Special Relativity Length, Momentum, ami Energy
he speed of light is the speed limit for all matter. Suppose that two spaceships are T both traveling at nearly the speed of light and they are moving directly toward each other. The realms of space-time for each spaceship differ in such a way that the relative speed of approach Mass converts to energy and is still less than the speed of light! For example, if both spaceships vice versa. are traveling toward each other at 80% the speed of light with respect to Earth, an observer on each spaceship would measure the speed of approach of the other spaceship as 98% the speed of light. There are no circumstances where the relative speeds of any material objects surpass the speed of light. Why is the speed of light the universal speed limit? To under- stand this, we must know how motion through space affects the length, momentum, and energy of moving objects.
16.1 Length Contraction
For moving objects, space as well as time undergoes changes. When viewed by an outside observer, moving objects appear to contract along the direction of motion. The amount of contraction is related to the amount of time dilation. For everyday speeds, the amount of contraction is much too small to be measured. For relativistic speeds, the contraction would be noticeable. A meter stick aboard a Figure 16.1 1 spaceship whizzing past you at 87% the speed of light, for example, A meter stick traveling at 87% would appear to you to be only 0.5 meter long. If it whizzed past at the speed of light relative to an 99.5% the speed of light, it would appear to you to be contracted to observer would be measured as one tenth its original length. The width of the stick, perpendicular to only half as long as normal. the direction of travel (Figure 16.1), doesn't change. As relative speed gets closer and closer to the speed of light, the measured lengths of objects contract closer and closer to zero.
232 Chapter 16 Special Relativity—Length, Momentum, and Energy Do people aboard the spaceship also see their meter sticks—and everything else in their environment—contracted? The answer is no. People in the spaceship see nothing at all unusual about the lengths of things in their own reference frame. If they did, it would violate the first postulate of relativity Recall that all the laws of physics are the same in all uniformly moving reference frames. Besides, there is no relative speed between the people on the spaceship and the things they observe in their own reference frame. However, there is a relative speed between themselves and our frame of reference, so they will see our meter sticks contracted and us as well. A rule of relativity is that changes due to alterations of space-time are always en in the frame of reference of the "other guy."
Figure 16.2 A to the frame of reference of the meter stick, its length is 1 meter. Observers m this frame see our meter sticks contracted. The effects of relativity are always attributed to "the other guy."
The contraction of speeding objects is the contraction of space tself. Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same in the two frames of reference. So if an object is moving hori- zontally, no contraction takes place vertically (Figure 16.3).
0 is. 0.87c u-.0.995c =0.999c o=C (1)
gure 16.3 A relative speed increases, contraction in the direction of motion increases. gths in the perpendicular direction do not change.
233 Relativistic length contraction is stated mathematically: L= Lol /1— (v2 I c2) In this equation, v is the speed of the object relative to the observer, „ LINK TO BIOLOGY c is the speed of light, L is the length of the moving object as mea- r sured by the observer, and Lo is the measured length of the object at rest.* Suppose that an object is at rest, so that v = 0. When 0 is substi- tuted for v in the equation, we find L= Lo, as we would expect. It was stated earlier that if an object were moving at 87% the speed of light, it would contract to half its length. When 0.87c is substituted for v in the equation, we find L = 0.5L0. Or when 0.995c is substituted for v, we find L= 0.1L0, as stated earlier. If the object could reach the speed c, its length would contract to zero. This is one of the reasons that the .c e st speed of light is the upper limit for the speed of any material object. ztor Eart,, s s . . U rise eramlifeti .e
secoridel-seemingly;too4',„: III Question .f.brietto.reach the groun A spacewoman travels by a spherical planet so fast that it appears below before decaying.;,, to her to be an ellipsoid (egg shaped). If she sees the short diameter But because niuons Move'. as half the long diameter, what is her speed relative to the planet? at nearly the speed of light, length contraction dramatically shortens their distance to Earth. You are hit by hundreds of muons every second! ?Awn impact, like that of 16.2 Momentum and Inertia all high-speed elementary in Relativity 1particles, causes biologi- cal mutations. So we see If we push an object that is free to move, it will accelerate. If we a link between the effects maintain a steady push, it will accelerate to higher and higher of relativity and the evo- speeds. If we push with a greater and greater force, we expect the lution of living creatures acceleration in turn to increase. It might seem that the speed should on Earth. increase without limit, but there is a speed limit in the universe— the speed of light. In fact, we cannot accelerate any material object enough to reach the speed of light, let alone surpass it. We can understand this from Newton's second law, which Newton originally expressed in terms of momentum: F= Amy/At (which reduces to the familiar F = ma, or a = Fl m). The momentum form, interestingly, remains valid in relativity theory. Recall from Chapter 7 that the change of momentum of an object is equal to the
III Answer The spacewoman passes the spherical planet at 87% the speed of light
This equation land those that follow) is simply stated as a "guide to thinking" about the ideas of special relativity. The equations are given here without any explanation as to how they are derived.
234 Chapter 16 Special Relativity—Length, Momentum, and Energy fl
impulse applied to it. Apply more impulse and the object acquires more momentum. Double the impulse and the momentum doubles. Apply ten times as much impulse and the object gains ten times as ver, much momentum. Does this mean that momentum can increase without any limit, even though speed cannot? Yes, it does. We learned that momentum equals mass times velocity In equa- tion form, p = my (we use p for momentum). To Newton, infinite ti- momentum would mean infinite speed. Not so in relativity. Einstein was showed that a new definition of momentum is required. It is ght, I) in mu •o, peed it the where v is the speed of an object and cis the speed of light. Notice ect. that the square root in the denominator looks just like the one in the formula for time dilation in the previous chapter. It tells us that the relativistic momentum of an object of mass m and speed v is larger than mu by a factor of 1/V1— (v2/c2). At relativistic speed, momentum increases dramatically. As v approaches c, the denominator of the equation approaches zero. el This means that the momentum approaches infinity! An object pushed to the speed of light would have infinite momentum and would require an infinite impulse, which is clearly impossible. So nothing that has mass can be pushed to the speed of light, much less beyond it. Here is another reason that c is the speed limit in the universe. What if v is much less than c? Then the denominator of the equation is nearly equal to I and pis nearly equal to mu. Newton's definition of momentum is valid at low speed. We often say that a particle pushed close to the speed of light acts as if its mass were increasing, because its momentum—its "iner- tia in motion"—increases more than its speed increases. The quan- tity m in the equation above is called the rest mass of the object. It is the a true constant, a property of the object no matter what speed it has. iould Subatomic particles are routinely pushed to nearly the speed of light. The momenta of such particles may be thousands of times Oct more than the Newton expression mu predicts. One way to look at the momentum of a high-speed particle is in terms of the "stiffness" of its trajectory The more momentum it has, the harder it is to At deflect it—the "stiffer" is its trajectory. If it has a lot of momentum, it gum more greatly resists changing course. om to the ELECTROMAGNETS Figure 16.4 If the momentum of the elec- trons were equal to the Newtonian value my, the beam would follow the dashed line. But because the relativistic momentum, or inertia in motion, about is greater, the beam follows the iation as SCREEN "stiffer" trajectory shown by the ELECTRON BEAM) solid line.
235 This can be seen when a beam of electrons is directed into a mag- netic field. Charged particles moving in a magnetic field experience a force that deflects them from their normal paths. For a particle with a small momentum, the path curves sharply For a particle with a large momentum, the path curves only a little—its trajectory is "stiffer" (Figure 16.4). Even though one particle may be moving only a little faster than another one—say 99.9% of the speed of light instead of 99% of the speed of light—its momentum will be considerably greater and it will follow a straighter path in the magnetic field. Through such experiments, physicists working with subatomic particles at atomic accelerators verify every day the correctness of the relativistic defini- don of momentum and the speed limit imposed by nature.
16.3 Equivalence of Mass and Energy
The most remarkable insight of Einstein's special theory of relativity is his conclusion that mass is simply a form of energy A piece of matter, even if at rest and t:.% en if not interacting with anything else, has "energy of being." This is called it‘ rest energy. Einstein con- cluded that it takes energy to make mass and that energy is released when mass disappears. Rest mass is, in effect, a kind of potential energy Mass stores energy. iust as a boulder rolled to the top of a hill stores energy. When the mass of something decreases, as it can do in nuclear reactions, energy is released, just as the boulder rolling to the bottom of the hill releases energy. The amount of rest energy E,, is related to the mass in by the most celebrated equation of the twentieth century, Eo = nic2 where c is again the speed of light. This equation gives the total energy content of a piece of stationary matter of mass in. In ordinary units of measurement, the speed of light c is a large quantity and its square is even larger. This means that a small amount of mass stores a large amount of energy. The quantity c2 is a "conversion factor." It converts the measurement of mass to the measurement of equivalent energy. Or it is the ratio of rest energy to mass: Eo l m = c2. Its appearance in either form of this equation has _nothing to do with light and nothing to do with motion. The magni- tude of c2 1s90 quadrillion (9 x 1016) joules per kilogram. One kilo- gram of matter has an "energy of being" equal to 90 quadrillion joules. Even a speck of matter with a mass of only 1 milligram has a rest energy of 90 billion joules. Rest energy, like any form of energy, can be converted to other forms, When we strike a match, for example, a chemical reaction occurs and heat is released. Phosphorus atoms in the match head rearrange themselves and combine with oxygen in the air to form new molecules. The resulting molecules have very slightly less mass than the separate phosphorus and oxygen molecules. From a mass stand- point, the whole is slightly less than the sum of its parts, but not by
236 Chapter 16 Special Relativity—Length, Momentum, and Energy Figure 16.5 Saying that a power plant delivers 90 million megajoules of energy to its consumers is equivalent to saying that it delivers 1 gram of energy to its consumers, because mass and energy are equivalent. very much—by only about one part in a billion. For all chemical reac- tions that give off energy there is a corresponding decrease in mass. In nuclear reactions, the decrease in rest mass is considerably more than in chemical reactions—about one part in a thousand. This decrease of mass in the sun by the process of thermonuclear fusion bathes the solar system with radiant energy and nourishes life. The present stage of thermonuclear fusion in the sun has been going on for the past 5 billion years, and there is sufficient hydrogen fuel for fusion to last another 5 billion years. It is nice to have such a big sun! The equation E.= mc2 is not restricted to chemical and nuclear reactions. A change in energy of any object at rest is accompanied by a change in its mass. The filament of a lightbulb has more mass when it is energized with electricity than when it is turned off. A hot cup of tea has more mass than the same cup of tea when cold. A Figure 16.6 A wound-up spring clock has more mass than the same clock when In one second, 4.5 million tons of unwound. But these examples involve incredibly small changes in rest mass are converted to radi- mass—too small to be measured by conventional methods. No won- ant energy in the sun. The sun is der the fundamental relationship between mass and energy was not so massive, however, that in a discovered until this century. million years only one ten-mil- The equation E.= mc2 is more than a formula for the conversion lionth of the sun's rest mass will have been converted to radiant of rest mass into other kinds of energy, or vice versa. It states that energy. energy and mass are the same thing. Mass is simply congealed energy. If you want to know how much energy is in a system, mea- sure its mass. For an object at rest, its energy is its mass. Shake a massive object back and forth; it is energy itself that is hard to shake.
• Question Can we look at the equation E, = mc2 in another way and say that matter transforms into pure energy when it is traveling at the speed of light squared?
• Answer No, no, no! Matter cannot be made to move at the speed of light, let alone the speed of light squared (which is not a speed!). The equation E0 = mc2 simply means that energy and mass are "two sides of the same coin." 16.4 Kinetic Energy in Relativity 16
Einstein dealt also with the energy of moving matter. His formula for If a nE the total energy of a moving piece of matter of mass m is of the regior rnc2 E= This r V1- (v2I c2) advar Newt Notice the by-now familiar square root in the denominator. If the ory ai object is at rest, we can set v equal to 0 and find that the denomina- newt tor is then equal to 1, leading to the famous rest-mass formula Newt E0 = mc 2. But if the object is moving, the denominator is less than less tl 1 and the total energy E is greater than mc2. Consider what happens when a subatomic particle—or some and n possible future spacecraft—moves at a speed close to the speed of light. Then the denominator becomes quite small and the total energy E becomes much greater than the rest energy mc2. If the speed v could be pushed to the speed of light. E would become infi- nite. Once again we see why no piece of matter can be made to ma% el at the speed of light. It would take infinite energy to do it No scien tist can imagine how the "warp speeds" of science fiction will ever become reality. Since the revised formula for total energy applies to objects that are moving, it is natural to associate kinetic energy with the differerve We c; between total energy and rest energy. So kinetic energy is value mc2 (v/c)2 KB- - mc- The V 1 - (v2/c2)
This equation looks a bit complicated and certainly quite different from the equation KB = 4 mv2. Yet it can be demonstrated mathemat- ically that, for ordinary low speeds, this relativistic equation for kinetic energy reduces to the familiar ICE = mv2. But at higher speeds, the actual kinetic energy is greater than 4 mv2. There are two important points in the discussion of the last few pages.
(1) Even at rest, an object has energy, which is locked So fc in its mass. obje for ri 121 As the speed Of ail object areirn e.s tH.e.cnc,nci pc kleic light, both its momentum and its energy approach char infinity, so there is no way that the speed of light can char 2. Dove, op be reached.* spec of 2 Concept-Development Practice Book 16-1 the 3 Problem-Solving * At least one thing reaches the speed of light—light itself! But light has no rest mass. new Exercises in The equations that apply to it are different. The theory of relativity tells us that light Physics 8-2 travels always at the same speed. A material particle can never be brought to the php speed of light. Light can never be brought to rest. imp
238 Chapter 16 Special Relativity—Length, Momentum, and Energy 16.5 The Correspondence Principle
If a new theory is to be valid, it must account for the verified results of the old theory New theory and old must overlap and agree in the region where the results of the old theory have been fully verified. This requirement is known as the correspondence principle. It was advanced as a principle by Niels Bohr earlier in this century when Newtonian mechanics was being challenged by both quantum the- ory and relativity. If the equations of special relativity (or any other new theory) are to be valid, they must correspond to those of Newtonian mechanics—classical mechanics—when speeds much less than the speed of light are considered. The relativity equations for time dilation, length contraction, and momentum are to t = VI -( v 2/ c
= L.V1 - (v2/ c
my P = 1_(v2/c2)
We can see that each of these equations reduces to a Newtonian value for speeds that are very small compared with c. Then, the ratio (rid is very small, and for everyday speeds may be taken to be zero. The relativity equations become
L = L0V1 -0 = L0
So for everyday speeds, the time scales and length scales of moving objects are essentially unchanged. Also, the Newtonian equation for momentum holds true (and so does the Newtonian equation for icinetic'energy). but when the speed of light is apploaciied, diiuga change dramatically. Near the speed of light Newtonian mechanics change completely. The equations of special relativity hold for all speeds, although they are significant only for speeds near the speed of light. So we see that advances in science take place not by discarding the current ideas and techniques, but by extending them to reveal new implications. Einstein never claimed that accepted laws of physics were wrong, but instead showed that the laws of physics implied something that hadn't before been appreciated.
239 SCIENCE, TECHNOLOGY, AND SOCIETY Scientists and Social Responsibility a "r he atomic bomb was an outcome of Einstein's physics. Einstein and other sci- entists were horrified by the bomb's destruc- tive power, and they spoke out about it. Their work resulted in the bomb, so they felt partly responsible for its creation and use. Many scientists today feel responsible for the consequences of their work. But other scientists feel the scientist's role is to focus on science alone. They say scientists have no particular expertise in matters of public policy and that society is best served when scientific investigation remains unrestricted.
The power of science to change the world is vast That power can be used wisely or foolishly, which often involves distinguishing between short-range and long-range benefits, policies, and goals. The scientist is the channel through which the power of science flows. Critical Thinking To what extent do you think scientists are obligated to consider social consequences of their work? Should Dr. J. R. Oppenheimer (left) (director of the Los scientists be considered more qualified than Alamos lab during the atomic bomb project) and other citizens to guide public policy? Major General Leslie Groves (right) view the test site of the first atomic bomb explosion.
Einstein's theory of relativity has raised many philosophical questions. What, exactly, is time? Can we say it is nature's way of see- ing to it that everything does not all happen at once? And why does time seem to move in one direction? Has it always moved forward? Are there other parts of the universe where time moves backward? Perhaps these unanswered questions will be answered by the physi- cists of tomorrow. How exciting!
240 Chapter 16 Special Relativity—Length, Momentum, and Energy Chapter Assessment
' Go For: Study and Review Review Questions Check Concepts 1 Online Visit: PHSchool.com masa. Web Code: csd -1160 1. If we witness events in a frame of reference moving past us, time appears to be stretched out (dilated). How do the lengths of objects in that frame appear? (16.1) Concept Summary 2. How long would a meter stick appear if it were thrown like a spear at 99.5% the speed When an object moves at very high speed relative of light? (16.1) to an observer, its measured length in the direc- tion of motion is contracted. 3. How long would a meter stick appear if it were traveling at 99.5% the speed of light, but When an object moves at very high speed relative with its length perpendicular to its direction to an observer, its momentum-its "inertia of of motion? (Why are your answers to this and motion"-is greater than the Newtonian value mu. the last question different?) (16.1) Mass and energy are equivalent-anything with 4. If you were traveling in a high-speed space- mass also has energy. Rest energy is given by the ship, would meter sticks on board appear equation E.= mc2. contracted to you? Defend your answer. (16.1) Only when the release of energy is very great is 5. What would be the momentum of an object if the release of mass large enough to be detected. it were pushed to the speed of light? (16.2) During any reaction, the total energy remains 6. What is meant by rest mass? (16.2) constant if the energy equivalent of the change in rest mass is accounted for. 7. What relativistic effect is evident when a beam of high-speed charged particles bends When the speed of an object approaches the in a magnetic field? (16.2) speed of light, both the momentum and the total energy of the object approach infinity. 8. What is meant by the equivalence of mass and energy? That is, what does the equation According to the correspondence principle, the Eo = mc2 mean? (16.3) equations of relativity must agree with the well- tested equations of Newtonian mechanics when 9. What is the numerical quantity of the ratio the speed involved is small compared with the rest energy/rest mass? (16.3) speed of light. 10. Does the equation E.= mc2 apply only to reac- tioAsIthat involve the atomic nucleus? Explain. n Key Terms 11. What evidence is there for the equivalence of correspondence principle (16.5) mass and energy? (16.3) relativistic momentum (16.2) 12. What effect does solar energy have on the rest energy (16.3) mass of the sun? (16.3) rest mass (16.2) 13. An object at rest has an "energy of being," '`• E.= mc2. When the same object is moving, is its total energy the same as, more than, or less than E.= mc2? (16.4)
241 14. Does the relativistic equation for kinetic energy 24. The electrons that illuminate your TV screen mathematically reduce to the Newtonian equa- travel at about 0.25c. At this speed, relativity tion for kinetic energy at low speeds? Explain. gives them an extra momentum that can be (16.4) interpreted as an effective mass increase of about 3%. Does this relativistic effect tend to 15. How much kinetic energy would a particle increase, decrease, or have no effect on your have if it could move at the speed of light? electric bill? (16.4) 25. Since there is an upper limit on the speed of a 16. What is the correspondence principle? (16.5) particle, does it follow that there is therefore 17. What results when low everyday speeds are an upper limit on its kinetic energy or used in the relativistic equations for time and momentum? Defend your answer. length? (165) 26. A friend says that the equation E'0 = mc2 has 18. Do the equations of Newton and Einstein relevance to nuclear power plants, but not to overlap, or is there a sharp break between fossil fuel power plants. Another friend looks them? (16.5) to see if you agree. What do you say? 27. Is this label on a consumer product cause for alarm? CAUTION: The mass of this product Think and Explain Think Critically contains the energy equivalent of 3 million tons of TNT per gram. 19. Suppose your spaceship passes Earth at nearly the speed of light, and Earth observers 28. Give three reasons why we say that there is a tell you that your ship appears to be con- speed limit for particles in the universe. tracted. Comment on the idea of checking their observation by measuring the space- ship yourself. Think and Solve All 20. From Earth, the distance to the center of our Develop Problem-Solving Skills galaxy is 24 000 light-years. From the frame of 29. Josie takes a ride in a spaceship moving at 0.8c reference of a photon of light traveling from to a star 4 light-years away. From Josie's frame Earth to the center of our galaxy what is this of reference, what distance in light-years does distance? she travel to the distant star? 21. According to Newton's laws, if you apply an 30. An electron, with a rest mass of 9.11 x 10 -31 kg, impulse to an object, acceleration occurs. shoots down a 1-km-long accelerator at an What prevents an acceleration to and beyond average speed of 0.95c. the speed of light? a. From the electron's frame of reference, how 22. The two-mile-long linear accelerator at long is the accelerator? Stanford University in California is less than a meter long to the electrons that travel in it. b. From the electron's frame of reference, how Explain. long does it take to make the trip? 23. Pretend you can travel with the electrons in 31. A 100-watt lightbulb consumes 100 joules of the Stanford accelerator after they are acceler- energy every second. How long could you ated and are coasting toward their target at burn that lightbulb from the energy in one nearly the speed of light. penny, which has a mass of 0.003 kg? (Assume• all the penny's mass is converted to energy) a. What would the momentum of an electron be in your frame of reference? Its energy? b. What could you say about the length of the accelerator in your frame of reference? What could you say about the motion of the target?
242 Chapter 16 Special Relativity—Length, Momentum, and Energy