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全是臭！帮你臭大家！ September 22, 1994SPECIAL RELATIVITY: LORENTZ TRANSFORMATIONSOur goal today is to derive the Lorentz transformations which are the foundation of Einstein's special theory of relativity. (Einstein's general theory of relativity, which describes the effects of gravitation, will not be discussed in this class.) Lorentz transformations take the place of Galilean transformations, which are not valid in the special theory. 1. The Constancy of the Speed of LightBy the beginning of the twentieth century, it was recognized that the speed of light does not depend on the velocity of the source. How was this known? Astronomers had discovered countless double stars by this time; pairs of stars that rotate around one another due to their mutual attraction. If the speed of light depended on the velocity of the star, then the image of the star approaching us would be seen substantially before the image of the star retreating from us. This would result in weird errors in the appearance of these systems, apparent violations of Newtonian mechanics. Since no such discrepancies were seen, it was concluded that the speed of light was independent of the speed of the source. At first, this may seem surprising. After all, the speed of a bullet from a gun depends on the velocity of the gun. If the bullet has a muzzle velocity of  , and the gun itself is moving the same direction at velocity  , the total bullet velocity was  . However, by this time it was already understood that light is a wave. Sound waves from a passing train, for example, travel at the velocity of sound compared to the air, whether the train is motionless or moving at 100 m/s. So it was argued that the velocity of light was some constant value  , compared to the medium which was sustaining the electromagnetic waves, called the ether. Although the velocity of light didn't depend on the velocity of the source, it was expected that it would depend on the velocity of the observer. After all, if the observer is moving toward an oncoming light beam with a velocity , and light was moving toward him at a velocity , then the observer would see the light moving at an apparent rate, compared to his coordinate system, of  . The next logical experiment was to determine the velocity of the earth relative to the ether. This experiment, which was performed by Michelson and Morley, came to the surprising conclusion that the earth was not moving with respect to the ether; that is, the apparent velocity of light was the same in all directions. This is surprising since the earth is rotating, revolving around the sun, and the solar system is itself moving around the galaxy at quite a respectable velocity. Even if the experiment were momentarily at rest with respect to the ether, one would expect that twelve hours (or six months) later, some of the velocity components of the earth would have reversed, resulting in a large velocity compared to the ether. What went wrong with these experiments? The fundamental error was in the addition of velocities formula, which in turn was derived directly from the formula for Galilean transformations. Einstein had the genius to recognize this was the problem, and by doing away with these transformations in favor of Lorentz transformations, he was able to abolish the ether theory and give us a whole new insight into the nature of space and time. I would now like to work my way up to the Lorentz transformations. I will start with the ideas of time dilation and Lorentz contraction, and then I will do the full calculation of Lorentz transformations. The radical assumptions I will build on are the following: (a) The speed of light in vacuum is the same in all reference frames, no matter what. (b) All uniformly moving reference frames are equally valid. I will not assume these reference frames are related by Galilean transformations. For reference, the speed of light in vacuum is approximately 
2. Time DilationConsider a moving reference frame (primed) moving with velocity in the  -direction with respect to a stationary reference frame (unprimed). Clocks are adjusted so that the the point  and  corresponds to the point  and  (this choice will be used throughout this lecture). A device consisting of a flash bulb and photodetector are set up at the origin as shown in the primed coordinate system. A mirror is set up on the  -axis at  . All of these components are attached to the moving reference frame, and are moving in the  direction with velocity . At , a burst of light leaves the flash bulb. Since light moves at velocity in this reference frame, it arrives at the mirror after a time  . It is then reflected back into the photodetector, arriving at time  . The previous paragraph describes what is seen by the observer in the primed coordinate system. What is seen by the observer in the unprimed coordinate system? Since the two observers agree on the origin, the unprimed observer also sees a burst of light leaving the origin at time . However, because the primed frame is moving at velocity , the unprimed observer must see the burst of light moving both in the positive and the positive  direction. After a time  , the light will bounce off the mirror, which has moved to the right by a distance  (this is just the definition of velocity). After a total time  , the light will impinge on the photodetector, which has moved to the right by a distance  . The total distance travelled by the light is 
Since the speed of light is still in this coordinate system, we must have 
Since  , this can be rewritten as 
Solving for , this says 
We have found a relation between  and , but note that we have only done so at a particular point, when . It is sometimes said that moving clocks run slower in special relativity, but this is a tricky point, and we must be careful about where we are talking about when we make this statement. For now, we will stick to the formula above and apply it only for the special case . When we get to the full Lorentz transformations, we can apply them everywhere. The coefficient of in the above expression appears so often in special relativity that it is given the name  ; that is, 
In terms of , then, the previous equation would be simply written  . Example: A muon, when at rest, lasts an average of  s. How far does it go if it is moving at 99.9% of the speed of light? Answer: In the rest frame of the particle (a frame moving with the particle) the particle is motionless ( always). If it starts its life at , then it will decay at a time  . In the laboratory frame (not moving with the particle), the particle is moving with roughly  , so the time it lasts will be  and the distance 
The velocity discussed here, by the way, is easily achievable in a modern particle accelerator. 3. Lorentz ContractionThe distortion of time discussed in the preceding section is strange, but things will get stranger still. The next oddity of special relativity is Lorentz contraction, the apparent distortion of distances. To calculate this effect, we will turn the previous experiment on its side, still putting the flash bulb and photodetector at the origin, but this time putting the mirror on the  -axis at the position  . We must be careful this time to distinguish the separation of the components of the system in the primed coordinates from the unprimed coordinates. At , the light leaves the flash bulb, travels to , and then returns to the origin. Since the speed of light is always , the light gets back at a time  . What is seen in the unprimed coordinate system? Well, let's call the separation of the bulb-photodetector and the mirror  . The light leaves the photodetector at and travels to the right at the velocity . The mirror is travelling to the right at velocity . It is easy to see that the difference between their positions is decreasing at a rate  . (Note: This is a statement about how things appear in the unprimed coordinate system, and does not involve changing coordinates. Hence this calculation is still correct, and does not assume galilean transformations.) Since a distance must be traversed, this takes a time  . On the way back, the light is travelling with velocity and the photocell is moving toward it with a velocity , so the return trip takes a time  . Hence the total time when the light returns to the photocell is 
Since this return time is occurring at the origin in the primed coordinates (), we may use the relation 
Combining this with the expressions for and , we find 
It is sometimes said that this implies that moving objects are shortened, or Lorentz contracted. It would be more accurate to say that objects that are moving are measured to be shorter in the direction of motion by an observer who is not moving. This leads to numerous apparent paradoxes which we will attempt to clarify in the next lecture. Example: A lead nucleus at rest is roughly a spherical blob of nucleons, as shown below. What do two lead nuclei look like as they are about to make a head-on collision at 99.9% of the speed of light? Answer: Once again, we have , so there is a 22-fold contraction in the direction of motion. The picture is shown above. Note that contraction is only in the direction of motion. 4. Lorentz TransformationsWe are now prepared to deal with the problem of how to relate an arbitrary and coordinate in an unmoving frame to an arbitrary and coordinate in a moving frame. For this we imagine an experiment in which we get rid of the mirror altogether, and simply have the flash bulb at and the photodetector at . The flash bulb flashes at an arbitrary time  at  , and the light is received at time at . By choosing the initial flash time carefully, it is possible to arrange an arbitrary time at which the light is received. Since the light travels at speed , we have 
That is what happens in the primed frame. What happens in the unprimed frame? The light starts at some time  and travels to the final point at time . Since the light travels at , and the photodetector is moving away from the flash at velocity , and the length of the separation is in this reference frame, the time it takes is given by 
At the end of the experiment, the flash bulb has moved a distance from the origin, and the photodetector is a distance beyond that, so 
Our goal is to obtain equations relating the coordinates and to and . We already have three equations, but we have five unknowns: , , , , and . We need two more equations. Fortunately, since the initial flash was at , we can use the time dilation relation 
and the Lorentz contraction relation 
to find the desired relationships. From here on out it is all algebra. First, combine equations (E) and (C) to find a relation for : 
That was relatively easy. Much harder is finding the expression for . We start by solving equation (A) for : 
We then use (D) to substitute for and (E) to solve for : 
We now solve (B) for , yielding 
If we now substitute equation (C) for , we have an expression for exclusively in terms of and , namely 
We have the transformation properties of and in the two coordinate systems. What we have not been careful about is the transformations of the other coordinates and  . Since the motion is only in the direction, it is not surprising that these coordinates are not affected by Lorentz transformations. So the full equations for the Lorentz transformations are given by 
These equations will be discussed at some length in the next lecture. 5. Transverse Directions: A Prelude to ParadoxThus far, we have simply assumed that the directions and are not transformed under Lorentz Transformations. For example, in the section on time dilation, we assumed that the transverse separation of distances is identical in both systems. How can we assume this? Is it possible to convince yourself that there is not a similar Lorentz contraction in the - and -directions? Suppose that a bullet of radius  is fired at a wall with a hole of radius in it. Imaging there is some sort of Lorentz contraction, so that the bullet is narrow as it passes through the hole. Then clearly the bullet can easily fit through the hole without even scraping the sides. Now, consider the picture in the rest frame of the bullet. In this frame, the bullet is motionless, but the wall is moving toward it at high velocity. We have already assumed that all observers are equally valid, so it must be that there is also Lorentz contraction of the hole in this coordinate frame. Hence when the bullet reaches the hole, it will be too wide to fit through, and will smash a ragged tear through the wall, breaking off bits as it goes. But this is inconsistent with what the observer in the wall's reference frame sees! The only way to avoid this inconsistency is to assume there is no Lorentz contraction in the transverse direction. A similar argument will convince you there cannot be Lorentz expansion either. The only logical conclusion is that transverse directions are neither shrunk nor increased. There is no alternative. Hence our equations  and  must be right. 6. A Sample Paradox ProblemIt is interesting and instructive to consider a similar problem concerning length in the direction of motion (longitudinal dimension). Suppose I am holding a measuring stick of length . A rocket ship has an identical measuring stick of length , pointing along the direction of motion, and is moving at a velocity  (so  ). Since distances are Lorentz contracted, I perceive their stick to have a length  , or substantially shorter than mine. However, since in the reference frame of the rocket, we are moving and the rocket is at rest, an astronaut on the rocket perceives our measuring stick as having a length  , shorter than theirs. How can we reconcile these apparent discrepancies? We communicate with the astronaut, and resolve to experimentally determine whose stick is really longer. The astronaut fastens paint brushes to both ends of the measuring stick, and then flies low over the ground. As she passes my measuring stick, she briefly lowers her stick to the ground, making two spots of paint on the ground. Then she asks me to measure them, and we find out who is right. What happens? Well, we can figure it all out using the Lorentz transformations. We will work (and make pictures) in unprimed coordinates, and imaging that we have chosen coordinates that match at the origin of space-time. Let us suppose that our stick has one end sitting at and one at  . In other words, the equations for the ends of our stick are 
This is graphed in the figure below. In contrast, the other stick we will imagine to have one end each at and at  . Using the Lorentz equation for , this is 
which is the same as 
This is also graphed in the figure below. As we can see, at every time the separation of the two ends of the moving meter stick are separated by only , which is just the Lorentz contraction. Now, we need to figure out where the two spots of paint end up. The astronaut will drop both brushes simultaneously in her reference frame, so let's say she does so at . For the left end of the brush, this is also at , and since this is the origin (of space and time), it corresponds to and . Since the dab of paint persists once it is dropped, and it doesn't move, it appears as a ray corresponding to 
Now, the other paint drop was dropped at time but at location . The Lorentz transformations give us two equations in two unknowns, namely 
Solving these gives the space-time coordinates of the brush drop, namely 
Since the dab persists, the equation for this ray is 
Both of these dabs are included in the figure above. Note first that the after the experiment, we both agree that the dabs of paint are farther apart than the length of my measuring stick. However, note that in the opinion of the observer on the ground, the two brushes were not dropped simultaneously. In other words, according to the ground observer, the astronaut won the argument by cheating. This illustrates the resolution of most apparent paradoxes of special relativity: the loss of any concept of simultaneity. Observers cannot agree on the meaning of ``at the same time'', and consequently interpret different experiments in different ways. Indeed, time can get so mixed up that ``past'' and ``future'' can get confused, in a way that we will try to untangle soon. First, however I would like to talk a little more about Lorentz transformations, and how they are similar to (and different from) Galilean boosts. 7. Slow Velocities, or How We Almost Got it RightA natural question to ask about the Lorentz transformations is why they were not noticed before. Until the time of Einstein, physicists regularly used the formulae for Galilean boosts and never noticed what was going wrong. Why not? The answer is, of course, that the speed of light is so large. In particular, suppose we consider what these equations become if we take the limit  . Specifically, if we substitute  , then the equations of the Lorentz transformation become 
This is, of course, just a Galilean boost. Sometimes it is useful to have formulas which tell us approximately how wrong the standard formulas and concepts are. To calculate this, it is necessary to take a brief excursion into math. Recall that Taylor's theorem allows one to calculate a function in terms of a nearby point by using the formula 
Specifically, this can be used to get the expansion for  around the point ; namely 
Now, if we let  and  , this gives us an approximate expression for : 
This series converges quickly for small velocities; i.e., for  . Example: A man spends his whole 100 year lifespan travelling in a plane at mach 1 ( m/s). How much does his high velocity increase his lifespan? The 100 years is measured in his own reference frame. Answer: Of course, in his own reference frame, his lifetime is 100 years. In his own reference frame, he isn't moving. So naturally in his own reference frame, and we can use the expression  . We want to know the difference between his lifetime in our reference frame and his reference frame, or  . Using the low velocity expansion for , this is about 
Now, you might wonder why we used an approximation, when we have the exact answer. You might try doing the exact calculation on your calculator, and see what you get. We suggest that this is not an effective way to lengthen your life span, especially since there is no subjective increase. We conclude this section with a simple comment: the errors of ignoring the effects of special relativity are always fractional errors of the order of  , where is the characteristic velocity of something in the problem. So if your velocities are high, or your need for precision is great, you should probably be including the effects of relativity.
Joe Watson
Fri Sep 30 18:34:19 EDT 1994 | 
indicates a constant; for, according to (3), the disappearance of (x – ct) involves the disappearance of (x' – ct').
where 
x in (7) must be equal to 
= 1; for (11) is a consequence of (8a) and (9), and hence also of (8) and (9). We have thus derived the Lorentz transformation.