Relativistic

The relativistic character of the laws of physics began to be apparent very early
in the evolution of classical physics. Even before the time of Galileo and
Newton, Nicolaus Copernicus 1 had shown that the complicated and imprecise
Aristotelian method of computing the motions of the planets, based on the assumption
that Earth was located at the center of the universe, could be made much simpler,
though no more accurate, if it were assumed that the planets move about the Sun
instead of Earth. Although Copernicus did not publish his work until very late in
life, it became widely known through correspondence with his contemporaries and
helped pave the way for acceptance a century later of the heliocentric theory of
planetary motion. While the Copernican theory led to a dramatic revolution in human
thought, the aspect that concerns us here is that it did not consider the location of
Earth to be special or favored in any way. Thus, the laws of physics discovered
on Earth could apply equally well with any point taken as the center — i.e., the
same equations would be obtained regardless of the origin of coordinates. This
invariance of the equations that express the laws of physics is what we mean by the
term relativity.
We will begin this chapter by investigating briefly the relativity of Newton’s
laws and then concentrate on the theory of relativity as developed by Albert Einstein
(1879–1955). The theory of relativity consists of two rather different theories, the
special theory and the general theory. The special theory, developed by Einstein and
others in 1905, concerns the comparison of measurements made in different frames
of reference moving with constant velocity relative to each other. Contrary to popu-
lar opinion, the special theory is not difficult to understand. Its consequences, which
can be derived with a minimum of mathematics, are applicable in a wide variety of
situations in physics and engineering. On the other hand, the general theory, also
developed by Einstein (around 1916), is concerned with accelerated reference frames
and gravity. Although a thorough understanding of the general theory requires more
sophisticated mathematics (e.g., tensor analysis), a number of its basic ideas and
important predictions can be discussed at the level of this book. The general theory
is of great importance in cosmology and in understanding events that occur in the
1-1 The Experimental
Basis of
Relativity 4
1-2 Einstein’s
Postulates 11
1-3 The Lorenz
Transformation 17
1-4 Time Dilation
and Length
Contraction 29
1-5 The Doppler
Effect 41
1-6 The Twin
Paradox and
Other Surprises 45
Relativity I
CHAPTER 1
4 Chapter 1 Relativity I
y
S
x
z
y
S
x
z
v
vicinity of very large masses (e.g., stars) but is rarely encountered in other areas of
physics and engineering. We will devote this chapter entirely to the special theory
(often referred to as special relativity) and discuss the general theory in the final
section of Chapter 2, following the sections concerned with special relativistic
mechanics.
1-1 The Experimental Basis of Relativity
Classical Relativity
In 1687, with the publication of the Philosophiae Naturalis Principia Mathematica,
Newton became the first person to generalize the observations of Galileo and others
into the laws of motion that occupied much of your attention in introductory physics.
The second of Newton’s three laws is
1-1
where is the acceleration of the mass m when acted upon by a net force F.
Equation 1-1 also includes the first law, the law of inertia, by implication: if ,
then also, i.e., . (Recall that letters and symbols in boldface type are
vectors.)
As it turns out, Newton’s laws of motion only work correctly in inertial reference
frames, that is, reference frames in which the law of inertia holds.2 They also have the
remarkable property that they are invariant, or unchanged, in any reference frame that
moves with constant velocity relative to an inertial frame. Thus, all inertial frames are
equivalent — there is no special or favored inertial frame relative to which absolute
measurements of space and time could be made. Two such inertial frames are illus-
trated in Figure 1-1, arranged so that corresponding axes in S and are parallel and
moves in the direction at velocity v for an observer in S (or S moves in the xxS
S
a 0dv>dt 0
F0
dv>dt a
F m dv
dt ma
Figure 1-1 Inertial reference frame S is attached to Earth (the palm tree) and S to the cyclist.
The corresponding axes of the frames are parallel, and S moves at speed v in the x direction
of S.
1-1 The Experimental Basis of Relativity 5
Figure 1-2 A mass suspended by a cord from the roof of a railroad boxcar illustrates the
relativity of Newton’s second law, F ma. The only forces acting on the mass are its weight mg
and the tension T in the cord. (a) The boxcar sits at rest in S. Since the velocity v and the
acceleration a of the boxcar (i.e., the system S) are both zero, both observers see the mass
hanging vertically at rest with F F 0. (b) As S moves in the x direction with v constant,
both observers see the mass hanging vertically but moving at v with respect to O in S and at rest
with respect to the S observer. Thus, F F 0. (c) As S moves in the x direction with
a 0 with respect to S, the mass hangs at an angle 0 with respect to the vertical. However,
it is still at rest (i.e., in equilibrium) with respect to the observer in S, who now “explains” the
angle by adding a pseudoforce Fp in the x direction to Newton’s second law.

x´
y´
z´
x´x
y´
y
z´
zS
S´ O´
O´
O
a=0
→
v=0
→
v>0
→
v
→
S´
→
a>0
ϑ
a
→
(
c)
x
y
zSO
x´
x
y´
y
z´
z
O´
O
S´
a=0
→
v>0
→
v
→
S
(
b)(
a)
Figure 1-3 A geosynchronous satellite has an orbital angular velocity
equal to that of Earth and, therefore, is always located above a particular
point on Earth; i.e., it is at rest with respect to the surface of Earth. An
observer in S accounts for the radial, or centripetal, acceleration a of the
satellite as the result of the net force FG . For an observer O at rest on
Earth (in S), however, a 0 and FG
ma. To explain the acceleration
being zero, observer O must add a pseudoforce Fp FG .

x´
y
S
Satellite
Earth Geosynchronous
orbit
ω
ω
x
z
S
´
y´
z´
direction at velocity for an observer in ). Figures 1-2 and 1-3 illustrate the con-
ceptual differences between inertial and noninertial reference frames. Transformation
of the position coordinates and the velocity components of S into those of is the
Galilean transformation, Equations 1-2 and 1-3, respectively.
1-2
1-3uœ
x u x v uœ
y u y uœ
zuz
x x vt y y z z t t
S
Sv
Classical Concept Review
The concepts of classical
relativity, frames of
reference, and coordinate
transformations — all
important background to
our discussions of special
relativity — may not have
been emphasized in many
introductory courses. As an
aid to a better
understanding of the
concepts of modern
physics, we have included
the Classical Concept Review
on the book’s Web site. As
you proceed through
Modern Physics, the icon
in the margin will alert
you to potentially helpful
classical background
pertinent to the adjacent
topics.
6 Chapter 1 Relativity I
Figure 1-4 The observers in S and S see identical electric fields 2k y1 at a distance
from an infinitely long wire carrying uniform charge per unit length. Observers in both S and
S measure a force 2kq y1 on q due to the line of charge; however, the S observer measures
an additional force due to the magnetic field at arising from the motion of
the wire in the x direction. Thus, the electromagnetic force does not have the same form in
different inertial systems, implying that Maxwell’s equations are not invariant under a Galilean
transformation.
yœ
1
0 v2q>(2 y1)
>
y1 yœ
1
>
x´
x
y´
y
y1
q
Sv
z´
z
S´
Notice that differentiating Equation 1-3 yields the result since for
constant v. Thus, This is the invariance referred to above. Generalizing
this result:
Any reference frame that moves at constant velocity with respect to an iner-
tial frame is also an inertial frame. Newton’s laws of mechanics are invariant
in all reference systems connected by a Galilean transformation.
Speed of Light
In about 1860 James Clerk Maxwell summarized the experimental observations of
electricity and magnetism in a consistent set of four concise equations. Unlike
Newton’s laws of motion, Maxwell’s equations are not invariant under a Galilean
transformation between inertial reference frames (Figure 1-4). Since the Maxwell
equations predict the existence of electromagnetic waves whose speed would be a par-
ticular value, , the excellent agreement between this
number and the measured value of the speed of light 3 and between the predicted po-
larization properties of electromagnetic waves and those observed for light provided
strong confirmation of the assumption that light was an electromagnetic wave and,
therefore, traveled at speed c. 4
That being the case, it was postulated in the nineteenth century that electromagnetic
waves, like all other waves, propagated in a suitable material medium. The implication
of this postulate was that the medium, called the ether, filled the entire universe,
including the interior of matter. (The Greek philosopher Aristotle had first suggested that
the universe was permeated with “ether” 2000 years earlier.) In this way the remarkable
opportunity arose to establish experimentally the existence of the all-pervasive ether by
measuring the speed of light relative to Earth as Earth moved relative to the ether at
speed v, as would be predicted by Equation 1-3. The value of c was given by the
Maxwell equations, and the speed of Earth relative to the ether, while not known, was
assumed to be at least equal to its orbital speed around the Sun, about 30 km s. Since
the maximum observable effect is of the order and given this assumption
, an experimental accuracy of about 1 part in 108 is necessary in order
to detect Earth’s motion relative to the ether. With a single exception, equipment and
v2>c2 108
v2>c2
>
c
c 1> 1 0P0 3.00 10 8 m>s
F ma Fœ.
dv>dt 0a a
1-1 The Experimental Basis of Relativity 7
Figure 1-5 Light source, mirror, and observer are moving with speed v relative to the ether.
According to classical theory, the speed of light c, relative to the ether, would be c v relative
to the observer for light moving from the source toward the mirror and c v for light
reflecting from the mirror back toward the source.
Observer
Light source Mirror
B
AL
v
c+
v
c–
v
Albert A. Michelson, here
playing pool in his later
years, made the first
accurate measurement of
the speed of light while an
instructor at the U.S. Naval
Academy, where he had
earlier been a cadet. [AIP
Emilio Segrè Visual Archives.]
techniques available at the time had an experimental accuracy of only about 1 part in
10 4
, woefully insufficient to detect the predicted small effect. That single exception was
the experiment of Michelson and Morley.5
Questions
1. What would the relative velocity of the inertial systems in Figure 1-4 need to be
in order for the S observer to measure no net electromagnetic force on the
charge q?
2. Discuss why the very large value for the speed of the electromagnetic waves
would imply that the ether be rigid, i.e., have a large bulk modulus.
The Michelson-Morley Experiment
All waves that were known to nineteenth-century scientists required a medium in
order to propagate. Surface waves moving across the ocean obviously require the
water. Similarly, waves move along a plucked guitar string, across the surface of a
struck drumhead, through Earth after an earthquake, and, indeed, in all materials acted
upon by suitable forces. The speed of the waves depends on the properties of the
medium and is derived relative to the medium. For example, the speed of sound waves
in air, i.e., their absolute motion relative to still air, can be measured. The Doppler ef-
fect for sound in air depends not only on the relative motion of the source and listener,
but also on the motion of each relative to still air. Thus, it was natural for scientists of
that time to expect the existence of some material like the ether to support the propa-
gation of light and other electromagnetic waves and to expect that the absolute mo-
tion of Earth through the ether should be detectable, despite the fact that the ether had
not been observed previously.
Michelson realized that although the effect of Earth’s motion on the results of any
“out-and–back” speed of light measurement, such as shown generically in Figure 1-5,
would be too small to measure directly, it should be possible to measure v2 c2 by a dif-
ference measurement, using the interference property of the light waves as a sensitive
“clock.” The apparatus that he designed to make the measurement is called the
Michelson interferometer. The purpose of the Michelson-Morley experiment was to
measure the speed of light relative to the interferometer (i.e., relative to Earth), thereby
detecting Earth’s motion through the ether and thus verifying the latter’s existence. To
illustrate how the interferometer works and the reasoning behind the experiment, let us
first describe an analogous situation set in more familiar surroundings.
>
8 Chapter 1 Relativity I
EXAMPLE 1-1 A Boat Race Two equally matched rowers race each other over
courses as shown in Figure 1-6a. Each oarsman rows at speed c in still water; the
current in the river moves at speed v. Boat 1 goes from A to B, a distance L, and
back. Boat 2 goes from A to C, also a distance L, and back. A, B, and C are marks
on the riverbank. Which boat wins the race, or is it a tie? (Assume c v.)
SOLUTION
The winner is, of course, the boat that makes the round trip in the shortest time,
so to discover which boat wins, we compute the time for each. Using the classical
velocity transformation (Equations 1-3), the speed of 1 relative to the ground is
, as shown in Figure 1-6b; thus the round-trip time t1 for boat 1 is
1-4
where we have used the binomial expansion. Boat 2 moves downstream at speed
relative to the ground and returns at , also relative to the ground. The
round-trip time t2 is thus
1-5
2L
c
1
1 v2
c2
2L
c a1 v2
c2 Á b
t2 L
cvL
c v 2Lc
c2 v2
c vc v
2L
c A1 v2
c2
2L
c a1 v2
c2 b
1/2
2L
c a1 1
2
v2
c2 Á b
t1 tASB tBSA L
2c2 v2 L
2c2 v2 2L
2c2 v2
(c2 v2)1>2
Ground
Ground
River C
B
A
1
2
L
L
v
(
a)
(
b)
c2–
v2
v
A→
B
c
c2–
v2
v
B→
A
c
Figure 1-6 (a) The rowers both row at speed c in still water. (See Example 1-1.) The current in
the river moves at speed v. Rower 1 goes from A to B and back to A, while rower 2 goes from A to
C and back to A. (b) Rower 1 must point the bow upstream so that the sum of the velocity vectors
c v results in the boat moving from A directly to B. His speed relative to the banks (i.e., points A
and B) is then The same is true on the return trip.(c2 v2
)1>2 .
1-1 The Experimental Basis of Relativity 9
Figure 1-7 Drawing of Michelson-Morley apparatus used in their 1887
experiment. The optical parts were mounted on a 5 ft square sandstone slab,
which was floated in mercury, thereby reducing the strains and vibrations
during rotation that had affected the earlier experiments. Observations
could be made in all directions by rotating the apparatus in the horizontal
plane. [From R. S. Shankland, “The Michelson-Morley Experiment,” Copyright ©
November 1964 by Scientific American, Inc. All rights reserved.]
Light source
Telescope
Mirrors
Adjustable
mirror Silvered
glass plate
Unsilvered
glass plate
Mirrors Mirrors
123
4
5
The Results Michelson and Morley carried out the experiment in 1887, repeating
with a much-improved interferometer an inconclusive experiment that Michelson
alone had performed in 1881 in Potsdam. The path length L on the new interferom-
eter (Figure 1-7) was about 11 meters, obtained by a series of multiple reflections.
Michelson’s interferometer is shown schematically in Figure 1-8a. The field of view
seen by the observer consists of parallel alternately bright and dark interference
bands, called fringes, as illustrated in Figure 1-8b. The two light beams in the inter-
ferometer are exactly analogous to the two boats in Example 1-1, and Earth’s motion
through the ether was expected to introduce a time (phase) difference as given by
which, you may note, is the same result obtained in our discussion of the speed of
light experiment in the Classical Concept Review.
The difference ¢t between the round-trip times of the boats is then
1-6
The quantity is always positive; therefore, t2 t1 and rower 1 has the
faster average speed and wins the race.
Lv2>c3
¢t t2 t1 2L
c a1 v2
c2 b 2L
c a1 1
2
v2
c2 b Lv2
c3