Relativity
You can measure lengths with
a ruler or meter stick. You can
time events with a stopwatch.
Einstein’s special theory of
relativity questions deeply
held assumptions about the
nature of space and time.
Chapter Goal: To understand
how Einstein’s theory of
relativity changes our
concepts of space and time.
Chapter 37. Relativity
Topics:
• Relativity: What’s It All About?
• Galilean Relativity
• Einstein’s Principle of Relativity
• Events and Measurements
• The Relativity of Simultaneity
• Time Dilation
• Length Contraction
• The Lorentz Transformations
• Relativistic Momentum
• Relativistic Energy
Reading Quizzes
In relativity, the Galilean
transformations are replaced
by the
A. Einstein tranformations.
B. Lorentz transformations.
C. Feynman transformations.
D. Maxwell transformations.
E. Laplace tranformations.
In relativity, the Galilean
transformations are replaced
by the
A. Einstein tranformations.
B. Lorentz transformations.
C. Feynman transformations.
D. Maxwell transformations.
E. Laplace tranformations.
A physical activity that takes place
at a definite point in space and time
is called
A. a gala.
B. a sport.
C. an event.
D. a happening.
E. a locale.
A physical activity that takes place
at a definite point in space and time
is called
A. a gala.
B. a sport.
C. an event.
D. a happening.
E. a locale.
Basic Content and Examples
Since relativity theory plays a major
role in the study of atomic and
nuclear physics, we shall discuss it
in some detail.
Einstein’s Principle of Relativity
The Galilean Transformations
Consider two reference frames S and S'. The coordinate
axes in S are x, y, z and those in S' are x', y', z'. Reference
frame S' moves with velocity v relative to S along the xaxis. Equivalently, S moves with velocity −v relative to S'.
The Galilean transformations of position are:
The Galilean transformations of velocity are:
INVARIANCE OF CLASSICAL MECHANICS
UNDER GALILEAN TRANSFORMATIONS
• Conservation of Linear Momentum.
• Invariance of Newton's Second Law.
• Conservation of Energy.
Conservation of Linear Momentum
• We suppose that an observer in system S2
and S1 watches a head-on collision between
two particles of respective masses m and
M(Fig.a and b)
For the observer in the inertial systemS2
Using the galilean velocity transformations
Invariance of Newton's Second Law.
• Consider two bodies, of respective masses m and M, that interact with one
another as a result of some force, such as the gravitational force. If no net
external force is applied to the system (the two bodies), the system is
isolated, and the total linear momentum of the system must remain
constant.
• Observer S1 states
which is Newton's third law. In a similar fashion observer S2would write
since
therefore
• From the invariance of Newton's second law and from the fact that the forces
and acceleration are unchanged it immediately follows that, if S1 is an
inertial system, then S2, which represents any coordinate system moving
with respect to S1 at a constant velocity v, is also an inertial system. From
the point of view of Newton's second law, all inertial systems, of which there
are an infinite number, are equivalent and indistinguishable. Clearly,
any coordinate frame of reference that is accelerated with respect to some
inertial system cannot itself be an inertial system, because no longer does
hold.
Conservation of Energy.
• Observer S2
Kinetic energy before collision = kinetic energy after collision
observer S1
Hence using the invariance of the conservation of momentum law
We have found that the laws of classical mechanics (the conservation of momentum,
Newton's law of motion, and the conservation of energy) are all invariant under a
Galilean transformation.
THE FAILURE OF GALILEAN
TRANSFORMATIONS
• One might well ask whether the laws of electromagnetism are
invariant under a Galilean transformation?
THE FAILURE OF GALILEAN
TRANSFORMATIONS
• In general, the measured speed of a sound pulse depends on the relative
speed obtaining between the observer and the medium through which the
pulse travels.
• This result is confirmed by experiments with sound 'waves.
• However, the measured velocity of the sound pulse in the system S1,in
which the air is at rest, does not depend on the velocity of a source of the
sound. Of course, if a source of sound generates sinusoidal variations in the
air pressure, rather than a pulse, the frequency and wavelength of the sound
will, according to the Doppler effect, depend on the relative motion of the
source and the medium, but the velocity of propagation of the disturbance
will be independent of the source's relative motion.
• A pulse of light travels to the right with respect to the medium through which it is
propagated (ether ) at a speed
.
•
Because nineteenth-century physicists were firmly convinced that all physical
phenomena were ultimately mechanical in origin, it was for them unthinkable that
an electromagnetic disturbance could be propagated in empty space. Thus, the ether
concept was invented.
•
From Galilean transformation
• Therefore, the speed of light is certainly not invariant under galilean transformations
• It is to measure the speed of light in a variety of inertial systems, noting whether the
measured speed is different in the different systems and, most especially, whether
there is evidence of a single, unique inertial system, "the ether," in which the speed
of light is c.
Michelson and Morley:
There was a change in the measured speed of light as the earth drifted through
a conjectured ether in its axial rotation and its revolution about the sun.
• It may be assumed that v/c << 1.
• the maximum fractional change in the round-trip time interval for
reorientation at 90° is
• Performing this experiment many times, at various times of the year and in
various locations, Michelson and Morley always found that
was
zero; that is, the result was always null.
Any inertial system S2 behaves as if it were the unique inertial system S1 ;
or the measured speed of light in every inertial system is found to be
the same, namely c, for all directions and for all observers.
Einstein’s Principle of Relativity
• Maxwell’s equations are true in all inertial reference frames.
• Maxwell’s equations predict that electromagnetic waves, including
light, travel at speed c = 3.00 × 108 m/s.
• Therefore, light travels at speed c in all inertial reference frames.
• Every experiment has found that light travels at 3.00 × 108 m/s in every
inertial reference frame, regardless of how the reference frames are moving
with respect to each other.
• The first postulate, the principle
of relativity, is basic also to
classical,
or
Newtonian,
mechanics; the second postulate,
the constancy of the speed of
light, is at seeming variance with
it and also with Postulate I, if the
classical concepts of space and
time are adhered to.
Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
Time Dilation
Time Dilation
The time interval between two events that occur at the same
position is called the proper time Δτ. In an inertial
reference frame moving with velocity v = βc relative to the
proper time frame, the time interval between the two events
is
The “stretching out” of the time interval is called time
dilation.
EXAMPLE 37.5 From the sun to Saturn
QUESTIONS:
EXAMPLE 37.5 From the sun to Saturn
EXAMPLE 37.5 From the sun to Saturn
EXAMPLE 37.5 From the sun to Saturn
EXAMPLE 37.5 From the sun to Saturn
EXAMPLE 37.5 From the sun to Saturn
EXAMPLE 37.5 From the sun to Saturn
Length Contraction
The distance L between two objects, or two points on one
object, measured in the reference frame S in which the
objects are at rest is called the proper length ℓ. The
distance L' in a reference frame S' is
NOTE: Length contraction does not tell us how an object
would look. The visual appearance of an object is
determined by light waves that arrive simultaneously at the
eye. Length and length contraction are concerned only with
the actual length of the object at one instant of time.
EXAMPLE 37.6 The distance from the sun to
Saturn
QUESTION:
EXAMPLE 37.6 The distance from the sun to
Saturn
EXAMPLE 37.6 The distance from the sun to
Saturn
EXAMPLE 37.6 The distance from the sun to
Saturn
The Lorentz Transformations
Consider two reference frames S and S'. An event occurs at
coordinates x, y, z, t as measured in S, and the same event
occurs at x', y', z', t' as measured in S' . Reference frame S'
moves with velocity v relative to S, along the x-axis.
The Lorentz transformations for the coordinates of one
event are:
The Lorentz Velocity Transformations
Consider two reference frames S and S'. An object moves at
velocity u along the x-axis as measured in S, and at velocity
u' as measured in S' . Reference frame S' moves with
velocity v relative to S, also along the x-axis.
The Lorentz velocity transformations are:
NOTE: It is important to distinguish carefully between v,
which is the relative velocity between two reference frames,
and u and u' which are the velocities of an object as
measured in the two different reference frames.
EXAMPLE 37.10 A really fast bullet
QUESTION:
EXAMPLE 37.10 A really fast bullet
EXAMPLE 37.10 A really fast bullet
EXAMPLE 37.10 A really fast bullet
Relativistic Momentum
The momentum of a particle moving at speed u is
where the subscript p indicates that this is γ for a particle,
not for a reference frame.
• If u << c, the momentum approaches the Newtonian value
of p = mu. As u approaches c, however, p approaches
infinity.
• For this reason, a force cannot accelerate a particle to a
speed higher than c, because the particle’s momentum
becomes infinitely large as the speed approaches c.
EXAMPLE 37.11 Momentum of a subatomic
particle
QUESTION:
EXAMPLE 37.11 Momentum of a subatomic
particle
EXAMPLE 37.11 Momentum of a subatomic
particle
EXAMPLE 37.11 Momentum of a subatomic
particle
EXAMPLE 37.11 Momentum of a subatomic
particle
Relativistic Energy
The total energy E of a particle is
This total energy consists of a rest energy
and a relativistic expression for the kinetic energy
This expression for the kinetic energy is very nearly ½mu2
when u << c.
EXAMPLE 37.12 Kinetic energy and total
energy
EXAMPLE 37.12 Kinetic energy and total
energy
EXAMPLE 37.12 Kinetic energy and total
energy
EXAMPLE 37.12 Kinetic energy and total
energy
Conservation of Energy
The creation and annihilation of particles with mass,
processes strictly forbidden in Newtonian mechanics, are
vivid proof that neither mass nor the Newtonian definition
of energy is conserved. Even so, the total energy—the
kinetic energy and the energy equivalent of mass—remains
a conserved quantity.
Mass and energy are not the same thing, but they are
equivalent in the sense that mass can be transformed into
energy and energy can be transformed into mass as long as
the total energy is conserved.
Chapter 37. Summary Slides
General Principles
Important Concepts
Important Concepts
Important Concepts
Important Concepts
Important Concepts
Important Concepts
Applications
Applications
Applications
Chapter 37. Clicker Questions
Which of these is an inertial
reference frames (or a very
good approximation)?
A. A rocket being launched
B. A car rolling down a steep hill
C. A sky diver falling at terminal speed
D. A roller coaster going over the top of a hill
E. None of the above
Which of these is an inertial
reference frames (or a very
good approximation)?
A. A rocket being launched
B. A car rolling down a steep hill
C. A sky diver falling at terminal speed
D. A roller coaster going over the top of a hill
E. None of the above
Ocean waves are approaching the beach
at 10 m/s. A boat heading out to sea
travels at 6 m/s. How fast are the waves
moving in the boat’s reference frame?
A. 4 m/s
B. 6 m/s
C. 16 m/s
D. 10 m/s
Ocean waves are approaching the beach
at 10 m/s. A boat heading out to sea
travels at 6 m/s. How fast are the waves
moving in the boat’s reference frame?
A. 4 m/s
B. 6 m/s
C. 16 m/s
D. 10 m/s
A carpenter is working on a house two
blocks away. You notice a slight delay
between seeing the carpenter’s
hammer hit the nail and hearing the
blow. At what time does the event
“hammer hits nail” occur?
A. Very slightly after you see the hammer hit.
B. Very slightly after you hear the hammer hit.
C. Very slightly before you see the hammer hit.
D. At the instant you hear the blow.
E. At the instant you see the hammer hit.
A carpenter is working on a house two
blocks away. You notice a slight delay
between seeing the carpenter’s
hammer hit the nail and hearing the
blow. At what time does the event
“hammer hits nail” occur?
A. Very slightly after you see the hammer hit.
B. Very slightly after you hear the hammer hit.
C. Very slightly before you see the hammer hit.
D. At the instant you hear the blow.
E. At the instant you see the hammer hit.
A tree and a pole are 3000 m apart. Each is
suddenly hit by a bolt of lightning. Mark, who is
standing at rest midway between the two, sees
the two lightning bolts at the same instant of
time. Nancy is at rest under the tree. Define
event 1 to be “lightning strikes tree” and event 2
to be “lightning strikes pole.” For Nancy, does
event 1 occur before, after or at the same time as
event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
A tree and a pole are 3000 m apart. Each is
suddenly hit by a bolt of lightning. Mark, who is
standing at rest midway between the two, sees
the two lightning bolts at the same instant of
time. Nancy is at rest under the tree. Define
event 1 to be “lightning strikes tree” and event 2
to be “lightning strikes pole.” For Nancy, does
event 1 occur before, after or at the same time as
event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
A tree and a pole are 3000 m apart. Each is
suddenly hit by a bolt of lightning. Mark, who is
standing at rest midway between the two, sees the
two lightning bolts at the same instant of time.
Nancy is flying her rocket at v = 0.5c in the direction
from the tree toward the pole. The lightning hits the
tree just as she passes by it. Define event 1 to be
“lightning strikes tree” and event 2 to be “lightning
strikes pole.” For Nancy, does event 1 occur before,
after or at the same time as event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
A tree and a pole are 3000 m apart. Each is
suddenly hit by a bolt of lightning. Mark, who is
standing at rest midway between the two, sees the
two lightning bolts at the same instant of time.
Nancy is flying her rocket at v = 0.5c in the direction
from the tree toward the pole. The lightning hits the
tree just as she passes by it. Define event 1 to be
“lightning strikes tree” and event 2 to be “lightning
strikes pole.” For Nancy, does event 1 occur before,
after or at the same time as event 2?
A. before event 2
B. after event 2
C. at the same time as event 2
Molly flies her rocket past Nick at constant
velocity v. Molly and Nick both measure the
time it takes the rocket, from nose to tail, to
pass Nick. Which of the following is true?
A. Nick measures a shorter time interval
than Molly.
B. Molly measures a shorter time interval
than Nick.
C. Both Molly and Nick measure the same
amount of time.
Molly flies her rocket past Nick at constant
velocity v. Molly and Nick both measure the
time it takes the rocket, from nose to tail, to
pass Nick. Which of the following is true?
A. Nick measures a shorter time interval
than Molly.
B. Molly measures a shorter time interval
than Nick.
C. Both Molly and Nick measure the same
amount of time.
Beth and Charles are at
rest relative to each
other. Anjay runs past at
velocity v while holding a
long pole parallel to his
motion. Anjay, Beth, and
Charles each measure
the length of the pole at
the instant Anjay passes
Beth. Rank in order,
from largest to smallest,
the three lengths LA, LB,
and LC.
A.
B.
C.
D.
E.
LA = LB = LC
LB = LC > LA
LA > LB = LC
LA > LB > LC
LB > LC > LA
Beth and Charles are at
rest relative to each
other. Anjay runs past at
velocity v while holding a
long pole parallel to his
motion. Anjay, Beth, and
Charles each measure
the length of the pole at
the instant Anjay passes
Beth. Rank in order,
from largest to smallest,
the three lengths LA, LB,
and LC.
A.
B.
C.
D.
E.
LA = LB = LC
LB = LC > LA
LA > LB = LC
LA > LB > LC
LB > LC > LA
An electron moves through the lab at 99%
the speed of light. The lab reference frame
is S and the electron’s reference frame is S´.
In which reference frame is the electron’s
rest mass larger?
A. Frame S, the lab frame
B. Frame S´, the electron’s frame
C. It is the same in both frames.
An electron moves through the lab at 99%
the speed of light. The lab reference frame
is S and the electron’s reference frame is S´.
In which reference frame is the electron’s
rest mass larger?
A. Frame S, the lab frame
B. Frame S´, the electron’s frame
C. It is the same in both frames.