#### Transcript Slide 1

```Special Relativity
Topics
Frames of reference
Einstein’s development of special relativity
Proper time and length
Relativistic time
Death of simultaneity
Relativistic length
Relativistic momentum
Matter-energy equivalence
Can you break the speed limit?
Muons, twins, and antimatter
Einstein the man
Motivation
Discover the origins of the Special Theory of Relativity.
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A short mathematical exploration
Before we start talking about Special Relativity, I want to get the hardest bit of math out
of the way so it won’t be a distraction later on.
Say hello to gamma:
It makes sense to track this function, using “v/c” as the variable.
v/c = β
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A short mathematical exploration
β (v/c)
0.1
0.3
0.5
0.7
0.8
0.9
0.95
0.97
0.99
0.999
γ
1.005
1.048
1.155
1.400
1.667
2.294
3.203
4.113
7.089
22.366
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That was the hard stuff!
If you’re still worried about the math, don’t be.
You’ve already passed the hard stuff—and
really, it wasn’t that bad, was it?
The challenge with Einstein is the mind-bending
change in perspective it requires.
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Understanding reference frames
Recall our definitions of reference frames, from the first
week of class.
An inertial reference frame is one in which Newton’s
First Law applies:
LAW #1: An object at rest stays at rest, unless acted upon
by an external, unbalanced force.
Similarly, an object in motion continues in motion, unless
acted upon by an external, unbalanced force.
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Understanding reference frames
Let us examine more carefully the idea of inertial, and
noninertial reference frames.
We will time travel to 1960, Ontario Canada, to the
studios of Drs. Hume and Ivey for guidance.
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Einstein begins
Einstein began his ruminations with the
observation that, whether they were
correct or not, Newton’s Laws did not
depend upon your location in the
Universe, orientation, or even velocity.
Said differently, one cannot detect
motion.
Newton’s Laws were equally true in every inertial reference frame.
The same could also be said for Maxwell’s equations for light, or every law of
physics, for that matter!
Example: Newton’s second law is not:
F=ma(1+V/Vz + X/Xsc)
Vz=vertical speed out of disk of galaxy;
Xsc=distance from the center of the Virgo Supercluster.
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Einstein begins
Einstein also noted that everyone seemed to measure the same value for the
speed of light. He explained this by simply adopting it as an assumption.
This is how Einstein interpreted Michelson’s inability to detect the
lumeniferous aether: He said that light always travelled at the same speed, and
that ultimately, there was no need to invoke a mysterious aether!
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Einstein’s postulates
The assumptions of relativity, therefore, are:
1. The laws of physics work for all observers.
(or, absolute motion cannot be detected)
2. The speed of light (c) is the same for all observers.
A modern perspective is to say that the constancy of “c” is a law
of physics, thus reducing the number of assumptions from two to
one:
The laws of physics work the same for everyone.
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Einstein’s postulates
1. The laws of physics work for all observers.
(or, absolute motion cannot be detected)
2. The speed of light (c) is the same for all observers.
On first glance there is nothing here that seems particularly difficult to
accept, but weird things await us.
Much of the weirdness from Einstein’s theory comes from the finite
speed of light.
Indeed, Einstein claims to have been haunted for years by the question,
“What do you see if you travel with a photon?”
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Einstein’s postulates
1. The laws of physics work for all observers.
(or, absolute motion cannot be detected)
2. The speed of light (c) is the same for all observers.
Consider two observers. One is stationary, the other is moving at 0.5c.
A pulse of light passes them…what does each see?
Both observers measure the pulse of light traveling at “c”.
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Annus mirabilis: 1905
On a Heuristic Viewpoint Concerning the Production and
Transformation of Light
Explaining the photoelectric effect via the particle nature of light.
On the Motion of Small Particles Suspended in a Stationary
Liquid, as Required by the Molecular Kinetic Theory of Heat
A study of Brownian motion, essentially solidifying the concept of atoms. Previously,
atoms had been thought of as useful conceptual tools, that might not have
corresponding physical reality.
On the Electrodynamics of Moving Bodies
Special relativity.
Does the Inertia of a Body Depend Upon Its Energy Content?
Simply stated, the formulation of E=mc2.
The 1922 telegram by the Swedish Secretary of Sciences clearly stated Einstein’s Nobel
Prize was for his accomplishments, but that the theory of relativity was not one of them!
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Being proper
Before we proceed, it is important to become
familiar with the relativistic use of the adjective
“proper.”
Proper Distance, Proper Length
A distance or length is said to be “proper” if (and only if)
it is measured in the frame of reference in which the object being measured
is not moving.
Proper Time Interval
A time interval is said to be “proper” if (and only if) the time interval is
measured at the same place (in some reference frame).
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Examples of properness
Suppose road-side Rhonda is standing at a
roadside bus stop, accompanied by her
handi-dog.
Road-side Rhonda can measure the proper
length of a parked car, or how long it takes
for her dog to fall asleep (the proper time).
Meanwhile, Bertha is on a bus, and is about
to have lunch.
Bus-rider Bertha can measure the proper
length of the moving bus, and how long it
takes for her to eat her lunch (the proper
time.
Examples of properness
But…
moving bus’s proper length, nor the
proper time for bus-rider Bertha to eat
her lunch.
Similarly, bus-rider Bertha cannot
measure the parked car’s proper length,
nor the proper time for road-side
Rhonda’s dog to nap.
This raises the question: what happens
when you measure things in a reference
frame moving with respect to yours?
The light clock
Let us follow Einstein in another thought experiment.
two mirrors, separated by a distance (D). How long (∆t′) does it take for
a photon to make a round trip?
Use the Rate Equation
v = ∆x/∆t
→ c = 2D/∆t′
→ ∆t′ = 2D/c
Notes:
1. The prime symbol is on ∆t′ because it is traditional.
2. ∆t′ is measured from the same bottom point, so it is a proper time.
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The light clock
Suppose, now, that the light clock is moving to the right at velocity (v)
with respect to a stationary observer. The clock moves from A to B to C.
What does the stationary observer, sitting by the clock dials, measure as
the clock-box zips to the right?
In the time (∆t) it takes for the beam to take the round trip, it travels to
the right as it moves up and down. The beam does all this, of course, at
the speed of light.
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The light clock
In half of the time for the entire round trip—(∆t/2)—the photon travels
from the bottom mirror to the top mirror. In this time, it travels to the
right, from position “A” to position “B”.
The horizontal distance is ∆x = v(∆t/2).
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Relativistic effect #1: time dilation
During this time (∆t/2), the photon travels (at c) along the long side of the triangle,
covering a distance c(∆t/2).
Since:
L2 = Δx2 + Δy2
Therefore:
(cΔt/2)2 = (vΔt/2)2 + D2
Solving for Δt…
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Relativistic effect #1: time dilation
Solving for Δt :
Since:
THEN….
and:
∆t = γ∆t′
- or -
∆t′ = 2D/c (from before)
∆t′ = ∆t/γ
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Relativistic effect #1: time dilation
∆t = γ∆t′
- or -
∆t′ = ∆t/γ
What does this mean?
The proper time interval measured by a person moving with the clock is ∆t′.
A second person, watching the clock zip by, experiences the time interval γ∆t′.
If the person with the clock experiences 10 seconds, the second person experiences
more than 10 seconds!
Time, in a frame of reference that is moving with respect
to yours, progresses at a slower rate!
Examples:
The ISS’s speed ≈ 7706 m/s (0.0000257c).
Astronauts on the ISS experience time at 0.99999999967× the rate you do.
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Relativistic effect #1: time dilation
What about speeds near the speed of light?
Consider a round trip to α Centauri (8.6 LY). How long does it take?
v/c
0.1
0.5
0.8
0.95
0.99
0.999
Δt/Δt′
0.995
0.866
0.600
0.312
0.141
0.045
Time (Earth)
86.0 y
17.2 y
10.8 y
9.1 y
8.7 y
8.6 y
Time (Astronaut)
85.7 y
14.9 y
6.5 y
2.8 y
1.2 y
0.38 y = 4.6 months
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Relativistic effect #2: relativistic simultaneity
What does it mean for two events to be “simultaneous”?
It means the two events happened at the same time.
Suppose a double-lightning bolt hits the ground in two places.
To an observer at the midpoint between the strike points, the flashes from both events
will arrive at the same time.
The observer declares that the two lightning strikes occurred simultaneously.
Now, let’s make the situation a little more complicated.
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Relativistic effect #2: relativistic simultaneity
Suppose a double-lightning bolt hits the front and back ends of a moving train car, and
as a result the wheels instantly spark and scorch the tracks.
An observer on the ground, at the midpoint between the scorch marks, sees the light
from both flashes at the same time.
This observer declares that the two lightning strikes occurred simultaneously.
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Relativistic effect #2: relativistic simultaneity
But…to the observer on the train, the situation is different. The train is moving to the
right, so the passenger moves towards the photons in front of him, at the same time
those photons are moving towards him.
Flash!
Photons from right
Photons from left
The passenger says the lightning struck the front of the train before it strikes the back.
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Relativistic effect #2: relativistic simultaneity
L
What the trainside observer sees
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What the train passenger sees
Relativistic effect #2: relativistic simultaneity
How much of an effect is this?
L
If the two lightning strikes are separated by a distance L, and the train has a velocity of
“v”, the passenger will measure a time (∆t) between arrival times of the pulses, given
by:
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Relativistic effect #2: relativistic simultaneity
Who is correct? Were the events really simultaneous, or did one really happen before
the other?
They both are! Remember, in relativity, both reference frames are equally valid.
Consider two spaceships passing each other in the opaque and mysterious Mutaran
Nebula. One ship is struck by two asteroid chunks. Whether the strikes were
simultaneous or not, depends upon your frame of reference.
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Relativistic effect #3: relativistic length
The endpoints of a train station are marked by points at x1 and x2.
The proper length of the train station Lo, measured by a dog by the station, is given by
Lo = x2 - x1
A person on a train moves by at velocity (v).
The dog measures the time (Δt ) for this person to travel from x1 to x2, and so:
Lo = v(Δt).
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Relativistic effects on length
From the perspective of the train passenger, the station is sliding by with velocity (v).
The moving passenger measures the station to move past in the time interval Δt′.
This means that the length of the station from the perspective of the passenger is:
L′ = v(Δt′).
Since (from before): Lo = v(Δt)
then…
Lo/L′ = v(Δt)/v(Δt′) = Δt/Δt′
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Relativistic Effects on Length
From the previous page:
Lo/L′ = v(Δt)/v(Δt′) = Δt/Δt′
But from our observations on time dilation, where moving reference frames have been
shown to have a slower rate of time-flow:
∆t = γ∆t′ - so Lo/L′ = γ
-or -
L′ =Lo/γ
This is called Lorentz contraction.
Examples: Lo = 10 m
v/c
0.1
0.5
0.8
0.95
0.99
0.999
L′
9.95 m
8.66 m
6.00 m
3.12 m
1.41 m
0.45 m
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Relativistic Effects on Length
What does this mean? Does the object REALLY become shortened?
No—there are no forces acting upon
the object to shorten it.
Furthermore, consider a massive
neutron star—if you shifted into a
frame of reference moving rapidly,
you would see the neutron star
become compressed in one dimension.
If you insisted the neutron star really were
compressed, you could force it to become a
black hole! This is obviously not possible.
L′ =Lo/γ
This business of Lorentz contraction, ultimately comes from a disagreement in where
the front and back ends of a travelling object lie, in a simultaneous moment of time, that
you must use when you measure its length.
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THE Equation
The most famous result of relativistic physics is:
E=mc2
This means that matter and energy are both aspects of the same thing: mass-energy.
The energy from this equation is called its “rest energy.”
Mass-energy can be switched back and forth in form, like ice to water to steam. This
powers stars and nuclear power plants.
We saw hints that matter and energy were interchangeable when we saw that both
entities have wavelengths, and were subject to particle-wave duality.
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Momentum and Energy
For Newton
For Einstein:
Momentum (p)
mv
γmv
Energy (E)
½mv2
γmc2
This means that as objects move faster and faster, the amount of
momentum (or energy) needed to accelerate it to even greater
speeds is harder and harder.
To move a particle to the speed of light would require infinite
energy and infinite momentum. You can’t do this.
The speed of light is a cosmic limit.
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Just to round things out… What happens when you add two velocities?
Newton
v1 + v2
Einstein
Suppose a rocket was moving at a velocity V, and it shot a missile
forwards at the same velocity. How fast would the missile move past
you?
V
Newton
Einstein
0.1 c
0.5 c
0.8 c
0.95 c
0.99 c
0.999 c
0.2 c
1.0 c
1.6 c
1.9 c
1.98 c
1.998 c
0.198 c
0.800 c
0.976 c
0.999 c
0.9999 c
0.999999 c
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Einstein’s Interpretation of the Speed Limit
Einstein: Matter and energy is interchangeable. Furthermore, matter and photons all
travel through space-time at the speed of light.
The rate you travel through space-time is divvied into your spatial velocity, and your
temporal velocity.
A stationary object (zero spatial velocity) has all its motion through space-time
expressed in the time component. It travels through time at 1 sec/sec.
A moving object has some of its space-time motion in its spatial velocity, and has less
to dedicate to its time component. It travels through time at less than 1 sec/sec.
A photon has all its space-time motion concentrated in its spatial velocity, and has no
time to dedicate to a time component. Photons do not experience time.
Nothing can exceed a spatial velocity of “c”, because that would be more speed than
anything/everything has.
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Antimatter
Recall the relativistic equations for momentum and energy?
p = γmv
E = γmc2
It is easy to rearrange these, to write that the total energy is a combination
of its momentum (p) and its rest energy (mc2).
E2 = (pc)2 + (mc2)2
In 1928, Paul Dirac noted that when there is no motion (p=0), this
equation can be solved two ways:
E = +mc2
E2 = (mc2)2
E = -mc2
Since energy is a positive value, and c2 must be positive, the second
solution predicted “negative mass,” or antimatter.
The anti-electron (positron) was discovered by Carl Anderson, in 1932.
Antihydrogen and antihelium-3 have been created artificially.
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Superluminal Motion
Amazingly, in active galactic nuclei jets, blobs of matter
have been observed coming out at velocities greater than the
speed of light! Velocities up to about 10c have been seen!
What is really going on…
The blobs are coming almost directly towards us. So in our
frame, the blob is almost keeping up with the light it emits.
The light from every episode of the blob’s entire life reaches
us at almost the same time.
As the blob moves sideways, we receive all of its sidewaysmoving images at nearly the same time!
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In transparent matter (water, air, gas), photons travel at the speed Vp:
Vp = c/n
where n = the index of refraction.
Air
1.0003
H2O
1.33
Glass
~1.6
High-energy electrons from reactors pass through the water used to
shield the reactor core. Many travel at speeds higher than Vp. As they
disturb the electric fields of the H2O molecules, they emit a forwardfacing shock cone of photons (mostly UV) that slows the particle to
sub-light speeds.
Pretty cool.
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Calculations: Relativistic Effects
An object is travelling at 1×108 m/s with respect to you.
Calculate β and γ.
If this object is 10 m long, how long does it look to you?
An object is travelling at 2.5×108 m/s with respect to you.
Calculate β and γ.
In the time this object experiences 60 seconds, how much time do
you experience?
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Calculations: Spaceship Probes
A spaceship travelling at 1×108 m/s (with respect to you) fires a
probe that leaves the ship at a launch speed of 2.5×108 m/s. How
fast does the probe travel through space, as viewed by you?
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A spaceship travelling at an extremely high speed (v) shoots out a
beam of light. How fast does the beam of light travel, as viewed
by a person on the ground?
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Calculations: Photons Fly
Note that if you watch two photons travelling in opposite
directions, as they pass each other, they have a relative velocity of
2c.
But what do the photons see? (i.e., v1 = v2 = c)
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Calculation: Muons
Cosmic particles strike the troposphere (h=7000 m), forming π-mesons,
which then decay into high-energy muons (v=0.998c).
Muons decay in about 2×10-6 sec. How far can they travel?
Non-relativistic calculation
D = v×t = (0.998) × (3×108m/s) × (2×10-6 sec) ≈ 600 m.
But they are easily detected at the Earth’s surface.
How do they reach the surface before they decay???
Relativistic calculation
D = v×(γt) = (0.998) × (3×108m/s) × (15.8) × (2×10-6 sec) ≈ 9500 m.
What do muons “see”? They see a Lorentz-contracted atmosphere:
= 15.8
Datm′ = Datm/γ = 7000 m/15.8 = 440 m
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Spacestation-Sally and Rocket-Rhonda are both 20 years old.
Rhonda flies at 0.8c to a point 8 LY away. She reaches the point, then immediately
returns, flying again at 0.8c.
Spacestation-Sally’s Perspective
Rocket-Rhonda flew away for 10 years (t = 8 LY/0.8c), and then back for 10 years—a
total of 20 years. Spacestation-Sally is therefore 40 years old when Rocket-Rhonda
returns.
But because of time dilation, Rocket-Rhonda aged only 20/γ years (12 y) during her
high velocity trip. When the two twins reunite, one’s age is 40, and the other’s is 32.
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Rocket-Rhonda’s Perspective
Traveling at 0.8c, the distance to the turn-around point is only 4.8 LY—the 8 LY is
Lorentz contracted (i.e., 8/γ LY). It takes only 6 years (4.8 LY/0.8c) for each leg of her
trip—hence she is 20+6+6=32 years old upon her return. This agrees with SpacestationSally’s perspective.
But….in Rocket-Rhonda’s frame, Spacestation-Sally is the one who is moving.
Indeed, during her 12-year trip, Rocket-Rhonda observes Spacestation-Sally to age only
7.2 years (12/γ years).
Upon their reunion, 32-year-old Rocket-Rhonda expects to find her sister to be only
20+7.2 = 27.2 years old. Imagine her surprise, when she meets a woman who is 40!
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