Relativity

Chapter 2: The Speed of Light

In 1862, Maxwell calculated the speed of light. As we have seen, this seemed a very strange result at the time. In Galilean relativity, the speed of light should depend on the speed of the observer and the speed of the source.

Physicists thought about this for some time, and came up with an explanation: they decided that all of space was filled with a substance, which they called the Lumeniferous Ether. Light was then thought to be a disturbance of this ether, just as water waves are a disturbance of the surface of the water. This idea also explained the prediction of the speed of light - the speed of light was thought to be relative to this ether.

In 1887, Albert Michelson decided to try to prove the existence of this ether substance. He noticed that as the Earth revolves around the sun, the Earth travels through space at about 20 miles per second, about 70,000 mph. The speed of light was known to be about 186,000 miles per second, so the orbital speed of the Earth is about .01% the speed of light. Michelson decided he should be able to detect this - he would compare the travel time of two beams of light, one which traveled along the direction of the Earth's orbit, and a second beam which traveled sideways compared to Earth's orbit. Today, we would also add in that the Sun is revolving around the center of the Galaxy, with an orbital velocity of about 200 miles per second. This is about .1% the speed of light. So, actually, Michelson's experiment was about ten times more sensitive than he knew.

Michelson's daughter says he explained his experiment to her like this:

Suppose we have a river 100 feet wide flowing at 3 feet per second, and two swimmers who both swim at 5 feet per second. The swimmers have a race. One swims upstream 100 feet, then swims back to the start. The other swims directly across the river, then turns around and swims back. Who wins?

The swimmer going upstream is easiest to analyze. Going against the current, the swimmer makes only 5-3=2 feet per second, so the 100 feet takes 50 second. Coming back he's going 5+3=8 feet per second, so it takes him 12.5 seconds. His total time is 62.5 seconds for the 200 foot swim.

The swimmer going across the river has a different job. As he swims across the flow at 5 feet per second, the river is carrying him downstream at 3 feet per second. So, he has to swim at an angle in order to make it straight across the river. His net speed is the hypotenuse of a 3,4,5 triangle, so his net speed is 4 feet per second. He swims the 100 foot width in 25 seconds, then takes another 25 seconds to swim back, for a total time of 50 seconds for the 200 foot swim. So, this swimmer wins.


Figure 2.1 - a swimmer in a river.

Michelson realized the this exact same argument should apply to light moving along and against the flow of the Ether, and across the flow of the Ether. The math is pretty much the same as swimming across the stream. Let's suppose one light beam travels with and against the Earth's motion over a distance L each way. The other light beam travels normal (sideways) to the Earth's motion a distance L each way. The Earth is moving at a velocity v around the Sun.

The time required for the first beam is L, the length the light travels, divided by the speed of the light beam. Michelson figured the speed would be c + v going one way, and c - v going the other.

L L 2 c L 2 L
T1 = ------- + -------  =  ---------  =  ---------------
c + v c - v c2 - v2 c (1 - v2 / c2)

The light beam going normal to the Earth's motion is following a path which is like the verticle leg of the triangle above. The three legs of the triangle now have length c (replaces 5), v (replaces 3), and Ö( c2 - v2 ) (replacing 4), so it's travel time is

2 L
T2  =  ------------------
c Ö(1 - v2 / c2)


Figure 2.2 - The light that travels normal to the Earth's velocity

Michelson set up an experiment and tried it. He found no effect - the two beams of light took exactly the same amount of time. As you can see in the picture below, Michelson's experiment was mounted on a big bearing. What he actually did was rotate the entire table as he was measuring, looking for a direction where the light took longer to travel in one direction than in the other. He never found such a direction - the light always took exactly the same amount of time to travel down either leg.


Michelson and Morley's Interferometer

Of course, Michelson immediately realized that perhaps the Ether was moving compared to the Sun, and on the day of his experiment the Earth happened to be precisely standing still in the Ether. So, he repeated his experiments every couple of months for a year, and continued to find no effect. Michelson also wondered if perhaps the Earth was dragging the Ether around with it, so he repeated his experiment on top of a mountain. Again, no effect. Michelson also reasoned that if the Ether was being dragged along with the Earth, we should see the apparent position of stars move, depending on the angle their light entered the Earth's ether. No such effect has ever been observed.

Everyone found this result very confusing - as we have seen, the time taken by the light should be longer when the beam is aligned with the Earth's motion than when the beam is at 90 to the Earth's motion. The understanding of the day was that the velocity of the Earth's motion should add and subtract from the speed of light, depending on the direction of the light. But, what was found was that the speed of light seemed to never change.

Hendrik Lorentz and George FitzGerald analyzed the Michelson-Morley experiment. They decided to postulate that when something is moving, it shrinks in the direction it is moving. This effect is called the Lorentz-FitzGerald contraction. This contraction can be calculated to exactly compensate for the velocity of the Earth. In other words, Lorentz and FitzGerald decided that the reason the beam aligned with the Earth's motion took the same time as the other beam was that it had a slightly shorter distance to traverse. So, they decided that Michelson's table shrunk in the direction of the Earth's motion, by exactly the right amount so that the two beams of light tied in their race.

We see immediately that if we were to multiply T1 by Ö(1 - v2 / c2), then these equations give the same result. So, Lorentz and FitzGerald decided to assume that the table had contracted by this factor, Ö (1 - v2 / c2), in the direction of Earth's motion. Then the math gave the right answer, which is that the travel time is the same in both directions.

What does this mean? To see this, we're going to learn how to make what are called Space-Time diagrams. This is an ordinary graph, but with time as the vertical axis. It's very inconvenient to label a graph with seconds running up the vertical axis, and units of 186,000 miles on the horizontal axis, so we're going to work in different units. We'll measure distance in feet, and time in nanoseconds. One nanosecond is one billionth of a second. One billion nanoseconds is one second. The speed of light is almost precisely one foot per nanosecond. That is, 186,000 miles times 5280 feet per mile is almost exactly one billion feet. On a space-time diagram, light is always drawn at a 45° angle, because light always moves at 1 foot per nanosecond.

By the way, you could ask "Why does light always move at one foot per nanosecond?" The answer is, we have not even the slightest clue. It just does. We've checked this a zillion times in as many ways as we can think of for over 125 years now, and it's always been true. That's why Einstein took this as an axiom. He could not prove it, he could not justify it, he could not even motivate it. He could only say that this seems to be true in our universe, so lets assume it's always true and see what the ramifications are. I suppose I should mention that perhaps you've heard of these people who have slowed light down to walking speeds, and wonder what this means. Well, while this is a very important and interesting result, it's a trick from our current perspective. This light is traveling through a very peculiar medium in a very peculiar fashion. When we speak of the speed of light being a constant, we mean in a vacuum. If there are atoms nearby, the light can interact with the electrons and protons and start doing strange quantum things. It's these strange quantum things that cause rainbows and make lenses work and make the sky blue and make your eyes work. But, we're not studying quantum things in this book, so we're just going to think about the vacuum.

Below in figure 2.2 is a space-time diagram, with a bunch of things drawn in it. Remember, the horizontal (X) axis is position, and the vertical (Y) axis is time. The units are feet and nanoseconds. We see three rays of light, one starting at x = 0, t = -2, and moving to the right. You can tell it's a ray of light because it's drawn at a 45° angle. Now, this is a space-time diagram, so there's something important to notice here: this particular ray of light comes into existence at -2 nanoseconds, and evaporates at +2 nanoseconds. There's another ray of light which starts at x = 6, t = -3, and ends at x = 3, t = 0. Another ray of lights starts at x = 0, t = 2 and goes to at least x = -3, t = 5. There's a particle moving very quickly, at half the speed of light, from x = 1, t = 2 to x = 2, t = 4. This particle only exists for a short time. There's another particle at x = -5. This particle is not moving at all, but it exists for at least as long as this graph exists. Finally, at about x = -2.5, t = 1.5, there's one of our new favorite characters, a rocket ship. This is not a book on art, so comments on the aesthetics of this particular rocket ship are not welcome.


Figure 2.3 - a Space-Time Diagram

So, we see that in a space-time diagram we know both where and when things are. This is a very different perspective than you are used to. For example, on a space-time diagram, you would look like a long pink tube, with one end attached to your mother and the other end just stopping somewhere about 75 years later. In between, the tube that is you twists and turns and wiggles to reflect where you went while you were alive. If you're female and have children, your children would start out as small pink tubes which branch off from you.

Space-time diagrams show "now" as the x-axis, and they show the past and the future below and above the x-axis. Things which are not moving are vertical lines. According to the rules, all lines which represent the motion of something with mass must be tipped from vertical less than 45°. Light is always at 45°. If a line is drawn which is tipped at more than 45°, it would represent something moving faster than light. We have a name for such objects: we call them tachyons. We also have names for things like "nice lawyers" and "honest politicians," but we've never actually seen any of these things, so don't get too excited.

Now, using our space-time diagram, we're going to try to understand what it means to say where and when something is. We're on unfamiliar ground here, so we're going to try to see how to do this without making any assumptions. For example, you might look up in the sky and see an airplane fly by, maybe 3 miles up, going maybe 500 mph. You could look at your watch, and say "That airplane was right over my head at noon." However, this is an assumption that we're not going to make. What you can say is, "I saw the airplane at noon." We will not assume that we know how to tell time at places far away from us. In fact, what we would really like would be to have a clock hanging in the air 3 miles right above our head, and when the airplane flies past the clock, we can see the airplane and the clock next to each other and read the time off of that clock.

Now, we have the problem of trying to synchronize this clock 3 miles away with our wrist watch. How can we do this? Well, first we'll design a new clock. Our new clock has hands, just like you're used to. Of course, the second hand ticks off nanoseconds, so to us mere humans it looks like a blur, but that's no big deal. Next, we'll add a flashing light to the top of the clock. The light will flash, say, every micro-second, that is every 1,000 nanoseconds. Now, anyone anywhere can synchronize their clock with the flash from our clock. When you see the flash, you know the second hand is pointing straight up. Of course, we're trying to synchronize clocks here, so we have to account for the speed of light. If you're 100 feet away from my clock, and you see the flash, you know that my clock emitted the flash 100 nanoseconds ago. But, no problem, you just make sure your clock's second hand hits 100 nanoseconds exactly as you see the flash from my clock. The clock that's 3 miles up in the air is about 15,000 feet away, so that clock will be set 15,000 nanoseconds ahead of the flash. Of course, all our clocks have this flash feature, so I can tell if I'm synchronized with you, too. Now, our clocks are synchronized. Remember, we're going to have a lot of clocks, strung out all over the place, all synchronized. We will know where each clock is. When we see something happen, we'll know where it happened and when it happened - the where is from knowing the position of the nearest clock, and the when is from reading the time off that clock, and no other. We have a special word for something happening, we call this an event. An event is something that happens at a particular time and place.

Here's a space-time diagram of us synchronizing a couple of clocks. At x = 0 (that's where we're standing) and t = 0, we send out a flash. The flash travels at the speed of light, which is a 45° line. Two feet away from us is another clock. Our trusty graduate student is standing there with another clock. He sees the flash at t = 2 nanoseconds, sets his clock, and sends a flash back. At t = 4 nanoseconds, we see our grad student's flash come back. Graduate students, by the way, are a very important part of physics: they're smart, educated, do what they're told, and work nearly for free. Without graduate students, all of science would come to a screeching halt.


Figure 2.4 - Synchronizing a pair of clocks with light flashes

Ok, now we have the basic tools, and we're ready to do some real, live special relativity. Here's the situation we're going to imagine. Suppose there's some guy, George, who has a small lab. He has two clocks and he wants to synchronize them. So, we know how he's going to do this. He's going to have flashers on his clocks to help set the clocks. When the clocks are synchronized, he's going to just sit back and watch the clocks flash at each other - it's going to look like they're bouncing a little light ball back and forth between them, like they're playing ping-pong with light. Now, here's the trick. George and his small lab are in a rocket ship, flying by us at half the speed of light. What do we see? Conveniently, his rocket ship is transparent, just like Wonder Woman's jet airplane, so we can see inside. We can turn on our clocks however we wish, so we'll agree that George will set the clock closest to him to read t = 0 exactly as he passes us. We'll also set our clock to read t = 0 just as George passes us. So, at the instant t = 0 our clock and one of George's clocks are right next to each other and read the same thing. Our job is to figure out what happens next. By the way, George's clocks happen to be bolted down to a large solid table which is 11.55 feet long.

Below is a space-time diagram of this situation. I've changed the scale to 5 feet and 5 nanoseconds per tick. George has two clocks. One of them flies right past us, so that's the line that goes through the origin. George is flying at half the speed of light, so in 10 nanoseconds he moves 5 feet - half as far as light would move. George has a second clock which is 11.55 feet away from him. But, we have to remember the Lorentz- FitzGerald contraction. Although George has carefully measured out 11.55 feet, we see his two clocks as being closer to each other. The distance is contracted by the factor Ö(1 - v 2 / c2). He's moving a c/2, so v2 / c2 = 1/4. Ö(3/4) = .866. .866 * 11.55 feet = 10 feet (I picked the 11.55 feet for just this reason). So we see George's two clocks as being 10 feet apart. In our space-time diagram, George's second clock is on a parallel line 10 feet away, just as I've drawn it. At our t = 0, we see one of George's clocks right on top of us, and one which is 10 feet away from us. I've also drawn a couple of light flashes emitted by George's clocks, as they look to us. Light always goes at 45°. When George's clock reads t = 0, it flashes. To us, it looks like it takes 20 nanoseconds for that flash to reach the clock at the other end of George's table. That clock then flashes, and it looks to us like it takes about 6.67 nanoseconds for that flash to reach George's first clock.


Figure 2.5 - George and his clocks fly past us at half the speed of light

Now, remember, George has carefully set his clocks. He doesn't think there's anything strange about his lab, he's just setting his clocks to read the right thing. So, when his second clock sees the flash from the first clock, it's reading 11.55 nanoseconds. After all, George has carefully synchronized his clocks. When the flash from that clock reaches George's first clock, the first clock is reading 23 nanoseconds. How can this be? To see how this works, we'll draw another space-time diagram where we show the light flashes that happened immediately before these flashes.


Figure 2.6 - George's clocks flash as they fly by us at half the speed of light

On this space-time diagram, I've also noted the time shown on George's clocks when they see a flash of light. We quickly notice some things. George's clocks are running slow. The clock nearest George read 23 nanoseconds when our clock reads 26.67 nanoseconds, and his reads -23 nanoseconds when ours reads -26.67 nanoseconds. The other thing we notice is that George's clock which is ten feet away from us at closest approach is not only running slow, but is also off. Halfway between the points where George's second clock reads -11.55 and 11.55 nanoseconds happens when our clocks are reading 7.33 nanoseconds. So, George's second clock reads 0 when our clocks read 7.33 nanoseconds. When our clocks read 0 and we see George's first clock as reading 0, we see George's second clock is reading -5.75 nanoseconds. So, this is big: George's clocks and our clocks are not synchronized. We see our clocks as synchronized, and George sees his clocks as synchronized, but we don't see George's clocks as synchronized, and he does not see our clocks as synchronized.

This special relativity idea has now cost us one of our most fundamental intuitions. It is not possible to synchronize clocks unambiguously. Or, we can say, there's no such thing as simultaneous. We see George's distant clock as being 5.75 nanoseconds behind his first clock, so events that George sees as simultaneous, like his two clocks both reading 0, we see as happening at very different times. We see our two clocks as reading zero at the same time - we worked very hard to arrange that - but George sees our two clocks as off by 5.75 nanoseconds.

The next thing we notice is that George's clocks are running slow by exactly the same factor as we think his table has contracted. That is, 26.67 / 23 = 11.5 / 10. This effect is called Lorentz-FitzGerald time dilation. We see George's clocks as running Ö (1 - v2 / c2) as fast as our clocks.

This time dilation effect is the source of the "twin's paradox." If you stay on Earth, and your twin gets in a rocket ship and flies away at a very high speed for a year, then turns around and flies back to Earth, he has aged two years, and you've aged more than two years.

Problems

2.1:

2.2:

2.3:

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Copyright © 2002 Mark Lawrence. All rights reserved. Reproduction is strictly prohibited.