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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.
b / The atomic clock has its own ticket and its own seat.
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 of the U.S. Naval Observatory brought atomic clocks aboard commercial airliners, b, and went around the world, once from east to west and once from west to east. (The clocks had their own tickets, and occupied their own seats.) 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 at the lab in Washington. The east-going clock lost time, ending up off by -59±10 nanoseconds, while the west-going one gained 273±7 ns.
This establishes that time doesn't work the way Newton believed it did when he wrote that “Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external...” We are used to thinking of time as absolute and universal, so it is disturbing to find that it can flow at a different rate for observers in different frames of reference. Nevertheless, the effect that Hafele and Keating observed were small. This makes sense: Newton's laws have already been thoroughly tested by experiments under a wide variety of conditions, so a new theory like relativity must agree with Newton's to a good approximation, within the Newtonian theory's realm of applicability. This requirement of backward-compatibility is known as the correspondence principle.
c / Newton's laws do not distinguish past from future. The football could travel in either direction while obeying Newton's laws.
It's also 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.
Causality is like a water-hungry front-yard lawn in Los Angeles: we know we want it, but it's not easy to explain why. Even in plain old Newtonian physics, there is no clear distinction between past and future. In figure c, number 18 throws the football to number 25, and the ball obeys Newton's laws of motion. If we took a video of the pass and played it backward, we would see the ball flying from 25 to 18, and Newton's laws would still be satisfied. Nevertheless, we have a strong psychological impression that there is a forward arrow of time. I can remember what the stock market did last year, but I can't remember what it will do next year. Joan of Arc's military victories against England caused the English to burn her at the stake; it's hard to accept that Newton's laws provide an equally good description of a process in which her execution in 1431 caused her to win a battle in 1429. There is no consensus at this point among physicists on the origin and significance of time's arrow, and for our present purposes we don't need to solve this mystery. Instead, we merely note the empirical fact that, regardless of what causality really means and where it really comes from, its behavior is consistent. Specifically, experiments show that if an observer in a certain frame of reference observes that event A causes event B, then observers in other frames agree that A causes B, not the other way around. This is merely a generalization about a large body of experimental results, not a logically necessary assumption. If Keating had gone around the world and arrived back in Washington before he left, it would have disproved this statement about causality.
d / 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.
e / The correspondence principle requires that the relativistic distortion of time become small for small velocities.
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. Making up some simple numbers to keep the arithmetic transparent, suppose that the earth rotates from west to east at 1000 km/hr, and that the planes fly at 300 km/hr. Then the speed of the clock on the ground is 1000 km/hr, the speed of the clock on the east-going plane is 1300 km/hr, and that of the west-going clock 700 km/hr. Since the speeds of 700, 1000, and 1300 km/hr have equal spacing on either side of 1000, we would expect the discrepancies of the moving clocks relative to the one in the lab to be equal in size but opposite in sign.
In fact, the two effects are unequal in size: -59 ns and 273 ns. This implies that there is a second effect involved, simply due to the planes' being up in the air. Relativity predicts that time's rate of flow also changes with height in a gravitational field. Einstein didn't figure out how to incorporate gravity into relativity until 1915, after much frustration and many false starts. The simpler version of the theory without gravity is known as special relativity, the full version as general relativity. We'll restrict ourselves to special relativity in this book, and that means that what we want to focus on right now is the distortion of time due to motion, not gravity.
We can now see in more detail how to apply the correspondence principle. 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 ∼10-15 --- 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 ∼10-10. 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. Special 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, which was actually more important as a test of general relativity, is that it is conceptually very direct.
a / 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.
b / A change of units distorts an x-t graph. This graph depicts exactly the same events as figure a. 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.
c / A convention we'll use to represent a distortion of time and space.
d / A Galilean version of the relationship between two frames of reference. 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 position axis that goes straight up.
e / A transformation that leads to disagreements about whether two events occur at the same time and place. This is not just a matter of opinion. Either the arrow hit the bull's-eye or it didn't.
f / A nonlinear transformation.
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.
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 are familiar by now, 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 a. 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 b.
We're going to have quite a few examples of this type, so I'll adopt the convention shown in figure c for depicting them. Figure c summarizes the relationship between figures a and b in a more compact form. The gray rectangle represents the original coordinate grid of figure a, while the grid of black lines represents the new version from figure b. 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 c 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. I want to emphasize that there is no purely logical way of deciding on this list of properties. The ones I'll list are simply a summary of the patterns observed in the results from a large body of experiments. Furthermore, some of them are only approximate. For example, property 1 below is only a good approximation when the gravitational field is weak, so it is a property that applies to special relativity, not to general relativity.
Most of these are not very subversive. Properties 1 and 2 date back to the time when Galileo and Newton started applying the same universal laws of motion to the solar system and to the earth; this contradicted Aristotle, who believed that, for example, a rock would naturally want to move in a certain special direction (down) in order to reach a certain special location (the earth's surface). Property 3 is the reason that Einstein called his theory “relativity,” but Galileo and Newton believed exactly the same thing to be true, as dramatized by Galileo's run-in with the Church over the question of whether the earth could really be in motion around the sun. Property 4 would probably surprise most people only because it asserts in such a weak and specialized way something that they feel deeply must be true. The only really strange item on the list is 5, but the Hafele-Keating experiment forces it upon us.
If it were not for property 5, we could imagine that figure d 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.
Figure d has familiar, comforting, and eminently sensible behavior, but it also happens to be wrong, because it violates property 5. 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.
It might seem as though we still had a huge amount of wiggle room available for the correct form of the distortion. It turns out, however, that properties 1-5 are sufficient to prove that there is only one answer, which is the one found by Einstein in 1905. To see why this is, let's work by a process of elimination.
Figure e shows a transformation that might seem at first glance to be as good a candidate as any other, but it violates property 3, that motion is relative, for the following reason. In observer 2's frame of reference, some of the grid lines cross one another. This means that observers 1 and 2 disagree on whether or not certain events are the same. For instance, suppose that event A marks the arrival of an arrow at the bull's-eye of a target, and event B is the location and time when the bull's-eye is punctured. Events A and B occur at the same location and at the same time. If one observer says that A and B coincide, but another says that they don't, we have a direct contradiction. Since the two frames of reference in figure e give contradictory results, one of them is right and one is wrong. This violates property 3, because all inertial frames of reference are supposed to be equally valid. To avoid problems like this, we clearly need to make sure that none of the grid lines ever cross one another.
The next type of transformation we want to kill off is shown in figure f, in which the grid lines curve, but never cross one another. The trouble with this one is that it violates property 1, the uniformity of time and space. The transformation is unusually “twisty” at A, whereas at B it's much more smooth. This can't be correct, because the transformation is only supposed to depend on the relative state of motion of the two frames of reference, and that given information doesn't single out a special role for any particular point in spacetime. If, for example, we had one frame of reference rotating relative to the other, then there would be something special about the axis of rotation. But we're only talking about inertial frames of reference here, as specified in property 3, so we can't have rotation; each frame of reference has to be moving in a straight line at constant speed. For frames related in this way, there is nothing that could single out an event like A for special treatment compared to B, so transformation f violates property 1.
The examples in figures e and f show that the transformation we're looking for must be linear, meaning that it must transform lines into lines, and furthermore that it has to take parallel lines to parallel lines. Einstein wrote in his 1905 paper that “... on account of the property of homogeneity [property 1] which we ascribe to time and space, the [transformation] must be linear.”1 Applying this to our diagrams, the original gray rectangle, which is a special type of parallelogram containing right angles, must be transformed into another parallelogram. There are three types of transformations, figure g, that have this property. Case I is the Galilean transformation of figure d on page 377, which we've already ruled out.
g / Three types of transformations that preserve parallelism. Their distinguishing feature is what they do to simultaneity, as shown by what happens to the left edge of the original rectangle. In I, the left edge remains vertical, so simultaneous events remain simultaneous. In II, the left edge turns counterclockwise. In III, it turns clockwise.
Case II can also be discarded. Here every point on the grid rotates counterclockwise. What physical parameter would determine the amount of rotation? The only thing that could be relevant would be v, the relative velocity of the motion of the two frames of reference with respect to one another. But if the angle of rotation was proportional to v, then for large enough velocities the grid would have left and right reversed, and this would violate property 4, causality: one observer would say that event A caused a later event B, but another observer would say that B came first and caused A.The only remaining possibility is case III, which I've redrawn in figure h with a couple of changes. This is the one that Einstein predicted in 1905. The transformation is known as the Lorentz transformation, after Hendrik Lorentz (1853-1928), who partially anticipated Einstein's work, without arriving at the correct interpretation. The distortion is a kind of smooshing and stretching, as suggested by the hands. Also, we've already seen in figures a-c on page 376 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.
As in our discussion of the Galilean transformation, slopes are interpreted as velocities, and the slope of the near-horizontal lines in figure i is interpreted as the relative velocity of the two observers. The difference between the Galilean version and the relativistic one is that now there is smooshing happening from the other side as well. Lines that were vertical in the original grid, representing simultaneous events, now slant over to the right. This tells us that, as required by property 5, different observers do not agree on whether events that occur in different places are simultaneous. The Hafele-Keating experiment tells us that this non-simultaneity effect is fairly small, even when the velocity is as big as that of a passenger jet, and this is what we would have anticipated by the correspondence principle. The way that this is expressed in the graph is that if we pick the time unit to be the second, then the distance unit turns out to be hundreds of thousands of miles. In these units, the velocity of a passenger jet is an extremely small number, so the slope v in figure i is extremely small, and the amount of distortion is tiny --- it would be much too small to see on this scale.
The only thing left to determine about the Lorentz transformation is the size of the transformed parallelogram relative to the size of the original one. Although the drawing of the hands in figure h may suggest that the grid deforms like a framework made of rigid coat-hanger wire, that is not the case. If you look carefully at the figure, you'll see that the edges of the smooshed parallelogram are actually a little longer than the edges of the original rectangle. In fact what stays the same is not lengths but areas. The proof of this fact is straightforward, but a little lengthy, so I've relegated it to page 846, following it with some remarks that may be of interest to teachers. Oversimplifying a little, the basic idea of the proof is that it wouldn't make sense if the area was increased by the Lorentz transformation, because then area would have to be decreased by a Lorentz transformation corresponding to motion in the opposite direction, and this would violate property 2 on page 377, which states that all directions in space have the same properties.
j / The γ factor.
l / Muons accelerated to nearly c undergo radioactive decay much more slowly than they would according to an observer at rest with respect to the muons. The first two data-points (unfilled circles) were subject to large systematic errors.
n / In the garage's frame of reference, 1, the bus is moving, and can fit in the garage. In the bus's frame of reference, the garage is moving, and can't hold the bus.
With a little algebra and geometry (homework problem 7, page 397), one can use the equal-area property to show that the factor γ (Greek letter gamma) defined in figure j is given by the equation
If you've had good training in physics, the first thing you probably think when you look at this equation is that it must be nonsense, because its units don't make sense. How can we take something with units of velocity squared, and subtract it from a unitless 1? But remember that this is expressed in our special relativistic units, in which the same units are used for distance and time. In this system, velocities are always unitless. This sort of thing happens frequently in physics. For instance, before James Joule discovered conservation of energy, nobody knew that heat and mechanical energy were different forms of the same thing, so instead of measuring them both in units of joules as we would do now, they measured heat in one unit (such as calories) and mechanical energy in another (such as foot-pounds). In ordinary metric units, we just need an extra convension factor c, and the equation becomes
Here's why we care about γ. Figure j defines it as the ratio of two times: the time between two events as expressed in one coordinate system, and the time between the same two events as measured in the other one. The interpretation is:
A clock runs fastest in the frame of reference of an observer who is at rest relative to the clock. An observer in motion relative to the clock at speed v perceives the clock as running more slowly by a factor of γ.
Since the Lorentz transformation treats time and distance entirely symmetrically, we could just as well have defined γ using the upright x axis in figure j, and we therefore have a similar interpretation in terms of space:
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 γ.
self-check: What is γ when v=0? What does this mean? (answer in the back of the PDF version of the book)
k / Apparatus used for the test of relativistic time dilation described in example 1. The prominent black and white blocks are large magnets surrounding a circular pipe with a vacuum inside.
(c) 1974 by CERN.
Particles called muons (named after the Greek letter μ, “myoo”) were produced by an accelerator at CERN, near Geneva. A muon is essentially a heavier version of the electron. Muons undergo radioactive decay, lasting an average of only 2.197 μs before they evaporate into an electron and two neutrinos. The 1974 experiment was actually built in order to measure the magnetic properties of muons, but it produced a high-precision test of time dilation as a byproduct. Because muons have the same electric charge as electrons, they can be trapped using magnetic fields. Muons were injected into the ring shown in figure k, circling around it until they underwent radioactive decay. At the speed at which these muons were traveling, they had γ=29.33, so on the average they lasted 29.33 times longer than the normal lifetime. In other words, they were like tiny alarm clocks that self-destructed at a randomly selected time. Figure l shows the number of radioactive decays counted, as a function of the time elapsed after a given stream of muons was injected into the storage ring. The two dashed lines show the rates of decay predicted with and without relativity. The relativistic line is the one that agrees with experiment.
Figure m 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.
m / Colliding nuclei show relativistic 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 paradox arises when we shut the door and then quickly slam on the brakes of the bus. An observer in the garage's frame of reference will claim that the bus fit in the garage because of its contracted length. 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. The person in the garage's frame can shut the door at an instant he perceives to be simultaneous with the front bumper's arrival 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.
o / 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.
Discussion question B
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 relativistic units, but that slope corresponds to c in ordinary metric units. We already know that the relativistic distance unit must be extremely large compared to the relativistic time unit, so c must be extremely large. 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. (The same thing happens with -c.)
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.
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 396). 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 o.
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.
p / The Michelson-Morley experiment, shown in photographs, and drawings from the original 1887 paper. 1. A simplified drawing of the apparatus. A beam of light from the source, s, is partially reflected and partially transmitted by the half-silvered mirror h1. The two half-intensity parts of the beam are reflected by the mirrors at a and b, reunited, and observed in the telescope, t. If the earth's surface was supposed to be moving through the ether, then the times taken by the two light waves to pass through the moving ether would be unequal, and the resulting time lag would be detectable by observing the interference between the waves when they were reunited. 2. In the real apparatus, the light beams were reflected multiple times. The effective length of each arm was increased to 11 meters, which greatly improved its sensitivity to the small expected difference in the speed of light. 3. In an earlier version of the experiment, they had run into problems with its “extreme sensitiveness to vibration,” which was “so great that it was impossible to see the interference fringes except at brief intervals ... even at two o'clock in the morning.” They therefore mounted the whole thing on a massive stone floating in a pool of mercury, which also made it possible to rotate it easily. 4. A photo of the apparatus.
◊ 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?
◊ Figure m from page 383, depicting the collision of two nuclei at the RHIC accelerator, is reproduced below. 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?
q / 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?
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=mc2.
Consider the following 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?
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.
a / 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.
Figure a/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 a/1? (answer in the back of the PDF version of the book)
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 a/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, the technique used in problem 1 can be used to show 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 |
b / 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 a/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 b, 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 a/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 b 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 b, 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
At very low velocities, γ is close to 1, and the result is very nearly mv, as demanded by the correspondence principle. But at very high velocities, γ gets very big --- the small ball in figure b has a γ of 5.0, and therefore has five times more inertia than we would expect nonrelativistically.
This also explains the answer to another paradox often posed by beginners at relativity. Suppose you keep on applying a steady force to an object that's already moving at 0.9999c. Why doesn't it just keep on speeding up past c? The answer is that force is the rate of change of momentum. At 0.9999c, an object already has a γ of 71, and therefore has already sucked up 71 times the momentum you'd expect at that speed. As its velocity gets closer and closer to c, its γ approaches infinity. To move at c, it would need an infinite momentum, which could only be caused by an infinite force.
d / A New York Times headline from November 10, 1919, describing the observations discussed in example 4.
e / Top: A PET scanner. Middle: Each positron annihilates with an electron, producing two gamma-rays that fly off back-to-back. When two gamma rays are observed simultaneously in the ring of detectors, they are assumed to come from the same annihilation event, and the point at which they were emitted must lie on the line connecting the two detectors. Bottom: A scan of a person's torso. The body has concentrated the radioactive tracer around the stomach, indicating an abnormal medical condition.
Now we're ready to see why mass and energy must be equivalent as claimed in the famous E=mc2. 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.3
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 mγ v, and the more rapidly moving atoms in the hot object have higher values of γ. 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 γ 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=mc2 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=mc2 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=mc2 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 c2. This means that even a small amount of mass is equivalent to a very large amount of energy.
c / Example 4, page 392.
A star with sufficiently strong gravity can prevent light from leaving. Quite a few black holes have been detected via their gravitational forces on neighboring stars or clouds of gas and dust.
You've learned about conservation of mass and conservation of energy, but now we see that they're not even separate conservation laws. As a consequence of the theory of relativity, mass and energy are equivalent, and are not separately conserved --- one can be converted into the other. Imagine that a magician waves his wand, and changes a bowl of dirt into a bowl of lettuce. You'd be impressed, because you were expecting that both dirt and lettuce would be conserved quantities. Neither one can be made to vanish, or to appear out of thin air. However, there are processes that can change one into the other. A farmer changes dirt into lettuce, and a compost heap changes lettuce into dirt. At the most fundamental level, lettuce and dirt aren't really different things at all; they're just collections of the same kinds of atoms --- carbon, hydrogen, and so on. Because mass and energy are like two different sides of the same coin, we may speak of mass-energy, a single conserved quantity, found by adding up all the mass and energy, with the appropriate conversion factor: E+mc2.
◊ The energy will appear as heat, which will be lost to the environment. The total mass-energy of the cup, water, and iron will indeed be lessened by 0.5 MJ. (If it had been perfectly insulated, there would have been no change, since the heat energy would have been trapped in the cup.) The speed of light is c=3×108 meters per second, so converting to mass units, we have
The change in mass is too small to measure with any practical technique. This is because the square of the speed of light is such a large number.
Natural radioactivity in the earth produces positrons, which are like electrons but have the opposite charge. A form of antimatter, positrons annihilate with electrons to produce gamma rays, a form of high-frequency light. Such a process would have been considered impossible before Einstein, because conservation of mass and energy were believed to be separate principles, and this process eliminates 100% of the original mass. The amount of energy produced by annihilating 1 kg of matter with 1 kg of antimatter is
which is on the same order of magnitude as a day's energy consumption for the entire world's population!
Positron annihilation forms the basis for the medical imaging technique called a PET (positron emission tomography) scan, in which a positron-emitting chemical is injected into the patient and mapped by the emission of gamma rays from the parts of the body where it accumulates.
One commonly hears some misinterpretations of E=mc2, one being that the equation tells us how much kinetic energy an object would have if it was moving at the speed of light. This wouldn't make much sense, both because the equation for kinetic energy has 1/2 in it, KE=(1/2)mv2, and because a material object can't be made to move at the speed of light. However, this naturally leads to the question of just how much mass-energy a moving object has. We know that when the object is at rest, it has no kinetic energy, so its mass-energy is simply equal to the energy-equivalent of its mass, mc2,
where the symbol stands for mass-energy. (You can write this symbol yourself by writing an E, and then adding an extra line to it. Have fun!) The point of using the new symbol is simply to remind ourselves that we're talking about relativity, so an object at rest has , not E=0 as we'd assume in classical physics.
Suppose we start accelerating the object with a constant force. A constant force means a constant rate of transfer of momentum, but p=mγ v approaches infinity as v approaches c, so the object will only get closer and closer to the speed of light, but never reach it. Now what about the work being done by the force? The force keeps doing work and doing work, which means that we keep on using up energy. Mass-energy is conserved, so the energy being expended must equal the increase in the object's mass-energy. We can continue this process for as long as we like, and the amount of mass-energy will increase without limit. We therefore conclude that an object's mass-energy approaches infinity as its speed approaches the speed of light,
Now that we have some idea what to expect, what is the actual equation for the mass-energy? As proved in my book Simple Nature, it is
self-check: Verify that this equation has the two properties we wanted. (answer in the back of the PDF version of the book)
◊ The speed of light is a very big number, so mc2 is a huge number of joules. The object has a gigantic amount of energy because of its mass, and only a relatively small amount of additional kinetic energy because of its motion.
Another way of seeing this is that at low speeds, γ is only a tiny bit greater than 1, so is only a tiny bit greater than mc2.
◊ Show that the equation obeys the correspondence principle.
◊ As we accelerate an object from rest, its mass-energy becomes greater than its resting value. Classically, we interpret this excess mass-energy as the object's kinetic energy,
Expressing γ as and making use of the approximation (1+ε)p≈ 1+pε for small ε, we have γ≈ 1+v2/2c2, so
which is the classical expression. As demanded by the correspondence principle, relativity agrees with classical physics at speeds that are small compared to the speed of light.
1. The figure illustrates a Lorentz transformation using the conventions employed in section 7.2. 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) Describe the motion of the other two ships according to Alice, who is aboard
ship A.
(b) Give the description according to Betty, whose frame of reference is ship B.
(c) Do the same for Cathy, aboard ship C.
3. What happens in the equation for γ when you put in a negative number for v? Explain what this means physically, and why it makes sense.
4.
The Voyager 1 space probe, launched in 1977, is moving faster relative to the earth than
any other human-made object, at 17,000 meters per second.
(a) Calculate the probe's γ.(answer check available at lightandmatter.com)
(b) Over the course of one year on earth, slightly less than one year passes on the probe.
How much less? (There are 31 million seconds in a year.)(answer check available at lightandmatter.com)
5. In example 1 on page 382, I remarked that accelerating a macroscopic (i.e., not microscopic) object to close to the speed of light would require an unreasonable amount of energy. Suppose that the starship Enterprise from Star Trek has a mass of 8.0×107 kg, about the same as the Queen Elizabeth 2. Compute the kinetic energy it would have to have if it was moving at half the speed of light. Compare with the total energy content of the world's nuclear arsenals, which is about 1021 J.(answer check available at lightandmatter.com)
6. The earth is orbiting the sun, and therefore is contracted relativistically in the direction of its motion. Compute the amount by which its diameter shrinks in this direction.
7. In this homework problem, you'll fill in the steps of the algebra required in order to find the equation for γ on page 380. To keep the algebra simple, let the time t in figure j equal 1, as suggested in the figure accompanying this homework problem. The original square then has an area of 1, and the transformed parallelogram must also have an area of 1. (a) Prove that point P is at x=vγ, so that its (t,x) coordinates are (γ,vγ). (b) Find the (t,x) coordinates of point Q. (c) Find the length of the short diagonal connecting P and Q. (d) Find the coordinates of the midpoint C of the parallelogram, and then find its distance OC from the origin. (e) Find the area of the parallelogram by computing twice the area of triangle PQO. [Hint: You can take PQ to be the base of the triangle.] (f) Set this area equal to 1 and solve for γ to prove .(answer check available at lightandmatter.com)
8. (a) A free neutron (as opposed to a neutron bound into an atomic nucleus) is unstable, and undergoes beta decay (which you may want to review). The masses of the particles involved are as follows:
| neutron | kg |
| proton | kg |
| electron | kg |
| antineutrino | kg |
Find the energy released in the decay of a free neutron. (answer check available at lightandmatter.com)
(b) Neutrons and protons make up essentially all of the mass of the ordinary
matter around us. We observe that the universe around us has no free neutrons, but
lots of free protons
(the nuclei of hydrogen, which is the element that 90% of the universe
is made of). We find neutrons only inside nuclei along with other neutrons and
protons, not on their own.
If there are processes that can convert neutrons into protons, we might imagine that there could also be proton-to-neutron conversions, and indeed such a process does occur sometimes in nuclei that contain both neutrons and protons: a proton can decay into a neutron, a positron, and a neutrino. A positron is a particle with the same properties as an electron, except that its electrical charge is positive (see chapter 7). A neutrino, like an antineutrino, has negligible mass.
Although such a process can occur within a nucleus, explain why it cannot happen to a free proton. (If it could, hydrogen would be radioactive, and you wouldn't exist!)
9.
(a) Find a relativistic equation for the velocity of an
object in terms of its mass and momentum (eliminating
γ).(answer check available at lightandmatter.com)
(b) Show that your result
is approximately the same as the classical value, p/m, at
low velocities.
(c) Show that very large momenta result in
speeds close to the speed of light.
10.
(a) Show that for v=(3/5)c, γ comes out to be a simple fraction.
(b) Find another value of v for which γ is a simple fraction.
11.
An object moving at a speed very close to the speed of light is referred to as
ultrarelativistic. Ordinarily (luckily) the only ultrarelativistic objects
in our universe are subatomic particles, such as cosmic rays or particles
that have been accelerated in a particle accelerator.
(a) What kind of number is γ for an ultrarelativistic particle?
(b) Repeat example 8 on page 394,
but instead of very low, nonrelativistic speeds, consider ultrarelativistic speeds.
(c) Find an equation for the ratio
. The speed may be relativistic, but don't
assume that it's ultrarelativistic.(answer check available at lightandmatter.com)
(d) Simplify your answer to part c for the case where the speed is ultrarelativistic.(answer check available at lightandmatter.com)
(e) We can think of a beam of light as an ultrarelativistic object --- it certainly moves at a speed
that's sufficiently close to the speed of light! Suppose you turn on a one-watt flashlight, leave it
on for one second, and then turn it off. Compute the momentum of the recoiling flashlight, in units
of kg⋅m/s.(answer check available at lightandmatter.com)
(f) Discuss how your answer in part e relates to the correspondence principle.
12. As discussed in
chapter 6, the speed at which a disturbance travels along
a string under tension is given by
, where μ is the mass per unit
length, and T is the tension.
(a) Suppose a string has a density ρ, and a cross-sectional
area A. Find an expression for the maximum tension that could possibly exist in the string
without producing v>c, which is impossible according to relativity. Express your answer in
terms of ρ, A, and c. The interpretation is that relativity puts a limit on how
strong any material can be.
(b) Every substance has a tensile strength, defined as the force
per unit area required to break it by pulling it apart. The tensile strength is measured in
units of N/m2, which is the same as the pascal (Pa), the mks unit of pressure.
Make a numerical estimate of the maximum tensile strength allowed by relativity in the case where
the rope is made out of ordinary matter, with a density on the same order of magnitude as
that of water. (For comparison, kevlar has a tensile strength of about 4×109 Pa,
and there is speculation that fibers made from carbon nanotubes could have
values as high as 6×1010 Pa.)
(c) A black hole is a star that has collapsed and become very dense, so that
its gravity is too strong for anything ever to escape from it. For instance, the escape
velocity from a black hole is greater than c, so a projectile can't be shot out of it.
Many people, when they hear this description of a black hole in terms of an escape velocity
greater than c, wonder why it still wouldn't be possible to extract an object from a black
hole by other means than launching it out as a projectile.
For example, suppose we lower an astronaut into a black hole on a rope, and then pull him
back out again. Why might this not work?
13. (a) A charged particle is surrounded by a uniform electric field.
Starting from rest, it is accelerated by the field to speed v after
traveling a distance d. Now it is allowed to continue for a further
distance 3d, for a total displacement from the start of 4d.
What speed will it reach,
assuming classical physics?
(b) Find the relativistic result for the case of v=c/2.
14. In the equation for the relativistic addition of velocities u and v, consider the limit in which u approaches 1, but v simultaneously approaches -1. Give both a physical and a mathematical interpretation.
15. Expand the equation K = m(γ-1) in a Taylor series, and find the first two nonvaninishing terms. Show that the first term is the classical expression for kinetic energy.
16. Expand the relativistic equation for momentum in a Taylor series, and find the first two nonvaninishing terms. Show that the first term is the classical expression.
In this exercise you will analyze the Michelson-Morley
experiment, and find what the results should have been
according to Galilean relativity and Einstein's theory of
relativity. A beam of light coming from the west (not shown)
comes to the half-silvered mirror A. Half the light goes
through to the east, is reflected by mirror C, and comes
back to A. The other half is reflected north by A, is
reflected by B, and also comes back to A. When the beams
reunite at A, part of each ends up going south, and these
parts interfere with one another. If the time taken for a
round trip differs by, for example, half the period of the
wave, there will be destructive interference.
The point of the experiment was to search for a difference in the experimental results between the daytime, when the laboratory was moving west relative to the sun, and the nighttime, when the laboratory was moving east relative to the sun. Galilean relativity and Einstein's theory of relativity make different predictions about the results. According to Galilean relativity, the speed of light cannot be the same in all reference frames, so it is assumed that there is one special reference frame, perhaps the sun's, in which light travels at the same speed in all directions; in other frames, Galilean relativity predicts that the speed of light will be different in different directions, e.g., slower if the observer is chasing a beam of light. There are four different ways to analyze the experiment:
Groups 1-4 work in the sun's frame of reference according to Galilean relativity.
Group 1 finds time AC. Group 2 finds time CA. Group 3 finds time AB. Group 4 finds time BA.
Groups 5 and 6 transform the lab-frame results into the sun's frame according to Einstein's theory.
Group 5 transforms the x and t when ray ACA gets back to A into the sun's frame of reference, and group 6 does the same for ray ABA.
Discussion:
Michelson and Morley found no change in the interference of the waves between day and night. Which version of relativity is consistent with their results?
What does each theory predict if v approaches c?
What if the arms are not exactly equal in length?
Does it matter if the “special” frame is some frame other than the sun's?
In Slowlightland, the speed of light is 20 mi/hr ≈ 32 km/hr ≈ 9 m/s. Think of an example of how relativistic effects would work in sports. Things can get very complex very quickly, so try to think of a simple example that focuses on just one of the following effects:
- relativistic momentum
- relativistic kinetic energy
- relativistic addition of velocities
- time dilation and length contraction
- Doppler shifts of light
- equivalence of mass and energy
- time it takes for light to get to an athlete's eye
- deflection of light rays by gravity
