Chapter 26
Relativity
INFO: Relativity Tutorial
Units of Chapter 26
Classical Relativity and the Michelson–Morley
Experiment
The Postulates of Special Relativity and the
Relativity of Simultaneity
The Relativity of Length and Time: Time
Dilation and Length Contraction
Relativistic Kinetic Energy, Momentum, Total
Energy, and Mass–Energy Equivalence
The General Theory of Relativity
Relativistic Velocity Addition
26.1 Classical Relativity and the
Michelson–Morley Experiment
We expect that the laws of physics should be
the same in any inertial reference frame—that
is, any frame that is not accelerating.
In an accelerating frame, this may not be true;
accelerating frames have “pseudoforces” that
replace the acceleration, such as the
centrifugal force in circular motion.
26.1 Classical Relativity and the
Michelson–Morley Experiment
This is the principle of classical relativity,
known to Newton:
The laws of mechanics are the same in all inertial
reference frames.
26.1 Classical Relativity and the
Michelson–Morley Experiment
When the equations of
electromagnetism were
found to predict a
specific speed for
electromagnetic waves,
classical relativity had a
problem—this could not
be true in all inertial
reference frames.
26.1 Classical Relativity and the
Michelson–Morley Experiment
In addition, there was no obvious medium that
propagated electromagnetic waves.
Both of these problems would be solved by
the existence of a “luminiferous ether.” This
all-permeating substance would carry the
electromagnetic waves, and would also
establish a preferred reference frame in which
the speed of light would have its calculated
value.
26.1 Classical Relativity and the
Michelson–Morley Experiment
Michelson and Morley devised an experiment
to detect the ether by measuring the
difference in the speed of light in two
perpendicular directions, produced by the
Earth’s motion through the ether.
All their measurements showed no difference
in the speed of light in any direction.
26.2 The Postulates of Special Relativity
and the Relativity of Simultaneity
Einstein thought that all the laws of physics,
not just the laws of mechanics, should be the
same in all inertial reference frames. This led
him to his postulates of relativity.
Postulate I (principle of relativity): All the laws of
physics are the same in all inertial reference frames.
Postulate II (constancy of the speed of light): The
speed of light in a vacuum has the same value in all
inertial systems.
26.2 The Postulates of Special Relativity
and the Relativity of Simultaneity
The first postulate is classical relativity,
extended to all laws of physics.
The second seems
counterintuitive—
how could the
speed of light not
add the same way
as other speeds?
26.2 The Postulates of Special Relativity
and the Relativity of Simultaneity
However, in science observation trumps
theory, and the constancy of the speed of light
has been confirmed many times over.
So, what are the consequences of the speed of
light being the same in all reference frames?
26.2 The Postulates of Special Relativity
and the Relativity of Simultaneity
First, events that
are simultaneous
in one frame are
not necessarily
simultaneous in
another.
26.2 The Postulates of Special Relativity
and the Relativity of Simultaneity
We see events occurring simultaneously in
our reference frame. However, another
observer moving with respect to us will not
see the events as simultaneous—the light
from one will reach them before the light from
the other.
Events that are simultaneous in one inertial reference
frame may not be simultaneous in a different inertial
frame.
26.2 The Postulates of Special Relativity
and the Relativity of Simultaneity
These effects are vanishingly small at
ordinary speeds; this is why the constancy
of the speed of light is not part of our
everyday experience.
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
A clock may be made from a light beam and
mirrors. As the speed of light is the same for
all observers, an observer in another reference
frame will see the clock running slow.
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
The time between pulses as seen by a moving
observer is:
This effect is called time dilation. The time
as measured in an object’s own reference
frame is called its proper time.
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
The equations of special relativity become
much less cumbersome to write if we define
the following shorthand:
Using this, the equation for time dilation
becomes:
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
The value of γ becomes
significantly different from 1
only when the relative speed
is a substantial fraction of the
speed of light.
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
Suppose we made a clock that operated in a
way that had nothing to do with the speed of
light. Would time dilation still occur?
Yes, it would—otherwise different clocks in the
same reference frame would behave differently
depending on that frame’s speed with respect
to another, violating the first principle of
relativity.
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
How do we measure the length of an object that
is moving with respect to us?
One way is to mark both ends simultaneously
and then measure the distance between them;
another is to measure the time it takes the entire
object to pass by a given point.
However, we already know that moving
observers disagree on simultaneity and on time
intervals!
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
Therefore, we measure
the length to be
shorter than in the
object’s own reference
frame.
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
An object’s length is largest
when measured by an observer
at rest with respect to it (the
“proper” observer). If the
object is moving relative to an
inertial observer, that observer
measures a smaller length
than the proper observer.
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
The contracted length is given by:
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
The twin paradox: Suppose one of a pair of
identical twins goes on a spaceship ride close
to the speed of light. When she returns to
Earth, after 20 years have passed, she has
aged only two years due to time dilation.
But wait! From her frame, the Earth was
moving with respect to her, and her Earth twin
should be the younger one.
What’s wrong here?
26.3 The Relativity of Length and Time:
Time Dilation and Length Contraction
The answer is that the traveling twin, in order to
return to Earth, has to turn around and come
back. Therefore, her frame is not inertial, and she
is indeed younger than her twin when she comes
home.
26.4 Relativistic Kinetic Energy,
Momentum, Total Energy, and
Mass–Energy Equivalence
Calculations show that relativistic kinetic
energy is given by:
26.4 Relativistic Kinetic Energy,
Momentum, Total Energy, and
Mass–Energy Equivalence
For speeds close to c, the
relativistic kinetic energy
increases rapidly, and
approaches infinity as the
speed approaches c.
26.4 Relativistic Kinetic Energy,
Momentum, Total Energy, and
Mass–Energy Equivalence
The relativistic momentum is then:
The momentum also becomes infinitely
large as speeds approach c.
26.4 Relativistic Kinetic Energy,
Momentum, Total Energy, and
Mass–Energy Equivalence
The total energy is then:
Note that this is not zero even when v = 0.
26.4 Relativistic Kinetic Energy,
Momentum, Total Energy, and
Mass–Energy Equivalence
When v = 0,
This equation (which you may recognize)
gives the rest energy of an object, also
called its mass energy. It is possible to
convert this energy to other forms of
energy, through nuclear reactions.
26.4 Relativistic Kinetic Energy,
Momentum, Total Energy, and
Mass–Energy Equivalence
So, when do we need to take relativistic
effects into account, and when can we ignore
them?
For speeds below 10% of the speed of light or kinetic
energies less than 0.5% of an object’s rest energy, the
error in using the nonrelativistic formulas is less than
1%, and it is then usually acceptable to use the
nonrelativistic expressions.
26.5 General Relativity
General relativity applies to accelerating
systems. It begins with the principle of
equivalence:
An inertial reference frame in a uniform
gravitational field is physically equivalent to a
reference frame that is not in a gravitational field,
but that is in uniform linear acceleration.
Equivalently,
No experiment performed in a closed system can
distinguish between the effects of a gravitational field
and the effects of an acceleration.
26.5 General Relativity
This is true for a spaceship
that is either on the ground
or accelerating with a = g.
26.5 General Relativity
It is also true for either an
isolated spaceship or one
in free fall near a massive
object.
26.5 General Relativity
This means that light must bend in a
gravitational field, just as it bends while
accelerating.
26.5 General Relativity
This bending was first observed in the early 20th
century during a total eclipse of the Sun.
26.5 General Relativity
This bending also means that an intervening
mass can serve as a “lens” for a more distant
object. Numerous examples have been
observed in astronomical photographs.
26.5 General Relativity
It is possible to have an object with a
gravitational field strong enough that the escape
speed is equal to, or greater than, the speed of
light. Such an object is called a black hole.
The radius within which the escape speed is c or
greater is called the Schwarzschild radius:
An object is a black hole only if the mass M is
within the radius R.
26.6 Relativistic Velocity Addition
Clearly, the relativistic addition of velocities
cannot be as simple as it is classically, if all
speeds are to remain less than the speed of
light.
26.6 Relativistic Velocity Addition
The correct form for velocity addition is given
by:
If both v and u’ are small compared to c,
this gives the classical formula u = v + u’.
Relativity Animations
Review of Chapter 26
Classical relativity supported the existence of
an “ether” to carry electromagnetic waves. No
evidence for the ether was ever found.
Principle of relativity: All the laws of physics are
the same in all inertial reference frames.
Principle of the constancy of the speed of light:
The speed of light in a vacuum has the same
value in all inertial systems.
Review of Chapter 26
Relativistic effects in moving systems include
time dilation, length contraction, and the
relativity of simultaneity.
Relativistic energy:
Rest energy:
Review of Chapter 26
Principle of equivalence:
An inertial reference frame in a uniform
gravitational field is physically equivalent to a
reference frame that is not in a gravitational
field, but that is in uniform linear acceleration.
General relativity predicts black holes and the
bending of light in a gravitational field.