Transformaciones de Lorentz para las velocidades
Sistema de referencia S
(en reposo)
Sistema de referencia S
(se mueve hacia la derecha con velocidad )
Transformaciones de velocidad de Galileo
Un objeto cuya velocidad se aproxime a con respecto a un
observador situado en S, tambin tendr una velocidad que se aproxime
a con respecto a otro observador situado en S, independientemente
del movimiento relativo de S y S
*
96 C H A P T E R 4 Motion in Two Dimensions
4.6 Relative Velocity and Relative Acceleration
In this section, we describe how observations made by different
observers in differentframes of reference are related to each
other. We find that observers in differentframes of reference may
measure different positions, velocities, and accelerations for
agiven particle. That is, two observers moving relative to each
other generally do notagree on the outcome of a measurement.
As an example, consider two observers watching a man walking on
a moving belt-way at an airport in Figure 4.21. The woman standing
on the moving beltway will seethe man moving at a normal walking
speed. The woman observing from the stationaryfloor will see the
man moving with a higher speed, because the beltway speed com-bines
with his walking speed. Both observers look at the same man and
arrive at differ-ent values for his speed. Both are correct; the
difference in their measurements is dueto the relative velocity of
their frames of reference.
Suppose a person riding on a skateboard (observer A) throws a
ball in such a waythat it appears in this persons frame of
reference to move first straight upward and
while the tangential acceleration vector has magnitude0.300 m/s2
and is horizontal. Because a ! ar " at , the mag-nitude of a is
! 0.309 m/s2
a ! a 2r " a 2t ! (#0.072 0)2 " (0.300)2 m/s2
If $ is the angle between a and the horizontal, then
This angle is measured downward from the horizontal. (SeeFigure
4.20b.)
#13.5$ ! tan#1arat
! tan#1! # 0.072 0 m/s20.300 m/s2 " !
at = 0.300 m/s2at
vv = 6.00 m/s
ar
at
a
(b)(a)
Figure 4.20 (Example 4.9) (a) A car passes over a rise that is
shaped like a circle.(b) The total acceleration vector a is the sum
of the tangential and radial accelerationvectors at and ar .
Figure 4.21 Two observers measure the speed of a man walking on
a moving beltway.The woman standing on the beltway sees the man
moving with a slower speed than thewoman observing from the
stationary floor.
S E C T I O N 4 . 6 Relative Velocity and Relative Acceleration
97
then straight downward along the same vertical line, as shown in
Figure 4.22a. An ob-server B on the ground sees the path of the
ball as a parabola, as illustrated in Figure4.22b. Relative to
observer B, the ball has a vertical component of velocity
(resultingfrom the initial upward velocity and the downward
acceleration due to gravity) and ahorizontal component.
Another example of this concept is the motion of a package
dropped from an air-plane flying with a constant velocitya
situation we studied in Example 4.6. An ob-server on the airplane
sees the motion of the package as a straight line downward to-ward
Earth. The stranded explorer on the ground, however, sees the
trajectory of thepackage as a parabola. Once the package is
dropped, and the airplane continues tomove horizontally with the
same velocity, the package hits the ground directly beneaththe
airplane (if we assume that air resistance is neglected)!
In a more general situation, consider a particle located at
point ! in Figure 4.23.Imagine that the motion of this particle is
being described by two observers, one in ref-erence frame S, fixed
relative to Earth, and another in reference frame S !, moving tothe
right relative to S (and therefore relative to Earth) with a
constant velocity v0. (Rel-ative to an observer in S !, S moves to
the left with a velocity " v0.) Where an observerstands in a
reference frame is irrelevant in this discussion, but for purposes
of this dis-cussion let us place each observer at her or his
respective origin.
(a) (b)
Path seenby observer B
AA
Path seenby observer A
B
Figure 4.22 (a) Observer A on a moving skateboard throws a ball
upward and sees itrise and fall in a straight-line path. (b)
Stationary observer B sees a parabolic path forthe same ball.
S
rr
v0t
S
O Ov0
!
Figure 4.23 A particle located at ! is de-scribed by two
observers, one in the fixedframe of reference S, and the other in
theframe S !, which moves to the right with aconstant velocity v0.
The vector r is the par-ticles position vector relative to S, and
r! isits position vector relative to S !.
S E C T I O N 4 . 6 Relative Velocity and Relative Acceleration
97
then straight downward along the same vertical line, as shown in
Figure 4.22a. An ob-server B on the ground sees the path of the
ball as a parabola, as illustrated in Figure4.22b. Relative to
observer B, the ball has a vertical component of velocity
(resultingfrom the initial upward velocity and the downward
acceleration due to gravity) and ahorizontal component.
Another example of this concept is the motion of a package
dropped from an air-plane flying with a constant velocitya
situation we studied in Example 4.6. An ob-server on the airplane
sees the motion of the package as a straight line downward to-ward
Earth. The stranded explorer on the ground, however, sees the
trajectory of thepackage as a parabola. Once the package is
dropped, and the airplane continues tomove horizontally with the
same velocity, the package hits the ground directly beneaththe
airplane (if we assume that air resistance is neglected)!
In a more general situation, consider a particle located at
point ! in Figure 4.23.Imagine that the motion of this particle is
being described by two observers, one in ref-erence frame S, fixed
relative to Earth, and another in reference frame S !, moving tothe
right relative to S (and therefore relative to Earth) with a
constant velocity v0. (Rel-ative to an observer in S !, S moves to
the left with a velocity " v0.) Where an observerstands in a
reference frame is irrelevant in this discussion, but for purposes
of this dis-cussion let us place each observer at her or his
respective origin.
(a) (b)
Path seenby observer B
AA
Path seenby observer A
B
Figure 4.22 (a) Observer A on a moving skateboard throws a ball
upward and sees itrise and fall in a straight-line path. (b)
Stationary observer B sees a parabolic path forthe same ball.
S
rr
v0t
S
O Ov0
!
Figure 4.23 A particle located at ! is de-scribed by two
observers, one in the fixedframe of reference S, and the other in
theframe S !, which moves to the right with aconstant velocity v0.
The vector r is the par-ticles position vector relative to S, and
r! isits position vector relative to S !.
39.1 The Principle of Galilean Relativity
To describe a physical event, we must establish a frame of
reference. You should recallfrom Chapter 5 that an inertial frame
of reference is one in which an object is observedto have no
acceleration when no forces act on it. Furthermore, any system
moving withconstant velocity with respect to an inertial frame must
also be in an inertial frame.
There is no absolute inertial reference frame. This means that
the results of anexperiment performed in a vehicle moving with
uniform velocity will be identical tothe results of the same
experiment performed in a stationary vehicle. The formalstatement
of this result is called the principle of Galilean relativity:
1246 C H A P T E R 3 9 Relativity
The laws of mechanics must be the same in all inertial frames of
reference.
Let us consider an observation that illustrates the equivalence
of the laws ofmechanics in different inertial frames. A pickup
truck moves with a constant velocity,as shown in Figure 39.1a. If a
passenger in the truck throws a ball straight up, and if aireffects
are neglected, the passenger observes that the ball moves in a
vertical path. Themotion of the ball appears to be precisely the
same as if the ball were thrown by aperson at rest on the Earth.
The law of universal gravitation and the equations ofmotion under
constant acceleration are obeyed whether the truck is at rest or
inuniform motion.
Both observers agree on the laws of physicsthey each throw a
ball straight up andit rises and falls back into their hand. What
about the path of the ball thrown by theobserver in the truck? Do
the observers agree on the path? The observer on theground sees the
path of the ball as a parabola, as illustrated in Figure 39.1b,
while, asmentioned earlier, the observer in the truck sees the ball
move in a vertical path.Furthermore, according to the observer on
the ground, the ball has a horizontalcomponent of velocity equal to
the velocity of the truck. Although the two observersdisagree on
certain aspects of the situation, they agree on the validity of
Newtons lawsand on such classical principles as conservation of
energy and conservation of linearmomentum. This agreement implies
that no mechanical experiment can detect anydifference between the
two inertial frames. The only thing that can be detected is
therelative motion of one frame with respect to the other.
(b)(a)
Figure 39.1 (a) The observer in the truck sees the ball move in
a vertical path whenthrown upward. (b) The Earth observer sees the
path of the ball as a parabola.
Principle of Galilean relativity
Quick Quiz 39.1 Which observer in Figure 39.1 sees the balls
correct path?(a) the observer in the truck (b) the observer on the
ground (c) both observers.
39.1 The Principle of Galilean Relativity
To describe a physical event, we must establish a frame of
reference. You should recallfrom Chapter 5 that an inertial frame
of reference is one in which an object is observedto have no
acceleration when no forces act on it. Furthermore, any system
moving withconstant velocity with respect to an inertial frame must
also be in an inertial frame.
There is no absolute inertial reference frame. This means that
the results of anexperiment performed in a vehicle moving with
uniform velocity will be identical tothe results of the same
experiment performed in a stationary vehicle. The formalstatement
of this result is called the principle of Galilean relativity:
1246 C H A P T E R 3 9 Relativity
The laws of mechanics must be the same in all inertial frames of
reference.
Let us consider an observation that illustrates the equivalence
of the laws ofmechanics in different inertial frames. A pickup
truck moves with a constant velocity,as shown in Figure 39.1a. If a
passenger in the truck throws a ball straight up, and if aireffects
are neglected, the passenger observes that the ball moves in a
vertical path. Themotion of the ball appears to be precisely the
same as if the ball were thrown by aperson at rest on the Earth.
The law of universal gravitation and the equations ofmotion under
constant acceleration are obeyed whether the truck is at rest or
inuniform motion.
Both observers agree on the laws of physicsthey each throw a
ball straight up andit rises and falls back into their hand. What
about the path of the ball thrown by theobserver in the truck? Do
the observers agree on the path? The observer on theground sees the
path of the ball as a parabola, as illustrated in Figure 39.1b,
while, asmentioned earlier, the observer in the truck sees the ball
move in a vertical path.Furthermore, according to the observer on
the ground, the ball has a horizontalcomponent of velocity equal to
the velocity of the truck. Although the two observersdisagree on
certain aspects of the situation, they agree on the validity of
Newtons lawsand on such classical principles as conservation of
energy and conservation of linearmomentum. This agreement implies
that no mechanical experiment can detect anydifference between the
two inertial frames. The only thing that can be detected is
therelative motion of one frame with respect to the other.
(b)(a)
Figure 39.1 (a) The observer in the truck sees the ball move in
a vertical path whenthrown upward. (b) The Earth observer sees the
path of the ball as a parabola.
Principle of Galilean relativity
Quick Quiz 39.1 Which observer in Figure 39.1 sees the balls
correct path?(a) the observer in the truck (b) the observer on the
ground (c) both observers.
39.1 The Principle of Galilean Relativity
To describe a physical event, we must establish a frame of
reference. You should recallfrom Chapter 5 that an inertial frame
of reference is one in which an object is observedto have no
acceleration when no forces act on it. Furthermore, any system
moving withconstant velocity with respect to an inertial frame must
also be in an inertial frame.
There is no absolute inertial reference frame. This means that
the results of anexperiment performed in a vehicle moving with
uniform velocity will be identical tothe results of the same
experiment performed in a stationary vehicle. The formalstatement
of this result is called the principle of Galilean relativity:
1246 C H A P T E R 3 9 Relativity
The laws of mechanics must be the same in all inertial frames of
reference.
Let us consider an observation that illustrates the equivalence
of the laws ofmechanics in different inertial frames. A pickup
truck moves with a constant velocity,as shown in Figure 39.1a. If a
passenger in the truck throws a ball straight up, and if aireffects
are neglected, the passenger observes that the ball moves in a
vertical path. Themotion of the ball appears to be precisely the
same as if the ball were thrown by aperson at rest on the Earth.
The law of universal gravitation and the equations ofmotion under
constant acceleration are obeyed whether the truck is at rest or
inuniform motion.
Both observers agree on the laws of physicsthey each throw a
ball straight up andit rises and falls back into their hand. What
about the path of the ball thrown by theobserver in the truck? Do
the observers agree on the path? The observer on theground sees the
path of the ball as a parabola, as illustrated in Figure 39.1b,
while, asmentioned earlier, the observer in the truck sees the ball
move in a vertical path.Furthermore, according to the observer on
the ground, the ball has a horizontalcomponent of velocity equal to
the velocity of the truck. Although the two observersdisagree on
certain aspects of the situation, they agree on the validity of
Newtons lawsand on such classical principles as conservation of
energy and conservation of linearmomentum. This agreement implies
that no mechanical experiment can detect anydifference between the
two inertial frames. The only thing that can be detected is
therelative motion of one frame with respect to the other.
(b)(a)
Figure 39.1 (a) The observer in the truck sees the ball move in
a vertical path whenthrown upward. (b) The Earth observer sees the
path of the ball as a parabola.
Principle of Galilean relativity
Quick Quiz 39.1 Which observer in Figure 39.1 sees the balls
correct path?(a) the observer in the truck (b) the observer on the
ground (c) both observers.
39.1 The Principle of Galilean Relativity
To describe a physical event, we must establish a frame of
reference. You should recallfrom Chapter 5 that an inertial frame
of reference is one in which an object is observedto have no
acceleration when no forces act on it. Furthermore, any system
moving withconstant velocity with respect to an inertial frame must
also be in an inertial frame.
There is no absolute inertial reference frame. This means that
the results of anexperiment performed in a vehicle moving with
uniform velocity will be identical tothe results of the same
experiment performed in a stationary vehicle. The formalstatement
of this result is called the principle of Galilean relativity:
1246 C H A P T E R 3 9 Relativity
The laws of mechanics must be the same in all inertial frames of
reference.
Let us consider an observation that illustrates the equivalence
of the laws ofmechanics in different inertial frames. A pickup
truck moves with a constant velocity,as shown in Figure 39.1a. If a
passenger in the truck throws a ball straight up, and if aireffects
are neglected, the passenger observes that the ball moves in a
vertical path. Themotion of the ball appears to be precisely the
same as if the ball were thrown by aperson at rest on the Earth.
The law of universal gravitation and the equations ofmotion under
constant acceleration are obeyed whether the truck is at rest or
inuniform motion.
Both observers agree on the laws of physicsthey each throw a
ball straight up andit rises and falls back into their hand. What
about the path of the ball thrown by theobserver in the truck? Do
the observers agree on the path? The observer on theground sees the
path of the ball as a parabola, as illustrated in Figure 39.1b,
while, asmentioned earlier, the observer in the truck sees the ball
move in a vertical path.Furthermore, according to the observer on
the ground, the ball has a horizontalcomponent of velocity equal to
the velocity of the truck. Although the two observersdisagree on
certain aspects of the situation, they agree on the validity of
Newtons lawsand on such classical principles as conservation of
energy and conservation of linearmomentum. This agreement implies
that no mechanical experiment can detect anydifference between the
two inertial frames. The only thing that can be detected is
therelative motion of one frame with respect to the other.
(b)(a)
Figure 39.1 (a) The observer in the truck sees the ball move in
a vertical path whenthrown upward. (b) The Earth observer sees the
path of the ball as a parabola.
Principle of Galilean relativity
Quick Quiz 39.1 Which observer in Figure 39.1 sees the balls
correct path?(a) the observer in the truck (b) the observer on the
ground (c) both observers.
S E C T I O N 3 9 . 1 The Principle of Galilean Relativity
1247
Suppose that some physical phenomenon, which we call an event,
occurs and isobserved by an observer at rest in an inertial
reference frame. The events location andtime of occurrence can be
specified by the four coordinates (x, y, z, t). We would liketo be
able to transform these coordinates from those of an observer in
one inertialframe to those of another observer in a frame moving
with uniform relative velocitycompared to the first frame. When we
say an observer is in a frame, we mean that theobserver is at rest
with respect to the origin of that frame.
Consider two inertial frames S and S! (Fig. 39.2). The frame S!
moves with aconstant velocity v along the common x and x! axes,
where v is measured relative to S.We assume that the origins of S
and S! coincide at t " 0 and that an event occursat point P in
space at some instant of time. An observer in S describes the
eventwith spacetime coordinates (x, y, z, t), whereas an observer
in S! uses the coordinates(x!, y!, z!, t!) to describe the same
event. As we see from the geometry in Figure 39.2,the relationships
among these various coordinates can be written
x! " x # vt y! " y z! " z t! " t (39.1)
These equations are the Galilean spacetime transformation
equations. Note thattime is assumed to be the same in both inertial
frames. That is, within the frameworkof classical mechanics, all
clocks run at the same rate, regardless of their velocity, sothat
the time at which an event occurs for an observer in S is the same
as the time forthe same event in S!. Consequently, the time
interval between two successive eventsshould be the same for both
observers. Although this assumption may seem obvious, itturns out
to be incorrect in situations where v is comparable to the speed of
light.
Now suppose that a particle moves through a displacement of
magnitude dx alongthe x axis in a time interval dt as measured by
an observer in S. It follows fromEquations 39.1 that the
corresponding displacement dx! measured by an observer in S!is dx!
" dx # v dt, where frame S! is moving with speed v in the x
direction relative toframe S. Because dt " dt!, we find that
or
u!x " ux # v (39.2)
where ux and u!x are the x components of the velocity of the
particle measured byobservers in S and S!, respectively. (We use
the symbol u for particle velocity ratherthan v, which is used for
the relative velocity of two reference frames.) This is theGalilean
velocity transformation equation. It is consistent with our
intuitive notionof time and space as well as with our discussions
in Section 4.6. As we shall soon see,however, it leads to serious
contradictions when applied to electromagnetic waves.
dx !dt !
"dxdt
# v
y
0 x
y
0 x
xvt x
P (event)
v
S S
Figure 39.2 An event occurs at apoint P. The event is seen by
twoobservers in inertial frames S andS!, where S! moves with a
velocity vrelative to S.
Galilean transformationequations
! PITFALL PREVENTION 39.1 The Relationship
Between the S andS! Frames
Many of the mathematical repre-sentations in this chapter are
trueonly for the specified relationshipbetween the S and S! frames.
Thex and x! axes coincide, exceptthat their origins are
different.The y and y! axes (and the z andz! axes), are parallel,
but do notcoincide due to the displacementof the origin of S! with
respect tothat of S. We choose the timet " 0 to be the instant at
whichthe origins of the two coordinatesystems coincide. If the S!
frameis moving in the positive x direc-tion relative to S, v is
positive;otherwise it is negative.
Quick Quiz 39.2 A baseball pitcher with a 90-mi/h fastball
throws a ballwhile standing on a railroad flatcar moving at 110
mi/h. The ball is thrown in thesame direction as that of the
velocity of the train. Applying the Galilean velocitytransformation
equation, the speed of the ball relative to the Earth is (a) 90
mi/h(b) 110 mi/h (c) 20 mi/h (d) 200 mi/h (e) impossible to
determine.
The Speed of Light
It is quite natural to ask whether the principle of Galilean
relativity also applies toelectricity, magnetism, and optics.
Experiments indicate that the answer is no.Recall from Chapter 34
that Maxwell showed that the speed of light in free space isc "
3.00 $ 108 m/s. Physicists of the late 1800s thought that light
waves moved through amedium called the ether and that the speed of
light was c only in a special, absolute frame
at rest with respect to the ether. The Galilean velocity
transformation equation wasexpected to hold for observations of
light made by an observer in any frame moving atspeed v relative to
the absolute ether frame. That is, if light travels along the x
axis andan observer moves with velocity v along the x axis, the
observer will measure the light tohave speed c ! v, depending on
the directions of travel of the observer and the light.
Because the existence of a preferred, absolute ether frame would
show that lightwas similar to other classical waves and that
Newtonian ideas of an absolute frame weretrue, considerable
importance was attached to establishing the existence of the
etherframe. Prior to the late 1800s, experiments involving light
traveling in media moving atthe highest laboratory speeds
attainable at that time were not capable of detectingdifferences as
small as that between c and c ! v. Starting in about 1880,
scientistsdecided to use the Earth as the moving frame in an
attempt to improve their chancesof detecting these small changes in
the speed of light.
As observers fixed on the Earth, we can take the view that we
are stationaryand that the absolute ether frame containing the
medium for light propagation movespast us with speed v. Determining
the speed of light under these circumstances isjust like
determining the speed of an aircraft traveling in a moving air
current, orwind; consequently, we speak of an ether wind blowing
through our apparatus fixedto the Earth.
A direct method for detecting an ether wind would use an
apparatus fixed to theEarth to measure the ether winds influence on
the speed of light. If v is the speed ofthe ether relative to the
Earth, then light should have its maximum speed c " v
whenpropagating downwind, as in Figure 39.3a. Likewise, the speed
of light should have itsminimum value c # v when the light is
propagating upwind, as in Figure 39.3b, and anintermediate value
(c2 # v2)1/2 in the direction perpendicular to the ether wind, as
inFigure 39.3c. If the Sun is assumed to be at rest in the ether,
then the velocity of theether wind would be equal to the orbital
velocity of the Earth around the Sun, whichhas a magnitude of
approximately 3 $ 104 m/s. Because c % 3 $ 108 m/s, it isnecessary
to detect a change in speed of about 1 part in 104 for measurements
in theupwind or downwind directions. However, while such a change
is experimentallymeasurable, all attempts to detect such changes
and establish the existence of theether wind (and hence the
absolute frame) proved futile! We explore the classicexperimental
search for the ether in Section 39.2.
The principle of Galilean relativity refers only to the laws of
mechanics. If it isassumed that the laws of electricity and
magnetism are the same in all inertial frames, aparadox concerning
the speed of light immediately arises. We can understand this
byrecognizing that Maxwells equations seem to imply that the speed
of light always hasthe fixed value 3.00 $ 108 m/s in all inertial
frames, a result in direct contradiction towhat is expected based
on the Galilean velocity transformation equation. According
toGalilean relativity, the speed of light should not be the same in
all inertial frames.
To resolve this contradiction in theories, we must conclude that
either (1) the lawsof electricity and magnetism are not the same in
all inertial frames or (2) the Galileanvelocity transformation
equation is incorrect. If we assume the first alternative, then
apreferred reference frame in which the speed of light has the
value c must exist and themeasured speed must be greater or less
than this value in any other reference frame, inaccordance with the
Galilean velocity transformation equation. If we assume the
secondalternative, then we are forced to abandon the notions of
absolute time and absolutelength that form the basis of the
Galilean spacetime transformation equations.
39.2 The MichelsonMorley Experiment
The most famous experiment designed to detect small changes in
the speed of light wasfirst performed in 1881 by Albert A.
Michelson (see Section 37.7) and later repeatedunder various
conditions by Michelson and Edward W. Morley (18381923). We state
atthe outset that the outcome of the experiment contradicted the
ether hypothesis.
1248 C H A P T E R 3 9 Relativity
c + v
(a) Downwind
(b) Upwind
(c) Across wind
vc
v
c v
c
v
cc 2 v 2
Figure 39.3 If the velocity of theether wind relative to the
Earthis v and the velocity of light relativeto the ether is c, then
the speedof light relative to the Earth is(a) c " v in the
downwinddirection, (b) c # v in the upwinddirection, and (c) (c2 #
v2)1/2in the direction perpendicular tothe wind.
S E C T I O N 3 9 . 4 Consequences of the Special Theory of
Relativity 1253
At the Active Figures linkat http://www.pse6.com, youcan observe
the bouncing ofthe light pulse for variousspeeds of the train.
Einsteins thought experiment demonstrates that two observers can
disagree on thesimultaneity of two events. This disagreement,
however, depends on the transittime of light to the observers and,
therefore, does not demonstrate the deepermeaning of relativity. In
relativistic analyses of high-speed situations, relativity
showsthat simultaneity is relative even when the transit time is
subtracted out. In fact, all ofthe relativistic effects that we
will discuss from here on will assume that we are
ignoringdifferences caused by the transit time of light to the
observers.
Time Dilation
We can illustrate the fact that observers in different inertial
frames can measuredifferent time intervals between a pair of events
by considering a vehicle moving to theright with a speed v, such as
the boxcar shown in Figure 39.6a. A mirror is fixed to theceiling
of the vehicle, and observer O! at rest in the frame attached to
the vehicle holdsa flashlight a distance d below the mirror. At
some instant, the flashlight emits a pulse oflight directed toward
the mirror (event 1), and at some later time after reflecting
fromthe mirror, the pulse arrives back at the flashlight (event 2).
Observer O! carries a clockand uses it to measure the time interval
"tp between these two events. (The subscript pstands for proper ,
as we shall see in a moment.) Because the light pulse has a speed
c, thetime interval required for the pulse to travel from O! to the
mirror and back is
(39.5)
Now consider the same pair of events as viewed by observer O in
a second frame, asshown in Figure 39.6b. According to this
observer, the mirror and flashlight are moving tothe right with a
speed v, and as a result the sequence of events appears entirely
different.By the time the light from the flashlight reaches the
mirror, the mirror has moved to theright a distance v "t/2, where
"t is the time interval required for the light to travel fromO! to
the mirror and back to O! as measured by O. In other words, O
concludes that,because of the motion of the vehicle, if the light
is to hit the mirror, it must leave the
"tp #distance traveled
speed#
2dc
two events that are simultaneous in one reference frame are in
general notsimultaneous in a second frame moving relative to the
first. That is, simultaneity isnot an absolute concept but rather
one that depends on the state of motion ofthe observer.
v
O O O
x O
y
vt
(b)
d
vt2
ct2
(c)
vMirror
y
x
d
O
(a)
Active Figure 39.6 (a) A mirror is fixed to a moving vehicle,
and a light pulse is sentout by observer O! at rest in the vehicle.
(b) Relative to a stationary observer O standingalongside the
vehicle, the mirror and O! move with a speed v . Note that what
observerO measures for the distance the pulse travels is greater
than 2d . (c) The right trianglefor calculating the relationship
between "t and "tp .
In other words,
S E C T I O N 3 9 . 4 Consequences of the Special Theory of
Relativity 1253
At the Active Figures linkat http://www.pse6.com, youcan observe
the bouncing ofthe light pulse for variousspeeds of the train.
Einsteins thought experiment demonstrates that two observers can
disagree on thesimultaneity of two events. This disagreement,
however, depends on the transittime of light to the observers and,
therefore, does not demonstrate the deepermeaning of relativity. In
relativistic analyses of high-speed situations, relativity
showsthat simultaneity is relative even when the transit time is
subtracted out. In fact, all ofthe relativistic effects that we
will discuss from here on will assume that we are
ignoringdifferences caused by the transit time of light to the
observers.
Time Dilation
We can illustrate the fact that observers in different inertial
frames can measuredifferent time intervals between a pair of events
by considering a vehicle moving to theright with a speed v, such as
the boxcar shown in Figure 39.6a. A mirror is fixed to theceiling
of the vehicle, and observer O! at rest in the frame attached to
the vehicle holdsa flashlight a distance d below the mirror. At
some instant, the flashlight emits a pulse oflight directed toward
the mirror (event 1), and at some later time after reflecting
fromthe mirror, the pulse arrives back at the flashlight (event 2).
Observer O! carries a clockand uses it to measure the time interval
"tp between these two events. (The subscript pstands for proper ,
as we shall see in a moment.) Because the light pulse has a speed
c, thetime interval required for the pulse to travel from O! to the
mirror and back is
(39.5)
Now consider the same pair of events as viewed by observer O in
a second frame, asshown in Figure 39.6b. According to this
observer, the mirror and flashlight are moving tothe right with a
speed v, and as a result the sequence of events appears entirely
different.By the time the light from the flashlight reaches the
mirror, the mirror has moved to theright a distance v "t/2, where
"t is the time interval required for the light to travel fromO! to
the mirror and back to O! as measured by O. In other words, O
concludes that,because of the motion of the vehicle, if the light
is to hit the mirror, it must leave the
"tp #distance traveled
speed#
2dc
two events that are simultaneous in one reference frame are in
general notsimultaneous in a second frame moving relative to the
first. That is, simultaneity isnot an absolute concept but rather
one that depends on the state of motion ofthe observer.
v
O O O
x O
y
vt
(b)
d
vt2
ct2
(c)
vMirror
y
x
d
O
(a)
Active Figure 39.6 (a) A mirror is fixed to a moving vehicle,
and a light pulse is sentout by observer O! at rest in the vehicle.
(b) Relative to a stationary observer O standingalongside the
vehicle, the mirror and O! move with a speed v . Note that what
observerO measures for the distance the pulse travels is greater
than 2d . (c) The right trianglefor calculating the relationship
between "t and "tp .
In other words,
S E C T I O N 3 9 . 4 Consequences of the Special Theory of
Relativity 1253
At the Active Figures linkat http://www.pse6.com, youcan observe
the bouncing ofthe light pulse for variousspeeds of the train.
Einsteins thought experiment demonstrates that two observers can
disagree on thesimultaneity of two events. This disagreement,
however, depends on the transittime of light to the observers and,
therefore, does not demonstrate the deepermeaning of relativity. In
relativistic analyses of high-speed situations, relativity
showsthat simultaneity is relative even when the transit time is
subtracted out. In fact, all ofthe relativistic effects that we
will discuss from here on will assume that we are
ignoringdifferences caused by the transit time of light to the
observers.
Time Dilation
We can illustrate the fact that observers in different inertial
frames can measuredifferent time intervals between a pair of events
by considering a vehicle moving to theright with a speed v, such as
the boxcar shown in Figure 39.6a. A mirror is fixed to theceiling
of the vehicle, and observer O! at rest in the frame attached to
the vehicle holdsa flashlight a distance d below the mirror. At
some instant, the flashlight emits a pulse oflight directed toward
the mirror (event 1), and at some later time after reflecting
fromthe mirror, the pulse arrives back at the flashlight (event 2).
Observer O! carries a clockand uses it to measure the time interval
"tp between these two events. (The subscript pstands for proper ,
as we shall see in a moment.) Because the light pulse has a speed
c, thetime interval required for the pulse to travel from O! to the
mirror and back is
(39.5)
Now consider the same pair of events as viewed by observer O in
a second frame, asshown in Figure 39.6b. According to this
observer, the mirror and flashlight are moving tothe right with a
speed v, and as a result the sequence of events appears entirely
different.By the time the light from the flashlight reaches the
mirror, the mirror has moved to theright a distance v "t/2, where
"t is the time interval required for the light to travel fromO! to
the mirror and back to O! as measured by O. In other words, O
concludes that,because of the motion of the vehicle, if the light
is to hit the mirror, it must leave the
"tp #distance traveled
speed#
2dc
two events that are simultaneous in one reference frame are in
general notsimultaneous in a second frame moving relative to the
first. That is, simultaneity isnot an absolute concept but rather
one that depends on the state of motion ofthe observer.
v
O O O
x O
y
vt
(b)
d
vt2
ct2
(c)
vMirror
y
x
d
O
(a)
Active Figure 39.6 (a) A mirror is fixed to a moving vehicle,
and a light pulse is sentout by observer O! at rest in the vehicle.
(b) Relative to a stationary observer O standingalongside the
vehicle, the mirror and O! move with a speed v . Note that what
observerO measures for the distance the pulse travels is greater
than 2d . (c) The right trianglefor calculating the relationship
between "t and "tp .
In other words,
S E C T I O N 3 9 . 4 Consequences of the Special Theory of
Relativity 1253
At the Active Figures linkat http://www.pse6.com, youcan observe
the bouncing ofthe light pulse for variousspeeds of the train.
Einsteins thought experiment demonstrates that two observers can
disagree on thesimultaneity of two events. This disagreement,
however, depends on the transittime of light to the observers and,
therefore, does not demonstrate the deepermeaning of relativity. In
relativistic analyses of high-speed situations, relativity
showsthat simultaneity is relative even when the transit time is
subtracted out. In fact, all ofthe relativistic effects that we
will discuss from here on will assume that we are
ignoringdifferences caused by the transit time of light to the
observers.
Time Dilation
We can illustrate the fact that observers in different inertial
frames can measuredifferent time intervals between a pair of events
by considering a vehicle moving to theright with a speed v, such as
the boxcar shown in Figure 39.6a. A mirror is fixed to theceiling
of the vehicle, and observer O! at rest in the frame attached to
the vehicle holdsa flashlight a distance d below the mirror. At
some instant, the flashlight emits a pulse oflight directed toward
the mirror (event 1), and at some later time after reflecting
fromthe mirror, the pulse arrives back at the flashlight (event 2).
Observer O! carries a clockand uses it to measure the time interval
"tp between these two events. (The subscript pstands for proper ,
as we shall see in a moment.) Because the light pulse has a speed
c, thetime interval required for the pulse to travel from O! to the
mirror and back is
(39.5)
Now consider the same pair of events as viewed by observer O in
a second frame, asshown in Figure 39.6b. According to this
observer, the mirror and flashlight are moving tothe right with a
speed v, and as a result the sequence of events appears entirely
different.By the time the light from the flashlight reaches the
mirror, the mirror has moved to theright a distance v "t/2, where
"t is the time interval required for the light to travel fromO! to
the mirror and back to O! as measured by O. In other words, O
concludes that,because of the motion of the vehicle, if the light
is to hit the mirror, it must leave the
"tp #distance traveled
speed#
2dc
two events that are simultaneous in one reference frame are in
general notsimultaneous in a second frame moving relative to the
first. That is, simultaneity isnot an absolute concept but rather
one that depends on the state of motion ofthe observer.
v
O O O
x O
y
vt
(b)
d
vt2
ct2
(c)
vMirror
y
x
d
O
(a)
Active Figure 39.6 (a) A mirror is fixed to a moving vehicle,
and a light pulse is sentout by observer O! at rest in the vehicle.
(b) Relative to a stationary observer O standingalongside the
vehicle, the mirror and O! move with a speed v . Note that what
observerO measures for the distance the pulse travels is greater
than 2d . (c) The right trianglefor calculating the relationship
between "t and "tp .
In other words,
flashlight at an angle with respect to the vertical direction.
Comparing Figure 39.6a and b,we see that the light must travel
farther in (b) than in (a). (Note that neither observerknows that
he or she is moving. Each is at rest in his or her own inertial
frame.)
According to the second postulate of the special theory of
relativity, both observersmust measure c for the speed of light.
Because the light travels farther according to O,it follows that
the time interval !t measured by O is longer than the time interval
!tpmeasured by O". To obtain a relationship between these two time
intervals, it is conve-nient to use the right triangle shown in
Figure 39.6c. The Pythagorean theorem gives
Solving for !t gives
(39.6)
Because !tp # 2d/c, we can express this result as
(39.7)
where
(39.8)
Because $ is always greater than unity, this result says that
the time interval !tmeasured by an observer moving with respect to
a clock is longer than the timeinterval !tp measured by an observer
at rest with respect to the clock. This effectis known as time
dilation.
We can see that time dilation is not observed in our everyday
lives by consideringthe factor $. This factor deviates
significantly from a value of 1 only for very highspeeds, as shown
in Figure 39.7 and Table 39.1. For example, for a speed of 0.1c,
thevalue of $ is 1.005. Thus, there is a time dilation of only 0.5%
at one-tenth the speed oflight. Speeds that we encounter on an
everyday basis are far slower than this, so we donot see time
dilation in normal situations.
The time interval !tp in Equations 39.5 and 39.7 is called the
proper timeinterval. (In German, Einstein used the term Eigenzeit ,
which means own-time.) In
$ #1
1 % v2c 2
!t #!tp
1 % v2c 2# $ !tp
!t #2d
c 2 % v2#
2d
c 1 % v2c 2
! c !t2 "2
# ! v !t2 "2
& d2
1254 C H A P T E R 3 9 Relativity
Time dilation
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
10
15
20
5
1
v(108 m/s )
Figure 39.7 Graph of $ versus v. As the speed approaches that of
light, $ increasesrapidly.
v/c "
0.001 0 1.000 000 50.010 1.000 050.10 1.0050.20 1.0210.30
1.0480.40 1.0910.50 1.1550.60 1.2500.70 1.4000.80 1.6670.90
2.2940.92 2.5520.94 2.9310.96 3.5710.98 5.0250.99 7.0890.995
10.010.999 22.37
Approximate Values for "at Various Speeds
Table 39.1
flashlight at an angle with respect to the vertical direction.
Comparing Figure 39.6a and b,we see that the light must travel
farther in (b) than in (a). (Note that neither observerknows that
he or she is moving. Each is at rest in his or her own inertial
frame.)
According to the second postulate of the special theory of
relativity, both observersmust measure c for the speed of light.
Because the light travels farther according to O,it follows that
the time interval !t measured by O is longer than the time interval
!tpmeasured by O". To obtain a relationship between these two time
intervals, it is conve-nient to use the right triangle shown in
Figure 39.6c. The Pythagorean theorem gives
Solving for !t gives
(39.6)
Because !tp # 2d/c, we can express this result as
(39.7)
where
(39.8)
Because $ is always greater than unity, this result says that
the time interval !tmeasured by an observer moving with respect to
a clock is longer than the timeinterval !tp measured by an observer
at rest with respect to the clock. This effectis known as time
dilation.
We can see that time dilation is not observed in our everyday
lives by consideringthe factor $. This factor deviates
significantly from a value of 1 only for very highspeeds, as shown
in Figure 39.7 and Table 39.1. For example, for a speed of 0.1c,
thevalue of $ is 1.005. Thus, there is a time dilation of only 0.5%
at one-tenth the speed oflight. Speeds that we encounter on an
everyday basis are far slower than this, so we donot see time
dilation in normal situations.
The time interval !tp in Equations 39.5 and 39.7 is called the
proper timeinterval. (In German, Einstein used the term Eigenzeit ,
which means own-time.) In
$ #1
1 % v2c 2
!t #!tp
1 % v2c 2# $ !tp
!t #2d
c 2 % v2#
2d
c 1 % v2c 2
! c !t2 "2
# ! v !t2 "2
& d2
1254 C H A P T E R 3 9 Relativity
Time dilation
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
10
15
20
5
1
v(108 m/s )
Figure 39.7 Graph of $ versus v. As the speed approaches that of
light, $ increasesrapidly.
v/c "
0.001 0 1.000 000 50.010 1.000 050.10 1.0050.20 1.0210.30
1.0480.40 1.0910.50 1.1550.60 1.2500.70 1.4000.80 1.6670.90
2.2940.92 2.5520.94 2.9310.96 3.5710.98 5.0250.99 7.0890.995
10.010.999 22.37
Approximate Values for "at Various Speeds
Table 39.1
For example, suppose that a meter stick moves past a stationary
Earth observer withspeed v, as in Figure 39.11. The length of the
stick as measured by an observer in a frameattached to the stick is
the proper length Lp shown in Figure 39.11a. The length of thestick
L measured by the Earth observer is shorter than Lp by the factor
(1 ! v2/c2)1/2.Note that length contraction takes place only along
the direction of motion.
The proper length and the proper time interval are defined
differently. The properlength is measured by an observer for whom
the end points of the length remain fixed inspace. The proper time
interval is measured by someone for whom the two events takeplace
at the same position in space. As an example of this point, let us
return to thedecaying muons moving at speeds close to the speed of
light. An observer in the muonsreference frame would measure the
proper lifetime, while an Earth-based observer wouldmeasure the
proper length (the distance from creation to decay in Figure 39.8).
In themuons reference frame, there is no time dilation but the
distance of travel to the surfaceis observed to be shorter when
measured in this frame. Likewise, in the Earth observersreference
frame, there is time dilation, but the distance of travel is
measured to be theproper length. Thus, when calculations on the
muon are performed in both frames, theoutcome of the experiment in
one frame is the same as the outcome in the other framemore muons
reach the surface than would be predicted without relativistic
effects.
S E C T I O N 3 9 . 4 Consequences of the Special Theory of
Relativity 1259
Lp
y
O (a)
x
Ly
O(b)
x
v
Active Figure 39.11 (a) A meterstick measured by an observer in
aframe attached to the stick (that is,both have the same velocity)
has itsproper length Lp. (b) The stickmeasured by an observer in a
framein which the stick has a velocity vrelative to the frame is
measured tobe shorter than its proper lengthLp by a factor (1 !
v2/c2)1/2.
At the Active Figures linkat http://www.pse6.com, youcan view
the meter stick fromthe points of view of twoobservers to compare
themeasured length of the stick.
Quick Quiz 39.6 You are packing for a trip to another star.
During thejourney, you will be traveling at 0.99c. You are trying
to decide whether you should buysmaller sizes of your clothing,
because you will be thinner on your trip, due to lengthcontraction.
Also, you are considering saving money by reserving a smaller cabin
to sleepin, because you will be shorter when you lie down. Should
you (a) buy smaller sizes ofclothing, (b) reserve a smaller cabin,
(c) do neither of these, or (d) do both of these?
Quick Quiz 39.7 You are observing a spacecraft moving away from
you. Youmeasure it to be shorter than when it was at rest on the
ground next to you. You alsosee a clock through the spacecraft
window, and you observe that the passage of time onthe clock is
measured to be slower than that of the watch on your wrist.
Compared towhen the spacecraft was on the ground, what do you
measure if the spacecraft turnsaround and comes toward you at the
same speed? (a) The spacecraft is measured to belonger and the
clock runs faster. (b) The spacecraft is measured to be longer and
theclock runs slower. (c) The spacecraft is measured to be shorter
and the clock runsfaster. (d) The spacecraft is measured to be
shorter and the clock runs slower.
SpaceTime Graphs
It is sometimes helpful to make a spacetime graph, in which ct
is the ordinate andposition x is the abscissa. The twin paradox is
displayed in such a graph in Figure 39.12
World-line of Speedo
World-line of light beamWorld-lineof Goslo
ct
x
Figure 39.12 The twin paradox on aspacetime graph. The twin who
stays onthe Earth has a world-line along the ct axis.The path of
the traveling twin throughspacetime is represented by a
world-linethat changes direction.
For example, suppose that a meter stick moves past a stationary
Earth observer withspeed v, as in Figure 39.11. The length of the
stick as measured by an observer in a frameattached to the stick is
the proper length Lp shown in Figure 39.11a. The length of thestick
L measured by the Earth observer is shorter than Lp by the factor
(1 ! v2/c2)1/2.Note that length contraction takes place only along
the direction of motion.
The proper length and the proper time interval are defined
differently. The properlength is measured by an observer for whom
the end points of the length remain fixed inspace. The proper time
interval is measured by someone for whom the two events takeplace
at the same position in space. As an example of this point, let us
return to thedecaying muons moving at speeds close to the speed of
light. An observer in the muonsreference frame would measure the
proper lifetime, while an Earth-based observer wouldmeasure the
proper length (the distance from creation to decay in Figure 39.8).
In themuons reference frame, there is no time dilation but the
distance of travel to the surfaceis observed to be shorter when
measured in this frame. Likewise, in the Earth observersreference
frame, there is time dilation, but the distance of travel is
measured to be theproper length. Thus, when calculations on the
muon are performed in both frames, theoutcome of the experiment in
one frame is the same as the outcome in the other framemore muons
reach the surface than would be predicted without relativistic
effects.
S E C T I O N 3 9 . 4 Consequences of the Special Theory of
Relativity 1259
Lp
y
O (a)
x
Ly
O(b)
x
v
Active Figure 39.11 (a) A meterstick measured by an observer in
aframe attached to the stick (that is,both have the same velocity)
has itsproper length Lp. (b) The stickmeasured by an observer in a
framein which the stick has a velocity vrelative to the frame is
measured tobe shorter than its proper lengthLp by a factor (1 !
v2/c2)1/2.
At the Active Figures linkat http://www.pse6.com, youcan view
the meter stick fromthe points of view of twoobservers to compare
themeasured length of the stick.
Quick Quiz 39.6 You are packing for a trip to another star.
During thejourney, you will be traveling at 0.99c. You are trying
to decide whether you should buysmaller sizes of your clothing,
because you will be thinner on your trip, due to lengthcontraction.
Also, you are considering saving money by reserving a smaller cabin
to sleepin, because you will be shorter when you lie down. Should
you (a) buy smaller sizes ofclothing, (b) reserve a smaller cabin,
(c) do neither of these, or (d) do both of these?
Quick Quiz 39.7 You are observing a spacecraft moving away from
you. Youmeasure it to be shorter than when it was at rest on the
ground next to you. You alsosee a clock through the spacecraft
window, and you observe that the passage of time onthe clock is
measured to be slower than that of the watch on your wrist.
Compared towhen the spacecraft was on the ground, what do you
measure if the spacecraft turnsaround and comes toward you at the
same speed? (a) The spacecraft is measured to belonger and the
clock runs faster. (b) The spacecraft is measured to be longer and
theclock runs slower. (c) The spacecraft is measured to be shorter
and the clock runsfaster. (d) The spacecraft is measured to be
shorter and the clock runs slower.
SpaceTime Graphs
It is sometimes helpful to make a spacetime graph, in which ct
is the ordinate andposition x is the abscissa. The twin paradox is
displayed in such a graph in Figure 39.12
World-line of Speedo
World-line of light beamWorld-lineof Goslo
ct
x
Figure 39.12 The twin paradox on aspacetime graph. The twin who
stays onthe Earth has a world-line along the ct axis.The path of
the traveling twin throughspacetime is represented by a
world-linethat changes direction.
The Relativistic Doppler Effect
Another important consequence of time dilation is the shift in
frequencyfound for light emitted by atoms in motion as opposed to
light emitted by atomsat rest. This phenomenon, known as the
Doppler effect, was introduced in Chapter17 as it pertains to sound
waves. In the case of sound, the motion of the sourcewith respect
to the medium of propagation can be distinguished from the motionof
the observer with respect to the medium. Light waves must be
analyzeddifferently, however, because they require no medium of
propagation, and nomethod exists for distinguishing the motion of a
light source from the motion ofthe observer.
If a light source and an observer approach each other with a
relative speed v, thefrequency fobs measured by the observer is
(39.10)
where f source is the frequency of the source measured in its
rest frame. Note that thisrelativistic Doppler shift equation,
unlike the Doppler shift equation for sound,depends only on the
relative speed v of the source and observer and holds for
relativespeeds as great as c. As you might expect, the equation
predicts that fobs ! f source whenthe source and observer approach
each other. We obtain the expression for the case inwhich the
source and observer recede from each other by substituting negative
valuesfor v in Equation 39.10.
The most spectacular and dramatic use of the relativistic
Doppler effect is themeasurement of shifts in the frequency of
light emitted by a moving astronomicalobject such as a galaxy.
Light emitted by atoms and normally found in the extremeviolet
region of the spectrum is shifted toward the red end of the
spectrum for atomsin other galaxiesindicating that these galaxies
are receding from us. The Americanastronomer Edwin Hubble
(18891953) performed extensive measurements of this redshift to
confirm that most galaxies are moving away from us, indicating that
theUniverse is expanding.
39.5 The Lorentz Transformation Equations
Suppose an event that occurs at some point P is reported by two
observers, oneat rest in a frame S and another in a frame S" that
is moving to the right withspeed v as in Figure 39.14. The observer
in S reports the event with spacetimecoordinates (x, y, z, t),
while the observer in S" reports the same event using
thecoordinates (x", y", z", t"). If two events occur at P and Q ,
Equation 39.1predicts that #x $ #x", that is, the distance between
the two points in space
f obs $1 % v/c1 & v/c
f source
1262 C H A P T E R 3 9 Relativity
What If? What if this trip is observed with a very
powerfultelescope by a technician in Mission Control on Earth?
Atwhat time will this technician see that the astronaut hasarrived
at Sirius?
Answer The time interval that the technician will measurefor the
astronaut to arrive is
In order for the technician to see the arrival, the light
fromthe scene of the arrival must travel back to Earth and
enter
#t $dv
$8 ly0.8c
$ 10 yr
the telescope. This will require a time interval of
Thus, the technician sees the arrival after 10 yr % 8 yr $18 yr.
Notice that if the astronaut immediately turnsaround and comes back
home, she arrives, according tothe technician, 20 years after
leaving, only 2 years after hesaw her arrive! In addition, she
would have aged by only12 years.
#t $dv
$8 ly
c$ 8 yr
y y vS S
O x x
P (event)
O
x
Q (event)
vtx
x x
Figure 39.14 Events occur atpoints P and Q and are observed byan
observer at rest in the S frameand another in the S" frame, whichis
moving to the right with a speed v.
