Einstein's Theory of Special Relativity, proposed in 1905, teaches us about the motion of objects travelling at near the speed of light. Actually, it tells us about objects travelling at any speed, but it only predicts bizarre results for the ultra-fast. The theory has incredible importance in the development of science, but not because knowledge of the ultra-fast is very important. What the theory really teaches us is a new way of looking at the universe, a way which challenges our preconceived notions of what time is and what space is. It teaches us to look very carefully at assumptions that seem so basic and obvious that no one bothers to challenge them. It teaches us a method of examining predictions that can't feasibly be tested experimentally.

We aren't going to start right off with Einsein's theory, he certainly didn't just sit down one day and write it out. Instead, we will look at Galilean Relativity, the accepted theory before Einstein.

What is 9 plus 5?

What is 9 apples plus 5 apples?

GR1 - I assume you answered 14 and 14 apples, respectively, for these two questions. The real question is this - are you sure? Did you make any assumptions in answering the questions?

A bus is moving forward at 9 m/s. Inside the bus, Sheila runs forward at 5 m/s with respect to the seats in the bus. That is, another student in the bus measures Sheila's speed to be 5 m/s. What does Sheila's dad, standing on the ground outside the bus, measure for her speed?

Well, every second the bus moves forward 9 meters and Sheila moves an additional 5 meters. So Sheila is moving at 14 m/s with respect to her dad. This is Galilean Relativity - simply add the velocities. What if Sheila ran backwards in the bus at 5 m/s? What if she ran across the bus at 5 m/s? Galilean Relativity can handle these, too. A diagram is helpful:

GR2

Adding vectors is important in many areas of physics. Here, as in most of this course, the vectors are either parallel, antiparallel, or perpendicular.

GR3 - Are you sure of your answers? Did you (did Galileo) make any assumptions in this?

A large spacecraft passes by Earth and is measured to move at a speed of .90c, or 90 percent of the speed of light. (c is the accepted symbol for the speed of light. We will measure it in an experiment, but in class we will use the value c = 3.0 x 108 m/s.) Inside it, a small rocket is shot and the riders on the large spacecraft measure it moving forward at .50c. What speed will we, on Earth, measure for the small rocket?

Surprisingly, the answer is not 1.4c. Galilean Relativity predicts the wrong answer here.

GR4 - Why? What is different about this situation? What about Galilean Relativity could possibly be wrong?

GR5 - This is a sketch of a river flowing to the right at speed v. In it is a boat, starting at position a. If the river is still (v = 0), the boat is able to move at speed c.

For each of the following situations, determine how much time it takes the boat to complete its task.

1. v= 0, the boat travels from a to c and back to a.

2. v= 0, the boat travels from b to c.

3. v = 2 m/s, c = 5 m/s, d = D = 40 m, the boat travels from a to c.

4. same conditions as 3, the boat travels from c to a.

5. same conditions as 3. If the boat ignores the river current and heads across the river, what will be its total speed? It will end up to the right of b. How much time will it take to reach the opposite bank of the river?

6. same conditions as 3. If it wants to reach b, it must aim upstream some. What will be its total velocity? How long will it take to go from a to b? b to a?

7. Is question 3 possible if v > c? question 4? question 5? question 6?

8. Repeat questions 3, 4, and 6, using the variables v, c, D, and d instead of the given numbers.

The accepted model of light in the 1800's was that it was a wave. In fact, they "knew" that it was an "electromagnetic wave", meaning that it consisted of oscillating electric fields and magnetic fields. These terms should be at least slightly familiar to you. Even though we aren't going to use the electromagnetic theory much, let's do a real quick review of it and of waves.

Electric fields are caused by electric charges (electrons or protons, mainly). They point toward negative charges and away from positive charges. The direction in which the electric field points tells you the direction a positive charge would be forced if placed there.

Magnetic fields are caused by electric currents (which is sort of what there is inside bar magnets, the kind you are familiar with). The direction of a magnetic field tells you the direction the north pole of a magnet gets forced if placed there.

Magnetic fields put a force on moving charges, but not stationary charges.

Changing a magnetic field creates an electric fielf.

In 1854, James Clerk Maxwell hypothesized that changing electric fields create magnetic fields. In doing so, he figured out what light is - vibrating charges create changing electric fields, those changing fields create changing magnetic fields, which create changing electric fields, and so on. This pattern of changing fields propagates outward as a wave at a speed of 3.0 x 108 m/s.

As a wave, it can display the phenomenon of interference - constructive and destructive. When two light beams of the same wavelength meet, if crest (perhaps an upward electric field) meets crest and trough (perhaps a downward electric field) meets trough, they will interfere constructively and the light will be brighter. If crest meets trough, they can (if they are the same intensity) completely cancel. This phenomenon of interference is a very important property of waves - light plus light equals dark (if done properly). One device built to display this phenomenon is called an interferometer...which brings us to the

Read the explanation of the Michelson-Morley Experiment at this location:

http://www.phys.virginia.edu/classes/252/michelson.html

Be sure to follow the links to the apparatus diagram and photo of the actual apparatus.

Einstein claims that he wasn't influenced by the results of the Michelson-Morley Experiment. They were widespread and it is hard to believe that he didn't know about the results, but we will never really know if they influenced his thinking. Einstein claims that he already knew that something was wrong with the accepted theories of light. Here, and throughout his lifetime, he made up a thought experiment (Gedanken experiment). Gedanken expermiments cannot actually be performed, but nevertheless are a useful tool for testing a theory. Einstein imagined a person travelling at the speed of light and asked how that person would perceive the light wave. If light is a wave (which Maxwell's equations said it was and nobody at that time considered otherwise) then the light shouldn't gain or lose on that person. That is, he/she should look at the light and notice that it consists of an electric field pointing (let's say) up. As he/she moves along, that electric field doesn't change, at that location to his/her side, it just points up. Now Maxwell's equations tell us that the only things that could possibly produce a steady upward electric field are negative charges above the location or positive charges below it. BUT there aren't any charges around. The thought experiment produced a contradiction. That means that something is seriously wrong - either with Maxwell's equations or with the thought experiment itself.

Most people when faced with this situation would say that it must be Maxwell's equations that were wrong, as what could possibly be wrong with the situation? Fortunately, Einstein didn't accept that possibility - let's remember that he was a genius, and geniuses are likely to question the "obvious interpretation". Einstein asked what if it was impossible to travel at the speed of light? What if our picture of Galilean Relativity wasn't right? What if our entire picture of space and time wasn't right?

He started from scratch by writing down two postulates. From these two postulates, proposed in his paper "On the Electrodynamics of Moving Bodies" published in 1905, he would reinvent everything else.

1st Postulate - The Relativity Principle: The laws of physics have the same form in all inertial reference frames.

2nd Postulate: Light propagates through empty space with a definite speed c independent of the speed of the source or observer.

An inertial reference frame is a reference frame which is not accelerating.

E1 - If you are riding on a bus or train at a constant velocity, how do you know if you are moving forwards, or if all the trees and cars around you are moving backwards?

E2 - You are in a vehicle with no windows. You drop a ball and it doesn't fall straight down, but moves in a parabola on its way down. What can you conclude?

E3 - You are in a spaceship travelling at .90c and turn on your headlights. At what speed (relative to you) will the light move away from you?

E4 - You are on Earth. A spaceship is travelling toward you at .90c with its headlights on. What will you measure for the speed of that light?

Perhaps the first postulate isn't so surprising, but the second is certainly puzzling. How could he postulate something that makes so little intuitive sense?

E5 - Why not? What makes the second postulate seem "wrong"? Was there any evidence against it? For it?

What does it mean to say that two events are "simultaneous"? The easy answer is that they happen at the same time. However, what if those two events occur far away from each other - so far that light takes a non-trivial amount of time to go from one location to the other? If this is the case, you might see one happen before the other.

In this diagram, Bruce and Sheila are at rest with respect to each other. Bruce sees 2 bolts of lightning at the same moment. Since he measures the distance to each of them as d, he concludes that not only did he see them simultaneously, but they must have occurred simultaneously.

Sheila, on the other hand, sees the left bolt of lightning before she sees the right bolt. She also measures the distance to the left bolt and the distance to the right bolt. Knowing the distance the light had to travel and the speed of light, she calculates how long each bolt took to reach her. Amazingly, she discovers that the bolts actually occurred at the same time.

This is important - Sheila didn't see them simulataneously, however, she agrees with Bruce in concluding that they occurred simultaneously.

Nothing terribly surprising so far, just making a distinction between what we see and what we conclude.

Now we add Michael into the picture. According to Bruce, Michael is in a train traveling at a speed v to the right. Also according to Bruce, when the lightning strikes, both ends of Michael's train get toasted simultaneously.

What does Michael see? More importantly, what what does he conclude?

Figure S3 shows things from Bruce's perpective. According to Bruce, when the lightning bolts hit (simultaneously), Michael is just passing him. Since Michael is moving to the right, he receives the light from the right bolt first. He will move further to the right before receiving light from the left bolt.

Things are different from Michael's perspective. He sees the right bolt of lightning hit the end of his train car first. Sometime later, he sees the left bolt. That is what he sees - the real question is what does he conclude?

Michael will recognize that both bolts hit the ends of his train car and that he is in the middle, equidistant from where the bolts hit. Since they travelled the same distance (according to him) and travelled at the same speed, they each must take the same amount of time to reach him. Michael must conclude that the right bolt hit first since it reached him first.

Bruce says the lightning bolts hit simultaneously, Michael says they did not! Special Relativity tells us that two observers don't necessarily agree on simultaneity of two events - and they are both "right".

S1 - In what ways were the 2 postulates of Special Relativity applied to this situation?

S2 - Will two observers at rest with respect to each other always agree on the simultaneity of two events?

S3 - If Michael was at a different location in his train car, it is possible that he would have seen the two bolts simultaneously. Approximately where? Would he then conclude that the two flashes were simultaneous?

S4 - Two events occur at the same location. One observer concludes they were simultaneous. Will an observer moving with respect to the first agree?

In the previous section, let's say Bruce concluded that both flashes occured at 1:00 exactly. Michael disagrees, perhaps the right flash occured at 1:00, but the left flash was later, let's say at 1:01. The concept of an "absolute" measure of time (which is probably hidden somewhere in our subconscious) is shown to be invalid. Different observers, moving with respect to one another, have different measures of time - different clocks essentially. Michael's clock was not "behind" Bruce's, his ideas about time and what happenned before what were different. This section will take that concept a step further; we will show that one clock runs at a different rate than the other! Not only that, but rulers shrink!

Go to the following web address:

http://www.theory.caltech.edu/people/patricia/st101.html

Link to "Why was Einstein's Special Theory of Relativity needed?". Read all 3 sections. The mathematical derivation for time dilation is in the second, "Time must be relative". It is extremely important and will be repeated in class. I recommend writing the math steps in your notes while you go through the web site.

(You might also want to read "What is spacetime?" and "How does Einstein's Special Theory of Relativity work?". They are both excellent, althought not necessary for what we are doing here.)

Let's look at the time dilation formula:

What does it tell us? One observer measures a time t as the time between two events, the other observer, travelling at speed u with respect to the first measures the time between the same two events as t'.

Notice:

The real difficulty in applying the formula comes in deciding who measures t and who measures t'. We call t' the proper time. Don't confuse it with the "correct" time, since both observers measurements are correct for the individuals that made them.

Look back at the original situation in the derivation to see who measures the proper time. The laser pulse device was in the blue car and they called the blue time measurement t'. The red car measured t. What is generalizable about this? The two events being timed here were the laser pulse being emitted and the pulse being received. Notice that both of these events occured at the same location according to the observer in the blue car. That is why the blue car measured the proper time. According to the observer in the red car, the light was emitted and received at different spatial locations.

Formally stated - proper time is the time measured by an observer for whom the two events being timed occur at the same location.

The passenger in the red car said that t seconds ticked off while the light travelled, but the blue clock itself measured t' seconds. Since t>t', the passenger in the red car says that the blue car's clock was running slowly.

Here's the tricky part. What would the passenger in the blue car say about the red car's clock? If there was a laser pulse clock in the red car, the blue car passenger would say that it (the red car's clock) ran slowly. Red says that blue's clock is slow and blue says that red's clock is slow - and both are correct. The fact is that they don't need to agree in this situation. After going through length contraction and some example problems, we will look at a situation in which observers must agree upon the outcome.

Once you have time dilation, length contraction comes easily. Look back at the website for the derivation. Remember that the observers agree upon speed. Since speed is distance divided by time, they are going to disagree about distances travelled.

Again, who measures L and who measures Lo? It is the observer for whom the length in question is at rest. That is, if you measure the length of your own car, you call that Lo (sometimes known as the rest length), but if you measure the length of a car moving by you, you call that L.

Note that the same observer doesn't necessarily measure both t' and Lo. Also note that L<Lo. This means that as objects move by you, you measure their length to be less than their rest length, so lengths are contracted.^*

Example:

An "exon" is measured to have a half life of 3.0 years when at rest (this means that after 3.0 years, half of them will graduate). A group of exons flies by you at a speed of .80c.

a. What will you measure to be the half life of the exons?

b. If these exons fly by a soccer field (100 meters), how long will the field be according to them? How much time will they take to fly by i) according to them, ii) according to an observer at rest on the field?

Solution:

a. First answer the question - is 3.0 years t or t'? When we say the half life is 3.0 years when at rest, that means that the start of the measurement of 3.0 years was made at the same location as the end (the exons didn't move during the measurement). Therefore we conclude that the proper time, t' = 3.0 years.

It's algebra from here, but there are some simplifications possible. Since you were given the speed in terms of c, don't convert it to m/s.

Now the c's cancel.

bi. The soccer field has a rest length, Lo, of 100 meters. The length according to the exons is:

Travelling at a speed of .80c, the soccer field will fly by the exons in:

bii. In the reference frame of the soccer field, the exons travel of distance of 100 meters at a speed of .80c.

Notice how T and T' were used here. The exons time the passing of the soccer field by starting the timer when one end passes and stopping when the other end passes. Both measurements are made in one location - where they are - so they measure T'. If we are standing on the soccer field, we start the timer as the exons pass one end, and stop it when they pass the other end - two different locations. Also notice that T and T' are again related through the time dilation formula. In fact, that could be used to solve the problem, rather than using the length contraction formula.
 
 

TD1 - Spaceman Spiff is flying in his 10 meter diameter saucer. He passes the Starship Voyager at a relative speed of .90 c.

a. How long is Spiff's saucer according to Captain Janeway (of Voyager)?

b. Spiff measures 1.5x10-6 seconds for Voyager to pass by him. How long does he think Voyager is?

c. How long is Voyager according to its passengers?

TD2 - In an hour, 500 bluons traveling at 99.0% of the speed of light are detected atop Mt. Everest (9,000 meters above sea level). At sea level, 250 per hour are detected. What is the half-life of a bluon AT REST?

TD3 - Suppose we want to send an astronaut to a star 10 light years away and we construct a spacecraft capable of travelling at .95c.

a. How long would the trip there take according to us?

b. How long would it take according to the astronaut?
 
 

Question TD3 brings up an interesting dilemma. The time measured by the astronaut is less than the time measured by us. But, from the astronaut's perspective, isn't the Earth-star system moving by and shouldn't it therefore measure the smaller time? Hmm... Maybe. We need to formalize the situation a little more in order to answer it concretely. The thought experiment is known as the Twin Paradox. It was suggested soon after Special Relativity was introduced and was first intended to show the theory was flawed. However, it isn't really a paradox and the theory handles it adequately.

A pair of twins, Sandy and Andy, are born on Earth. Sandy is immediately sent out on a spaceship travelling at near the speed of light. On her 15th birthday, the spaceship turns around and returns to Earth on her 30th birthday. What does she find?

a. Andy is more than 30, let's say 90 years old.

b. Andy is less than 30, let's say 10 years old.

c. Sandy says Andy is 10, Andy claims he is 90.

d. Andy is also 30 years old.

The situation seems pretty symmetrical. That is, Andy says Sandy is moving, therefore she should have the slow clock, that suggests answer a. Sandy says Andy is moving, therefore he should have the slow clock and answer b would be correct. Maybe both are correct from their own perspective and it is answer c. Maybe it is a combination of both and answer d is correct. We can't use our standard method of determining who has the proper time, since both Andy and Sandy make both of their measurements at the same location, Earth.

Let's first eliminate answer c. Sandy returns and looks at Andy, she can certainly tell if he is 10, 30, or 90 years old. She won't stand in front of a gray haired old man and claim that he is "wrong", that he is really only a child of 10. We have set up the situation in a way that they must agree. Not all situation are like this.

The resolution of the paradox comes in noticing that it really is not symmetrical? Sandy can not say that Andy is the one who is moving. The reason is that Andy did not accelerate, Sandy did. When Sandy turned the spaceship around, she had to accelerated. Einstein's first postulate tells us that we can't tell who is moving if two observers are both in inertial reference frames. But we can tell who is accelerating, as they will feel the forces on them. Sandy definitely accelerated (and therefore definitely moved). Sandy has the slow clock. She returns to Earth to find a very old twin brother.
 
 

AP1 - Another famous apparent paradox is known as the Ladder and Barn Dilemma. Homer owns a 15 meter long barn and a 20 meter ladder. He wants to put the ladder in the barn (not diagonally), but obviously can't. He tells his son Bart to run with the ladder at a very high speed. The ladder will contract. As soon as the ladder is completely within the barn, Homer will slam the barn door. At least for an instant, the ladder will be completely within the barn (before smashing through the back wall).

Being a lazy son, Bart says "no way, Pops. You are thinking about it wrong. The barn will be moving and will therefore contract. It will be much shorter than the ladder. The ladder is going to smash through the back wall before you shut the door."

What happens when they try it?
 
 

AP2 - A child , Sue, is born on Earth. At the instant of her birth, a spaceship flies by at a speed of .90c. Inside the spaceship, also at that same moment, a child named Lou is born. When Sue is 20 years old (by her measurement), she broadcasts a photo of herself. Lou does the same. The signals take some time to travel and reach the other party. Examine this situation. Is there any paradox here? Do they agree on who has the slow clock? Can you make it quantitive?
 
 

Go to the following web address. Read the whole page, there are no further links. This is a good review of what we have done. It also adds in a formula for velocity addition (underived, but we can use it anyway) and a visual representation of what things look like when we travel at very high speeds.

http://www.duke.edu/~set2/physics.html

The following link gives a more detailed solution of the Ladder and Barn and Twin paradoxes. It's complicated, but worthwhile.

http://www.phys.virginia.edu/classes/252/srel_twins.html

Consider a box on a frictionless track, with a laser at one end. What happens when we fire the laser and the light is absorbed by the other end of the box? It turns out that light carries momentum, and this makes the box move slightly, just like the box would move if someone inside threw a ball from one end to the other.

This picture is exagerrated, as ordinary light doesn't carry much momentum. The box starts at rest, moves at speed v while the light is travelling, and stops when the light hits the other end, after travelling a distance _x. Let's related these quantities mathematically.

Light has momentum: 

Since the total momentum of the box and light must be zero, since it started at zero, the momentum of the box to the left must equal the momentum of the light to the right. Assume the box has a mass M and the light an energy E:

It takes the light a time _t to travel L (slight approximation here, do you know what it is?) and reach the absorber. In that time, the box moves:

Rearranging the previous three equations, we get:

The center of mass of the box moved by M_x to the left. There is a principle that says that the center of mass of an object can never move due to internal forces, but it seems to have moved in this case. However, if we consider energy and mass to be equivalent, the problem can be resolved. The left side lost energy and the right side gained energy. We can interpret this as a mass change, too.

The equivalent mass, m, shifted a distance L to the right:

rearrange to get:

Perhaps the most famous equation in physics, it tells us that when the energy of an object changes, its mass changes as well. This is true in general, any kind of energy gain or loss will be accompanied by a mass gain or loss.

Two examples can help to clarify this famous equation:

Example 1. A typical flashlight gives off about 10 Watts (or 10 Joules per second). What mass change accompanies this?

Solution: Each second, the flashlight's (chemical) energy is reduced by 10 Joules.

That's pretty tiny. Leave the flashlight on for a week, and it loses 6.71x10-11kg, still an unmeasurably small amount.

Example 2. If a nuclear reactor uses fission of uranium to produce electrical energy, how much energy will be produced if the mass of the fuel is reduced by 1.0 kg?

Solution:

This is approximately the output of a small nuclear reactor in one year. In order to get this much mass change, it is necessary to use about 1000 kg of uranium, as the mass is only reduced by about .10% the nuclear reactions.
 
 

Notice that the derivation avoids the question of whether light itself has mass. This is a tricky question. In this case it seems to have acted as if it does have mass. However, things with mass should be able to accelerate, which light can't, and shouldn't be able to reach the speed of light, which light obviously does. Physicists today say that light has no mass. It does, however, have momentum - sometimes this gets confused with mass.

Let's say we see 2 rockets fly by us; we measure rocket A to be travelling at .80c to the left and rocket B at .90c to the right. What would rocket A measure for the velocity of rocket B? Uh oh, we don't want to say 1.7c, do we? If we follow the time and distance formulas carefully, toss in a bit of calculus, we will get the Lorentz Velocity Addition formula:

Where u is the velocity (careful with directions) of rocket A with respect to us, v is the speed of rocket B with respect to us, and v' is the velocity of rocket B as measured by rocket A.

A rocket flies by us at a speed of .80c. Inside it, Marlene uses a spring scale to pull a 1.0 kg mass forward. She pulls it with a force of 25 Newtons. According to Marlene, the force is 25 Newtons, the mass is 1.0 kg, so the mass should accelerate at 25 m/s2. It does, according to her. But it accelerates at less than that, according to us, because we disagree on its original velocity, final velocity, and how long it takes to pull it to change that velocity. Think about the details.

What do we conclude? We have a choice - we can disagree about the force, the mass, or Newtons Second Law, F=ma. Physicists have chosen the mass, as the quantity which changes when something is moving at a very high speed.

Where mo is the rest mass, invariant mass or the mass of the object measured by an observer in the reference frame in which the object is at rest. (Note, the concept of relativistic mass is going out of fashion. Not that it is wrong, but it is usually the relativistic momentum or energy of an object that we are interested in.)

Assigned Reading

(in order of assignment)

http://www.phys.virginia.edu/classes/252/michelson.html

http://math.ucr.edu/home/baez/physics/relativity.html