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a / The clock took up two seats, and two tickets were bought for it under the name of “Mr. Clock.

Chapter 4. Relativity

a / This Global Positioning System (GPS) system, running on a smartphone attached to a bike's handlebar, depends on Einstein's theory of relativity. Time flows at a different rates aboard a GPS satellite than it does on the bike, and the GPS software has to take this into account.

4.1 Relativity According To Einstein

b / All three clocks are moving to the east. Even though the west-going plane is moving to the west relative to the air, the air is moving to the east due to the earth's rotation.

c / The correspondence principle requires that the relativistic distortion of time become small for small velocities. The effects are so small that we have to describe them in scientific notation (p. 12). For example, $$10^{-15}$$ means $$0.000000000000001$$, which is a hundred thousand times smaller than $$10^{-10}$$.

d / Two events are given as points on a graph of position versus time. Joan of Arc helps to restore Charles VII to the throne. At a later time and a different position, Joan of Arc is sentenced to death.

e / A change of units distorts an $$x$$-$$t$$ graph. This graph depicts exactly the same events as figure d. The only change is that the $$x$$ and $$t$$ coordinates are measured using different units, so the grid is compressed in $$t$$ and expanded in $$x$$.

f / A convention we'll use to represent a distortion of time and space.

g / A Galilean version of the relationship between two frames of reference, as introduced in figure e, p. 43. As in all such graphs in this chapter, the original coordinates, represented by the gray rectangle, have a time axis that goes to the right, and a distance axis that goes straight up.

h / In the units that are most convenient for relativity, the transformation has symmetry about a 45-degree diagonal line.

i / Interpretation of the Lorentz transformation. The slope indicated in the figure gives the relative velocity of the two frames of reference. Events A and B that were simultaneous in frame 1 are not simultaneous in frame 2, where event A occurs to the right of the $$t=0$$ line represented by the left edge of the grid, but event B occurs to its left.

j / The $$\gamma$$ factor.

k / An example in which the $$\gamma$$ factor is numerically simple to work out. The bottom edge of the parallelogram rises 3 units and goes 5 units to the right, so its slope is 3/5. This slope represents the speed of one frame of reference relative to the other. It's easy to verify that the square and the parallelogram have the same area, because one diagonal of the square has been stretched to twice its original length, the other smooshed down by a half. The bottom-right corner of the square is at a time of 4 units, while the corresponding corner of the parallelogram is at 5. As defined in figure j, the ratio of these times is the value of $$\gamma=5/4$$.

Time is not absolute

So far we've been discussing relativity according to Galileo and Newton, but there is also relativity according to Einstein. When Einstein first began to develop the theory of relativity, around 1905, the only real-world observations he could draw on were ambiguous and indirect. Today, the evidence is part of everyday life. For example, every time you use a GPS receiver, a, you're using Einstein's theory of relativity. Somewhere between 1905 and today, technology became good enough to allow conceptually simple experiments that students in the early 20th century could only discuss in terms like “Imagine that we could...” A good jumping-on point is 1971. In that year, J.C. Hafele and R.E. Keating, shown in the photo above, brought atomic clocks aboard commercial airliners, and went around the world, once from east to west and once from west to east. The clocks were capable of keeping time to within a few nanoseconds. (A nanosecond, abbreviated ns, is one billionth of a second.) Hafele and Keating observed that there was a discrepancy between the times measured by the traveling clocks and the times measured by similar clocks that stayed home at the U.S. Naval Observatory in Washington. The east-going clock lost time, ending up off by $$-59$$ nanoseconds, while the west-going one gained $$273$$ ns.

Causality

It reassuring that the effects on time were small compared to the three-day lengths of the plane trips. There was therefore no opportunity for paradoxical scenarios such as one in which the east-going experimenter arrived back in Washington before he left and then convinced himself not to take the trip. A theory that maintains this kind of orderly relationship between cause and effect is said to satisfy causality.

Time affected by motion and gravity

Hafele and Keating were testing specific quantitative predictions of relativity, and they verified them to within their experiment's error bars. Let's work backward instead, and inspect the empirical results for clues as to how time works.

The two traveling clocks experienced effects in opposite directions, and this suggests that the rate at which time flows depends on the motion of the observer. The east-going clock was moving in the same direction as the earth's rotation, so its velocity relative to the earth's center was greater than that of the clock that remained in Washington, while the west-going clock's velocity was correspondingly reduced. The fact that the east-going clock fell behind, and the west-going one got ahead, shows that the effect of motion is to make time go more slowly. This effect of motion on time was predicted by Einstein in his original 1905 paper on relativity, written when he was 26.

If this had been the only effect in the Hafele-Keating experiment, then we would have expected to see effects on the two flying clocks that were equal in size. In fact, the two effects are unequal in size: $$-59$$ ns and 273 ns. This turns out to be because there was a second effect involved, a gravitational one, simply due to the planes' being up in the air. The gravitational effects are beyond the scope of this book.

The correspondence principle

The effects that Hafele and Keating observed were small. This makes sense: the version of relativity worked out by Galileo (sections 2.2 and 2.3, pp. 39-53) had already been thoroughly tested by experiments under a wide variety of conditions, so a new theory like Einstein's relativity must agree with Galileo's to a good approximation, within the Galilean theory's realm of applicability. This is an example of the correspondence principle (p. 31). The behavior of the three clocks in the Hafele-Keating experiment shows that the amount of time distortion increases as the speed of the clock's motion increases. Newton lived in an era when the fastest mode of transportation was a galloping horse, and the best pendulum clocks would accumulate errors of perhaps a minute over the course of several days. A horse is much slower than a jet plane, so the distortion of time would have had a relative size of only $$\sim10^{-15}$$ --- much smaller than the clocks were capable of detecting. At the speed of a passenger jet, the effect is about $$10^{-12}$$, and state-of-the-art atomic clocks in 1971 were capable of measuring that. A GPS satellite travels much faster than a jet airplane, and the effect on the satellite turns out to be $$\sim10^{-10}$$. The general idea here is that all physical laws are approximations, and approximations aren't simply right or wrong in different situations. Approximations are better or worse in different situations, and the question is whether a particular approximation is good enough in a given situation to serve a particular purpose. The faster the motion, the worse the Newtonian approximation of absolute time. Whether the approximation is good enough depends on what you're trying to accomplish. The correspondence principle says that the approximation must have been good enough to explain all the experiments done in the centuries before Einstein came up with relativity.

By the way, don't get an inflated idea of the importance of the Hafele-Keating experiment. Relativity had already been confirmed by a vast and varied body of experiments decades before 1971. The only reason I'm giving such a prominent role to this experiment is that it is conceptually very direct.

Distortion of time and space

Relativity says that when two observers are in different frames of reference, each observer considers the other one's perception of time to be distorted. We'll also see that something similar happens to their observations of distances, so both space and time are distorted. What exactly is this distortion? How do we even conceptualize it?

The idea isn't really as radical as it might seem at first. We can visualize the structure of space and time using a graph with position and time on its axes. These graphs were introduced on p. 43 in figures d and e, but we're going to look at them in a slightly different way. Before, we used them to describe the motion of objects. The grid underlying the graph was merely the stage on which the actors played their parts. Now the background comes to the foreground: it's time and space themselves that we're studying. We don't necessarily need to have a line or a curve drawn on top of the grid to represent a particular object. We may, for example, just want to talk about events, depicted as points on the graph as in figure d. A distortion of the Cartesian grid underlying the graph can arise for perfectly ordinary reasons that Isaac Newton would have readily accepted. For example, we can simply change the units used to measure time and position, as in figure e.

We're going to have quite a few examples of this type, so I'll adopt the convention shown in figure f for depicting them; this convention was originally introduced in figure e on p. 43. Figure f summarizes the relationship between figures d and e in a more compact form. The gray rectangle represents the original coordinate grid of figure d, while the grid of black lines represents the new version from figure e. Omitting the grid from the gray rectangle makes the diagram easier to decode visually.

Our goal of unraveling the mysteries of special relativity amounts to nothing more than finding out how to draw a diagram like f in the case where the two different sets of coordinates represent measurements of time and space made by two different observers, each in motion relative to the other. Galileo and Newton thought they knew the answer to this question, but their answer turned out to be only approximately right. To avoid repeating the same mistakes, we need to clearly spell out what we think are the basic properties of time and space that will be a reliable foundation for our reasoning.

Experiments show that:

1. The laws of physics have translation symmetry (section 2.1), time symmetry (section 1.6), and rotational symmetry (p. 15 and section 3.3).
2. The principle of inertia holds (p. 16).
3. Causality holds, in the sense described on page 72.
4. Time depends on the state of motion of the observer.

If it were not for property 4, we could imagine that figure g would give the correct transformation between frames of reference in motion relative to one another. Let's say that observer 1, whose grid coincides with the gray rectangle, is a hitch-hiker standing by the side of a road. Event A is a raindrop hitting his head, and event B is another raindrop hitting his head. He says that A and B occur at the same location in space. Observer 2 is a motorist who drives by without stopping; to him, the passenger compartment of his car is at rest, while the asphalt slides by underneath. He says that A and B occur at different points in space, because during the time between the first raindrop and the second, the hitch-hiker has moved backward. On the other hand, observer 2 says that events A and C occur in the same place, while the hitch-hiker disagrees. The slope of the grid-lines is simply the velocity of the relative motion of each observer relative to the other. (Recall that slope is defined as the rise over the run. On these graphs of distance versus time, the slope is the distance traveled divided by the elapsed time.)

Figure g has familiar, comforting, and eminently sensible behavior, but it also happens to be wrong, because it violates property 4. The distortion of the coordinate grid has only moved the vertical lines up and down, so both observers agree that events like B and C are simultaneous. If this was really the way things worked, then all observers could synchronize all their clocks with one another for once and for all, and the clocks would never get out of sync. This contradicts the results of the Hafele-Keating experiment, in which all three clocks were initially synchronized in Washington, but later went out of sync because of their different states of motion.

Based on properties 1-4, there is only one possible way to modify g, which is the one shown in h.1 This distortion is the one that Einstein predicted in 1905, and is known as the Lorentz transformation, after Hendrik Lorentz (1853-1928). The distortion is a kind of smooshing and stretching, as suggested by the hands. Also, we've already seen in figures d-f on page 73 that we're free to stretch or compress everything as much as we like in the horizontal and vertical directions, because this simply corresponds to choosing different units of measurement for time and distance. In figure h I've chosen units that give the whole drawing a convenient symmetry about a 45-degree diagonal line. Ordinarily it wouldn't make sense to talk about a 45-degree angle on a graph whose axes had different units. But in relativity, the symmetric appearance of the transformation tells us that space and time ought to be treated on the same footing, and measured in the same units.

The exact size and shape of the parallelogram are controlled by the requirements that (i) the slope labeled in the figure corresponds properly to the velocity; (ii) the units are the special ones described above; and (iii) the area of the parallelogram is the same as the area of the original square.2

The $$\gamma$$ factor

We've seen the experimental evidence that motion changes the rate of flow of time, and this effect is correctly reproduced by the Lorentz transformation.

Time dilation

A clock runs fastest in the frame of reference of an observer who is at rest relative to the clock.

We define the factor $$\gamma$$ (Greek letter gamma) as in figure j. An observer in motion relative to the clock at speed $$v$$ perceives the clock as running more slowly by a factor of $$\gamma$$. For example, if $$\gamma$$ equals 2, then the observer says the clock runs at half its normal speed.

Figure k shows an example of how we can use properties (i)-(iii) on p. 75 to find the value of $$\gamma$$ for a given velocity $$v$$ of the clock and the observer relative to one another. By plotting many such points,3 we get the graph shown in figure l.

l / The behavior of the $$\gamma$$ factor. (The velocity is in the special units described on p. 75. More on these units in section 4.2.)

For small velocities, the graph is nearly flat at $$\gamma\approx 1$$, meaning that there is very little time dilation. This is required by the correspondence principle.

Distances are also distorted:

Length contraction

A meter-stick appears longest to an observer who is at rest relative to it. An observer moving relative to the meter-stick at $$v$$ observes the stick to be shortened by a factor of $$\gamma$$.

m / Example 1: In the garage's frame of reference, the bus is moving, and can fit in the garage due to its length contraction. In the bus's frame of reference, the garage is moving, and can't hold the bus due to its length contraction.

One of the most famous of all the so-called relativity paradoxes has to do with our incorrect feeling that simultaneity is well defined. The idea is that one could take a schoolbus and drive it at relativistic speeds into a garage of ordinary size, in which it normally would not fit. Because of the length contraction, the bus would supposedly fit in the garage. The driver, however, will perceive the garage as being contracted and thus even less able to contain the bus.

The paradox is resolved when we recognize that the concept of fitting the bus in the garage “all at once” contains a hidden assumption, the assumption that it makes sense to ask whether the front and back of the bus can simultaneously be in the garage. Observers in different frames of reference moving at high relative speeds do not necessarily agree on whether things happen simultaneously. As shown in figure m, the person in the garage's frame can shut the door at an instant B he perceives to be simultaneous with the front bumper's arrival A at the back wall of the garage, but the driver would not agree about the simultaneity of these two events, and would perceive the door as having shut long after she plowed through the back wall.

4.2 Speeds In Relativity

n / A proof that causality imposes a universal speed limit. In the original frame of reference, represented by the square, event A happens a little before event B. In the new frame, shown by the parallelogram, A happens after $$t=0$$, but B happens before $$t=0$$; that is, B happens before A. The time ordering of the two events has been reversed. This can only happen because events A and B are very close together in time and fairly far apart in space. The line segment connecting A and B has a slope greater than 1, meaning that if we wanted to be present at both events, we would have to travel at a speed greater than $$c$$ (which equals 1 in the units used on this graph). You will find that if you pick any two points for which the slope of the line segment connecting them is less than 1, you can never get them to straddle the new $$t=0$$ line in this funny, time-reversed way. Since different observers disagree on the time order of events like A and B, causality requires that information never travel from A to B or from B to A; if it did, then we would have time-travel paradoxes. The conclusion is that $$c$$ is the maximum speed of cause and effect in relativity.

o / A ring laser gyroscope.

Discussion question B

The universal speed $$c$$

Let's think a little more about the role of the 45-degree diagonal in the Lorentz transformation. Slopes on these graphs are interpreted as velocities. This line has a slope of 1 in our special relativistic units, but that slope corresponds to some number, call it $$c$$, in ordinary units of meters per second. Now note what happens when we perform a Lorentz transformation: this particular line gets stretched, but the new version of the line lies right on top of the old one, and its slope stays the same. In other words, if one observer says that something has a velocity equal to $$c$$, every other observer will agree on that velocity as well.

Velocities don't simply add and subtract.

This is counterintuitive, since we expect velocities to add and subtract in relative motion. If a dog is running away from me at 5 m/s relative to the sidewalk, and I run after it at 3 m/s, the dog's velocity in my frame of reference is 2 m/s. According to everything we have learned about motion, the dog must have different speeds in the two frames: 5 m/s in the sidewalk's frame and 2 m/s in mine. But velocities are measured by dividing a distance by a time, and both distance and time are distorted by relativistic effects, so we actually shouldn't expect the ordinary arithmetic addition of velocities to hold in relativity; it's an approximation that's valid at velocities that are small compared to $$c$$.

A universal speed limit

For example, suppose Janet takes a trip in a spaceship, and accelerates until she is moving at $$0.6c$$ relative to the earth. She then launches a space probe in the forward direction at a speed relative to her ship of $$0.6c$$. We might think that the probe was then moving at a velocity of $$1.2c$$, but in fact the answer is still less than $$c$$ (problem 1, page 86). This is an example of a more general fact about relativity, which is that $$c$$ represents a universal speed limit. This is required by causality, as shown in figure n.

Light travels at $$c$$.

Now consider a beam of light. We're used to talking casually about the “speed of light,” but what does that really mean? Motion is relative, so normally if we want to talk about a velocity, we have to specify what it's measured relative to. A sound wave has a certain speed relative to the air, and a water wave has its own speed relative to the water. If we want to measure the speed of an ocean wave, for example, we should make sure to measure it in a frame of reference at rest relative to the water. But light isn't a vibration of a physical medium; it can propagate through the near-perfect vacuum of outer space, as when rays of sunlight travel to earth. This seems like a paradox: light is supposed to have a specific speed, but there is no way to decide what frame of reference to measure it in. The way out of the paradox is that light must travel at a velocity equal to $$c$$. Since all observers agree on a velocity of $$c$$, regardless of their frame of reference, everything is consistent.

The Michelson-Morley experiment

The constancy of the speed of light had in fact already been observed when Einstein was an 8-year-old boy, but because nobody could figure out how to interpret it, the result was largely ignored. In 1887 Michelson and Morley set up a clever apparatus to measure any difference in the speed of light beams traveling east-west and north-south. The motion of the earth around the sun at 110,000 km/hour (about 0.01% of the speed of light) is to our west during the day. Michelson and Morley believed that light was a vibration of a mysterious medium called the ether, so they expected that the speed of light would be a fixed value relative to the ether. As the earth moved through the ether, they thought they would observe an effect on the velocity of light along an east-west line. For instance, if they released a beam of light in a westward direction during the day, they expected that it would move away from them at less than the normal speed because the earth was chasing it through the ether. They were surprised when they found that the expected 0.01% change in the speed of light did not occur.

Example 2: The ring laser gyroscope

If you've flown in a jet plane, you can thank relativity for helping you to avoid crashing into a mountain or an ocean. Figure o shows a standard piece of navigational equipment called a ring laser gyroscope. A beam of light is split into two parts, sent around the perimeter of the device, and reunited. Since the speed of light is constant, we expect the two parts to come back together at the same time. If they don't, it's evidence that the device has been rotating. The plane's computer senses this and notes how much rotation has accumulated.

Example 3: No frequency-dependence

Relativity has only one universal speed, so it requires that all light waves travel at the same speed, regardless of their frequency and wavelength. Presently the best experimental tests of the invariance of the speed of light with respect to wavelength come from astronomical observations of gamma-ray bursts, which are sudden outpourings of high-frequency light, believed to originate from a supernova explosion in another galaxy. One such observation, in 2009,4 found that the times of arrival of all the different frequencies in the burst differed by no more than 2 seconds out of a total time in flight on the order of ten billion years!

Example 4: An interstellar road trip
Because the distances between the stars are so vast, it's convenient to measure them in light-years rather than kilometers. A light-year is defined as the distance traveled by light in one year. If we adopt the year as our unit of time, and the light-year as our unit of distance, then the speed of light is 1, i.e., these units qualify as the kind of “special units” that we've been assuming in all the graphs.

Suppose that Alice stays on earth while her twin Betty heads off in a spaceship for Tau Ceti, a nearby star. Tau Ceti is 12 light-years away, so even though Betty travels at 87% of the speed of light, it will take her a long time to get there: 14 years, according to Alice.

p / Example 4.

Betty experiences time dilation. At this speed, her $$\gamma$$ is 2.0, so that the voyage will only seem to her to last 7 years. But there is perfect symmetry between Alice's and Betty's frames of reference, so Betty agrees with Alice on their relative speed; Betty sees herself as being at rest, while the sun and Tau Ceti both move backward at 87% of the speed of light. How, then, can she observe Tau Ceti to get to her in only 7 years, when it should take 14 years to travel 12 light-years at this speed?

We need to take into account length contraction. Betty sees the distance between the sun and Tau Ceti to be shrunk by a factor of 2. The same thing occurs for Alice, who observes Betty and her spaceship to be foreshortened.

Discussion Questions

A person in a spaceship moving at 99.99999999% of the speed of light relative to Earth shines a flashlight forward through dusty air, so the beam is visible. What does she see? What would it look like to an observer on Earth?

A question that students often struggle with is whether time and space can really be distorted, or whether it just seems that way. Compare with optical illusions or magic tricks. How could you verify, for instance, that the lines in the figure are actually parallel? Are relativistic effects the same, or not?

On a spaceship moving at relativistic speeds, would a lecture seem even longer and more boring than normal?

Mechanical clocks can be affected by motion. For example, it was a significant technological achievement to build a clock that could sail aboard a ship and still keep accurate time, allowing longitude to be determined. How is this similar to or different from relativistic time dilation?

The figure shows an artist's rendering of the length contraction for the collision of two gold nuclei at relativistic speeds in the RHIC accelerator in Long Island, New York, which went on line in 2000. The gold nuclei would appear nearly spherical (or just slightly lengthened like an American football) in frames moving along with them, but in the laboratory's frame, they both appear drastically foreshortened as they approach the point of collision. The later pictures show the nuclei merging to form a hot soup, in which experimenters hope to observe a new form of matter. What would the shapes of the two nuclei look like to a microscopic observer riding on the left-hand nucleus? To an observer riding on the right-hand one? Can they agree on what is happening? If not, why not --- after all, shouldn't they see the same thing if they both compare the two nuclei side-by-side at the same instant in time?

Discussion question E: colliding nuclei show relativistic length contraction.

If you stick a piece of foam rubber out the window of your car while driving down the freeway, the wind may compress it a little. Does it make sense to interpret the relativistic length contraction as a type of strain that pushes an object's atoms together like this? How does this relate to discussion question E?

4.3 Dynamics

So far we have said nothing about how to predict motion in relativity. Do Newton's laws still work? Do conservation laws still apply? The answer is yes, but many of the definitions need to be modified, and certain entirely new phenomena occur, such as the conversion of mass to energy and energy to mass, as described by the famous equation $$E=mc^2$$.

Momentum

Here's a flawed scheme for traveling faster than the speed of light. The basic idea can be demonstrated by dropping a ping-pong ball and a baseball stacked on top of each other like a snowman. They separate slightly in mid-air, and the baseball therefore has time to hit the floor and rebound before it collides with the ping-pong ball, which is still on the way down. The result is a surprise if you haven't seen it before: the ping-pong ball flies off at high speed and hits the ceiling! A similar fact is known to people who investigate the scenes of accidents involving pedestrians. If a car moving at 90 kilometers per hour hits a pedestrian, the pedestrian flies off at nearly double that speed, 180 kilometers per hour. Now suppose the car was moving at 90 percent of the speed of light. Would the pedestrian fly off at 180% of $$c$$?

q / An unequal collision, viewed in the center-of-mass frame, 1, and in the frame where the small ball is initially at rest, 2. The motion is shown as it would appear on the film of an old-fashioned movie camera, with an equal amount of time separating each frame from the next. Film 1 was made by a camera that tracked the center of mass, film 2 by one that was initially tracking the small ball, and kept on moving at the same speed after the collision.

To see why not, we have to back up a little and think about where this speed-doubling result comes from. For any collision, there is a special frame of reference, the center-of-mass frame, in which the two colliding objects approach each other, collide, and rebound with their velocities reversed. In the center-of-mass frame, the total momentum of the objects is zero both before and after the collision.

Figure q/1 shows such a frame of reference for objects of very unequal mass. Before the collision, the large ball is moving relatively slowly toward the top of the page, but because of its greater mass, its momentum cancels the momentum of the smaller ball, which is moving rapidly in the opposite direction. The total momentum is zero. After the collision, the two balls just reverse their directions of motion. We know that this is the right result for the outcome of the collision because it conserves both momentum and kinetic energy, and everything not forbidden is mandatory, i.e., in any experiment, there is only one possible outcome, which is the one that obeys all the conservation laws.

self-check:

How do we know that momentum and kinetic energy are conserved in figure q/1?

Let's make up some numbers as an example. Say the small ball has a mass of 1 kg, the big one 8 kg. In frame 1, let's make the velocities as follows:

 before the collision after the collision small ball -0.8 0.8 big ball 0.1 -0.1

Figure q/2 shows the same collision in a frame of reference where the small ball was initially at rest. To find all the velocities in this frame, we just add 0.8 to all the ones in the previous table.

 before the collision after the collision small ball 0 1.6 big ball 0.9 0.7

In this frame, as expected, the small ball flies off with a velocity, 1.6, that is almost twice the initial velocity of the big ball, 0.9.

If all those velocities were in meters per second, then that's exactly what happened. But what if all these velocities were in units of the speed of light? Now it's no longer a good approximation just to add velocities. We need to combine them according to the relativistic rules. For instance, reasoning similar to that in problem 1, page 86 tells us that combining a velocity of 0.8 times the speed of light with another velocity of 0.8 results in 0.98, not 1.6. The results are very different:

 before the collision after the collision small ball 0 0.98 big ball 0.83 0.76

r / An 8-kg ball moving at 83% of the speed of light hits a 1-kg ball. The balls appear foreshortened due to the relativistic distortion of space.

We can interpret this as follows. Figure q/1 is one in which the big ball is moving fairly slowly. This is very nearly the way the scene would be seen by an ant standing on the big ball. According to an observer in frame r, however, both balls are moving at nearly the speed of light after the collision. Because of this, the balls appear foreshortened, but the distance between the two balls is also shortened. To this observer, it seems that the small ball isn't pulling away from the big ball very fast.

Now here's what's interesting about all this. The outcome shown in figure q/2 was supposed to be the only one possible, the only one that satisfied both conservation of energy and conservation of momentum. So how can the different result shown in figure r be possible? The answer is that relativistically, momentum must not equal $$mv$$. The old, familiar definition is only an approximation that's valid at low speeds. If we observe the behavior of the small ball in figure r, it looks as though it somehow had some extra inertia. It's as though a football player tried to knock another player down without realizing that the other guy had a three-hundred-pound bag full of lead shot hidden under his uniform --- he just doesn't seem to react to the collision as much as he should. This extra inertia is described by redefining momentum as

$\begin{equation*} \text{momentum} = m \gamma v . \end{equation*}$

At very low velocities, $$\gamma$$ is close to 1, and the result is very nearly $$mv$$, as demanded by the correspondence principle. But at very high velocities, $$\gamma$$ gets very big --- the small ball in figure r has a $$\gamma$$ of 5.0, and therefore has five times more inertia than we would expect nonrelativistically.

Equivalence of mass and energy

Now we're ready to see why mass and energy must be equivalent as claimed in the famous $$E=mc^2$$. So far we've only considered collisions in which none of the kinetic energy is converted into any other form of energy, such as heat or sound. Let's consider what happens if a blob of putty moving at velocity $$v$$ hits another blob that is initially at rest, sticking to it. The nonrelativistic result is that to obey conservation of momentum the two blobs must fly off together at $$v/2$$. Half of the initial kinetic energy has been converted to heat.5

Relativistically, however, an interesting thing happens. A hot object has more momentum than a cold object! This is because the relativistically correct expression for momentum is $$m\gamma v$$, and the more rapidly moving atoms in the hot object have higher values of $$\gamma$$. In our collision, the final combined blob must therefore be moving a little more slowly than the expected $$v/2$$, since otherwise the final momentum would have been a little greater than the initial momentum. To an observer who believes in conservation of momentum and knows only about the overall motion of the objects and not about their heat content, the low velocity after the collision would seem to be the result of a magical change in the mass, as if the mass of two combined, hot blobs of putty was more than the sum of their individual masses.

Now we know that the masses of all the atoms in the blobs must be the same as they always were. The change is due to the change in $$\gamma$$ with heating, not to a change in mass. The heat energy, however, seems to be acting as if it was equivalent to some extra mass.

But this whole argument was based on the fact that heat is a form of kinetic energy at the atomic level. Would $$E=mc^2$$ apply to other forms of energy as well? Suppose a rocket ship contains some electrical energy stored in a battery. If we believed that $$E=mc^2$$ applied to forms of kinetic energy but not to electrical energy, then we would have to believe that the pilot of the rocket could slow the ship down by using the battery to run a heater! This would not only be strange, but it would violate the principle of relativity, because the result of the experiment would be different depending on whether the ship was at rest or not. The only logical conclusion is that all forms of energy are equivalent to mass. Running the heater then has no effect on the motion of the ship, because the total energy in the ship was unchanged; one form of energy (electrical) was simply converted to another (heat).

The equation $$E=mc^2$$ tells us how much energy is equivalent to how much mass: the conversion factor is the square of the speed of light, $$c$$. Since $$c$$ a big number, you get a really really big number when you multiply it by itself to get $$c^2$$. This means that even a small amount of mass is equivalent to a very large amount of energy.

We've already seen several examples of applications of $$E=mc^2$$, on page 30.

Homework Problems

s / Problem 5.

1. The figure illustrates a Lorentz transformation using the conventions described on p. 43. For simplicity, the transformation chosen is one that lengthens one diagonal by a factor of 2. Since Lorentz transformations preserve area, the other diagonal is shortened by a factor of 2. Let the original frame of reference, depicted with the square, be A, and the new one B. (a) By measuring with a ruler on the figure, show that the velocity of frame B relative to frame A is $$0.6c$$. (b) Print out a copy of the page. With a ruler, draw a third parallelogram that represents a second successive Lorentz transformation, one that lengthens the long diagonal by another factor of 2. Call this third frame C. Use measurements with a ruler to determine frame C's velocity relative to frame A. Does it equal double the velocity found in part a? Explain why it should be expected to turn out the way it does.(answer check available at lightandmatter.com)

2. Astronauts in three different spaceships are communicating with each other. Those aboard ships A and B agree on the rate at which time is passing, but they disagree with the ones on ship C.
(a) Alice is aboard ship A. How does she describe the motion of her own ship, in its frame of reference?
(b) Describe the motion of the other two ships according to Alice.
(c) Give the description according to Betty, whose frame of reference is ship B.
(d) Do the same for Cathy, aboard ship C.

3. Figure e on p. 43 shows a convention for representing a Lorentz transformation using a parallelogram. Recall that on these graphs, the slope of the parallelogram's bottom edge represents the velocity, and that special units are assumed in which the speed of light equals 1. What would happen to the diagram if the velocity equaled the speed of light?

Problem 4.

4. The figure shows a famous thought experiment devised by Einstein. A train is moving at constant velocity to the right when bolts of lightning strike the ground near its front and back. Alice, standing on the dirt at the midpoint of the flashes, observes that the light from the two flashes arrives simultaneously, so she says the two strikes must have occurred simultaneously. Bob, meanwhile, is sitting aboard the train, at its middle. He passes by Alice at the moment when Alice later figures out that the flashes happened. Later, he receives flash 2, and then flash 1. He infers that since both flashes traveled half the length of the train, flash 2 must have occurred first. How can this be reconciled with Alice's belief that the flashes were simultaneous? Explain using a graph. Note that the light from the flashes will move at velocity $$c$$ or $$-c$$, represented by lines at 45-degree angles.

5. The rod in the figure is perfectly rigid. At event A, the hammer strikes one end of the rod. At event B, the other end moves. Since the rod is perfectly rigid, it can't compress, so A and B are simultaneous. In frame 2, B happens before A. Did the motion at the right end cause the person on the left to decide to pick up the hammer and use it?

6. Suppose that the starship Enterprise from Star Trek has a mass of $$8.0\times10^7$$ kg, about the same as the Queen Elizabeth 2. Suppose that it was moving at half the speed of light. Read its $$\gamma$$ off of the graph in figure l on p. 76, and use this to compute its energy. Compare with the total energy content of the world's nuclear arsenals, which is about $$10^{21}$$ J.(answer check available at lightandmatter.com)

7. In the graph in figure l on p. 76, the $$\gamma$$ factor blows up to infinity as the velocity approaches the speed of light. Recall that force is the rate of change of momentum, and that relativistic momentum is given by $$m\gamma v$$. Based on these ideas, what would happen if we applied a constant force to an object for a very long time? Would it eventually go faster than the speed of light?

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

Footnotes
[1] For a proof that no other version is possible, see ch. 23 of my free online book Light and Matter.
[2] The equal-area property is proved in Light and Matter.
[3] To avoid the tedious work of drawing many figures like k, one can use algebra and geometry to derive the equation $$\gamma=1/\sqrt{1-v^2}$$.
[5] A double-mass object moving at half the speed does not have the same kinetic energy. Kinetic energy depends on the square of the velocity, so cutting the velocity in half reduces the energy by a factor of 1/4, which, multiplied by the doubled mass, makes 1/2 the original energy.