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Introduction to the Theory of Relativity Part II: Special Relativity

By epepke in Science
Thu Jun 19, 2003 at 11:44:08 AM EST
Tags: Science (all tags)
Science

This is the second of a series of elementary, informal, and mostly equation-free articles descibing the Theory of Relativity in physics. The series will have four installments:

  1. Part I: History
    This described the history of ideas in the development of relativity.
  2. Part II: Special Relativity
    This describes Einstein's Special Theory of Relativity. It assumes a familiarity with Part I but assumes very little in the way of prior understanding of physics.
  3. Part III: General Relativity
    This will give at least a taste of Einstein's General Theory of Relativity, which extends the Special Theory to cases involving acceleration and gravity.
  4. Part IV: Implications, Controversies, and Miscellany
    This will address implications of the Theory of Relativity, controversies both old and modern, and anything else that isn't covered in the first three installments.


Part I described the history of relativity and showed support for two educated guesses. One is that the principle of relativity is true, that there really is no way to determine or measure speed except relative to something else. From this we get the idea that the physics of motion works the same way for everyone no matter how they are moving (so long as they aren't accelerating). Of course, we also need some idea of what the physics of motion are, but despite the rocky history of the study of motion, most of this is intuitive, and we can show it in due course.

The other educated guess is that the speed of light depends neither on the speed of the light source nor on the speed of the observer.

These two premises don't seem so strange. Many philosophers argue that the relativity of motion is self-evident, which certainly seems consistent with experience. Sitting in an airplane in smooth skies feels like not moving. The independence of the speed of light from the speeds of both the source and observer is the only premise that Newton and Galileo didn't have. At least it doesn't seem to contradict any experience.

Following Einstein's theory of relativity, using these premises and reason, results in conclusions that seem very strange indeed. Here are a few, stated in informal form as they tend to be filtered through popular media:

  1. When objects go fast, time slows down.
  2. Nothing can go faster than light.
  3. When objects go fast, they become more massive.
  4. When objects go fast, they contract in the direction of travel.
  5. Even though it's the Theory of Relavitiy, the speed of light seems absolute.
  6. Time is the fourth dimension.
  7. E = mc2

Now, these do seem strange! Many people, seeing them, find them so bizarre and counterintuitive that they conclude that Special Relativity must be wrong. Part of the problem is how the ideas are popularly presented, and so we may find that they are subtly or even profoundly different from what Special Relativity actually says.

Remember also that if we assert that the implications of a scientific theory are wrong, then logically it must mean that either the reasoning in the theory is wrong or that one or more of the premises is wrong.

The rest of this article shows the reasoning behind Einstein's Special Theory of Relativity and how the conclusions follow from the premises. Many of the examples are adaptations of the thought experiments that Einstein himself used, but they are presented in an unusual way. The goal is to give an intuitive feel for the reasoning behind Special Relativity. It relies on insight, vision, informal logic, common sense, and pattern matching as much as possible, only using equations when there seems to be no other choice. When used, they are not derived, although links to the derivations are provided. It is a non-traditional approach, an attempt at a short-cut to understanding. Those who prefer traditional presentations or wish to supplement their intuition can use those links.

First Things First

Special relativity is "special" because it only deals with the special case of constant motion in a straight line. We'll imagine everything happening out in deep space in the middle of nowhere, far away from any gravity that might change the results. It doesn't matter exactly where we do it; anywhere is fine as long as it is far enough away from stars and planets that their influence does not matter. A light-year or two in any direction away from Earth there are lots of places that are more than good enough.

Einstein thought in terms of railroad cars, but we can use rocket ships. We'll use a simple square rocket ship, moving to the right. We'll concentrate on the interior of the ship, so we don't have to worry about the thickness of the hull. The engines are simple and always produce the same force. We won't have it carry any of its fuel so we don't have to worry about the change in mass as it is used; maybe there's another rocket ship going alongside with a big hose to give it fuel.

+--------------+
|              |
|              |
|              |
|              |   ->
|              |
|              |
|K             |
+--------------+

Kirk (K) is sitting in the rocket ship and will do experiments while we watch. Of course, we're also sitting in a rocket ship watching Kirk's ship. We'll make it the same size and mass as Kirk's ship. We'll do most of the thought experiments with Kirk's ship coasting at constant speed in a straight line to the right relative to us. Because of relativity, it would be as accurate to say that we are coasting to the left relative to Kirk or that we're both coasting at different speeds; only the relative motion is important.

This ship looks a bit like a coordinate system with Kirk sitting near the origin, where +X is toward the right and +Y is toward the top of the page. This coordinate system provides a frame of reference. It's called an "inertial" frame of reference because it is coasting from inertia.

Next, we need something to measure space. We'll use a stick, something like a meter stick, as long as the ship. We'll also need a clock to measure time. To make sure that everything that affects the stick also affects the clock, we'll make the clock from the stick. Since we're presuming that relative motion does not affect the speed of light, we'll use light. At one end of the stick we'll mount a light emitter, a light detector, and some electronics. At the other end of the stick we'll mount a mirror. The electronics will send out a pulse, which will travel up to the mirror. That's a tick. Then the light pulse will bounce back from the mirror and be detected. bounce back, and be detected. That's a tock. The round trip is a ticktock. Immediately, the electronics will send out another pulse of light toward the mirror. Every time the electronics send a light pulse toward the mirror, it will also send out another signal (maybe another flash of light), so that we can see it at a distance.

We'll make the electronics fast enough so that we can ignore its latency and think of a single pulse of light bouncing back and forth. We could also make it like a laser so that the next pulse is stimulated automatically. Now we have a lightstick that can measure space and time. One ticktock is the time it takes light to go twice the length of the stick, which is also the length or breadth of the ship. It also provides a natural way of measuring speed, so we don't have to use meters per second or furlongs per fortnight. Light speed is two sticks per ticktock.

Kirk can hold the stick vertically, transverse to the direction of travel:

+--------------+
|^v            |
|^v            |
|^v            |
|^v            |   ->
|^v            |
|^v            |
|^vK           |
+--------------+

or horizontally, along the axis of travel:

+--------------+
|              |
|              |
|              |
|              |   ->
|K             |
| > > > > > > >|
|< < < < < < < |
+--------------+

The symbols in the diagram suggest the direction the light pulse travels.

Now we have everything we need.

1. When objects go fast, time slows down.

We start our first experiment and tell Kirk to set up his lightstick vertically, across the breadth of the ship, and remain stationary relative to us. We ignore the rest of the ship and concentrate on the lightstick:

^v
^v
^v
^v
^v
^v
^v

So far, so good. A light pulse goes up; a light pulse goes down. Up and down, over and over again. We compare it with our own lightstick, and they're perfectly synchronized.

Then we tell Kirk to fly off to the left, speed up with his rockets, and then coast by us with his lightstick going. We're not so sure about speeds, yet, but using trial and error we figure out how long he has to run the engines so that we see the signals from his lightstick one stick apart. We'll call running the engines for this amount of time one impulse, and we'll put one button in the cockpit to deliver one impulse to the ship. We'll have his path always be at about the same distance from us, so that we don't have to worry about how long it takes for his signals to get to it. In our frame of reference, the pulse of Kirk's light stick traces this path:

      ^v            ^v
     ^  v          ^  v
    ^    v        ^    v
   ^      v      ^      v      ^  etc.
  ^        v    ^        v    ^
 ^          v  ^          v  ^
^            v^            v^

The light pulse goes in a zig-zag pattern. It still goes up and down, but also goes to the right. The path the light has to take is longer than when Kirk was stationary relative to us, because it has to move along a diagonal. We have presumed that the speed of light is independent of the speed of the source, so it can't go any faster simply because Kirk and the lightstick are moving fast past us. The distance that Kirk's light pulse travels and the distance that our light pulse travels must be the same.

Our clock makes one ticktock, like this:

^v
^v
^v
^v
^v
^v
^v

The light in Kirk's clock has to travel the same distance, about this much:

      ^v
     ^  v
    ^    v
   ^
  ^
 ^
^

Our lightstick has already made one ticktock. His light stick, with the path of light traveling a greater distance at the same speed, hasn't finished yet. We see Kirk's lightstick as going more slowly than ours.

Of course, Kirk doesn't see anything strange about his clock. In fact, when he looks at our clock, he sees it as going more slowly than his, in exactly the same way.

All of his clocks have to work at the same rate in his frame of reference. (Of course, they could be broken or not very good, but this can't depend on his speed relative to us.) Why? Well, if he had a pocketwatch that read a different time when he was moving, then he would be able to tell that he was moving by looking at the difference between his pocketwatch and the lightstick. If the premise of relativity is correct, that would be impossible. All of his clocks, provided they are all at the same location, also have to work the same when viewed from our frame of reference. If his lightstick sent out a red flash as a signal when it read 6:30, then Kirk could also send out a blue flash when his pocketwatch read 6:30. The flashes would come from the same place at the same time, and they both have to travel at the same speed of light, so we would see one purple flash.

Around now, someone suggests another idea. Maybe Kirk's ship got skinnier, contracted up and down transverse to the direction of travel. Things are already strange, so why not? Maybe as our lightstick made one ticktock, so did his, but more like this:

      ^v
    ^    v
  ^        v
^            v

That is, although the path is still stretched out horizontally, it is contracted vertically enough to make the total length the same.

This cannot be. To test it, we attach a hoop to our ship, just bigger than Kirk's ship and ask him to fly through it. He's a pretty good pilot, so he manages the trick with ease. We put the same hoop on his ship and have him fly the hoop around us. If he contracted transverse to his direction of travel, the hoop would contract as well, and it would break on our hull. Yet assuming that Kirk is as skillful this time, it would have to come out the same, because according to relativity, it is not possible to say which one of us is the one who is moving.

We're left with the conclusion that Kirk's clocks slow down relative to us when viewed from our frame of reference. Because all of his clocks slow down, in some sense time must slow down in his frame relative to ours. Of course, he sees time in our frame of reference slow down relative to his as well. Otherwise, it wouldn't be relative.

This is called "time dilation."

Those who like formulas and equations may notice that the path of Kirk's light pulses can be seen as forming two right triangles:

      /\
     /||\
    / || \
   /  ||  \
  /   ||   \
 /    ||    \
/-----++-----\

This observation plus a modified Pythagorean theorem and high-school algebra are enough to derive the Lorentz Transformations, the bulk of the mathematics of Special Relativity.

2. Nothing can go faster than light.

Kirk can't go faster than light in his frame of reference. If he sends out a light pulse forward, he will perceive it as going ahead of him at the speed of light. Since the speed of light does not depend on the speed of the observer, no matter how fast he tries to go, the light pulse will still be going away from him at the speed of light.

Can he go faster than the speed of light in our frame of reference? This wouldn't work, because then we'd see something different about the speed of light, which we have presumed is impossible. There's another way of looking at this. We have already seen what happens to Kirk's clock when he gives his engines one impulse:

      ^v
     ^  v
    ^    v
   ^      v
  ^        v
 ^          v
^            v

We tell him to give his engines two impulses:

            ^v
          ^    v
        ^        v
      ^            v
    ^                v
  ^                    v
^                        v

Three impulses:

                  ^v
               ^      v
            ^            v
         ^                  v
      ^                        v
   ^                              v
^                                    v

Four impulses:

                        ^v
                    ^        v
                ^                v
            ^                        v
        ^                                v
    ^                                        v
^                                                v

We might be tempted to say that Kirk is going double, triple, and quadruple his speed in the subsequent diagrams, but we don't really know that. All we can see are the light flashes from his clock in space, and we draw the triangles to trace out the path his light pulse makes in our frame of reference.

As Kirk tries to go faster and faster, the triangle becomes more and more stretched out. It can never stretch so far that it breaks. However fast Kirk tries to go, in our frame of reference, light still has to get there, and it must get there at the same speed. The shortest distance is a straight horizontal line. The zig-zag pattern of his lightstick will always be longer than that. He can never catch up to the speed of light.

This property of the speed of light as a maximum speed follows from the premise that the speed of light is constant. It isn't a barrier, like the speed of sound. It isn't the maximum speed because light happens to go at that speed. Rather, light goes at that speed because, as will be discussed later, the maximum speed is the only one at which light can go.

3. When objects go fast, they become more massive.

In our normal experience, masses resist being pushed. A car is more massive than a bicycle, and it's easier to push a bicycle than to push a car. Another way to look at this is to say that if we push a bicycle and a car with the same force, the bicycle will accelerate faster than the car.

When we push a massive object and accelerate it, it builds up momentum. Like velocity, momentum has an amount and a direction. Each impulse of Kirk's ship adds the same amount of momentum, as long as it is in the same direction.

Newton's three laws of motion describe how momentum works. The first law says that momentum remains the same without acceleration. The second law says that an applied force changes momentum proportional to the force and the time the force is applied, inversely proportional to the mass. The third law states that momentum is conserved. The conservation of momentum is also supported by the work of Emmy Noether, who showed that every symmetry in physics required a conservation law, and vice versa. We've already decided where exactly we do our experiments doesn't matter. This goes by the name of "translational symmetry." According to Noether's Theorem, it means that momentum is conserved as well.

In the previous section, we saw what happened when Kirk gave two, three, and four impulses. We saw that it was appealing to think that he went two, three, and four times as fast. After all, every impulse that Kirk puts out, he sees himself increasing in speed by the same amount. He can't tell any difference between impulse number 1 and impulse number 4. He also sees the blast from his rocket going away from him in the opposite direction, precisely fast enough and in the right direction to conserve momentum. So in his frame of reference, Kirk sees himself as speeding up by the same amount every impulse.

We see a different picture, but we also have to see momentum conserved. As has been shown, we can't see him going faster than light, so his impulses can't keep adding up speed. In our frame of reference, the faster he goes, the less difference an impulse makes on his speed. His inertia, his resistance to acceleration, is given by his mass. Since his resistance to acceleration increases with speed, we can say in some sense that as he goes faster relative to us, his mass increases in our frame of reference.

This is sometimes called the relativistic mass, as distinguished from the rest mass, the mass of Kirk's ship when it isn't moving relative to us.

Relativistic mass is related to rest mass by a factor called γ or "gamma." This factor appears in many equations in relativity and can be derived from the modified Pythagorean Theorem mentioned earlier. For our current purposes, suffice it to say that it is 1 for a relative speed of zero, climbs up slowly as the speed increases, and approaches infinity (actually, 1/0) as a limit as the speed aproaches the speed of light. It starts to climb so slowly that something has to be going more than 14% of the speed of light for the relativistic numbers to vary as much as 1% from the Galilean numbers. This is why Galilean/Newtonian motion works so well at low speed.

This shows another reason why an object cannot ever accelerate to the speed of light in any frame of reference. If it ever did, its mass would become infinite.

It's also possible to avoid the notion of relativistic mass, as long as we make sure that the factor of γ is in the right equations. Many physicists prefer to reserve the word "mass" only for the rest mass. There are many ways of formulating the equations, and which one to choose is a matter of convenience and preference.

4. When objects go fast, they contract in the direction of travel.

We give Kirk a second lightstick and ask him to hold it along the length or the ship and go past us at his original speed. The lightsticks are synchronized, so they flash at the same time. Since the speed of light is the same, the lengths of the paths must be the same, like this:

       ^v
      ^  v
     ^    v
    ^      v
   ^        v
  ^          v
 ^            v
  > > > > > > > > > >
               < < <
 1            3     2

Three events are marked on the diagram:

  1. A pulse of light is emitted from the lightstick at the stern of the ship.
  2. The pulse hits the mirror at the prow of the ship and bounces back.
  3. The pulse is detected at the stern of the ship.

Bringing back the pictures of the ship, we can see these three events as follows. The path of the light from the horizontal lightstick is still there as a guide, and the number of the illustrated event is underlined. Only the part of the ship we know about is drawn.

+--
|
|
|
|
|
|
|
+--
  > > > > > > > > > >
               < < <
 1            3     2

                   --+
                     |
                     |
                     |
                     |
                     |
                     |
                     |
                   --+
  > > > > > > > > > >
               < < <
 1            3     2

             +--
             |
             |
             |
             |
             |
             |
             |
             +--
  > > > > > > > > > >
               < < <
 1            3     2

At event 2, where is the stern of Kirk's ship in our frame of reference? If we assume that the ship does not contract in the direction of travel, we might guess this:

      +--------------+
      |              |
      |              |
      |              |
      |              |
      |              |
      |              |
      |              |
      +--------------+
  > > > > > > > > > >
               < < <
 1            3     2

This can't be right. The stern of the ship is as far from 3 as 3 is from 2. Since light has to come back from 2 to 3, and the stern cannot be going as fast as light, there will not be enough time for the stern to catch up. Instead, the picture must be more like this:

           +---------+
           |         |
           |         |
           |         |
           |         |
           |         |
           |         |
           |         |
           +---------+
  > > > > > > > > > >
               < < <
 1            3     2

Kirk's ship appears contracted in our frame of reference when it moves past us. This might be expected from the description of Lorentz contraction in Part I.

Some find ASCII diagrams unconvincing and are encouraged to try this at home or the office. Get a piece of string or yarn already stretched to its limit. Loop the ends with a slipknot, and using three pushpins, make a triangle like the path of the light on Kirk's vertical clock. The string is the path of the light. The wider and shorter the triangle, the more obvious the effect will be. Moving only the center pushpin, like drawing an ellipse, pull the string out to the right. This will be the path of the light on Kirk's horizontal clock.

We also notice something else from the horizontal path of the light on Kirk's lightstick: in our frame of reference, his ticks take longer than his tocks! If Kirk sat between the ends of the light stick, he would see the ticks and tocks as taking the same time. If he turned another synchronized lightstick the other way, he would see the signals as simultaneous. We would not see them as simultaneous.

The problems with simultaneity suggest that our common sense notions of "now" don't quite work with relativity. Barnard's Star is a bit more than four light years away, so in one sense, we see it as it "was" a bit more than four years ago. I wonder what it's like now. However, "now" doesn't really work with relativity. An observer the same distance between us and Barnard's Star might get some relative notion of now, but any other observer would disagree.

5. Even though it's the Theory of Relavitiy, the speed of light seems absolute.

We are used to seeing time and space as related through velocity. A moving object in our normal experience has a velocity that describes how it moves through space relative to time.

Velocity is a slightly different idea from speed. 40 mph is a speed. Velocity combines speed with direction. 40 mph due North is a velocity. The speed component relates distance (one measurement of space) and time, while the direction component adds more information related to space. In an important sense, the history of relativity described in Part I is the history of the increasing respect for the importance of speed.

Velocity, space, and time are all interrelated in the case of an object moving in a straight line at a constant speed. Given any two, it is easy to calculate the third in common experience. The most natural idea is that two of them are somehow basic or fundamental, and the third is a consequence of the two. Which two?

Space is pretty obvious; we can see it. On the other hand, we can ride a horse and watch the trees pass. Still, maps are useful after a couple of years, so perhaps we can count on space. Time we can't really see, but it seems basic in another way: it passes no matter what we do. It always seemed natural to define velocity and speed in terms of space and time.

Aristotle didn't consider velocity very important. He thought the cosmos was organized into concentric spheres, with the Earth at the center. Objects had their natural places, rock in the earth, and smoke in the air. An object moved away tried to return according to its natural motion. Based on Aristotle's ideas, medieval thinkers thought that invisible angels pushed the planets around the earth.

By the time of Galileo and then Newton, velocity was seen as more important. Galileo showed that the acceleration, understood as the change of velocity over time, of falling objects did not depend on the mass of the objects. Newton, with his first law of motion, showed that in the absence of acceleration objects kept the same velocity. Velocity seemed much more basic. As Richard Feynman pointed out, the medieval theory of invisible angels had to be modified; the angels only needed to push inward. The velocity around can be taken for granted; all that is needed is a change of the velocity toward another mass, such as the Sun.

During the 19th century, one particular speed, the speed of light, started to seem very important indeed. It is a speed that does not depend on the velocity of either the source or the observer. The speed of light seems, in a sense, absolute.

One way of looking at relativity is, when in doubt, respect the importance of speed, whether the constant speed of light or the relative speeds of ordinary objects. There are many different ways of looking at motion, but since speed seems so important, we try to use speed as a basic concept.

Just speed isn't enough; we need to understand where objects go as well as how fast. One number from speed cannot by itself explain the complexity of motion. Fortunately, velocity also has the idea of a direction in space. It also has a direction in time, which we usually ignore. Yet 1 meter per second North is the same as 2 meters per 2 seconds North. There are many ways to represent the same velocity as long as the change in space (called "translation") and the change in time is kept consistent.

Using these ideas, plus the idea of an observer to give a reference in space and time, we can define everything we can observe of motion around us, including relative space and time. It seems different from using space and time as the absolute bedrock of reality, but it works even better. It's the only way to look at motion that preserves the idea that the speed of light is constant.

Since speed is so important, we might intuitively expect it to work a bit differently from the direction component of the velocity. We have already seen that it does. It affects measurements of time, as in time dilation, and space, as in Lorentz contraction. We had Kirk going to the right, but it would have worked as well if he had moved to the left or top of the page or at an angle. We had him move in the plane of the page so that we didn't have to worry about the differences in distance from us. If he had gone toward or away from us, we would have had to take the differences in distance from us into account, which makes the math much harder, but the effects are the same.

The speed of light is presumed constant, but speed is not quite the same as velocity. We might expect that, although the speed cannot change, the other component of velocity, the direction, can change. This is true. The velocity of light can change, so long as it only changes direction and not speed. Other attributes of the light, such as the color (related to the energy and momentum), can also change. This is the source of the famous "red shift" of galaxies that are moving away from us. (Or we're moving away from them. We never can tell which; it's the basis of modern cosmology.)

Relativity also ties up one more loose end. Speed affects both time and space, but we normally use seemingly unconnected units for time and space, like seconds and meters or hours and miles. It would be nice to have a natural way to measure time and space in relation to each other. Fortunately, the speed of light is always constant no matter how we move, so we can use that speed to relate space and time. We have already used units where the speed of light is 2 sticks per ticktock. Physicists prefer to use units where the speed of light is 1. This is very close to measuring space in feet and time in nanoseconds.

6 . Time is the fourth dimension.

After so many seemingly bizarre implications, it's a relief to come across one that doesn't seem so odd. Since time and space are both affected by speed, it might be elegant to try to look as time and space as different aspects of the same thing.

We are used to seeing time used as a dimension with ordinary graphs. Consider sliding a ring randomly back and forth over a meter stick for two seconds. The path of the ring can be represented in a single graph like this:

1 m ^
    |      ***
    |  ** *   *
    | *  *     *
    |*          *
    |            *    **
    |             *  *
    |              **
0 m +------------------>
    0 s                2 s

In the previous diagram, space is represented vertically, and time is represented horizontally. When talking about relativity, it's conventional to show time vertically and space horizontally. Based on this, we can make a graph of space and time:

                           ^
                           |
                           |
                       Our | Future
                           |
                           |
                           |
                           |
To the Left <--------------+-------------> To the Right
                           |
                           |
                           |
                           |
                       Our | Past
                           |
                           |
                           v

The top half is our future and holds everything that can happen as a result of what we do. The bottom half is our past and holds everything in the past that can affect us. The left and the right show one direction in space. Of course, our space has three dimensions. We could make a model with sticks balls for two dimensions plus time, but we have to imagine a model with three dimensions plus time. In any event, additional dimensions in space work the same as the one drawn here.

We imagine ourselves sitting at the center, on the +. We always draw the diagram like this, with us at the origin, showing time up and down and space left and right. Other objects have paths in our diagram, but whatever we do, we use this diagram to look at the world. We always know that the + at the origin is where we are now. The other objects can construct their own diagrams.

This is a spacetime diagram of Galilean and Newtonian physics. It represents a world where speeds can be as fast as we like. Kirk, as well as any other object, traces a "world line" through this diagram. Objects that do not move relative us trace vertical world lines. Instantaneous light traces a horizontal world line. Objects moving between these speeds have world lines at an angle. As long as the object remains inertial and doesn't accelerate, the line is straight.

A diagram like this is fine, but is it real? All talk of dimensions is a human abstraction, but is it totally arbitrary, like plotting sales of ice cream in Boston against the price of tea in China, or is there some deeper physical connection? We'll try to look at this as an analogue of an ordinary map. In a map, we can find a point and call that a location on the map. In this diagram, we'll extend the idea of location to include time and call it an event.

In the Galilean/Newtonian spacetime diagram, our future consists of every event above the horizontal line. Our past consists of every event below the line.

The discovery that light had a finite speed poses a problem for this diagram. An infinite speed is horizontal, but we shouldn't expect to be able to draw a horizontal line for the speed of a real object. We need to modify the diagram by drawing two world lines for a pulse of light from where and when we are:

                           ^
             *             |             *
               *           |           *
                 *     Our | Future  *
                   *       |       *
                     *     |     *
                       *   |   *
                         * | *
To the Left <--------------+-------------> To the Right
                         * | *
                       *   |   *
                     *     |     *
                   *       |       *
                 *     Our | Past    *
               *           |           *
             *             |             *
                           v

In one dimension of space, we can only send a pulse of light in one of two directions. In two dimensions of space, we can send a flash of light in any direction on a 360 degree circle. The possible world lines of light form a cone, so this is sometimes called a light cone. In three dimensions of space, it becomes hard to draw.

Our future is now the area in the top, and our past is the area to the bottom. What of the areas to the left and the right? They are neither part of our past nor part of our future. Certainly there are events there, but we cannot see or affect them from here and now.

Using this diagram, we once again tell Kirk to pass us from left to right. We make a diagram of his world line, a snapshot relative the event where he is closest to us. It comes out like this:

                           ^
             *             |      /      *
               *           |     /     *
                 *         |    /    *
                   *       |   /   *
                     *     |  /  *
                       *   | / *
                         * |/*
            <--------------+-------------->
                         */| *
                       * / |   *
                     *  /  |     *
                   *   /   |       *
                 *    /    |         *
               *     /     |           *
             *      /      |             *
                           v

We might think, from looking at this diagram, that Kirk doesn't see light as receding from him in all directions at the same speed. After all, that / looks a lot closer to the * on right than it does to the * on the left in our future. Remember, though, that he sees us relative to him as we see him relative to us. He, too, can draw a spacetime diagram for what he sees of us moving relative to him. Assuming that we agree on the directions we call "left" and "right," his diagram looks like this:

                           ^
             *      \      |             *
               *     \     |           *
                 *    \    |         *
                   *   \   |       *
                     *  \  |     *
                       * \ |   *
                         *\| *
            <--------------+-------------->
                         * |\*
                       *   | \ *
                     *     |  \  *
                   *       |   \   *
                 *         |    \    *
               *           |     \     *
             *             |      \      *
                           v

It seems almost as if we saw Kirk's vertical line rotated relative to ours based on his speed relative to us, and he saw ours rotated in the opposite direction for the same reason. It can't be a simple rotation, because we can't rotate past the light cone: that would mean exceeding the speed of light. Still, the idea of rotation is appealing.

Ordinary maps have some interesting properties. Take a map small enough that the curvature of the Earth does not matter much, and we can pretend that it's flat. Say it's a map of Central Florida, and we want to fly from Tampa to Orlando. We draw an arrow from Tampa to Orlando on the map. A friend comes by and looks over our shoulders. Even though the point of view is different from ours, rotated and translated a bit, our friend still sees the same arrow going from Tampa to Orlando. It may be at a different angle, but the distance between Tampa and Orlando is the same as we rotate the map. If many people viewed the map around a round table, they would each see the arrow at a different angle, but they would also all agree on the distance.

It would be nice to have something like the distance that was the same in relativity for all observers, not only no matter what their position, but also no matter what their velocity. It turns out there is, and it's called the interval.

We're used to finding the distance between two points based on the Pythagorean Theorem. Consider this right triangle:

       b
      /|
     / |
  D /  | Dy
   /   |
  /    |
 a-----+
    Dx

The distance D between events a and b is related to Dx and Dy by the Pythagorean formula D2 = Dx2 + Dy2. Of course, it works in three dimensions as well. We want to put a difference of time, a Dt, in there somehow. Time already seems different from space. As Kirk's ship moves through space, his ship seems to get shorter, while his times seem to get longer, so we might expect it to be treated differently in the formula. It turns out that the interval is related to the distance in space and time as I2 = Dx2 - Dt2 in one dimension of space. The minus sign makes all the difference. This also works as well in three dimensions.

Another way of looking at this is that the minus sign comes from squaring Dt. If you square something and get a negative number, that means the original number was imaginary, some real number multiplied by i, the square root of -1.

Hermann Minkowski showed that the interval would not change using a coordinate system where the time axis was imaginary, scaled according to the speed of light. Furthermore, he showed that the Lorentz contraction and time dilation could be viewed as simple rotations in such a coordinate system. It turns out our intuition about rotation was correct.

Time can be considered the fourth dimension, in physically meaningful way that preserves the properties of maps, but only if it time is considered an imaginary number. This property also helps us understand why the speed of light is a constant. This spacetime diagram shows a few events:

                           ^
             *             |             *
               *           |           d
                 *         |    b    *
                   *       |       *
                     *     |     *
                       *   |   *    c
                         * | *
            <--------------a----------e--->
                         * | *
                       *   |    *
                     *     |      *
                   *       |        *
                 *         |          *
               *           |            *
             *             |              *
                           v

If this were a map of space, the distances would all be positive or zero, and there wouldn't be anything to distinguish those angles at 45 degrees. With the definition of an interval, things look more sensible. Going from a to b, there is a bigger difference in time than space, so the square of the interval is negative. We call this a timelike interval. Similarly, from a to c, there is a bigger difference in space than time, so square of the interval is positive. This is a spacelike interval. The interval from a to d is 0, and we call that a lightlike interval. The interval of 0 distinguishes those 45 degree angles.

Note that if there is no difference in time between events, as with events a and e, the interval is the same as the distance in space.

The problems with simultaneity describe earlier only apply to spacelike intervals. For every spacelike interval between two events e1 and e2, there are some observers who will see the events at the same time, some who will see e1 before e2, and some who will see e1 after e2. Fortunately, for timelike intervals, all observers will agree on the same order.

Many physicists prefer not to consider time imaginary and use different operators that handle the minus signs. Furthermore, it works as well to consider space imaginary and time real. Even better, there is a kind of number called a quaternion that has a single real number and three imaginary numbers and handles the right-hand rule by definition, but quaternions never really caught on. Mathematics has many different ways to express the same idea; which one to use is a matter of convention and preference.

7. E = mc2

This is an equation that most people know, so we need to use equations to understand it.

Remember the use of γ or "gamma" in the description of momentum and relativistic mass. Gamma depends on relative speed. Considering only massive objects, the momentum works out to be p = γm0v, where m0 is the rest mass of the object, and v is the velocity. This can also be written p = mrv, where mr is the relativistic mass, the same as γm0.

Moving objects have something other than momentum: kinetic energy. This is the amount of energy needed to get an object moving at a certain speed. For example, the amount the brakes of a car have to heat up to stop it is given by the kinetic energy of the car. There's another way to think about it. A force over a period of time imparts a change in momentum, while a force over a distance imparts a change in kinetic energy.

It's also possible to derive kinetic energy in relativity. It turns out to be Ek = (γ - 1)m0c2

That "- 1" seems out of place. This led Einstein to suggest that an object had some energy, not because it is moving, but simply because it is there and has mass. That energy would be the rest energy, given by the 1, and the kinetic energy would be the difference between that and the total energy when moving. This whole mess of equations can be summarized by one simple one: E = mrc2, or E = mc2 if we don't bother to write the "r". (Physicists seldom use this formulation, preferring to reserve m for the rest mass, m0.)

If we have the right kind of insight, we might notice a few things:

  • Energy uses one (scalar) number. Momentum uses three numbers.
  • Space is measured using three numbers. Time is measured using one number, even it it's imaginary.
  • Energy is changed by force over distance (space). Momentum is changed by force over time.
  • Space and time are related to each other, through Minkowski space or thinking of time as imaginary.
  • Maybe we can think of energy and momentum as related to each other in a similar way.

This is true. We can put time and energy into a 4-vector or quaternion, rotate it the same way according to relative motion, and get behavior similar to what we get from rotating intervals.

The only thing yet to understand is light. Light has energy. If it did not, it wouldn't cost anything to run a light bulb. Light also has momentum. At the speed of light, γ approaches 1/0.

We notice that γ is also multiplied by the rest mass for objects that have rest mass. If the rest mass is 0, then multiplying it by γ at the limit of the speed of light gives 0/0. 0/0 is undefined. It's the same as asking what number, when multiplied by 0, gives 0. Any number works, so 0/0 doesn't help us pick the right one. We have to conclude that light or anything going at the speed of light must have a rest mass of 0, and vice versa. Fortunately, quantum thinking and experiment give good values for the momentum and energy of light, which depends on its color and the number of photons.

Summary

Einstein's Special Theory of Relativity, which results in some seemingly weird implications, results from some rather simple and straightforward reasoning based on two premises given in Part I. Although a lot of the mathematics and equations have been left out of this description, it is still possible to get a pretty good quantitative idea of all the important aspects of the Special Theory.

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Related Links
o Part I: History
o Part II: Special Relativity
o Einstein
o Special relativity is "special"
o Pythagorea n theorem
o Lorentz Transformations
o Newton's three laws of motion
o Emmy Noether
o Noether's Theorem
o &gamma; or "gamma."
o Richard Feynman
o light cone
o interval
o Hermann Minkowski
o quaternion
o rest mass
o derive kinetic energy
o Light also has momentum.
o Also by epepke


Display: Sort:
Introduction to the Theory of Relativity Part II: Special Relativity | 84 comments (75 topical, 9 editorial, 0 hidden)
-1 I'm sorry but I hate all science (1.22 / 18) (#10)
by Tex Bigballs on Thu Jun 19, 2003 at 11:12:29 AM EST

after i was brutally gangraped by bill nye and mr. wizard.

Mr. Wizard was a bit creepy (3.00 / 4) (#11)
by epepke on Thu Jun 19, 2003 at 11:25:27 AM EST

I don't think that Bill Nye could remain still long enough to rape anybody, though.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Links, books, and audio (5.00 / 3) (#12)
by epepke on Thu Jun 19, 2003 at 11:28:20 AM EST


The truth may be out there, but lies are inside your head.--Terry Pratchett


Einstein once said... (3.25 / 4) (#13)
by Raindrop on Thu Jun 19, 2003 at 11:53:02 AM EST

...When asked to sum up the theory of relativity in one sentence, stated: "Time, space, and gravity have no seperate existance from one another." There now, wasn't that easier than a several-page long proof in ASCII art?
--
Many questions are unanswerable.
Many answers are questionable.

my favorite Physics book for non-physicists (5.00 / 5) (#14)
by modmans2ndcoming on Thu Jun 19, 2003 at 12:08:54 PM EST

(these books got me into physics in highschool)

Michio kaku:
Hyperspace

Michio Kaku & Jennifer Thomson:
Beyond Einstein

Kip S. Thorne:
Black Holes & Time Warps

and the grand daddy of em all......

Stephen Hawking:
A Brief History of Time (illustrated or not illustrated)

those were my favorites but there are a lot more I enjoyed reading rather than listen in my classes :-)


The complete equation for energy (5.00 / 4) (#15)
by Confusion on Thu Jun 19, 2003 at 12:18:24 PM EST

E = myc2

In practice, this formula is usually written as
E = ( m02c4 + p2c2 )1/2,
where p is the momentum of the particle. Otherwise, the formula isn't clear on what the energy of a photon is, since it has m0 = 0 (the relativistic mass term hides the complexity).
--
Any resemblance between the above and reality is purely coincidental.

Time Dilation (none / 0) (#16)
by PunkAssBitch on Thu Jun 19, 2003 at 01:25:55 PM EST

Great article, well written. Please give me a push down the straight and righteous path regarding time dilation:

The example with Kirk and his vertical light stick made sense - to us, Kirk seems to have slowed down.  But, how can it be possible that we seem to have slowed down to Kirk at the same time?  That would imply that from our and Kirk's respective perspectives, both our light sticks appear to be in conflicting states of tick-tock-ness - our light stick has completed a tick-tock, but Kirk thinks it has not; likewise, Kirk's stick has completed a tick-tock, but we think it has not.

It occurs to me that this could be due to the distance that's now between us ... Kirk's seeing our light stick as it was in the past due to the distance that his cruise past our ship has put between us.  But, if this is the case, then we can remove that distance by setting up the experiement such that Kirk passes our ship right when he sees his light stick as having completed one tick-tock, in which case it's tough to explain how we both appear to have slowed to the other.

Perhaps time seems to slow or quicken, depending on if you're receeding or approaching with respect to the observer?  I haven't quite thought it through, but this would seem to patch things up?

Answers (5.00 / 1) (#17)
by epepke on Thu Jun 19, 2003 at 01:51:40 PM EST

Please give me a push down the straight and righteous path regarding time dilation:

There is no one straight and righteous path. (You knew I was going to say that, didn't you?)

The example with Kirk and his vertical light stick made sense - to us, Kirk seems to have slowed down. But, how can it be possible that we seem to have slowed down to Kirk at the same time?

We hold a light stick up and down. Kirk sees it just the same.

That would imply that from our and Kirk's respective perspectives, both our light sticks appear to be in conflicting states of tick-tock-ness - our light stick has completed a tick-tock, but Kirk thinks it has not; likewise, Kirk's stick has completed a tick-tock, but we think it has not.

That's right. There is no one true absolute ticktock.

It occurs to me that this could be due to the distance that's now between us ... Kirk's seeing our light stick as it was in the past due to the distance that his cruise past our ship has put between us. But, if this is the case, then we can remove that distance by setting up the experiement such that Kirk passes our ship right when he sees his light stick as having completed one tick-tock, in which case it's tough to explain how we both appear to have slowed to the other.

That's right, too. Your intuition is correct. There are obvious effects due to distance and light travel, but you don't need relativity to understand them. We've set up Kirk to go past us at right angles to the direction of his travel so that, when he is crossing us, the differences in distance are small enough to be unimportant.

Perhaps time seems to slow or quicken, depending on if you're receeding or approaching with respect to the observer? I haven't quite thought it through, but this would seem to patch things up?

Your intuition is largely correct. There are effects depending on receding or approaching. There was a good description of this in the movie Young Einstein with Yahoo Serious. They can be used as good illustrations, if you have the right frame of mind. However, from Part I we have the premise that suggests that relative speed, not velocity with direction, is important for some of these effects. The idea of going toward or away from us doesn't patch things up.

Besides, we have Kirk going at right angles to us so the distances don't change. If the slight change in distance as he passes us seems a problem, we could just be farther away and make it as small as we like. The effect doesn't (and can't) change based on how far away we are.

That's why I put in the section about the importance of speed, which some people found out of place in the editing phase. I think it's an important concept for grokkage. The speed of light is constant, but the velocity is not.

This get more interesting with the so-called Twin Paradox. What happens when Kirk comes back, and we compare clocks? Obviously, we can't have ours reading earlier than his and his reading earlier than ours. I'll wait until somebody asks about that, though, because the part of my brain that tries to anticipate what people might find troublesome is kind of worn out. Also, there are people here who understand this as well or better than I, and they might have some ideas.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Twins (5.00 / 1) (#18)
by PunkAssBitch on Thu Jun 19, 2003 at 02:17:30 PM EST

You're right, the Twins Paradox does a better job of raising the same question - I feared making my first post unreadably long by including it as a way to rephrase the question.

Once the space twin arrives, both the clocks of the space twin and the earth twin can't read earlier than the other's at the same time. In fact, the earth twin is older, so he saw his space sibling's time slowing greatly, while the space sibling saw his earth counterpart's time speeding up ... right?  If so, then which components of motion can we use to explain who speeds up from the other's perspective and who slows down?

[ Parent ]

Twin paradox (5.00 / 3) (#21)
by epepke on Thu Jun 19, 2003 at 03:20:09 PM EST

OK, let's specify the twin paradox a little more carefully. The gravity of Earth can get in the way, so we'll have Kirk going away from Starbase One. Kirk leaves his evil twin Skippy here and flies off to Rigel, maybe to sleep with some green women or something. When he is going out, we see his clock going more slowly than ours. Then he gets enough and comes back, at the same speed. We also see his clocks going more slowly than ours. When he comes back, we think he should be younger than Skippy, and his clock should read less.

But wait! It's relative! On the way out and on the way back, he sees our clocks going slower than his. So, he should expect Skippy to be younger than he!

These cannot both be true, because we have to come up with some answer when we measure the clocks side-by-side. The answer is that Kirk is younger.

nce the space twin arrives, both the clocks of the space twin and the earth twin can't read earlier than the other's at the same time. In fact, the earth twin is older, so he saw his space sibling's time slowing greatly, while the space sibling saw his earth counterpart's time speeding up ... right?

Not quite. During the trip, he sees Skippy's clock slowing down relative to his. However, during the time he slows down, stops, and starts to come back, he's accelerating. Acceleration, unlike steady motion in a straight line, is not relative. He can tell he's accelerating without looking at something else because the rockets go off and he's pressed into his seat and everything's flying around the cockpit and hitting him in the face, and so on. During this period of time, he sees our clock speed way, way up relative to his. We might think that we can make the problem go away by having him accelerate more quickly, but the more quickly he does it, the more pronounced this effect is.

Because this deals with acceleration, which isn't completely specified until General Relativity (part III), most descriptions of the twin paradox put it off until GR.

However, there's also a way of looking at this using just Special Relativity. Thanks to dipierro and others for pointing this out. Because it's relative, all measurements that we make have to agree eventually no matter what the frame of reference.

First we'll try it in our frame of reference. We see Kirk's clock as moving more slowly both on the way out and on the way in. When Kirk comes back, he will be younger than Skippy.

Second, we'll try it in a frame of reference moving along with Kirk's rocket ship on the way out. We see Kirk's clock moving normally and Skippy's clock in Starbase One moving slowly. However, on the return trip, Kirk's ship is moving even faster relative to us. The curve of γ or "gamma" always slopes in a curve upward, so the effects of his greater speed relative to us on the way back more than compensate for any time Kirk gained on the way out. Kirk arrives younger.

Third, we'll try is in a frame of reference moving along with Kirk's ship on the way back. The situation is the same, except that Kirk is now going way fast compared to us on the trip out. The time he gains on the way back has already been more than compensated for on the way out.

All three agree, which we would expect, since it's relative. What they agree upon is that Kirk will be younger than Skippy.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Acceleration == The Secret Sauce (5.00 / 1) (#27)
by PunkAssBitch on Thu Jun 19, 2003 at 08:50:21 PM EST

Good answer.

It seems that the explanation based on GR acceleration effects you presented first could in fact be derived from a careful look at the special relativity cases you examined.  After all, the only difference between Kirk and Skippy's reference frames (aside from the hypothetical green booty-call that Kirk's travelling to make), is that Kirk's motion changes direction.  My point is that perhaps the special relativity conclusions reached when the direction of motion is reversed imply exactly the effect that acceleration MUST have on time dilation in order for thing to work out, since accleration is the only way to change direction.

Thanks, it's nice to have the ideas spoon-fed to me. Einstein and his homies were some sharp sons of bitches to put it all together the first time without even being able to ask anyone on K5 for answers.

[ Parent ]

I was under the impression (5.00 / 1) (#33)
by Ken Arromdee on Fri Jun 20, 2003 at 11:01:26 AM EST

that the answer wasn't really the acceleration, at least not directly. When Kirk is speeding by you, you can say that according in your reference frame you're motionless, he's moving, and he's got time dilation, but he can say that he's motionless and *you're* moving.

But if he turns around and comes back to you, he can't do that. He can claim that he's at rest on his trip out (so he had time dilation on his return trip), or that he's at rest on his trip back (so he had time dilation on his trip going out), but he can't claim he was at rest on both legs of the trip. Of course whichever he picks ends up giving him the same amount of time dilation as you get by assuming you're at rest all the time.

He doesn't have to lose any noticeable amount of time while accelerating, the acceleration just keeps him from treating both parts of the trip the same.

Disclaimer: I am not a physicist, but I hope some physicist can comment on this.

[ Parent ]

I think... (none / 0) (#50)
by hobbified on Sat Jun 21, 2003 at 11:13:19 AM EST

that it really amounts to the same thing. In either case, if he turns around, and you keep going at constant velocity (or at rest, whatever), then his acceleration "breaks the symmetry". What you just said makes sense as to why we couldn't see time measurements that don't hold up to relativity; GR tells us exactly how they come out. In either case, it's acceleration that makes it happen.

[ Parent ]
That's a good way to look at it (none / 0) (#57)
by epepke on Sat Jun 21, 2003 at 12:35:01 PM EST

The arguments behind Special Relativity are based on the idea of people moving around in constant-speed reference frames in a flat space. However, it turns out that SR is good for some of the effects.

In general, one can think of effects of acceleration in one of three ways:

  1. Effects that are equivalent to looking at Special Relativity in a slightly different way (such as here by using three moving frames of reference).
  2. Effects that are already handled by Maxwell's equations (which could be derived from SR but there's a lot of ugly math)
  3. Effects relating to gravity (which are better handled in GR)

In essence, Maxwell's Equations, SR, and GR are one big theory broken into chunks due to historical reasons but which also make them easier to understand.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
acceleration and the twin paradox (none / 0) (#73)
by tgibbs on Mon Jun 23, 2003 at 06:46:57 PM EST

The only problem with talking about acceleration is that people get the idea that somehow the acceleration itself slows the aging of the traveling twin. But the answer is the same if you assume that the traveling twin somehow magically switches from outgoing to incoming. The key point is that the traveling twin changes frames of reference (i.e. direction). In practice, of course, you can't change frames of reference without accelerating. But I'm sure what happens in a closed universe if the traveling twin goes all the way around without ever changing "direction." But then, I guess we can't escape getting General about it.

[ Parent ]
Good insight (5.00 / 3) (#35)
by epepke on Fri Jun 20, 2003 at 11:44:35 AM EST

It seems that the explanation based on GR acceleration effects you presented first could in fact be derived from a careful look at the special relativity cases you examined. After all, the only difference between Kirk and Skippy's reference frames (aside from the hypothetical green booty-call that Kirk's travelling to make), is that Kirk's motion changes direction. My point is that perhaps the special relativity conclusions reached when the direction of motion is reversed imply exactly the effect that acceleration MUST have on time dilation in order for thing to work out, since accleration is the only way to change direction.

Yes, that's exactly right. The arguments in General Relativity concerning the behavior of clocks, etc. according to acceleration are very similar to the arguments in the Special Theory and just as straightforward. GR goes beyond this to include gravity. Possibly, in an alternate universe, more of the arguments about acceleration would have been published in SR, reserving only gravity for GR.

Thanks, it's nice to have the ideas spoon-fed to me. Einstein and his homies were some sharp sons of bitches to put it all together the first time without even being able to ask anyone on K5 for answers.

Yes, they were. One thing I haven't harped on much but would like to convey is just how they were sharp. Einstein in particular, and to some extent Poincare, had a kind of innocent genius, an ability to stop thinking and forget what they already knew, which freed them up to look at things in a fresh, almost childlike way. It's part of the essence of true genius, and it's a terrible thing that education systems the world over do what seems to be their best in destroying this ability. Einstein pointed this out in a quote that is worth printing out in a large font: " is almost a miracle that modern teaching methods have not yet entirely strangled the holy curiousity of inquiry; for what this delicate little plant needs more than anything, besides stimulation, is freedom."

I have a lot of vitriol especially for the way science is normally taught in American public schools, as a dry collection of facts, suported by authority-based pedagogy. Mistakes happen, though. In High School, I was lucky to have two brilliant teachers in public High School. Mr. Malinsky, the biology teacher, often referred to natural selection as "sex and gambling." Mrs. Bradley, my physics teacher who taught us relativity and other "advanced topics", was particularly good. Once I remember we had done a classroom exercise involving trying to figure out some electron energy transitions for the Helium atom given its spectral lines. A classmate and I had worked up a theory. I don't even remember what it was, but that's not particularly important. We got some predictions out of it and handed them in. Later, we asked her, "C'mon, were we right?" Her answer, which I shall never forget, was "Nobody knows. But I've been keeping the results from classes over the years and have been looking to see if a pattern emerges." If she hadn't said this, I probably wouldn't know what science is. I'd probably think it is what most victims of the educational system think it is. But I got lucky.

My view, which an awful lot of people disagree with, is that genuis is not a matter of I.Q. (the two towering physics geniuses of the 20th Century, Einstein and Feynman, both tested modestly bright on I.Q. tests), or how much raw smarts you have in some other way, or how much you know, or how fast your brain works, but rather to an attitude, a kind of mental flexibility that lets you look at problems in a different way, that, if you will, lets you unlearn.

Normally, people suffer through years of classical mechanics and electrodynamics before getting to relativity. This is all useful, and of course there is a certain amount of "wax on, wax off" mental discipline that helps. But there's also a great danger of poisoning minds this way, of drilling in some ideas so effectively that it becomes impossibly difficult to turn away from them when you need to look at a problem in a new way.

All children are natural scientists. All of them need to learn how the world works. All play consists of scientific experiments, and all scientific experiments are a kind of play. Let's put this Tinkertoy/linear accelerator in with these Lego pieces/hadronic detectors and see what we get. This partly explains why some people are willing to pay a lot of money and endure enormous amounts of academic stupidity and ego attacks for a lifetime of genteel poverty in academia. It's the play.

It's a great shame that in his later years Einstein seemed to lose that ability. He was never quite able to free himself from thinking about God not playing dice with the universe. Otherwise, he might have been able to step aside long enough to be able to see the insight of a grand unified theory, the logical successor to General Relativity. Still, he did more in his life than the vast majority of people.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Cows (5.00 / 3) (#56)
by rusty on Sat Jun 21, 2003 at 12:34:58 PM EST

I suspect that a lot of high school teachers don't encourage the kind of thinking that makes good scientists because it can cause kids to be a huge pain in the ass.

In high school, my physics class stumbled on the general principle that all units of measurement are essentially arbitrary and convertible. I think our teacher was pleased that we grasped this, but perhaps less pleased that we decided our standard unit of reference would be the cow. We defined certain standards for the properties of the ideal cow, such as length, mass, and so forth, and used them as often as possible on tests and quizzes. Having only one name the cow does have the drawback that knowing which quantity you're describing is kind of contextual. So speed, for example, can be described as cows per second (where cow is a unit of length) but density could be expressed as kilocows per cubic cow (where the first cow is mass and the second cow is length cubed, or volume).

He put up with it, though, presumably on the principle that at least we were enjoying physics. I believe by the end of the year he was doing his answer sheets in both metric units and cows to make it easier to grade the tests.

My other great story from that class involved a test on classical motion. We had one question where you were at the top of a cliff shooting a cannon at someone at the bottom of the cliff and some distance away from it. Call the guy on top Bob, and the guy on the bottom Phill. So part one asks if Bob shoots his cannon at such an angle and gives the cannonball such an amount of force, will the ball hit Phill down at the bottom? So I did the math and the answer was that the cannon ball will land four meters short of Phill.

The second part was what Bob has to do in order to hit Phill. I thought about it for about three seconds and then wrote down "He needs to convince Phill to move four meters closer to the cliff."

Each part of the question was worth one point. When I got my test back, the second part was marked: "+1. I hate you."

____
Not the real rusty
[ Parent ]

We used toads in high school (none / 0) (#66)
by epepke on Sun Jun 22, 2003 at 01:13:15 PM EST

But when we got to quantum stuff, we used noses on blocks for h-bar.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Unlearning (none / 0) (#75)
by The Writer on Tue Jun 24, 2003 at 11:48:21 AM EST

Thank you, thank you, for pointing this out!

I've always disagreed with the notion of "smarter" == "higher IQ". My view is that nobody is born "stupid"; except perhaps if they have a medical condition. Everyone has the potential capacity to grok even the hardest subjects, in science or otherwise. The problem is that many people don't develop this capacity, which is, as you point out, the ability to unlearn and look at things in a brand new way. People tend to be lazy; after having learned things one way, they seem to resist dropping what they've learned to look at things in a new way. It's not that some people are "smarter" than others; it's just that some people refuse to unlearn what they've already acquired.

Like a friend of mine says, when you're in the middle of a chess game, you often fail to see things that are obvious to observers around you. That's because you are stuck in a certain mode of thinking, caused perhaps by the pressure to win the game. But the observers, being under no pressure, are freer in their perception, and thus can see things which are not immediately obvious to you. I find this very true; true insights often come when you stop what you're doing, take a step back, and look at the problem in a fresh new way.

Some people say that the reason math problems "work themselves out" if you take a break is because your "subconsciousness" is "working on it in the background". I disagree; I think that the reason this happens is precisely because you stopped thinking about it. Your mind has stopped working at it, and therefore, it is able to break out of your current mindset; and when you return to the problem, you are able to look at it in a fresh new way, which reveals insights that would never have occurred to you in your previous mindset.

In other words, you're right on, that "smartness" is not something people are born with; it's an attitude that people develop. An attitude that allows one to set aside learned conventions temporarily, so that a fresh new insight is able to impress itself upon you without being rejected outright by said conventions.

The ability to unlearn is a greater learning than the ability merely to learn.

[ Parent ]

So the next question obviously is... (5.00 / 3) (#28)
by TypographicalError on Thu Jun 19, 2003 at 11:34:01 PM EST

What if Kirk and Skippy are living in a hyperspherical universe, and Kirk needn't accelerate to get back to his departure point?

--
I had dreams of covering my entire body in light blue cottony sugar and chasing down young children. - Parent ]

Still ... (5.00 / 1) (#34)
by PunkAssBitch on Fri Jun 20, 2003 at 11:09:46 AM EST

Even if he takes a straight ride across (around?) the entire hypershperical universe, Kirk's got to accelerate to get going in the first place, and decelerate to park his ride once he gets back home to compare clocks.

[ Parent ]
Actually... (5.00 / 1) (#39)
by TypographicalError on Fri Jun 20, 2003 at 12:18:50 PM EST

The question here is not really the twins. It's the time dilation they experience. So what you could do is simply take note of the time in the "moving" body as it first passes something, then note the time when it laps it again. You could thusly maintain constant velocity. Then, you'd compare the time lapse there to the time lapse observed from the "stationary" body.

Basically, you're getting two lap times for the moving body. I'm not sure if it's necessary that they agree, but it seems like they should.

--
I had dreams of covering my entire body in light blue cottony sugar and chasing down young children. - Parent ]

To hell with twins, look at triplets! (none / 0) (#46)
by BlaisePascal on Fri Jun 20, 2003 at 05:17:34 PM EST

You don't have to be stopped relative to someone to compare clocks.  You just have to be in the same place at the same time -- or more specifically, know that one point (x,y,z,t) in your reference frame is (x',y',z',t') in another reference frame.

But to show that accelleration isn't necessary for the twin paradox, let's look at a three-person experiment instead.

No one is accellerating in this experiment.  Everyone is travelling at constant velocities, and they sync their clocks when they pass each other.  In Alice's reference frame, Bob is travelling to the right at 0.866c and Chuck is travelling to the left at 0.866c.  Both Bob and Chuck's clocks are going half-speed relative to Alices'.

Alice can watch Bob and Chuck via a telescope.  She knows how fast Bob and Chuck are going, and how far away they are, so she can do things like: when receiving a signal from Bob Bob sent when he was 1ly away figure out that Bob sent it 1 year (her time) prior to when she got it, and adjust her records accordingly.  Bob and Chuck are similarly equipt, and knowlegable.

There are three events of interest here.  An "Event" is a specific location in space-time, but the coordinates for the event will be different for different observers.

Event 1: Bob and Alice meet. Bob sets his clock to Alice's time.
Event 2: Bob and Chuck meet. Chuck sets his clock to Bob's time.
Event 3: Chuch and Alice meet.  The two of them compare clocks.

From Alice's viewpoint, the coordinates of those three events are:

E1 = (0ly,0y)  (0 light-years, 0 years)
E2 = (8.66ly, 10y)
E3 = (0ly,20y)

From Bob's viewpoint, the coordinates of those three events are:

E1 = (0ly,0y)
E2 = (0ly,Ty)
E3 = (-2Dly,2Ty)

From Chuck's viewpoint, the coordinates are

E1 = (-2Dly,0y)
E2 = (0ly,Ty)
E3 = (0ly,2Ty)

The question is...  What are T and D?  If spacetime was Gallilean as opposed to Minkowski, then T would be 10y, D would be 8.66ly, and Alice and Chuck's clocks would match at event 3.

Let's say that Bob is sending out a daily status report radio signal to Alice.  Based on his speed, Alice sees his clock slowed by a factor of 2, so she sees a new daily status report in a (distance-corrected) time of once every other day.  It's actually longer, since each "day's" status report is delayed by an additional 1.733 light-day's worth of distance.  She gets a new status report every 3.733 days, or a year's worth of reports every 3.733 years.  After 18.66 years (10 years travel time, 8.66 years light-speed delay) she gets a status report indicating that after 5 years (Bob time), Bob passed Chuck.  So T, in this case, is 5y.  And therefore, D has to be 4.16ly.

So, 5y after Bob passed Alice, Bob meets Chuck.  5 y after Chuck passes Bob, Chuck meets Alice.  From the point of view of Chuck and Bob, it's been 10 years.  However, Between Alice meeting Bob and Alice meeting Chuck, 20y have passed for her.

This is functionally equivalent to the Twin paradox, but doesn't involve any acceleration.  How does it work?


[ Parent ]

Ooh, nice one! (5.00 / 2) (#38)
by epepke on Fri Jun 20, 2003 at 12:02:13 PM EST

If the Universe is closed, then we would be living in a hyperspherical Minkowski manifold that is locally Minkowski space. Just looking a the spatial components, it would be a standard hypershperical manifold that is locally Euclidean. Special Relativity assumes that the times and distances are local, so that the overall curvature of such a manifold can be ignored. So, what you'd have to do is solve the equations of General Relativity, which does allow for a wide variety of manifolds in terms of this hyperspherical manifold and use them instead. The math is gonna be a lot harder.

'Bob' is said to have said once, "Bleeding head good. Healed head bad." For real cranial blood loss, try applying GR to manifolds with cusp points (a black hole might be viewable as a kind of cusp point), Lobachevskian hyperbolic manifolds, and even orbifolds. With an orbifold, we don't even need a twin, because we'll run into ourselves sooner or later.

This is quite a valuable thing to do, as it can be used for all sorts of cosmology. You might be able to get a paper out of it. There's also some fairly good science fiction based on these ideas.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Too bad I'm not a physicist. (5.00 / 1) (#40)
by TypographicalError on Fri Jun 20, 2003 at 12:22:35 PM EST

Though I suppose that theoretical physics is pretty much isomorphic to the navel-gazing mathematics that I'll be doing for the rest of my life anyways.

On the other hand.. I have problems comprehending time dilation (which is pretty much the critical point of SR, I think), and I've never tried GR.

--
I had dreams of covering my entire body in light blue cottony sugar and chasing down young children. - Parent ]

Wait.. (5.00 / 2) (#19)
by Kwil on Thu Jun 19, 2003 at 02:22:55 PM EST

..how can Kirk come back and compare clocks without relativistic effects taking hold again, but this time in reverse?

That Jesus Christ guy is getting some terrible lag... it took him 3 days to respawn! -NJ CoolBreeze


[ Parent ]
Excellent (5.00 / 1) (#20)
by epepke on Thu Jun 19, 2003 at 02:49:40 PM EST

how can Kirk come back and compare clocks without relativistic effects taking hold again, but this time in reverse?

This is why I enjoy talking about what is an admittedly geeky subject. Some people say that relativity is too weird for intuition and common sense. Not only is this patronizing and insulting, but in my experience it's wrong. I've found that common sense and intuition is actually very good, and your intuition is a case in point.

It isn't exactly right, though, because the relativistic effects are partly based on speed. When Kirk comes back, his velocity is reverse, but he has the same speed. (We'll presume he uses the same number of impulses to stop and accelerate back; otherwise the math gets hard.) But you're basically correct, that there's something that Kirk does when coming back that makes it all come out in the wash and makes us agree on what the clocks say. What he does is turn around and accelerate back. I'll describe this in more detail in another response.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Why I love relativity (5.00 / 1) (#55)
by rusty on Sat Jun 21, 2003 at 12:17:38 PM EST

Some people say that relativity is too weird for intuition and common sense. Not only is this patronizing and insulting, but in my experience it's wrong.

I suffered through a year and a half as a misplaced physics major in college. Ultimately the math defeated me, so it should be no surprise that relativity was my absolute favorite subject. The test we had after the relativity section of one course was basically an essay test. We had a few fairly classic examples of relativistic effects, and we had to figure out things like whether or not thieves carrying a ladder and running at relativistic speed would fit entirely inside a barn or not. The math is dead simple middle-school algebra, so I could handle that. The hard part of the test was that for each problem, you didn't just have to run the math, but you had to explain why the answer was so.

I was the only one in the class who got an A on that test. My average was dreadful, but when you gave me an easy math problem and something complex to explain in words, I was a star. That test went a pretty long way toward convincing me to change my major to something heavy on critical theory instead of math.

____
Not the real rusty
[ Parent ]

This is great ... but (none / 0) (#22)
by zenboy on Thu Jun 19, 2003 at 03:57:52 PM EST

...you do know you have to do one on Quantum Mechanics next. Someone out there has to explain to me why the Copenhagen interpretation is the only one worth taking. (Tried In Search of Schrodinger's Cat and that book suffers from it's own version of relativistic problems.)

Oog! Not now. (5.00 / 1) (#23)
by epepke on Thu Jun 19, 2003 at 04:15:51 PM EST

I've thought of descriptions of QM, but Feynman did such a great job in the QED in NZ tapes that I don't think I can add anything.

Besides, if I were to do it, I would have to say that which interpretation of QM you choose is a matter of taste and disposition and nothing else. When they make different predictions, then you can talk about them, but so long as they don't, who cares? Pick what you like. The whole idea of the interpretations is that they aren't scientifically differentiable, so it isn't a scientific question of which one to pick. Personally, I pick an interpretation different from all the common ones but which makes sense to me, but that's because my brain is very strange. I haven't had much luck communicating it to people.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Why we use the Copenhagen interpretation (5.00 / 2) (#25)
by BrentN on Thu Jun 19, 2003 at 04:44:42 PM EST

Because it conforms to what my graduate-level QM prof refers to as "The Principle of Least Silliness."

Translated, that means that the implications of the Copenhagen interpretation are less odd and difficult to work with than any others proposed.

[ Parent ]

It's not (none / 0) (#77)
by jbuck on Thu Jun 26, 2003 at 12:32:25 AM EST

The Many Worlds interpretation seems more workable for quantum computing, and gets rid of the mystical stuff that the new age folks like.

[ Parent ]
infinite relativistic rest mass (5.00 / 1) (#24)
by ethereal on Thu Jun 19, 2003 at 04:41:37 PM EST

Great series, by the way. I was really interested in this stuff when I was younger, but I'm glad to get the chance to revisit it and really understand how Einstein worked this stuff out. Thanks.

One question, though - if the relativistic mass of an object is divorced from its rest mass, as you mentioned in the previous article, then what exactly stops it from becoming infinite if the object reaches the speed of light? I accept the argument that the rest mass can't become infinite, just out of common sense, but by making the relativistic mass a much more nebulous entity, I'm not sure that it's as convincing to say that it cannot become infinite. Is it just because the concept of relativistic mass allows the rest of the theory to work so well?

I'm not really disputing special relativity; I just wonder how physicists really think through this without too much cognitive dissonance.

--

Stand up for your right to not believe: Americans United for Separation of Church and State

Well, ya know (5.00 / 2) (#26)
by epepke on Thu Jun 19, 2003 at 05:27:00 PM EST

One question, though - if the relativistic mass of an object is divorced from its rest mass, as you mentioned in the previous article, then what exactly stops it from becoming infinite if the object reaches the speed of light?

Let's say it's a golf ball, or a pea, or even a single electron. If you're pushing it, where are you going to get the energy to accelerate it to the speed of light? You need an infinite amount. The gas stations have already run out. And you could convert the entire rest of the universe into energy, and it still wouldn't be enough.

Is it just because the concept of relativistic mass allows the rest of the theory to work so well?

I have to reiterate that it's popular nowadays to use the term "mass" only to refer to the rest mass and use the term "energy" for the relativistic mass/energy. The idea works just as well so long as we understand E = mc2 or E = mrc2 or E = mγc2 or however you want to put it. I got called to task in the editing process for using the concept of a relativistic mass at all. Depending on how you like your equations, you can use it or not use it. I used it anyway, because in my experience the concept of pushing a very heavy bowling ball seems much more intuitive than the concept of momentum, especially for people who haven't taken a physics class in years.

I just wonder how physicists really think through this without too much cognitive dissonance.

In my experience, physicists tend to go for heavy English ales, except for string theorists, who tend to like espresso.

I know this does not seem like a very serious answer, but it's the kind of non-seriousness that's very serious underneath. There is a certain amount of pain involved in wrapping one's brain around these concepts. Cognitive dissonance is par for the course, because we have to give up some concepts that seem so obviously basic. After a while, it all collapses and makes sense. The only untoward side-effects are that your hair starts sticking out in all directions unless you use a lot of conditioner and a certain glassy eyeball look that you can control with some effort. But even Einstein got laid rather a lot, so there's still hope.

At least it's easier than quantum stuff, which pretty much consists of a lifetime commitment to a hurting brain. It's possible to get past this, too, but if you do, you learn to hide it very well so people don't put you in a rubber room.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
thanks for your explanations (5.00 / 1) (#41)
by ethereal on Fri Jun 20, 2003 at 01:43:37 PM EST

I think that makes sense, more or less. And thanks for your efforts to explain various concepts throughout your series of articles.

--

Stand up for your right to not believe: Americans United for Separation of Church and State
[ Parent ]

Definition of mass (5.00 / 3) (#43)
by Rich0 on Fri Jun 20, 2003 at 03:22:27 PM EST

Perhaps another way of explaining it might help.  Mass is nothing more than the property of an object that causes it to resist accelleration.  If I push a bowling ball and a car with the same force, the bowling ball will accellerate more.

As you increase in speed, your mass increases (at least when viewed from outside).  An outsider would observe that you would need more and more force to make smaller and smaller changes in velocity.  No amount of force would actually get you to the speed of light.

Actually, if an object with mass were created moving at the speed of light, this would be possible.  Of course, that object would essentially drag the entire universe behind it - and you wouldn't want to get in its way!  Since gravity propagates at the speed of light we're pretty sure there is nothing like this in the universe within a radius of the speed of light times the age of the universe.  But I don't think the laws of physics would be violated if somehow something like this popped into existance.  

[ Parent ]

Just a quick comment about mass (5.00 / 1) (#68)
by manobes on Mon Jun 23, 2003 at 02:35:26 AM EST

I have to reiterate that it's popular nowadays to use the term "mass" only to refer to the rest mass and use the term "energy" for the relativistic mass/energy.

"Popular" here should be taken to mean "universally taught and used by virtually all practioners". In high energy physics, relativistic mass is never used, ever. Mass=rest mass.

I just wonder how physicists really think through this without too much cognitive dissonance

They don't :) Those who use special relativity (by and large high energy physicists) use "mass=rest mass" only. Anyway, it's not that confusing since it basically amounts to a choice of where you're going to put the gamma.

At least it's easier than quantum stuff, which pretty much consists of a lifetime commitment to a hurting brain. It's possible to get past this, too, but if you do, you learn to hide it very well so people don't put you in a rubber room.

It's not that bad really. After working with it for a few years, most people adopt a pretty good working interpretation of quantum mechanics.

No one can defend creationism against the overwhelming scientific evidence of creationism. -- Big Sexxy Joe


[ Parent ]
What if Kirk points his light stick toward us (5.00 / 1) (#29)
by frankwork on Fri Jun 20, 2003 at 01:39:43 AM EST

Wow, that sounds dirty :P.

My physics prof mentioned in passing that Lorentz contraction doesn't work exactly like most textbooks describe it.

Let's say that Kirk's ship is going by at half the speed of light. It looks like it's been rotated 45 degrees, since the light coming off the far rear corner of the ship takes about as long to reach the front of the ship as it takes the front of the ship to travel half a stick forward. IOW, you see the back and the side of Kirk's ship.

Does this tie into Minkowski rotation other than that they both have "rotation" in the name? I guess this isn't really observable with stars, since they look pretty much the same from every angle.

Good point (5.00 / 1) (#44)
by epepke on Fri Jun 20, 2003 at 03:29:43 PM EST

There's a whole bunch of phenomena that I omitted which are not based on the relative speed but on differences in distance from us. One of the reasons that I omitted them was that these are all fairly straightforward and can be derived from ordinary Galilean insights. This is one of them.

Let's say that Kirk's ship is going by at half the speed of light. It looks like it's been rotated 45 degrees, since the light coming off the far rear corner of the ship takes about as long to reach the front of the ship as it takes the front of the ship to travel half a stick forward. IOW, you see the back and the side of Kirk's ship.

This is something that was studied much later, less than 40 years ago, first by Roger Penrose (I think; I don't have a link handy). When we take a picture of Kirk's ship, the picture looks like it's mooning us.

This isn't really the same thing as Lorentz contraction or Minkowsky rotation. For one thing, it really isn't a rotation; it's really a skew (which looks a little bit like rotation for small values, anybody who's ever used Open GL probably knows the difference). There would be a very slight percieved rotation in addition to this from some of the effects of the camera making the picture in perspective, but we could make this arbitrarily small by going farther away.

Furthermore, we would "see" the same thing if we used something much slower than light to do the measurements, such as sound. If we used sonar (of a special kind that used ambient sound, and we made a picture of the sounds that arrived at one time), we would see the submarine mooning us, too, even though it isn't going anywhere near fast enough for Lorentz contraction to matter. It's a separate effect, perfectly well predicted by classical mechanics.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
The most important question... (none / 0) (#30)
by Pseudonym on Fri Jun 20, 2003 at 01:42:03 AM EST

Where can I buy one of those trains with sealed windows which travel at relativistic speeds that physicists keep talking about? Is it from the same place as you buy the cylinders of ideal gas?



sub f{($f)=@_;print"$f(q{$f});";}f(q{sub f{($f)=@_;print"$f(q{$f});";}f});
Relativistic speeds (5.00 / 1) (#31)
by Ranieri on Fri Jun 20, 2003 at 06:59:42 AM EST

What you consider to be a "relativistic speed" largely depends on what you can measure. If your instruments are sensitive enough to detect relativistic effect while going at everyday speeds, then for all purposes it's a relativistic velocity.

So realistically all we need is a regular train with sealed windows and very very very precise measuring equipment. I have no idea where to get the equipment from, but I hear Kim Jung Il has got the train ...
--
Taste cold steel, feeble cannon restraint rope!
[ Parent ]

I hear (none / 0) (#36)
by i on Fri Jun 20, 2003 at 11:45:07 AM EST

Soyuz offers some tourist class seats. You'll need some mighty expensive (but by no means unavailable) clocks and rods, and a few million bucks for the ticket itself, but they say it's worth every penny.

and we have a contradicton according to our assumptions and the factor theorem

[ Parent ]
Doing the experiments (5.00 / 1) (#45)
by epepke on Fri Jun 20, 2003 at 03:38:05 PM EST

Actually, I think that given a couple of bars of invar metal, I think that one could probably put together some good enough lightsticks for not to many bucks of Radio Shack parts. We actually did the Michelson-Morley experiment in high school using lasers and mirrors and interference patterns. It would be even cheaper now, with semiconductor lasers. A $5 light pen and a half-inflated bicycle inner tube to get rid of vibration, and you're in business. I pity the fools who had to do this stuff with vats of mercury and gas lights and parts built by guys in handlebar moustaches on lathes run from belts connected to a shaft near the ceiling.

It also reminds me of an experiment done way back when, though not quite that way back. They got a cesium clock, put it on a plane, and flew it around the world. Today, you could probably ship it Fed Ex.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
I know a store that has both of those right down.. (none / 0) (#42)
by jforan on Fri Jun 20, 2003 at 03:08:45 PM EST

the imaginary street from me.  Unfortunately, both the ideal gas and the ticket on the ship cost an infinite amount of money.

Jeff

I hops to be barley workin'.
[ Parent ]

But (none / 0) (#53)
by rusty on Sat Jun 21, 2003 at 12:06:14 PM EST

If you need a Frictionless Plane, a Standard Horse, some Inextensible String, a Point Particle, or a Singularity, I know just the place.

____
Not the real rusty
[ Parent ]
Well... (none / 0) (#74)
by Pseudonym on Mon Jun 23, 2003 at 08:50:20 PM EST

...I'm still trying to find a place which sells tape for my Universal Turing Machine.



sub f{($f)=@_;print"$f(q{$f});";}f(q{sub f{($f)=@_;print"$f(q{$f});";}f});
[ Parent ]
Here it is (none / 0) (#47)
by epepke on Fri Jun 20, 2003 at 06:15:02 PM EST

I knew someone had to have done this.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Motion, Maxwell's equations, and a lonely electron (none / 0) (#32)
by zentara on Fri Jun 20, 2003 at 07:47:51 AM EST

All this talk of relativistic motion, reminds me of another "mind boggler". Imagine an electron floating alone in space. If you(the observer) remain "stationary" with respect to the electron, you will only see an electric field emanating from the electron. If you are in motion with repect to the electron, a magnetic field appears. And if the electron is oscillating back and forth with respect to you, you will see photons(electo-magnetic fields). If you can somehow match the motion of the electron, so you are "relativistically stationary" with respect to the electron, you will once again see nothing but electric field. How does motion(and by inference Time) change the electric field into a magnetic field, and possibly photons? I think our brains are limited by a 3-dimensional view of things, with Time being measured by "motion". The "higher beings" in the cosmos probably see things in 15 or 20 dimensions, and probably laugh at the way humans try to project their 3d view onto all of the cosmos. Om............

Magnetism is nothing but a relativistic effect (5.00 / 1) (#37)
by levsen on Fri Jun 20, 2003 at 11:54:58 AM EST

You are right, magnetism is a relativistic effect. It can be fully explained by the special theory of relativity. So far for saying that the theory of relativity has not meaning in the real world. If you work out the equations, the "magnetic force" is in reality just the relativistic component of the electric force.
This comment is printed on 100% recycled electrons.
[ Parent ]
Your .sig (5.00 / 1) (#52)
by rusty on Sat Jun 21, 2003 at 12:02:59 PM EST

Are the recycled electrons relativistic, classical, or quantum? I think you have to use different bins.

____
Not the real rusty
[ Parent ]
Dimensions (5.00 / 1) (#59)
by epepke on Sat Jun 21, 2003 at 06:39:41 PM EST

It's already been pointed out that electromagetism is just a relativistic effect.

The "higher beings" in the cosmos probably see things in 15 or 20 dimensions, and probably laugh at the way humans try to project their 3d view onto all of the cosmos.

Actually, there's good reason to think that there are 3, 7, 15, or 31 spatial dimensions, with no other numbers possible. (Or else 1 or 0, but that doesn't work.) String theorists aren't even remotely higher beings.

Four string theorists--one French, one Russian, and two American--get together at a conference. The French string theorist poured some champagne and throws the half-empy bottle out the window, explaining "In France we have so much fine champagne we can afford to waste it." The Russian dishes out some caviar and throws the rest out the window. "In Russia we have so much fine caviar we can afford to waste it." One American string theorist looks at the other and throws him out the window.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
some notes (none / 0) (#48)
by mveloso on Fri Jun 20, 2003 at 09:05:27 PM EST

Light exceeds the speed of light all the time.

Infinity in a mathematical equation doesn't mean that the value is immesurably large, it means that the value becomes undefined, which is not quite the same thing.

look up 'exceed' in the dictionary. [nt] (none / 0) (#49)
by fae on Sat Jun 21, 2003 at 01:17:27 AM EST



-- fae: but an atom in the great mass of humanity
[ Parent ]
Duh - pay attention to new stuff [nt] (none / 0) (#62)
by mveloso on Sun Jun 22, 2003 at 01:27:35 AM EST



[ Parent ]
Eh? (5.00 / 1) (#51)
by rusty on Sat Jun 21, 2003 at 12:01:09 PM EST

Light exceeds the speed of light all the time.

Light goes slower than c (which is specifically the speed of light in a vacuum, though this is often not mentioned) all the time. Like in any other medium than a vacuum. In fact, the way it does this is also particularly freaky and interesting.

But as far as I know, it doesn't exceed c.

Which makes me wonder why not actually. The reason anything with mass can't go faster than c is that it would require an infinite amount of energy to get it going that fast. So why is light or anything without mass bound by the same restriction? Any physicists have a quick answer to that?

____
Not the real rusty
[ Parent ]

Index of refraction (5.00 / 2) (#58)
by epepke on Sat Jun 21, 2003 at 01:25:48 PM EST

Light goes slower than c (which is specifically the speed of light in a vacuum, though this is often not mentioned) all the time. Like in any other medium than a vacuum. In fact, the way it does this is also particularly freaky and interesting.

Perhaps this is what you mean by freaky, but of course light doesn't slow down even then; it merely takes longer to pass through the solid. The truly predicitve-in-all-cases explanation requires Quantum Electrodynamics and is based on the interference of the amplitude of the electron with the amplitude of a photon resulting in maximum probability paths for the photon that look a bit like a slalom (except that all possible ways of doing the slalom are represented). An explanation that is easier to understand consists of modeling the interaction of the photons and the electrons in the solid by Feynman diagrams and saying that the light is absorbed by the electron and re-emitted some time later. (Superficially this seems like a different explanation, but if you use all the appropriate math, they're equivalent.) You could always shoot something like a neutrino through a piece of glass and have the neutrino outpace the photon in the long run, but that's just because it doesn't interact the same way with the electrons.

Which makes me wonder why not actually. The reason anything with mass can't go faster than c is that it would require an infinite amount of energy to get it going that fast. So why is light or anything without mass bound by the same restriction? Any physicists have a quick answer to that?

As was pointed out in the recent excellent article, infinity is tricky. What happens is a denominator goes to zero. You can plug in speeds greater than c, no problem, and the infinites go away. Except that the denominator becomes negative, and when taking the square root to get γ, it's an imaginary number. People have come up with a name for particles of this kind: tachyon. They'd be sort of like our normal matter and antimatter but on the other side of c. In any event, like our matter and antimatter, anything going at a tachyon-like speed would have to have a non-zero rest mass (possibly an imaginary one). Photons don't have rest masses, so they can't go faster than c for the same reason they can't go slower than c.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Yes (5.00 / 1) (#60)
by rusty on Sat Jun 21, 2003 at 08:29:42 PM EST

My fuzzy memory of this stuff has dredged up some concept that light doesn't actually slow down. I don't think I ever quite got that. I always thought the fact that it takes the path of least time through a solid was super cool though. Cue the inevitable drowning swimmer example. :-)

____
Not the real rusty
[ Parent ]
Least time (5.00 / 1) (#65)
by epepke on Sun Jun 22, 2003 at 11:47:21 AM EST

I always thought the fact that it takes the path of least time through a solid was super cool though.

It is an amazing thing. It's very easy to understand in terms of QED. What happens is that the you look at all the nearby paths that the light could possibly take. The least time path is the one such that the amplitudes nearby reinforce rather than destroy the amplitude of the least-time path.

Of course, one could imagine that if you could change that property, then light wouldn't take the least-time path any more. That you can do, by frosting or sandblasting the surface of the glass or, when you make the glass, putting a lot of chunks of something opaque in it. Then, of course, the light gets scattered all over the place.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Infinity (none / 0) (#64)
by mveloso on Sun Jun 22, 2003 at 01:35:20 AM EST

If you think of mathematics as a language that describes certain conditions, an infinite (or undefined) answer means exactly that - the results are, if not indeterminate, then uh, undefined. Basically the result is outside of the scope that can be described. Weird, huh?

[ Parent ]
Singularity (5.00 / 1) (#67)
by epepke on Sun Jun 22, 2003 at 01:22:17 PM EST

If you think of mathematics as a language that describes certain conditions, an infinite (or undefined) answer means exactly that - the results are, if not indeterminate, then uh, undefined. Basically the result is outside of the scope that can be described. Weird, huh?

That's one way that a singularity can come about, a situation where the mathematics can break down. It isn't really undefined in the sense that 0/0 is undefined, but it is usually undefined in the sense that the equation at that point can't tell you proper answers. This, of course, is true in relativity--the standard kinematic equations can't tell you anything about the momentum or energy of a photon. With γ, however, there's also a smooth, differentiable curve that has values for an arbitrarily small ε away from c. In this case, the intuitions about infinity match what you can see anywhere near that point. This is not always the case in all equations. And, of course, there are other ways for a value to be undefined, such as the sum 1 - 1 + 1 - 1 + 1...


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
A relay? (none / 0) (#76)
by symtry on Wed Jun 25, 2003 at 07:37:03 PM EST

This thread had me thinking about why a photon's speed is changed when it passes through a medium. The only explanation I could come up with is that the photon interacts with the matter the medium is composed of, which could set of a relay. The photon can pass through to a certain depth, but there's a high probability that it will end up interacting with an electron. This bumped up electron will then emit another photon and the process repeats itself. Does this explanation make any sense?

If the process is something like this, then how can we explain the fact that a beam of light more or less stays on the same path while going through the medium? Is the photon's direction conserved in the case where it is absorbed and emitted?

- *fap* *fap* *fap*
[ Parent ]

As good a metaphor as any (5.00 / 1) (#80)
by epepke on Fri Jun 27, 2003 at 01:11:46 PM EST

This thread had me thinking about why a photon's speed is changed when it passes through a medium. The only explanation I could come up with is that the photon interacts with the matter the medium is composed of, which could set of a relay. The photon can pass through to a certain depth, but there's a high probability that it will end up interacting with an electron. This bumped up electron will then emit another photon and the process repeats itself. Does this explanation make any sense?

That's as good a metaphor as any, and that's about how it is generally explained by the Feynman diagram approach.

If the process is something like this, then how can we explain the fact that a beam of light more or less stays on the same path while going through the medium? Is the photon's direction conserved in the case where it is absorbed and emitted?

This does seem like the puzzle. We would expect the light to be re-emitted in any old direction. There are cases where this is true. The phosphors of a CRT, for example, have electrons that are excited by other electrons and decay, giving off light. The inner coating of a fluorescent bulb has electrons that get excited by ultraviolet light and give off visible light. That glow-in-the-dark paint works roughly like that, too.

Some pretty heavy-duty minds, such as Richard Feynman, were puzzled by this as well. Your question cannot adequately be addressed with classical physics. In order to try to explain that property, we have to get into the murky realm of quantum behavior, which is a bit beyond the current series. However, I hint at it in one of the other responses.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
exceeding c (5.00 / 1) (#61)
by adiffer on Sun Jun 22, 2003 at 12:14:07 AM EST

There is a situation where this is being explored.  Imagine shining intense light at an opaque block.  There is always a chance for a photon to tunnel through the block and keep going.  Set up the situation right and there is no chance of ever observing the photon inside the block.  

The question then is for the case of a tunneling photon, do you count the space occupied by the block as distance the photon must travel when figuring its speed over the whole distance between where it is emitted and absorbed?  If not, the photon may get to the end of its trip in less time than one that makes the trip without the block being there.  If so, should there be a time span when there is no chance to observe the photon at all?
--BE The Alien!
[ Parent ]

Light goes faster than itself - link (5.00 / 1) (#63)
by mveloso on Sun Jun 22, 2003 at 01:31:39 AM EST

While no particle can exceed the speed of light in a vacuum, it is possible for particles to travel faster than light in certain mediums, such as water. The speed of light in a particular medium, v, is related to the speed of light in a vacuum, c, by the index of refraction, n, by v = c/n. Water has an index of refraction of 1.3, thus the speed of light in water is 2.3x108 meters per second. Therefore, beta particles with kinetic energies of 0.26 MeV travel at speeds in excess of 230 million m/s!

http://web.umr.edu/~reactor/cerenkov.html

Note that light is just radiation in a specific range of wavelengths.

[ Parent ]

Isn't it the case... (none / 0) (#78)
by onemorechip on Thu Jun 26, 2003 at 12:43:28 AM EST

Isn't it the case that, while the propagation of a wavefront of light through a medium must travel slower than c, the photons themselves must be moving at c? I picture it as photons getting absorbed into atoms, then released when the atom returns to a lower energy state, with the delay between absorption and release accounting for the difference in speed. I'm not sure if that's a physically correct explanation but it works for me.
--------------------------------------------------

I did my essay on mushrooms. It's about cats.
[ Parent ]

Older comments (none / 0) (#54)
by epepke on Sat Jun 21, 2003 at 12:11:23 PM EST

Here are the comments from when this was first submitted. Some of them are excellent. I'm linking them here because a) I promised to do so, and b) people haven't come up with too many complaints about this one, so we haven't had a chance to talk about a lot of interesting stuff.

Here is also a list of possible complaints that was in the original story but had to be removed because it was way too long:

What about the Twin Paradox?
What about objects moving fast in opposite directions?
What about the barn door paradox?
What about the index of refraction?
I still don't believe it, what about experimental verification?
Etc. and so on and so forth?

If you would like to complain but can't think of anything to complain about, perhaps one of these would be good.


The truth may be out there, but lies are inside your head.--Terry Pratchett


request (none / 0) (#69)
by manobes on Mon Jun 23, 2003 at 02:41:17 AM EST

Can you email me at manobes@sfu.ca ?


No one can defend creationism against the overwhelming scientific evidence of creationism. -- Big Sexxy Joe


[ Parent ]
Can't even nslook(sfu.ca)up (none / 0) (#70)
by epepke on Mon Jun 23, 2003 at 03:37:04 PM EST

Or was it a way to tell me to S(T)FU?


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
huh? (none / 0) (#71)
by manobes on Mon Jun 23, 2003 at 04:17:15 PM EST

It's a university, that's the overall name. Email to manobes@sfu.ca will get to me.


No one can defend creationism against the overwhelming scientific evidence of creationism. -- Big Sexxy Joe


[ Parent ]
OK (none / 0) (#72)
by epepke on Mon Jun 23, 2003 at 05:03:18 PM EST

I'm used to having domains have an alias for their www sites. Anyway, check your mail.


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Contraction and dilation (5.00 / 2) (#79)
by onemorechip on Thu Jun 26, 2003 at 12:52:59 AM EST

When objects go fast, they contract in the direction of travel.

As I see it, the object doesn't really contract; this is just a figure of speech.

Instead, this is like the effect of foreshortening in linear perspective -- only we are using the fourth dimension, not the third.

When we look at an object moving fast relative to us, the leading edge of that object is being observed at a different moment in time (in the objects frame of reference) than the trailing edge. This is due to the relativity of simultaneity. We are seeing an earlier instance of the leading edge, and a later instance of the trailing edge. The trailing edge has moved farther in the direction of travel than the leading edge, so the object appears foreshortened.

The section on Minkowski diagrams covers this but I thought this description might be useful. A similar idea explains the dilation of time.
--------------------------------------------------

I did my essay on mushrooms. It's about cats.

My favorite book ! (none / 0) (#81)
by UnknownReference on Wed Jul 02, 2003 at 06:49:10 AM EST

There is one book by Asimov on particle physics that details about the special and the general theory of relativity ! it's an amazing book ! whoa ! I havent quite understood one of the lorentz equations where in, if the speed of a particle approaches the speed of light, then the relative mass reaches infinity and not tending towards 0 as would be expected. Can anyone of you give me some explanation on this ?

Why does speed of light vary in different media? (none / 0) (#82)
by xeoatthermopylae on Sun Jul 27, 2003 at 02:56:38 PM EST

In the article the speed of light is treated as constant. But in optics it is commonly stated that light travels at different speeds in different media (water, plastic, glass, etc.). This causes refraction.

What mechanism causes the speed of light to vary in these media?

Answered elsewhere (none / 0) (#83)
by epepke on Mon Aug 04, 2003 at 02:48:20 AM EST

Here


The truth may be out there, but lies are inside your head.--Terry Pratchett


[ Parent ]
Challenge of Relativity (none / 0) (#84)
by losang on Sat Dec 27, 2003 at 12:00:41 PM EST

What is the reason that the speed of light is measured the same by all observers independent of their relative motion to the light's source?

Introduction to the Theory of Relativity Part II: Special Relativity | 84 comments (75 topical, 9 editorial, 0 hidden)
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