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Historical Development
• Late 1600s
• Newton discovered his laws of mechanics
• Applied to a wide variety of problems over the next two
centuries
• Worked well
• Late 1800s
• Maxwell’s equations explained the physics of
electromagnetism and light
• Early 1900s
• Relativity
• Quantum Mechanics
Introduction
Types of Relativity Theory
• Special relativity
• Concerned with objects and observers moving at a
constant velocity
• Topic of this chapter
• General relativity
• Applies to situations when the object or the observer is
accelerated
• Gravitational fields also are a form of acceleration
(take g, for example)
• GR is the world’s best classical theory of gravitation
• GR tells us how clocks slow down in a gravitational
potential well (Earth’s surface). Crucial to GPS !
Introduction
Reference Frames
• A reference frame can be thought of as a set of
coordinate axes
• Inertial reference frames move with a constant
velocity -- unaccelerated
• The principle of Galilean relativity is the idea that
the laws of motion should be the same in all inertial
frames
• If Newton’s Second Law is obeyed in one inertial
frame, it is obeyed in all inertial frames
• For example, an observer moving at a constant
velocity relative to an object does not see any change
in the acceleration of the object
Section 27.1
Relativity
• When the railroad car moves with a constant
velocity, Ted and Alice see different motions of the
ball, due to their constant relative velocity (and Ted
sees Alice, and the ground, moving backward
relative to him and the train at constant velocity!)
Section 27.1
Interpretation by Ted and Alice
• Ted observes that the ball’s motion is purely vertical
• Alice sees the ball also move with constant
•
•
•
•
horizontal velocity, to the right
Both agree that the ball’s acceleration is downward
with a magnitude g
Both agree the ball’s horizontal acceleration is zero
Both agree that the only force acting on the ball is
the force of gravity and that Newton’s second law is
obeyed
So: same laws of physics in both reference frames
Section 27.1
Galilean Relativity and Light
• According to Maxwell’s equations, the speed of light
•
•
•
•
is a constant, derived from the laws of electricity and
magnetism
Relativity requires the laws of physics to be the
same even if two inertial systems have relative
motion
Assume Ted is moving with a constant velocity v
relative to Alice when he turns on a flashlight
Newton’s mechanics predict that speed of the light
wave relative to Alice should be c + v
According to Maxwell’s theory, with relativity valid,
Ted and Alice should both observe the light wave to
move with speed c
Section 27.1
Galilean Relativity and Light, cont.
• Galilean Relativity and
electromagnetism predict
different results for
relative motion of two
frames
• Experiments showed that
Maxwell’s theory was
correct
• The speed of light in a
vacuum is always c
• Galilean relativity is
wrong
Section 27.1
Special Relativity
• Einstein developed a theory to accommodate
constancy of the speed of light, called special
relativity
• Postulates of Special Relativity
• All the laws of physics are the same in all inertial
reference frames
• The speed of light in a vacuum is a constant,
independent of the motion of the light source and all
observers
• (That’s nice, but it leads to very interesting
“contortions” in space and time!!!)
Section 27.2
Earth as a Reference Frame
• Since the Earth spins about its axis as it orbits the
Sun, “all” points on the Earth’s surface have a
nonzero acceleration (except at each pole)
• Technically, a person standing on the surface of the
Earth is not in an inertial reference frame
• However, the Earth’s acceleration is small enough
that it can generally be ignored
• In most situations we can consider the Earth to be
an inertial reference frame
Section 27.2
Light Clock
• The two postulates of
relativity lead to a
surprising result about
time
• A light clock keeps time
by using a pulse of light
that travels back and
forth between two
mirrors
• Call the time for one
clock “tick” = the time
needed for one round
trip: 2ℓ / c
2
t 0 
c
Section 27.3
Moving Light Clock
• The clock moves with a constant velocity v relative to
the ground
• From Ted’s reference frame, the light pulse travels
up and down between the two mirrors
2
t 0 
c
Section 27.3
Moving Clock, Alice’s Time
• For Alice, the time for one tick of the clock is
• Δt = 2z/c
•
•
•
•
since Alice sees the light travel at the
universal speed c
So z = c∆t/2 = √{l2 +(v∆t/2)2 }
c2∆t2/4 = l2 + v2∆t2/4
l2 = (c2-v2) ∆t2/4
∆to = 2l/c = (2/c) √(c2-v2) ∆t/2 = √{1-v2/c2} ∆t
Section 27.3
Moving Clock, Alice’s Time
• Alice observes the time for one tick of Ted’s clock:
Dt =
Dt0
1- v
2
c2
which is different (slower) than the time observed by
Ted
This effect is called time dilation
• For typical terrestrial speeds, the difference between
Δt and Δt0 is negligible
Section 27.3
Time Dilation
• When the speed is
small compared to c,
the factor 1  v 2 c 2 is
very close to 1
• For v ~ c, Newtonian
approximations may be
used in many terrestrial
cases
• As v approaches c, the
ratio approaches 1 and
the contents of the
square root approach
zero
1/sqrt  infinity
Section 27.3
It actually works!
• Experiments have shown that the time dilation
predicted by special relativity is correct
• The result applies to all clocks, even biological ones
• At Fermilab, pi mesons traveling close to c can
travel for several kilometers in the lab, even though
they typically decay in 26 ns (in their own “rest
frame”) and you would naively expect them to travel
only a distance d = c x 26E-9s where c = 3E8m/s.
d = 7.8 m
• There are many more experimental examples, and
they always work.
Section 27.3
Proper Time
• The time interval Δt0 is measured by the observer at
rest relative to the clock
• This quantity is called the proper time
• The time interval measured by a moving observer is
always longer than the proper time in the rest system
• The proper time is always the shortest possible time
that can be measured for a process, by any observer
Section 27.3
Twin Paradox
• An astronaut, Ted, visits a
nearby star, Sirius, and
returns to Earth
• Sirius is 8.6 light-years
from Earth
• Ted is traveling at 0.90 c
• Alice, Ted’s twin, stays on
Earth and monitors Ted’s
trip
Section 27.3
Twin Paradox, Times
• Alice measures the trip as taking 19 years
17.2 ly
Dt =
= 19 years
0.90 c
• Ted’s body measures the proper time of 8.3 years
Dt0 = Dt 1 - v c = (19 years ) 1 - (0.90c ) / c 2 = 8.3 years
2
2
2
• Alice concludes that Ted will be younger than she is
• By SYMMETRY, Ted naively calculates the Earth (and
Alice) move away from him at 0.90 c
• Ted concludes Alice will age 8.3 years while he ages 19
years
• Ted (incorrectly) concludes that Alice will be younger than
he is
Section 27.3
Twin Paradox, Resolution
• It appears that time dilation leads to contradictory
results
• Alice’s analysis is correct
• She remains in an inertial frame and so can apply the
results of special relativity
• Ted is incorrect
• He hugely accelerates to start his trip, and when he
turns around at Sirius, and to slow down at the end of
his return trip.
• Special relativity cannot be applied during this time
spent in an accelerating frame of reference
Section 27.3
Time Dilation and GPS
• Each GPS satellite
contains a very
accurate clock
• The satellite clocks are
moving in orbit, so they
experience time dilation
• They run slow by about
7µs per day
• To accurately determine
a position, the effect of
time dilation must be
accounted for
Section 27.3
Time Dilation and GPS
• GPS works by
determining distance
using time intervals
• The satellite clocks are
moving in orbit, so they
experience time dilation
• They run slow by about
7µs per day
• c = 1 foot/ns, to a close
approx.
• 7 µs  7,000 feet
• Need Special Relativity
for accuracy.
Section 27.3
Time Dilation and GPS
• But there’s more:
General Relativity says
clocks slow down when
they are down a
gravitational potential
well (such as at the
Earth’s surface).
• So there’s an extra
correction for time
differences, from GR in
addition to SR
Section 27.3
Gravitational redshift (blueshift)
Orbital altitude 20,183 km
Clock runs fast by 45.7 s per day
Time dilation
Satellite velocity 3.874 km/s
Clock runs slow by 7.1 s per day
Net secular effect (satellite clock runs fast)
Clock runs fast by 38.6 s per day
Basic science can pay off (hugely) in totally
unpredictable ways
Simultaneity
• Two events are simultaneous if they occur at the same
time
• Ted is standing in the middle of his railroad car
• He moves at a speed v relative to Alice
• Two lightning bolts strike the ends of the car and leave
burn marks on the ground which indicate the location of
Section 27.4
the two ends of the car where the bolts strike
Simultaneity, cont.
• Did the two lightning bolts strike simultaneously?
• According to Alice
• She is midway between the burn marks
• The light pulses reach her at the same time
• She sees the bolts as simultaneous
• According to Ted
• The light pulse from at B struck before the bolt at A
•
•
Since he is moving toward B and away from A
So by the time the light arrives, he’s farther from A than from B
• The two bolts are not simultaneous in Ted’s view
Section 27.4
(Lorentz) Length Contraction
• Alice marks two points
on the ground and
measures length L0
between them
• Ted travels in the xdirection at constant
velocity v and reads his
clock as he passes
point A and point B
• This is the proper time
interval of the motion
Section 27.5
Length Contraction, cont.
• Distance measured by Alice = L0 = v Δt
• Ground distance measured by Ted = L = v Δt0
• Since Δt ≠ Δt0, L ≠ L0
L = L0 1 - v 2 / c 2
• The difference is due to time dilation and
• The length measured by Ted is shorter than the
actual length on the ground, observed by Alice in the
“ground” reference frame. To put it another way, if
you see an object moving towards (or away from)
you, it will appear shorter than it actually is in its own
reference frame.
Section 27.5
Lorentz Length Contraction Equation
• Length contraction is
described by
L
v2
= 1- 2
L0
c
• When the speed is very
small, the contraction
factor is very close to 1
• This is the case for
typical terrestrial
speeds
If v/c=0.1 , (v/c)2 = 0.01 and sqrt(0.99) = 0.995
Section 27.5
Lorentz Length Contraction Equation
L
v2
= 1- 2
L0
c
• Let’s revisit the decaying pi mesons. They go a lot
farther in the lab than you’d predict from their restframe proper decay lifetime.
• From the point of view of the pions, the kilometerslong lab has Lorentz-contracted to a few meters
length! So the pion has no problem living long
enough to go that short distance.
• Both viewpoints (time dilation, length contraction) are
correct, and are in essence equivalent.
Section 27.5
Addition of Velocities
• Ted is traveling on a
railroad car at constant
speed vTA with respect to
Alice
• He throws an object with a
speed relative to himself
of vOT
• What is the velocity vOA of
the ball relative to Alice?
• Alice is at rest on the
ground
Section 27.6
Newton’s Addition of Velocities
• Newton and Galileo would predict that vOA = vOT + vTA
• The velocity of the object relative to Alice = the velocity
of the object relative to Ted + the velocity of Ted relative
to Alice
• This result is inconsistent with special relativity when
the speeds are very high
• For example, if the object’s speed relative to Ted is
0.9 c and the railroad car is moving at 0.9 c, then the
object would be traveling at 1.8 c relative to Alice
• Newton’s theory gives a speed faster than the speed of
light, which is forbidden by relativity
Section 27.6
Relativistic Addition of Velocities
• The result of special relativity for the addition of velocities
is
vOA
vOT + vTA
=
vOT vTA
1+
2
c
• Denominator is close to 1, unless v’s are large
• Special cases: c + c = c, relativistically
• More generally, c + anything = c
• Check it out at home – it’s fun, and instructive
Section 27.6
LECTURE QUIZ
• Add two velocities together,
• Each velocity = c/2
• Their relativistic sum is:
•
•
•
•
•
A)
B)
C)
D)
E)
vOA
vOT + vTA
=
vOT vTA
1+
2
c
0.65 c
0.75 c
0.80 c
0.85c
0.995 c
Section 27.6
Relativistic Addition of Velocities, cont.
• When the velocities vOT and vTA are much less than
the speed of light, you get nearly the same result as
the Newtonian equation
• For speeds less than approximately 10% the speed
of light, the Newtonian velocity equation works well
• For the example, with each speed being 0.9 c, the
relativistic result is 0.994 c
• Compared to 1.8 c from Newton’s prediction
• Experiments with particles moving at very high
speeds show that the relativistic result is correct
• For example when fast particles decay into photons,
the photons do NOT get a boost, they still travel at c
Section 27.6
Momentum
• According to Newton’s mechanics, a particle of
mass m0 moving with speed v has a momentum
given by
p = m0 v
• Conservation of momentum is one of the
fundamental conservation rules in physics and is
believed to be satisfied by all the laws of physics,
including the theory of special relativity
• The momentum of a single particle can also be
written as
Dx
p = m0
Dt
Section 27.7
Relativistic Momentum
• Einstein showed that you should use the proper time
to calculate momentum
• The result from special relativity is
p = m0
m0v
Dx
Dx
= m0
=
Dt0
Dt 1 - v 2 c 2
1- v 2 c 2
• This is the correct expression for momentum and
applies even for a particle moving at high speed,
close to the speed of light
Section 27.7
Newton’s vs. Relativistic Momentum
• As v approaches the
speed of light, the
relativistic result is very
different than Newton’s
• There is no limit to how
large the momentum
can be
• However, even when
the momentum is very
large, the particle’s
speed never quite
reaches the speed of
light
Section 27.7
Mass - Energy
• The total energy of a particle is also governed by the
•
•
•
•
“sqrt” factor, and is
E = m/sqrt(1-v2/c2)
This E includes both Kinetic Energy and Rest Mass
Energy
In the low-velocity approximation, v/c << 1
E = mc2 + ½ mv2
• In general, E grows without bound (just like
momentum) while v approaches but never quite
reaches c
• v=c gives infinite E --- ain’t gonna happen!
Section 27.8
• v>c gives imaginary number --- nope!
Mass
• Newton’s second law gives mass, m0, as the
constant of proportionality that relates acceleration
and force
• This works well as long as the object’s speed is
small compared with the speed of light
• At high speeds, though, Newton’s second law
breaks down
Section 27.8
Relativistic Mass
• When the postulates of special relativity are applied
to Newton’s second law, the mass needs to be
replaced with a relativistic factor
m0 ®
m0
(1 - v
2
c
2
)
3
2
• At low speeds, the relativistic term approaches m0
and the two acceleration equations will be the same
• When v ≈ c, the acceleration is very small even when
the force is very large
Section 27.8
Rest Mass
• When the speed of the mass is close to the speed of
light, the particle responds to a force as if it had a
mass larger than m0
• The same result happens with momentum where at
high speeds the particle responds to impulses and
forces as if its mass were larger than m0
• Rest mass is denoted by m0
• This is the mass measured by an observer who is
moving very slowly relative to the particle
• The best way to describe the mass of a particle is
through its rest mass
Section 27.8
Mass and Energy
• Relativistic effects need to be taken into account
when dealing with energy at high speeds
• From special relativity and work-energy,
KE =
m0c 2
1- v 2 c 2
- m0c 2
• For v << c, this gives KE ≈ ½ m0 v2 which is the
expression for kinetic energy from Newton’s results
Section 27.9
Kinetic Energy and Speed
• For small velocities, KE
is given by Newton’s
results
• As v approaches c, the
relativistic result has a
different behavior than
does Newton
• Although the KE can be
made very large, the
particle’s speed never
quite reaches the speed
of light
Section 27.9
Rest and Total Energies
• The kinetic energy can also be thought of as the
difference between the final and initial energies of
the particle
• The initial energy, m0c2, is a constant called the rest
energy of the particle
• A particle will have this much energy even when it is at
rest
• The total energy of the particle is the sum of the
kinetic energies and the rest energy
Section 27.9
Mass as Energy
• The rest energy equation implies that mass is a
form of energy
• It is possible to convert an amount of energy (m0c2)
into a particle of mass m0
• It is possible to convert a particle of mass m0 into an
amount of energy (m0c2)
• The principle of conservation of energy must be
extended to include this type of energy
Section 27.9
Speed of Light as a Speed Limit
• Several results of special relativity suggest that
speeds greater than the speed of light are not
possible
2
2
• The factor 1  v c that appears in time dilation and
length contraction is imaginary if v > c
• The relativistic momentum of a particle becomes
infinite as v → c
• This suggests that an infinite force or impulse is
needed for a particle to reach the speed of light
Section 27.9
Speed Limit, cont.
• The total energy of a particle becomes infinite as v
→c
• This suggests that an infinite amount of mechanical
work is required to accelerate a particle to the speed
of light
• The idea that c is a “speed limit” is not one of the
postulates of special relativity, it is a consequence
• Combining the two postulates of special relativity
leads to the conclusion that it is not possible for a
particle to travel faster than the speed of light
Section 27.9
Mass-Energy Conversions
• Conversion of mass into energy is important in
nuclear reactions, but also occurs in other cases
• A chemical reaction occurs when a hydrogen atom is
dissociated
• The mass of a hydrogen atom must be less than the
sum of the masses of an electron and proton
• The energy is lower by 13.6 eV when bound in the
atom
• Mass is not conserved when a hydrogen atom
dissociates
• Δm0 = 2.4 x 10-35 kg
• This is much less than the mass of a proton and can
be ignored
Section 27.9
Conservation Principles
• Conservation of mass
• Mass is a conserved quantity in Newton’s mechanics
•
The total mass of a closed system cannot change
• Special relativity indicates that mass is not conserved
•
The principle of conservation of energy must be extended to
include mass
• Momentum is conserved in collisions
• Use the relativistic expression for momentum
• Electric charge is conserved
• It is possible to create or annihilate charges as long as
the total charge does not change
Section 27.9
General Relativity
• A noninertial reference frame is one that has a
nonzero acceleration
• Physics in noninertial frames is describe by general
relativity
• General relativity is based on a postulate known as
the equivalence principle
• The equivalence principle states the effects of a
uniform gravitational field are identical to motion with
constant acceleration
Section 27.10
Equivalence Example
• Ted stands in an elevator
at rest (A)
• He feels the normal force
exerted by the floor on
his feet
• He concludes that he is in
a gravitational field
• The elevator is not in a
gravitational field and has
an acceleration of g (B)
• Since there is an
acceleration, Ted feels
the same force on his
feet
Section 27.10
Equivalence Principle, cont.
• According to the equivalence principle, there is no
way for Ted to tell the difference between the effects
of the gravitational field and the accelerated motion
• The equivalence principle has the following
consequences
• Inertial mass and gravitational mass are equivalent
• Light can be deflected by gravity
Section 27.10
Light and Gravity
• The light beam travels through the elevator while the elevator
is in distant space
• When a = 0, Ted sees the light beam travel in a straight line (A)
• When a ≠ 0, the light beam travels in a curved line relative to
the elevator (B)
• In a gravitational field, the light beam also curves (C)
Section 27.10
Deflection of Light by Sun
• The gravitational field of the Sun should deflect light
from a star
• Easiest to see during a solar eclipse
• Experiments in 1919 verified light passing near the
Sun during an eclipse was deflected by the predicted
amount
Section 27.10
Black Holes
• Black holes contain so
much mass that light is
not able to escape from
their gravitational
attraction
• A black hole can be
“seen” by its effect on the
motion of nearby objects
• Stars near a black hole
move in curved
trajectories and so the
mass and location of the
black hole can be
determined
Section 27.10
Gravitational Lensing
• If the black hole is between the star and the Earth, light
from the star can pass by either side of the black hole and
still be bent by gravity and reach the Earth
• The black hole acts as a gravitational lens
• Light from a single star can produce multiple images
• Analysis of the images can be used to deduce the mass
Section 27.10
of the black hole
Relativity and Electromagnetism
• Alice is at rest with the charged line and the point
charge
• Ted sees the line of charge and the point charge in
motion
• The moving charged line acts as a current
Section 27.11
Relativity and EM, cont.
• Ted says that there is an electric force and a
magnetic force on the particle
• Alice says there is only an electric force
• Both are correct
• They will agree on the total force acting on the
particle
• The larger electric force seen by Ted due to his motion
is canceled by the magnetic force produced
• Maxwell’s equations were already consistent with
special relativity
Section 27.11
Importance of Relativity
• The relation between mass and energy and the
possibility that mass can be converted to energy
(and energy to mass) mean that mass is not
conserved
• Instead we have a more general view of energy and its
conservation
• The three conservation principles in physics are
•
•
•
Conservation of energy
Conservation of momentum
Conservation of charge
• It is believed that all the laws of physics must obey
these three conservation principles
Section 27.12
Importance of Relativity, cont.
• The rest energy of a particle is huge
• This has important consequences for the amount of
energy available in processes such as nuclear
reactions
• Relativity changes our notion of space and time
• Time and position are two primary quantities in physics
but it is not possible to give precise definitions of such
quantities
• Our everyday intuition breaks down when applied to
special relativity
Section 27.12
Importance of Relativity, final
• Relativity plays a key role in understanding how the
universe was formed and how it is evolving
• Black holes can’t be understood without relativity
• Relativity shows that Newton’s mechanics is not an
exact description of the physical world
• Instead, Newton’s laws are only an approximation that
works very well in some cases, but not in others
• We shouldn’t discard Newton’s mechanics, but
understand its limits
Section 27.12