s58y, Flickr)
Learning Objectives
• State and explain both of Einstein’s postulates.
• Explain what an inertial frame of reference is.
• Describe one way the speed of light can be changed.
28.2. Simultaneity And Time Dilation
• Describe simultaneity.
• Describe time dilation.
• Calculate γ.
• Compare proper time and the observer’s measured time.
• Explain why the twin paradox is a false paradox.
• Describe proper length.
• Calculate length contraction.
• Explain why we don’t notice these effects at everyday scales.
28.4. Relativistic Addition of Velocities
• Calculate relativistic velocity addition.
• Explain when relativistic velocity addition should be used instead of classical addition of velocities.
• Calculate relativistic Doppler shift.
• Calculate relativistic momentum.
• Explain why the only mass it makes sense to talk about is rest mass.
• Compute total energy of a relativistic object.
• Compute the kinetic energy of a relativistic object.
• Describe rest energy, and explain how it can be converted to other forms.
• Explain why massive particles cannot reach C.
Introduction to Special Relativity
Have you ever looked up at the night sky and dreamed of traveling to other planets in faraway star systems? Would there be other life forms? What
would other worlds look like? You might imagine that such an amazing trip would be possible if we could just travel fast enough, but you will read in
1000 CHAPTER 28 | SPECIAL RELATIVITY
this chapter why this is not true. In 1905 Albert Einstein developed the theory of special relativity. This theory explains the limit on an object’s speed
and describes the consequences.
Relativity. The word relativity might conjure an image of Einstein, but the idea did not begin with him. People have been exploring relativity for many
centuries. Relativity is the study of how different observers measure the same event. Galileo and Newton developed the first correct version of
classical relativity. Einstein developed the modern theory of relativity. Modern relativity is divided into two parts. Special relativity deals with observers
who are moving at constant velocity. General relativity deals with observers who are undergoing acceleration. Einstein is famous because his
theories of relativity made revolutionary predictions. Most importantly, his theories have been verified to great precision in a vast range of
experiments, altering forever our concept of space and time.
Figure 28.2 Many people think that Albert Einstein (1879–1955) was the greatest physicist of the 20th century. Not only did he develop modern relativity, thus revolutionizing
our concept of the universe, he also made fundamental contributions to the foundations of quantum mechanics. (credit: The Library of Congress)
It is important to note that although classical mechanic, in general, and classical relativity, in particular, are limited, they are extremely good
approximations for large, slow-moving objects. Otherwise, we could not use classical physics to launch satellites or build bridges. In the classical limit
(objects larger than submicroscopic and moving slower than about 1% of the speed of light), relativistic mechanics becomes the same as classical
mechanics. This fact will be noted at appropriate places throughout this chapter.
28.1 Einstein’s Postulates
Figure 28.3 Special relativity resembles trigonometry in that both are reliable because they are based on postulates that flow one from another in a logical way. (credit: Jon
Oakley, Flickr)
Have you ever used the Pythagorean Theorem and gotten a wrong answer? Probably not, unless you made a mistake in either your algebra or your
arithmetic. Each time you perform the same calculation, you know that the answer will be the same. Trigonometry is reliable because of the certainty
that one part always flows from another in a logical way. Each part is based on a set of postulates, and you can always connect the parts by applying
those postulates. Physics is the same way with the exception that all parts must describe nature. If we are careful to choose the correct postulates,
then our theory will follow and will be verified by experiment.
Einstein essentially did the theoretical aspect of this method for relativity. With two deceptively simple postulates and a careful consideration of how
measurements are made, he produced the theory of special relativity.
Einstein’s First Postulate
The first postulate upon which Einstein based the theory of special relativity relates to reference frames. All velocities are measured relative to some
frame of reference. For example, a car’s motion is measured relative to its starting point or the road it is moving over, a projectile’s motion is
measured relative to the surface it was launched from, and a planet’s orbit is measured relative to the star it is orbiting around. The simplest frames of
reference are those that are not accelerated and are not rotating. Newton’s first law, the law of inertia, holds exactly in such a frame.
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Inertial Reference Frame
An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at a constant speed in a
straight line unless acted on by an outside force.
The laws of physics seem to be simplest in inertial frames. For example, when you are in a plane flying at a constant altitude and speed, physics
seems to work exactly the same as if you were standing on the surface of the Earth. However, in a plane that is taking off, matters are somewhat
more complicated. In these cases, the net force on an object, F , is not equal to the product of mass and acceleration, ma . Instead, F is equal to
ma plus a fictitious force. This situation is not as simple as in an inertial frame. Not only are laws of physics simplest in inertial frames, but they
should be the same in all inertial frames, since there is no preferred frame and no absolute motion. Einstein incorporated these ideas into his first
postulate of special relativity.
First Postulate of Special Relativity
The laws of physics are the same and can be stated in their simplest form in all inertial frames of reference.
As with many fundamental statements, there is more to this postulate than meets the eye. The laws of physics include only those that satisfy this
postulate. We shall find that the definitions of relativistic momentum and energy must be altered to fit. Another outcome of this postulate is the famous
equation E = mc 2 .
Einstein’s Second Postulate
The second postulate upon which Einstein based his theory of special relativity deals with the speed of light. Late in the 19th century, the major tenets
of classical physics were well established. Two of the most important were the laws of electricity and magnetism and Newton’s laws. In particular, the
laws of electricity and magnetism predict that light travels at c = 3.00×108 m/s in a vacuum, but they do not specify the frame of reference in
which light has this speed.
There was a contradiction between this prediction and Newton’s laws, in which velocities add like simple vectors. If the latter were true, then two
observers moving at different speeds would see light traveling at different speeds. Imagine what a light wave would look like to a person traveling
along with it at a speed c . If such a motion were possible then the wave would be stationary relative to the observer. It would have electric and
magnetic fields that varied in strength at various distances from the observer but were constant in time. This is not allowed by Maxwell’s equations.
So either Maxwell’s equations are wrong, or an object with mass cannot travel at speed c . Einstein concluded that the latter is true. An object with
mass cannot travel at speed c , and the further implication is that light in a vacuum must always travel at speed c relative to any observer. Maxwell’s
equations are correct, and Newton’s addition of velocities is not correct for light.
Investigations such as Young’s double slit experiment in the early-1800s had convincingly demonstrated that light is a wave. Many types of waves
were known, and all travelled in some medium. Scientists therefore assumed that a medium carried light, even in a vacuum, and light travelled at a
speed c relative to that medium. Starting in the mid-1880s, the American physicist A. A. Michelson, later aided by E. W. Morley, made a series of
direct measurements of the speed of light. The results of their measurements were startling.
Michelson-Morley Experiment
The Michelson-Morley experiment demonstrated that the speed of light in a vacuum is independent of the motion of the Earth about the Sun.
The eventual conclusion derived from this result is that light, unlike mechanical waves such as sound, does not need a medium to carry it.
Furthermore, the Michelson-Morley results implied that the speed of light c is independent of the motion of the source relative to the observer. That
is, everyone observes light to move at speed c regardless of how they move relative to the source or one another. For a number of years, many
scientists tried unsuccessfully to explain these results and still retain the general applicability of Newton’s laws.
It was not until 1905, when Einstein published his first paper on special relativity, that the currently accepted conclusion was reached. Based mostly
on his analysis that the laws of electricity and magnetism would not allow another speed for light, and only slightly aware of the Michelson-Morley
experiment, Einstein detailed his second postulate of special relativity.
Second Postulate of Special Relativity
The speed of light c is a constant, independent of the relative motion of the source and observer.
Deceptively simple and counterintuitive, this and the first postulate leave all else open for change. Some fundamental concepts do change. Among
the changes are the loss of agreement on the elapsed time for an event, the variation of distance with speed, and the realization that matter and
energy can be converted into one another. You will read about these concepts in the following sections.
Misconception Alert: Constancy of the Speed of Light
The speed of light is a constant c = 3.00×108 m/s in a vacuum. If you remember the effect of the index of refraction from The Law of
Refraction, the speed of light is lower in matter.
1002 CHAPTER 28 | SPECIAL RELATIVITY
Explain how special relativity differs from general relativity.
Solution
Special relativity applies only to unaccelerated motion, but general relativity applies to accelerated motion.
28.2 Simultaneity And Time Dilation
Figure 28.4 Elapsed time for a foot race is the same for all observers, but at relativistic speeds, elapsed time depends on the relative motion of the observer and the event that is observed. (credit: Jason Edward Scott Bain, Flickr)
Do time intervals depend on who observes them? Intuitively, we expect the time for a process, such as the elapsed time for a foot race, to be the
same for all observers. Our experience has been that disagreements over elapsed time have to do with the accuracy of measuring time. When we
carefully consider just how time is measured, however, we will find that elapsed time depends on the relative motion of an observer with respect to
the process being measured.
Simultaneity
Consider how we measure elapsed time. If we use a stopwatch, for example, how do we know when to start and stop the watch? One method is to
use the arrival of light from the event, such as observing a light turning green to start a drag race. The timing will be more accurate if some sort of
electronic detection is used, avoiding human reaction times and other complications.
Now suppose we use this method to measure the time interval between two flashes of light produced by flash lamps. (See Figure 28.5.) Two flash
lamps with observer A midway between them are on a rail car that moves to the right relative to observer B. The light flashes are emitted just as A
passes B, so that both A and B are equidistant from the lamps when the light is emitted. Observer B measures the time interval between the arrival of
the light flashes. According to postulate 2, the speed of light is not affected by the motion of the lamps relative to B. Therefore, light travels equal
distances to him at equal speeds. Thus observer B measures the flashes to be simultaneous.
Figure 28.5 Observer B measures the elapsed time between the arrival of light flashes as described in the text. Observer A moves with the lamps on a rail car. Observer B
receives the light flashes simultaneously, but he notes that observer A receives the flash from the right first. B observes the flashes to be simultaneous to him but not to A.
Simultaneity is not absolute.
CHAPTER 28 | SPECIAL RELATIVITY 1003
Now consider what observer B sees happen to observer A. She receives the light from the right first, because she has moved towards that flash
lamp, lessening the distance the light must travel and reducing the time it takes to get to her. Light travels at speed c relative to both observers, but
observer B remains equidistant between the points where the flashes were emitted, while A gets closer to the emission point on the right. From
observer B’s point of view, then, there is a time interval between the arrival of the flashes to observer A. Observer B measures the flashes to be
simultaneous relative to him but not relative to A. Here a relative velocity between observers affects whether two events are observed to be
simultaneous. Simultaneity is not absolute.
This illustrates the power of clear thinking. We might have guessed incorrectly that if light is emitted simultaneously, then two observers halfway
between the sources would see the flashes simultaneously. But careful analysis shows this not to be the case. Einstein was brilliant at this type of
thought experiment (in German, “Gedankenexperiment”). He very carefully considered how an observation is made and disregarded what might
seem obvious. The validity of thought experiments, of course, is determined by actual observation. The genius of Einstein is evidenced by the fact
that experiments have repeatedly confirmed his theory of relativity.
In summary: Two events are defined to be simultaneous if an observer measures them as occurring at the same time (such as by receiving light from
the events). Two events are not necessarily simultaneous to all observers.
Time Dilation
The consideration of the measurement of elapsed time and simultaneity leads to an important relativistic effect.
Time dilation
Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer.
Suppose, for example, an astronaut measures the time it takes for light to cross her ship, bounce off a mirror, and return. (See Figure 28.6.) How
does the elapsed time the astronaut measures compare with the elapsed time measured for the same event by a person on the Earth? Asking this
question (another thought experiment) produces a profound result. We find that the elapsed time for a process depends on who is measuring it. In
this case, the time measured by the astronaut is smaller than the time measured by the Earth-bound observer. The passage of time is different for the
observers because the distance the light travels in the astronaut’s frame is smaller than in the Earth-bound frame. Light travels at the same speed in
each frame, and so it will take longer to travel the greater distance in the Earth-bound frame.
Figure 28.6 (a) An astronaut measures the time Δ t 0 for light to cross her ship using an electronic timer. Light travels a distance 2 D in the astronaut’s frame. (b) A person on the Earth sees the light follow the longer path 2 s and take a longer time Δ t . (c) These triangles are used to find the relationship between the two distances 2 D and 2 s .
To quantitatively verify that time depends on the observer, consider the paths followed by light as seen by each observer. (See Figure 28.6(c).) The
astronaut sees the light travel straight across and back for a total distance of 2 D , twice the width of her ship. The Earth-bound observer sees the
light travel a total distance 2 s . Since the ship is moving at speed v to the right relative to the Earth, light moving to the right hits the mirror in this
frame. Light travels at a speed c in both frames, and because time is the distance divided by speed, the time measured by the astronaut is
(28.1)
Δ t 0 = 2 D
c .
This time has a separate name to distinguish it from the time measured by the Earth-bound observer.
1004 CHAPTER 28 | SPECIAL RELATIVITY
Proper Time
Proper time Δ t 0 is the time measured by an observer at rest relative to the event being observed.
In the case of the astronaut observe the reflecting light, the astronaut measures proper time. The time measured by the Earth-bound observer is
(28.2)
Δ t = 2 sc.
To find the relationship between Δ t 0 and Δ t , consider the triangles formed by D and s . (See Figure 28.6(c).) The third side of these similar triangles is L , the distance the astronaut moves as the light goes across her ship. In the frame of the Earth-bound observer,
(28.3)
L = vΔ t
2 .
Using the Pythagorean Theorem, the distance s is found to be
(28.4)
2
s = D 2 + ⎛ vΔ t⎞
⎝ 2 ⎠ .
Substituting s into the expression for the time interval Δ t gives
(28.5)
2
2 D 2 + ⎛ vΔ t⎞
Δ t = 2 s
⎝
⎠
c =
2
c
.
We square this equation, which yields
(28.6)
4⎛
⎞
⎝ D 2 + v 2(Δ t)2⎠
(Δ t)2 =
4
c 2
= 4 D 2
c 2 + v 2
c 2(Δ t)2.
Note that if we square the first expression we had for Δ t 0 , we get (Δ t 0 )2 = 4 D 2
c 2 . This term appears in the preceding equation, giving us a means
to relate the two time intervals. Thus,
(28.7)
(Δ t)2 = (Δ t 0)2 + v 2
c 2(Δ t)2.
Gathering terms, we solve for Δ t :
(28.8)
(Δ t)2⎛
⎞
⎝1 − v 2
c 2⎠ = (Δ t 0 )2.
Thus,
(28.9)
(Δ t)2 = (Δ t 0 )2.
1 − v 2
c 2
Taking the square root yields an important relationship between elapsed times:
(28.10)
Δ t = Δ t 0 = γΔ t 0,
1 − v 2
c 2
where
(28.11)
γ =
1 .
1 − v 2
c 2
This equation for Δ t is truly remarkable. First, as contended, elapsed time is not the same for different observers moving relative to one another,
even though both are in inertial frames. Proper time Δ t 0 measured by an observer, like the astronaut moving with the apparatus, is smaller than
time measured by other observers. Since those other observers measure a longer time Δ t , the effect is called time dilation. The Earth-bound
observer sees time dilate (get longer) for a system moving relative to the Earth. Alternatively, according to the Earth-bound observer, time slows in
the moving frame, since less time passes there. All clocks moving relative to an observer, including biological clocks such as aging, are observed to
run slow compared with a clock stationary relative to the observer.
Note that if the relative velocity is much less than the speed of light ( v<< c ), then v 2
c 2 is extremely small, and the elapsed times Δ t and Δ t 0 are
nearly equal. At low velocities, modern relativity approaches classical physics—our everyday experiences have very small relativistic effects.
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The equation Δ t = γΔ t 0 also implies that relative velocity cannot exceed the speed of light. As v approaches c , Δ t approaches infinity. This would imply that time in the astronaut’s frame stops at the speed of light. If v exceeded c , then we would be taking the square root of a negative
number, producing an imaginary value for Δ t .
There is considerable experimental evidence that the equation Δ t = γΔ t 0 is correct. One example is found in cosmic ray particles that continuously
rain down on the Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called
muons. The half-life (amount of time for half of a material to decay) of a muon is 1.52 µ s when it is at rest relative to the observer who measures the
half-life. This is the proper time Δ t 0 . Muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It
has been found that the muon’s half-life as measured by an Earth-bound observer ( Δ t ) varies with velocity exactly as predicted by the equation
Δ t = γΔ t 0 . The faster the muon moves, the longer it lives. We on the Earth see the muon’s half-life time dilated—as viewed from our frame, the
muon decays more slowly than it does when at rest relative to us.
Example 28.1 Calculating Δ t for a Relativistic Event: How Long Does a Speedy Muon Live?
Suppose a cosmic ray colliding with a nucleus in the Earth’s upper atmosphere produces a muon that has a velocity v = 0.950 c . The muon
then travels at constant velocity and lives 1.52 µ s as measured in the muon’s frame of reference. (You can imagine this as the muon’s internal
clock.) How long does the muon live as measured by an Earth-bound observer? (See