HPS 0410 Einstein for Everyone

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Special Theory of Relativity: The Basics

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

Background reading: J. Schwartz and M. McGuinness, Einstein for Beginners. New York: Pantheon.. pp. 66 - 151.

"On the Electrodynamics of Moving Bodies"

In June 1905, when Albert Einstein was still a patent examiner in Bern, Switzerland, he sent a paper with this title to the journal Annalen der Physik. It contained his special theory of relativity. He argued that altering our understanding of the behavior of space and time could resolve certain problems in electrodynamics. (See page one in German or English.)

To understand what these alterations were, we need some preliminary notions.

## Inertial and Accelerated Motion

 There is a preferred motion in space known as inertial motion. Any body left to itself in space will default to an inertial motion, which is just motion at uniform speed in a straight line. The easiest example to visualize is a huge spaceship with the engines turned off, gliding through space. At any point in space, many inertial motions are possible. They will be pointed in different directions and will be at different speeds. Any other motion is accelerated. This includes motion at uniform speed in a circle. While the speed stays the same, the direction does not. So the motion is accelerated. Sometimes we will talk of an "inertial observer," which is just an observer moving inertially. Such an observer might set up an elaborate system of measuing rods and other physical devices to fix the positions of events; and an elaborate system of clocks to fix their timing. Such a system is an inertial frame of reference.

This file has been donated to the Wikimedia Foundation and released into the public domain by Pearson Scott Foresman.http://commons.wikimedia.org/wiki/File:Spacecraft_(PSF).png

## Absolute versus Relative Motion

 Relative motion arises when one body moves with respect to another. For example, our spaceship might move relatively to a nearby planet.

 Correspondingly the planet moves relative to the spaceship.

Prior to Einstein, it was generally thought that there was another sense of motion, absolute motion. According to this sense, there is a fact of the matter as to whether the spaceship is moving, without regard to whether it moves relative to another object, such as a planet. There is an absolute state of rest in space, according to this earlier view. Either the spaceship is in this state and at rest; or it is not and it is moving. Einstein's theory did away with this notion of absolute rest and absolute motion.

Beware of a simple confusion. It is easy connect relativity theory with the popular slogan "It's all relative." or "All motion is relative." However that is not what we learn from relativity theory. The theory eradicates absolute motion. Something can only be at rest if it is at rest with respect to something else. So the better slogan would be "All rest is relative."

That relativity does not extend to inertial motion. Whether something is moving inertially does NOT require us to ask if it is moving inertially with respect to something else. Whether it is moving inertially is simply a fact about the motion. It is discoverable by an experiment within a closed body. If the body is accelerating, that will be revealed by effects within the body. For example, an airplane flies inertially until it hits an air pocket. Then the plane lurches--accelerates--and things inside are thrown about. Everyone inside has no doubt of the acceleration.

Einstein found it most convenient to base his theory of relativity on two postulates; once they were assumed it became an exercise in logic to develop the whole theory. The two postulates are
I. The Principle of Relativity and
II. The Light Postulate.

## I. The Principle of Relativity

All inertial observers find the same laws of physics.

What this says is just this: imagine two spaceships, each moving inertially in space but with different velocities. If we conduct experiments on either ship aimed at determining a law of physics, we will end up with the same law no matter which spaceship we are on.

Or, more simply, the laws of physics simply tell us which physical process can happen and which cannot. So if all inertial observers find the same laws, that just means that any process that can happen for one inertial observer can happen for any other.

Here are some important consequences of the principle:

No experiment aimed at detecting a law of nature can reveal the inertial motion of the observer.
Absolute velocity has no place in any law of nature.
No experiment can reveal absolute motion.

Notice that the principle of relativity is limited to inertial motions. In special relativity, this relativity of motion does not extend to accelerated motion. If something accelerates, then it does so absolutely; there is no need to say that it "accelerates with respect to..." A traditional indicator of accelertion is inertial forces. If you are in an airplane that flies uniformly in a straight line, you have no sense of motion. If the airplane hits turbulence and accelerates, you sense immediately the acceleration as inertial forces throw things around in the cabin.

## II. The Light Postulate

All inertial observers find the same speed for light.

That speed is 186,000 miles per second or 300,000 kilometers per second. Because this speed crops up so often in relativity theory, it is represented by the letter "c".

That Einstein should believe the principle of relativity should not come as such a surprise. We are moving rapidly on planet earth through space. But our motion is virtually invisible to us, as the principle of relativity requires.

Why Einstein should believe the light postulate is a little harder to see. We would expect that a light signal would slow down relative to us if we chased after it. The light postulate says no. No matter how fast an inertial observer is traveling in pursuit of the light signal, that observer will always see the light signal traveling at the same speed, c.

The principal reason for his acceptance of the light postulate was his lengthy study of electrodynamics, the theory of electric and magnetic fields. The theory was the most advanced physics of the time. Some 50 years before, Maxwell had shown that light was merely a ripple propagating in an electromagnetic field. Maxwell's theory predicted that the speed of the ripple was a quite definite number: c.

The speed of a light signal was quite unlike the speed of a pebble, say. The pebble could move at any speed, depending on how hard it was thrown. It was different with light in Maxwell's theory. No matter how the light signal was made and projected, its speed always came out the same.

The principle of relativity assured Einstein that the laws of nature were the same for all inertial observers. That light always propagated at the same speed was a law within Maxwell's theory. If the principle of relativity was applied to it, the light postulate resulted immediately.

## A Light Clock

One cannot have both of Einstein's postulates and leave everything else unchanged. We can only retain both without contradiction if we make systematic changes throughout our physics. Let us begin investigating these changes, which include our basic, classical presumptions about space and time. One of them is that we learn that a moving clock runs slower.

 To see how this comes about, we could undertake a detailed analysis of a real clock, like a wristwatch or a pendulum clock. That would be difficult and complicated--and unnecessarily so. All we need is to demonstrate the effect for just one clock and that will be enough, as we shall see shortly, to give it to us for all clocks. So let us pick the simplest design of clock imaginable, one specifically chosen to make our analysis easy. A light clock is an idealized clock that consists of a rod of length 186,000 miles with a mirror at each end. A light signal is reflected back and forth between the mirrors. Each arrival of the light signal at a mirror is a "tick" of the clock. Since light moves at 186,000 miles per second, it ticks once per second.
 Here are some light clocks ticking:

## Light Clocks are Slowed by Motion

What happens when light clock is set into rapid motion, close to the speed of light. It is easy to see without doing any sums that the light clock will be slowed down. That is, it will be slowed down in the judgment of someone who does not move with the light clock.

We will take the simple case first of a light clock whose motion is perpendicular to the rod. The light clock will function as before. But now there is an added complication. The light signal leaves one end of the rod and moves toward the other end. But since the rod is moving rapidly, the light signal must now chase after the other end as it flees. As a result, the light signal requires more time to reach the other end of the rod. That means that the moving light clock ticks more slowly than one at rest.

Remember the light postulate. It tells us that the light always goes the same speed. That the rod along which it bounces is moving rapidly will not alter the speed of the light.

Here's an animation that shows a light clock at rest and a second light clock that moves perpendicular to its rod. The light signal in the moving clock chases after the rod. To reach the other end, it covers more distance and, as a result, required more time.

Here's a smaller version in case the big one is too much for your screen.

 If you watch the animation carefully, you will see that the moving light clock ticks at exactly half the speed of the resting clock. That is because the light signal of the moving clock has to cover twice the distance to go from one end of the rod to the other. To get this doubling of the distance takes a careful adjustment of the speed of the moving clock. It turns out that the moving clock has to be traveling at 86.6% the speed of light.

Just how much slowing do we get for some particular speed? That question turns out to be easy to answer with a little geometry. The trick is to figure out how much distance the light signal has to travel to reach the other end of the rod. Once we know that distance, we know the time taken, since light always travels at 186,000 miles per second.

To make things interesting, let's take a very high speed: 99.5% the speed of light. (We'll write this compactly as "0.995c.") An observer traveling with the clock will still see the light signal bounce backwards and forwards between the mirrors as before. This process looks quite different from the perspective of an observer who stays behind and does not move with the clock. The path traveled by the light will now look like this:

That observer at rest will agree with one that moves with the rod: a light signal leaves one end of the rod and arrives at the other end. But the observer at rest judges that end to be rushing away from the light signal at 99.5% the speed of light. A quick calculation shows that that the signal will now take 10 seconds to reach the other end of the rod.

To see this, note that in ten seconds the rod will move 1,850,700 miles, as shown in the figure above. So to get to the end of the rod, the light signal must traverse the diagonal path shown. A little geometry tells us that a right angle triangle with sides 186,000 miles and 1,850,700 miles will have a diagonal of 1,860,000 miles.

 Pythagoras' theorem tells us the diagonal is 1,860,000 miles since 1,860,000 miles2 = 1,850,700 miles2 + 186,000 miles2

Since light moves at 186,000 miles per second, it will need ten seconds to traverse the diagonal.

Setting the arithmetic aside, the result is simple. Since the light signal must travel so much farther to traverse the rod of a moving clock, it takes much longer to do it. So a moving light clock ticks slower. In this case, for a clock moving at 99.5% the speed of light, it ticks once each ten seconds instead of once each second.

## All Moving Clocks Are Slowed by Motion

 A simple application of the principle of relativity shows that all clocks must be slowed by motion, not just light clocks. We set a clock of any construction next to a light clock at rest in an inertial laboratory. We notice that they both tick at the same rate. That must remain true when we set the laboratory into a different state of inertial motion. But since the light clock has slowed with the motion, the other clock must also slow if it is to keep ticking at the same rate as the light clock.

You might be tempted to say that the other clock would not keep pace with the light clock. But then you would have devised a device that detects absolute motion, in contradiction with the principle relativity. That device would pick out absolute rest as the only state in which the two clocks run at the same rate.

## Moving Rods Shrink in the Direction of Their Motion

So far, we have considered a light clock whose rod is perpendicular to the direction of its motion. If we now consider a light clock whose rod is oriented parallel to the direction of motion, we will end up concluding that its rod must shrink in the direction of its motion. To get this result, we proceed by reasoning just as we have before. Once again we have a light signal on a moving clock, chasing after an end of a rod that flees rapidly. We now add in the extra complication that the rod is parallel to the direction of motion of the clock. That extra complication will force us to conclude that the rod has shrunk.

Getting to this result uses no new ideas or methods. It is just messier, so if you are not too bothered by details piling up, work through what follows. Or, if you are not so brave, you can skip to the end and just read the final result.

To get to the result, we need two steps:

First Step: Light clocks oriented perpendicular to one another run at the same speed.

 Take the light clock considered above. Image a second, identical light clock with its rod oriented parallel to the direction of the motion. Once again the principle of relativity requires that both clocks run at the same speed. We could just leave it at that--an application of the earlier result. However it is reassuring to go through it from scratch. To begin, we don't need the principle of relativity to see that the clocks at rest run at the same rate. They will run at the same rate simply because they are the same clocks oriented in different directions. That just follows from the isotropy of space. All its directions are equivalent. So the orientation of the clock cannot affect its speed.

Now imagine that we take the entire system of the two clocks and set it into rapid motion at, say, 99.5% the speed of light, in the direction of one of the light clocks.

 An observer moving with the two light clocks must find them to continue to run at the same rate. We now do need the principle of relativity to establish this. Our earlier isotropy argument doesn't work anymore, since the two directions of the clocks are intrinsically different. One is perpendicular to the direction of motion; the other is parallel to it. The principle of relativity requires that they run at the same rate. For, if they ran at different rates, the device would be an experiment that could detect absolute motion. We could detect absolute motion just by taking two light clocks perpendicular to each other and checking if they run at the same rate. Only when we are rest would they run at the same rate. If they do not run at the same rate we would know we are moving absolutely. The principle of relativity prohibits an experiment that can do this. So the two clocks must run at the same rate.

Second Step: The rod oriented in the direction of motion must shrink.

We know from the earlier analysis that a light clock (indeed any clock) moving at 99.5% the speed of light is slowed so that it ticks only once in ten seconds. So now we know that the light clock oriented parallel to the direction of motion must tick once each ten seconds. But that cannot happen if everything is just as we describe it. Imagine the outward bound journey of the light signal.

 How do I get this? If you have to know, here are the details. The light signal chases at 100% c after the leading end of the rod. That end is initially 186,000 miles away and moving at 99.5% c. So the light signal approaches the end of the rod at 0.5% c, which is 930 miles per second. The distance to cover is 186,000 miles, so it takes 186,000/930 = 200 seconds. The light signal has to go from one end to the other of a 186,000 mile rod. The light moves at 186,000 miles per second. But the rod is also moving in the same direction at 99.5% the speed of light. So the light has to chase after a rapidly fleeing end and will need much more than a second to catch it. With a little arithmetic it turns out that the light will need 200 seconds to make the trip.
 But the light clock has to tick once every ten seconds! Something has gone badly wrong. What has gone wrong is our assumption that the rod parallel to the direction of motion retains its length. That is incorrect. That rod actually shrinks to 10% of original length, so the moving pair of clocks really looks more like:

Now the light signal has time to get from one end of the rod to the other and keep the clock ticking at once each ten seconds as expected. The signal just has far less distance to travel so now it can maintain the rate of ticking expected.

There are more details in this last calculation that I don't want to bother you with. But since some of you will ask, here they are--but only for those who really want them.

Overall it will turn out that the light signal now needs 20 seconds to complete the journey from the trailing end of the rod to the front and then back. That is what we expect. The round trip journal is "two ticks" and should take 2x10=20 seconds. The catch is that virtually all of the 20 seconds will be spent in the forward trip and virtually none of it in the rearward trip. This effect actually figures in the relativity of simultaneity which we will discuss at some length later.

If you want to see this for yourself you should redo the calculations. If you do, you'll need to undo my rounding off. The rod is not contracted exactly 10%--I rounded things off to keep life simple. It is 9.987%. The ticks are not exactly 10 seconds apart, but 10.0125 seconds. The forward trip will take 19.9750 seconds. The rearward trip will take 0.05 seconds. That gives a total round trip of 20.025 seconds = 2x10.0125 as expected.

The analysis is now complete. We have learned that a clock moving at 99.5% the speed of light, slows by a factor of ten. It ticks once each ten seconds instead of once each second. A rod, oriented in the direction of motion, shrinks to 10% of its length. Rods perpendicular to the direction of motion are unaffected.

The two effects are not noticeable as long as our speeds are far from that of light. They become marked when we get close to the speed of light. The closer we get the the speed of light, the closer clocks come to stopping completely and rods come to shrkinking to no length in the direction of motion. For more details of how the effects depend on speed, see What Happens at High Speeds.

## What you need to know:

• Inertial and accelerated motion.
• Absolute versus relative motion.
• Einstein's two postulates and how to apply them.
• What a light clock is and how it is affected by motion.
• Moving rods are shrunk in the direction of their motion.

Copyright John D. Norton. January 2001, August 30, 2002, July 20, 2006; January 8 2007, January 3, August 21, 27, 2008, January 6 10, 2011.