause the speed of light to come out the same in a moving frame. This conclusion could have been reached by the physicists of two generations before, but the attitudes about absolute space and time stated by Newton were so strongly ingrained that such a radical approach didn't occur to anyone before Einstein. In fact, George FitzGerald had suggested that the negative result of the Michelson-Morley experiment could be explained if the earth, and every physical object on its surface, was contracted slightly by the strain of the earth's motion through the ether, and Hendrik Lorentz had worked out the relevant mathematics, but they had not had the crucial insight that this it was space and time themselves that were being distorted, rather than physical objects.5

Homework Problems

1. Astronauts in three different spaceships are communicating with each other. Those aboard ships A and B agree on the rate at which time is passing, but they disagree with the ones on ship C.
(a) Alice is aboard ship A. How does she describe the motion of her own ship, in its frame of reference?
(b) Describe the motion of the other two ships according to Alice.
(c) Give the description according to Betty, whose frame of reference is ship B.
(d) Do the same for Cathy, aboard ship C.

2. (a) Figure g on page 78 is based on a light clock moving at a certain speed, v. By measuring with a ruler on the figure, determine v/c.
(b) By similar measurements, find the time contraction factor γ, which equals T/t.
(c) Locate your numbers from parts a and b as a point on the graph in figure h on page 79, and check that it actually lies on the curve. Make a sketch showing where the point is on the curve.\

3. This problem is a continuation of problem 2. Now imagine that the spaceship speeds up to twice the velocity. Draw a new triangle, using a ruler to make the lengths of the sides accurate. Repeat parts b and c for this new diagram.

4. What happens in the equation for γ when you put in a negative number for v? Explain what this means physically, and why it makes sense.

5. (a) By measuring with a ruler on the graph in figure m on page 83, estimate the γ values of the two supernovae.(answer check available at lightandmatter.com)
(b) Figure m gives the values of v/c. From these, compute γ values and compare with the results from part a.(answer check available at lightandmatter.com)

6. The Voyager 1 space probe, launched in 1977, is moving faster relative to the earth than any other human-made object, at 17,000 meters per second.
(a) Calculate the probe's γ.
(b) Over the course of one year on earth, slightly less than one year passes on the probe. How much less? (There are 31 million seconds in a year.)(answer check available at lightandmatter.com)

7. (a) Find a relativistic equation for the velocity of an object in terms of its mass and momentum (eliminating γ). For momentum, use the symbol p, which is standard notation.
(b) Show that your result is approximately the same as the classical value, p/m, at low velocities.
(c) Show that very large momenta result in speeds close to the speed of light.

8. (a) Show that for v=(3/5)c, γ comes out to be a simple fraction.
(b) Find another value of v for which γ is a simple fraction.

9. In Slowlightland, the speed of light is 20 mi/hr = 32 km/hr = 9 m/s. Think of an example of how relativistic effects would work in sports. Things can get very complex very quickly, so try to think of a simple example that focuses on just one of the following effects:

  • relativistic momentum
  • relativistic addition of velocities
  • time dilation and length contraction
  • equivalence of mass and energy
  • time it takes for light to get to an athlete

References

  1. The myth that he failed his elementary-school classes comes from a misunderstanding based on a reversal of the German numerical grading scale.
  2. Fred Jerome, The Einstein File, St. Martin's Press, 2002
  3. http://foia.fbi.gov/foiaindex/einstein.htm
  4. Actually there is some controversy on this historical point. The experiment in any case remained controversial until 40 years after it was first performed. Michelson and Morley themselves were uncertain about whether the result was to be trusted, or whether systematic and random errors were masking a real effect from the ether. There were a variety of competing theories, each of which could claim some support from the shaky data. For example, some physicists believed that the ether could be dragged along by matter moving through it, which inspired variations on the experiment that were conducted in tents with thin canvas walls, or with part of the apparatus surrounded by massive lead walls.
  5. See discussion question F.
  6. Readers frequently wonder why the effects of the decelerations don't cancel out the effects of the accelerations. There are a couple of subtle issues here. First, there's no clearcut way to decide whether the time distortion happens during the accelerations and decelerations, or during the long periods of constant-speed cruising in between. This is because simultaneity isn't well defined, so there's no well-defined answer if Earth-bound Emma asks, “Is my sister's time distorted right now ?” During the long period when spacefaring Sarah is cruising away from Earth at constant speed, Emma may observe that her sister's voice on the radio sounds abnormally slow, and conclude that the time distortion is in progress. Sarah, however, says that she herself is normal, and that Emma is the one who sounds slow. Each twin explains the other's perceptions as being due to the increasing separation between them, which causes the radio signals to be delayed more and more. The other thing to understand is that, even if we do decide to attribute the time distortion to the periods of acceleration and deceleration, we should expect the time-distorting effects of accelerations and decelerations to reinforce, not cancel. This is because there is no clear distinction between acceleration and deceleration that can be agreed upon by observers in different inertial frames. This is a fact about plain old Galilean relativity, not Einstein's relativity. Suppose a car is initially driving westward at 100 km/hr relative to the asphalt, then slams on the brakes and stops completely. In the asphalt's frame of reference, this is a deceleration. But from the point of view of an observer who is watching the earth rotate to the east, the asphalt may be moving eastward at a speed of 1000 km/hr. This observer sees the brakes cause an acceleration, from 900 km/hr to 1000 km/hr: the asphalt has pulled the car forward, forcing car to match its velocity.
  7. A double-mass object moving at half the speed does not have the same kinetic energy. Kinetic energy depends on the square of the velocity, so cutting the velocity in half reduces the energy by a factor of 1/4, which, multiplied by the doubled mass, makes 1/2 the original energy.

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