Mass converts to energy and vice versa.
Figure 16.1 1
A meter stick traveling at 87% the speed of light relative to an observer would be measured as only half as long as normal.
Special Relativity
Length,
Momentum, ami
Energy
T
he speed of light is the speed limit for all matter. Suppose that two spaceships are 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 is still less than the speed of light! For example, if both spaceships 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 spaceship whizzing past you at 87% the speed of light, for example, would appear to you to be only 0.5 meter long. If it whizzed past at 99.5% the speed of light, it would appear to you to be contracted to one tenth its original length. The width of the stick, perpendicular to 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.
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.
„ LINK TO BIOLOGY r
1
.c e st ztor Eart,, s s . . U rise eramlifeti .e secoridel-seemingly;too4',„: .f.brietto.reach the grounbelow before decaying.;,, But because niuons Move'. 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 all high-speed elementary particles, causes biologi-cal mutations. So we see a link between the effects of relativity and the evo-lution of living creatures on Earth.
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, c is the speed of light, L is the length of the moving object as mea-sured by the observer, and Lo is the meamea-sured 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 speed of light is the upper limit for the speed of any material object.
III Question
A spacewoman travels by a spherical planet so fast that it appears
to her to be an ellipsoid (egg shaped). If she sees the short diameter as half the long diameter, what is her speed relative to the planet?
16.2
Momentum and Inertia
in Relativity
If we push an object that is free to move, it will accelerate. If we maintain a steady push, it will accelerate to higher and higher speeds. If we push with a greater and greater force, we expect the acceleration in turn to increase. It might seem that the speed should 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.
fl
ver, ti-was ght, I) in •o, peed it the ect. el the iould Oct At gum om to the about iation asimpulse 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 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 momentum would mean infinite speed. Not so in relativity. Einstein showed that a new definition of momentum is required. It is
mu
where v is the speed of an object and cis the speed of light. Notice 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. 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 a true constant, a property of the object no matter what speed it has. Subatomic particles are routinely pushed to nearly the speed of light. The momenta of such particles may be thousands of times 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 deflect it—the "stiffer" is its trajectory. If it has a lot of momentum, it more greatly resists changing course.
ELECTROMAGNETS
ELECTRON BEAM)
SCREEN
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, is greater, the beam follows the
"stiffer" trajectory shown by the solid line.
This can be seen when a beam of electrons is directed into a mag-netic field. Charged particles moving in a magmag-netic 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
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 wound-up spring clock has more mass than the same clock when unwound. But these examples involve incredibly small changes in mass—too small to be measured by conventional methods. No won-der the fundamental relationship between mass and energy was not discovered until this century.
The equation E.= mc2 is more than a formula for the conversion of rest mass into other kinds of energy, or vice versa. It states that 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."
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.
Figure 16.6 A
In one second, 4.5 million tons of rest mass are converted to radi-ant energy in the sun. The sun is so massive, however, that in a million years only one ten-mil-lionth of the sun's rest mass will have been converted to radiant energy.
2. Dove op , 2 Concept-Development Practice Book 16-1 3 Problem-Solving Exercises in Physics 8-2
16.4
Kinetic Energy in Relativity
Einstein dealt also with the energy of moving matter. His formula for the total energy of a moving piece of matter of mass m is
rnc2 E=
V1- (v2I c2)
Notice the by-now familiar square root in the denominator. If the object is at rest, we can set v equal to 0 and find that the denomina-tor is then equal to 1, leading to the famous rest-mass formula E0 = mc2. But if the object is moving, the denominator is less than 1 and the total energy E is greater than mc2.
Consider what happens when a subatomic particle—or some 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 between total energy and rest energy. So kinetic energy is
mc2
KB- - mc-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 in its mass.
121 As the speed Of ail object areirn e.s tH.e.cnc,nci pc
light, both its momentum and its energy approach infinity, so there is no way that the speed of light can be reached.*
* At least one thing reaches the speed of light—light itself! But light has no rest mass. The equations that apply to it are different. The theory of relativity tells us that light travels always at the same speed. A material particle can never be brought to the speed of light. Light can never be brought to rest.
16
If a nE of the regior This r advar Newt ory ai newt Newt less tl and n We c; value (v/c)2 The So fc obje for ri kleic char char spec of the new php imp16.5
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
t =
VI -(
v 2 / cto
= 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.
a
SCIENCE, TECHNOLOGY, AND SOCIETY
Scientists and Social Responsibility
"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.
Dr. J. R. Oppenheimer (left) (director of the Los Alamos lab during the atomic bomb project) and Major General Leslie Groves (right) view the test site of the first atomic bomb explosion.
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 scientists be considered more qualified than other citizens to guide public policy?
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 everythsee-ing 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!
Chapter Assessment
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Concept Summary
When an object moves at very high speed relative to an observer, its measured length in the direc-tion of modirec-tion is contracted.
When an object moves at very high speed relative to an observer, its momentum-its "inertia of motion"-is greater than the Newtonian value mu. Mass and energy are equivalent-anything with mass also has energy. Rest energy is given by the equation E.= mc2.
Only when the release of energy is very great is the release of mass large enough to be detected. During any reaction, the total energy remains constant if the energy equivalent of the change in rest mass is accounted for.
When the speed of an object approaches the speed of light, both the momentum and the total energy of the object approach infinity.
According to the correspondence principle, the equations of relativity must agree with the well-tested equations of Newtonian mechanics when the speed involved is small compared with the speed of light.
Key Terms
correspondence principle (16.5) relativistic momentum (16.2) rest energy (16.3) rest mass (16.2) '`•Review Questions Check Concepts
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)2. How long would a meter stick appear if it were thrown like a spear at 99.5% the speed of light? (16.1)
3. How long would a meter stick appear if it were traveling at 99.5% the speed of light, but with its length perpendicular to its direction of motion? (Why are your answers to this and the last question different?) (16.1)
4. If you were traveling in a high-speed space-ship, would meter sticks on board appear contracted to you? Defend your answer. (16.1) 5. What would be the momentum of an object if
it were pushed to the speed of light? (16.2) 6. What is meant by rest mass? (16.2)
7. What relativistic effect is evident when a beam of high-speed charged particles bends in a magnetic field? (16.2)
8. What is meant by the equivalence of mass and energy? That is, what does the equation Eo = mc2 mean? (16.3)
9. What is the numerical quantity of the ratio rest energy/rest mass? (16.3)
10. Does the equation E.= mc2 apply only to reac-tioAsIthat involve the atomic nucleus? Explain. n
11. What evidence is there for the equivalence of mass and energy? (16.3)
12. What effect does solar energy have on the mass of the sun? (16.3)
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)
14. Does the relativistic equation for kinetic energy mathematically reduce to the Newtonian equa-tion for kinetic energy at low speeds? Explain. (16.4)
15. How much kinetic energy would a particle have if it could move at the speed of light? (16.4)
16. What is the correspondence principle? (16.5) 17. What results when low everyday speeds are
used in the relativistic equations for time and length? (165)
18. Do the equations of Newton and Einstein overlap, or is there a sharp break between them? (16.5)
Think and Explain
Think Critically19. Suppose your spaceship passes Earth at nearly the speed of light, and Earth observers tell you that your ship appears to be con-tracted. Comment on the idea of checking their observation by measuring the space-ship yourself.
20. From Earth, the distance to the center of our galaxy is 24 000 light-years. From the frame of reference of a photon of light traveling from Earth to the center of our galaxy what is this distance?
21. According to Newton's laws, if you apply an impulse to an object, acceleration occurs. What prevents an acceleration to and beyond the speed of light?
22. The two-mile-long linear accelerator at Stanford University in California is less than a meter long to the electrons that travel in it. Explain.
23. Pretend you can travel with the electrons in the Stanford accelerator after they are acceler-ated and are coasting toward their target at nearly the speed of light.
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?
24. The electrons that illuminate your TV screen travel at about 0.25c. At this speed, relativity gives them an extra momentum that can be interpreted as an effective mass increase of about 3%. Does this relativistic effect tend to increase, decrease, or have no effect on your electric bill?
25. Since there is an upper limit on the speed of a particle, does it follow that there is therefore an upper limit on its kinetic energy or momentum? Defend your answer.
26. A friend says that the equation E'0 = mc2 has relevance to nuclear power plants, but not to fossil fuel power plants. Another friend looks 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 contains the energy equivalent of 3 million tons of TNT per gram.
28. Give three reasons why we say that there is a speed limit for particles in the universe.
Think and Solve
Develop Problem-Solving Skills
29. Josie takes a ride in a spaceship moving at 0.8c to a star 4 light-years away. From Josie's frame of reference, what distance in light-years does she travel to the distant star?
All
30. An electron, with a rest mass of 9.11 x 10 -31 kg,
shoots down a 1-km-long accelerator at an average speed of 0.95c.
a. From the electron's frame of reference, how long is the accelerator?
b. From the electron's frame of reference, how long does it take to make the trip?
31. A 100-watt lightbulb consumes 100 joules of energy every second. How long could you burn that lightbulb from the energy in one penny, which has a mass of 0.003 kg? (Assume• all the penny's mass is converted to energy)
