Introduction

Special relativity is not hard to understand if you accept the idea that the measurements of  physical quantities
(speed, length, energy, time, momentum ..)
depends on the frame of reference it is measured in. The Math is very easy, you just need Algebra.
The difficult part is the conceptual part. It took the genius of Einstein to realize that speed
is not the only physical quantity to be relative. Time, lengths, energy.. also depends on the Frame of reference
it is measured in.  

A reference frame  includes a x,y,z coordinate system and a clock.
In the frame of reference of the lab (Earth based) the shuttle has the same x-coordinate and  there is no x-component of the speed. only a y-component.
In the moving frame of reference the x-coordinate of the shuttle decreases with time (they get closer) and the x-component of the speed is not zero.
(it is equal to the speed of the plane relative to the ground, the shuttle gets closer).
The big idea of SR is that not only speed and location are relative and depend on the frame of reference but also time, energy, mass, length ..
source image : Cutnell and Johnson, publisher Wiley

First let's review Galilean relativity

PART 1 : GALILEAN RELATIVITY

1) Read and fill the blanks

What is your speed right now ?
Maybe you said s0=0 . In the frame of reference of the lab this is true. But the Earth is spinning about its axis.
So in the frame of reference attached to the center of the EArth we also have a rotational speed.
Evaluate this speed. Suppose the the radius of the EArth is about 4,000 miles. Take Pi to be 3 .
Find the speed in miles per hour s1 = _______ .
So we are going around the axis of the EArth at about the same speed as an airplane. And we don't feel it !
(this is because the force of gravity largely balance the centrifugal force that we would otherwise feel
in the moving frame around the axis. Like the force you feel on a merry go round)

But Wait. The EArth is also moving around the Sun. So what is our speed in the frame of reference
attached to the Sun ? (don't take in account the rotational speed computed earlier).
Supposed it takes 1 year for the Earth to go around the Sun and the average distance EArth-Sun
is 93 million miles. Round to 100 million miles.  take 1 year = 3 10⁷ seconds
Find in miles per second. S2 = _________. That's really fast.

What about the solar system moving around the center the the milky way ?
If we attach the frame of reference in the center, the solar system goes at a speed of about 200km/s
or s3 = _______ miles per second.

But recently (80s) the speed of our galaxy in the frame of reference of other galaxies was computed
to be 1 million miles per hour = s4 = ___________ miles per second. galaxies move around relative
to each other. ours moves toward Andromeda and will eventually collapse with it.

So what is our speed now? is it so ?  s1 ? s2 ? s3? s4 ? Which speed is the right one ?
All of them, it depends of the frame of reference. Speed is relative and that was known
since Galileo

2) Le'ts take an example.
You are in a plane (600 miles per hour) and you hold a ball. In the frame of reference of the plane,
what is the speed of the plane ? ___________ (in the moving frame)
But in the frame of reference of the Earth, the speed is _________.

3) Like wise, Say you are firing a bullet from a gun while you are walking. The ball (relative to the gun) has a speed
of 1000 ft/s. You walk at speed of 3 ft/s . In the frame of reference attached to you (called the moving
frame ) the speed of the bullet is _________ but in the frame of reference of the lab the speed
of the bullet is ______________. The measurements depend on the frame of reference it is measured in.

4) You are in a truck pitching a baseball. You always pitch at 60km/h relative to the truck.

(source = Paul Hewitt, conceptual PhysicS)
Your friend is catching the ball. Your friend stays at rest. If the truck is not  moving,
in the frame of reference attached to your friend (to the ground) , the speed of the ball is _________.
In the frame of reference of the truck the speed of the ball is ________

But if the truck is moving toward you, at a speed of 40km/s, the speed of the ball is now _________
in the frame of reference of your friend (it will get to him/her faster) but the speed is
still ________ in the moving frame of reference.( the truck)/

If the truck is  moving away from your friend at a speed of 60km/h , the speed of the ball is now ______
in the frame of reference of your friend (takes longer to reach you, more distance to cover)
but still _________ in the moving frame of reference.

So speed in frame of reference at rest = speed of moving frame of reference + speed of ball

5) So the speed of an object is relative to the frame of reference and so is the trajectory.
Say you friend, still at rest, is watching you walking at a constant speed and throwing a ball straight up.
In the moving frame of reference( attached to you) what is the trajectory of the ball ?
Is is a curve or a line ? (what do you see).
In the frame of reference of the lab, your friend does not " see a line" but a ___________.
So the trajectory is relative to the frame of reference. Some times, it is convenient to
change the frame of reference to make the computations.


6) 2 cars are moving toward each other at 50mph (in the frame of reference attached to the ground).
In the moving frame of reference of  either car, the speed of the other car is __________
If both car are moving on the same line, at the same speed, one behind the other, then
the relative is __________. From the moving frame perspective the other car is not
getting closer.


7)  to demonstrate 5 use a moving gun that fire (spring system) a ball up.

In the moving frame, the ball goes up and down along a line. It is what would
describe an observer moving with the car.
But in the frame of reference of the lab, the ball goes along a _____________.
So in the moving frame, the ball's motion  has only 1 component (y-component)
but in the frame of reference of the lab it has 2 (x-component and y-component).
Here is the demonstration
(if you were to stand in the moving frame, you would see an up and down motion)

8) In the previous experience, if you were to stand in the car, all curtains closed, you would have no idea
if the car was moving or no (as long as there is no acceleration) . You see the ball going up and down. Likewise, when a  high quality
train pass another train, it is hard to tell which one is moving and sometimes we
feel going backward. (like when you are the passenger of a car, when you are passed
by another car, you think you are moving back).
 We can " feel" acceleration (see inertia principle. In the frame of reference of the car,
you feel a force )   but  you can't tell if you are in a
uniformly moving reference frame (constant speed and along line)  or no. The ball in its moving reference frame goes up and
down the same way it will in the frame of reference at rest, like in the lab.
If you are in the car , you are not aware of a parabola trajectory computed
in another frame. This is called the first postulate of special relativity.
There is no physical experiment we can perform to determine our state of uniform motion,
The laws of Physics within the uniformly moving car are the same as those in a stationary laboratory.
We can't detect the state of uniform motion. We can detect acceleration.

Now if you accept that speed can be relative, maybe other physical quantities like time, energy or lengths
are relative too. Their measurements depend on the frame of reference you are measuring them in.
This was not making sense until 1905 when Einstein formulated the second postulate of SR.
It all came to him because of the result of the experiment of Michelson-Morley

PARt2 measuring the speed of light = about 300,000 km/s or 186,000 miles per second or 1 foot 1 a billionth of a second
You can skip this part.

The first experiment done to measure the speed of light was done by Galileo Galilei but it failed.
Galileo Galilei opens his lantern and tries  to measure the time
the light takes to reach his student. The student  notifies Galileo that he sees
the light by opening his own lantern. But light goes too fast. It was not possible to
record any time difference.

In 1676 Roemer (danish) uses Astronomy and Geometry to estimate the speed to light.
he got c = 220,000 km/s instead of 300,000 km/s What is the percentage error ?
He used the satellite of Jupiter Io for his experiment.

Io is a satellite of Jupiter and it takes Io about 42.5 Earth hours to complete 1 rotation.
When Io gets behind Jupiter it is eclipsed during a given time from Earth point of view.
From Earth point of view Io disappears behind Jupiter during a time t..
Io is eclipsed by Jupiter. Roemer noticed that the time t  the eclipse lasts depends on the position
of the Earth. So the duration of the eclipse is not constant. (as thought by Kepler).
Roemer understood  that this was because the light had a finite speed.
If both A and B are looking at the eclipse, the eclipses will seem longer for B than for A
as it takes a longer time for the light to reach the observer.

Let's take the starting point of Io rotation around Jupiter as the immersion point. (just before the eclipse starts).
The light travels in a straight line to A (or to B). The observers on A and B compute the time it takes
IO to complete a rotation (immersion to the next immersion) and they notice that there is a time difference T.
(they are computing the period of rotation of the planet).
T is the time it took the ligth to cover AB = diameter of the orbit. Can you compute the speed
of light the same way Roemer did ?
Take T = 16 minutes 40 seconds (about)
The diameter of the orbit of the Earth is about 300 million km.
c = ________________--

At the time,  the diameter of the orbit of the Earth was not well known (between 68 and 138 million for the radius instead of 150 million)
and the time T was computed to be 22 minutes then 14 minutes by Roemer.

2) 
in 1849  the French Physicist Fizeau got a better estimate : 315,300 km/s





The beam of light go through a notch and comes back through the next notch.
The wheel included 720 notches so 720 x 2 angular sections all together.
For that to happen, Fizeau computed that the speed of the wheel has to be
12.6 turns per second (or 2 pi x 12.6 / 1   rad/s). So during the time t it takes the light to cover the distance 2d
the wheel turns therefore 2 pi / 2x720 rad.
The distance d was 8633 meters (between 2 mounts  Montmartre (Paris) et Mont Valerien (Suresnes) )
The speed of light c = 2d/t .
Can you compute c ?
(angular speed of the wheel in rad/s )
It was hard to estimate the speed of the wheel with precision.

In 1862 the trench Leon Foucault uses the same idea but uses rotating mirrors instead. (see below the set up).
he gets 298 000 km/s. And he shows that light travels faster in air than in water.

In 1873 Maxwell publishes his paper that unifies electricity and magnetism in 4 equations. He introduces
the " ether " the material that impermeate the universe (we call it the vacuum today) and in which EM waves travel.

Foucault  set up is even more improved by  Albert Michelson (using very expensive materials), an American Physicist. He gets a better estimate  in 1878.
He finds  299 798 km/s (± 4 km/s). Here is the set up below.
The wheel he used was made of mirrors. While the light travels from A to 2nd mirror and to C, the wheel turns 1/8 of 2pi.
The rotational speed to the wheel was 32,000 rev/minutes. the reflecting mirror was about  35 km from the wheel.
(he placed the 2 pieces on 2 Californian mountains). Estimate the speed of light.


Michelson was very good at building experiments that can achieve accurate measurements.
He traveled to Europe to study optics for 2 years . He came back to USA and with Edward Morley
built the famous Michelson-Morley experiment.  It was an interferometer designed to compute the
speed of light very accurately by comparing the interference patterns.

(note: Later other set up were designed to get a better precision. The set up used lasers.
in 1983 the speed of light was set to be 299 792,458 km/s .
here for more details )


PART3 experiment ofMichelson-Morley : The most famous failed experiment

see here for more detail

The sophisticated experiment involved mirrors, lenses, beam of coherent light and a screen to look at interference patterns,
If the 2 beams take 2 different paths, but move at the same speed (speed of light)
The experiment can be used to compute the speed to light.
SEE here for details.
In that case the experiment described above, the 2 beams of light don't take the same path and the phase shift between the beam (waves) is
 used to compute the speed of light. It is like the 2 slits experiments ( Below):
speed = difference in path / time delay. the phase shift is computed at a given point on the screen.


In his famous experiment, Michelson, uses the same idea but that time both beams take the same path.
If one beam has a larger speed than the other one,  then an interference pastern should appear.
(other wise the wave just reinforce them selves and you get a bright spot)
The interference pattern could be used to determine the difference in the speed of the beams.

It was the idea behind Michelson-Morley experiment.

The famous Michelson-Morley experiment was designed to detect the motion of the Earth using the
Ether wind. The light was supposed to have a larger speed when traveling with the wind/ The " ether"  was the name
given to the stuff that makes up the vacuum. The ether was thought to permeate the Universe.
An electromagnetic wave (like visible light) is a perturbation traveling in that ether.  

The motion of the ether was thought to be due to the motion of the EArth. If the Earth was immersed in the Ether
then the ether would stream past the Earth like water streams past a moving boat.
The set up is shown below. The speed of light should have been larger along the direction of the wind. (speed of light in the lab frame
= speed of light in the lab + speed of ether ).  The same way that
from the frame of reference attached to a shore, the speed of a boat = speed of river + speed of boat relative to the river.
Then a interference pattern was expected to be recorded on the screen.
 
The 2 beams of light cover the same distance but one travels along the " stream" of the ether
due to the rotation of the Earth and the other beam is travels perpendicularly to the stream.
Here is a very nice applet to understand what was expected to observe:
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6.htm

Surprising, No matter how hard they try, the speed of light didn't change with the direction of the beam.
Many scientists came up with different explanations. Maybe the ether was moving with the EArth like
the air moves with the EArth ? that's why the speed of sound is the same in all directions.
It took the genius of Einstein to come with an explanation. This is the 2nd postulate of relativity.

part 4: second postulate of special relativity and consequences

The second postulate says that the speed of light is the same in all frame of references.
For the postulate to be true then the distance and the time had to be relative.
c = distance in frame1/time in frame1 = distance in frame2/time in frame 2
His genius was to understand that not only speed (except speed of light) could depend on
the frame of reference it is measured in but also distance, time, energy, momentum ....

source: Cutnell and Johnson, Wiley
This postulate is non intuitive. If someone stands in the truck and shines a beam of light at you
and if you measures the speed of the light in your frame of reference, you should find c + 15
(like the baseball speed is 60km/s in the moving frame of reference but 60 + 40 in the Earth's frame of reference.
40 being the speed of the truck). But experiences have shown that Einstein was right and the speed
of light in the moving frame and in the frame at rest is C.
The speed of light does not depend on the frame of reference it is measured in.
h


1) time dilation

These strange effects have been measured. The life time of a moving  particle
could be 1 second as measured in the moving reference frame (attached to the particle) but 7  seconds in the frame of reference
of the lab. So from the point of view of the person making the measurement in the lab, the particle lives longer.
For the effect to be dramatic the speed of the moving frame has to be
very fast. (larger than 50% of the speed of light). In that case, the particle is moving at 99% of the speed of light.

We say the time is dilated in the frame of reference of the lab.
But the observers do not disagree.
See below. Consider an astronaut brushing his teeth. When he is on Earth, the brushing lasts to= 1 minute.
He leaves  EArth in a fast spaceship. In his moving frame of reference, the brushing still lasts to=1 minute.
But in the frame of reference attached to EArth, the brushing lasts more. Like if it happens in a slow motion.
if the speed V is 99.99% of the speed of light, the brushing lasts 1 hour when measured in the lab.
But as measured in the lab, the heart beats of the astronaut will also slow down and so his metabolism.
so every thing will be consistent.


But no matter if the astronaut is on Earth or in the moving frame, the brushing lasts 1 minute. they agree on that.
If the astronaut in his moving reference frame measures the time it takes for someone to brush its teeth on earth he notices
time dilation. Because from his frame, the Earth is moving away.

If the space ship go back to EArth there will be a delay between the clock' s. (labs and ship's) See below the twin paradox.

The table below show the time dilation as a function of the velocity (as a fraction of the speed of light). Comment on the table.

The math is very simple. In involves the gamma factor = sqrt ( 1 - V²/c²)
The ratio between the time measured in the lab frame and the time measured in the moving frame is :
1 / (gamma factor) , the factor is smaller than one.
 
And since you know the time is dilated( as measured in the lab) , then you divide by the gamma factor (smaller than 1) to find
t (time an event that happens in the spaceship lasts when measured in the lab). (since when you divide by a number smaller than 1
you get a number larger).

Try that :  T/To =2, find the V/c =    %
T/To = 0.1 , V/c = ________ %

Time really does pass more slowly in a moving system compared with one at relative rest.

This effect is a consequence of the speed of light being independent of the frames of reference.
 It is nicely illustrated by Paul Hewitt in " conceptual Physics " .

source: Paul Hewitt " conceptual Physics"
If the spaceship travels away from the earth at a speed of 87% the speed of light, the gamma factor is 2.
Say the space shift sends a flash every 6 minutes . In his moving frame the time between 2 flashes is 6 minutes.
But as it moves away, the light has more distance to travel. (because the light speed ignores the speed of the ship).
So the Earth 's observer will record a longer time between 2 flashes. If the speed if 87% of the speed of light,
the observer will get a flash every 12 minutes. So an event that last 6 minutes in the moving frame,
will have a duration for 12 minutes in the EArth's frame of reference.  

If the space ship moves toward the EArth, the opposite happens. A event that lasts 6 minutes in the moving frame, the event lasts
3 minutes in the EArth's frame of reference.


problems:

A) What will be the mean lifetime of a muon as measured in the laboratory if it is traveling at the v = 0.6 c
with respect to the laboratory ? Its mean life at rest is 2,2 10^-6s .
How far does a muon travel in the laboratory, on average, before decaying. (it has to be larger than expected since it lives longer)

B) Le'ts check dilation for everyday speeds. A car traveling 100km/h covers a certain distance in 10s according to the driver's watch.
What does an observer on eArth measure for the time interval.

2) length contraction


Above a moving refeference frame with coordinate system and clock.
The length of an object too is relative. Say you are superman and you zoom through the lecture hall carrying a 50cm stick.
(measured previously in the lab before superman accelerates and zoom along)
In the moving frame superman measures the length to be 50cm and nothing seems wrong to him.
But if you measure the length in the frame of reference of the lab, the length will be smaller.
The table above shows that at 90% the stick will be measured to be 25 cm.
So is it 50cm or 25 cm ? Both measurements are right. THe same way you speed right now
is 0 in the frame of reference of the lab, or 600 miles per second in the frame attached to EArth or
20 miles per second in the frame attached to the sun ... All these answers are right.
The math is still very easy. It still has the gamma factor. Since the length is measured as smaller
in the frame of the lab, then you multiply by the gamma factor. (that is smaller than 1).


Superman would feel physically ok but from your frame of reference, he would contract too and would look like a pancake.

So you expect him to still measure 50cm in his moving frame since his head also shrinks. So both observers agree.
25 cm in the frame at rest and 50cm in the moving frame.


http://en.wikibooks.org/wiki/Special_Relativity/Print_version

problems:
A rectangular painting measured 1.00m tall and 1.50 wide. It is hung on the side wall of a spaceship which is moving
past the EArth at a speed of 0.90c.
What are the dimensions of the picture according to the captain of the ship. What are the dimensions
as seen by an observer on THe Earth ?

3) mass increase
According to SR mass too depends on the frame of reference it is measured in.
As the speed increases, the mass too increases as measured in the lab's frame of reference.
Not only superman will look like a panckae but his mass will increase as well.
In the moving frame, the mass will be the same. Superman will have the same mass in his moving frame (unless he eats a lot).
So in as measured in the lab : mass = restmass / gamma factor
So the momentum will be also different:


According to SR no object can go faster than the speed of light. As a source of energy increases the speed of the object,
its mass increases as well. So it takes more energy to accelerate the object. As the speed gets close to the speed
of light, the mass increases goes to infinity.

problem:
calculate the mass of an electron when it has a speed of 4 10⁷m/s in the CRT of a television set. And 0.98c in an accelerator used for
cancer therapy.



4) deriving the gamma factor
Deriving the gamma factor is easy and is done in most textbooks.

source: Cutnell and Johnson, Wiley
In the moving frame of reference of the astronaut (a) , a beam of laser is shined toward a mirror,
The light bounces back. C = 2D/∆to   if the event lasts ∆to as measured in the moving frame of reference.
The same event lasts ∆t as measured in the frame of the lab. In the lab the speed of light is still c
instead of c + v and this is the key thing. so in the lab you have : c= 2 s /∆t   and v = 2L /∆t
(let's forget about ∆ to simplify notations)
so if you consider the right angle triangle and Pythagorean theorem you get : s² = L² + D²
(ct)² = (vt)² + (cto)²

Use your Algebra skills to derive the time dilation formula. derive t/to

5) simultaneity
Consider a fast  moving space ship zipping along in the lecture hall. from left to right.
In the moving frame, the astronaut hits a table simultanously with his 2 hands/ In his frame of reference
the 2 events happen at the same time (right and left hands hitting the table).
In the frame of reference of the lab, the events are not simultaneous anymore. The left hand hits the table before the right hand.
it makes sense because in the frame of reference of the lab,  events lasts longer (compared to when measured in the moving frame).

We can illustrate that by imaging the astronaut in the middle of the ship. He shines 2 beams of light in 2 opposite directions.

In the moving frame, the light reaches the sides at the same time/. But in the frame of reference of the lab,
the light reaches the left side first because there is less distance to cover. (consequence of the speed of light
ignoring the frame of reference).

6) twin paradox

This is not really a paradox. First you need to understand one thing. Let's take the example of an astronaut on a journey to
Proxima Centauri. (closest star). In the frame of reference of the lab, the ship is shorter compared to his length  measured in the
moving frame. (or measured on EArth before take off). An event lasts longer when measured in the lab's frame.
But from the astronaut point of view, it is the EArth that is zooming away and the star that is zooming toward him.
So he too experiences length contraction or time dilation.

So if the astronaut leaves for the star and then come back , there should not be a shift between his clock
and yours. Since in the moving frame of the astronaut,
time passes more slowly on Earth compared to his own time. (the astronaut is at relative rest compared to EArth).
But we know this is not true. When the astronaut comes back, he would have aged less than his friends on Earth.
If the astronaut takes a round-trip journey for 1 year and if he travels at 87% the speed of light when
he comes back 2 years will have elapsed on EArth. Why is that ?
This is well explained by Paul Hewitt in " conceptual Physics " . The shift in the clocks is again a consequence of the speed
light being independent of the frame of reference:
Suppose a spaceship lives for a 2hours round trip at 87% the speed of light. the gamma factor is 2.
She spaceship emits a flash every 6 minutes. In his frame of reference,  he sends 20 flashes. (6 x 20 = 2 hours).
In the EArth's frame of reference, the flashes are recorded. 20 of them. When the ship moves away, the Earth
gets a flash every 12 minutes . (time dilation). Then when the ship comes back, Earth records flashes every 3 minutes.
so the EArth gets 10 flashes every 12 minutes and 10 flashes every 3 minutes that is 150 minutes altogether
from EArth's point of view. So Earth's people have aged a half hour more than the twin aboard the spaceship.






7) Energy

Since mass depends on the frame of reference it is measured in, kinetic energy is also relative.
Einstein showed that the kinetic energy of a particle is given by:
KE = mc² - moc²  = (m-mo)c² with mo the mass at rest.  moc² is the rest energy.
So the total energy of a particle is E = mc² = moc² + KE
So for a particle at rest in a given reference frame, its total energy is Eo= moc²   which we have called the rest energy.
For this idea to have any meaning, then mass ought to be convertible to energy and vice versa. This has been experimentally confirmed.
In a collider, protons collide and turn to pure energy then to matter again. In a fission reaction mass is turned to energy.
Electromagnetic radiation under certain condition can be converted to material particle as electrons. The radiant energy we receive
from the Sun is an example of E = mc². The Sun's mass is continually decreasing as it radiated electromagnetic energy outward.
Problem:
A meson (mo = 2.4 10^-28 kg) travels at the speed of v = 0.8c . WHat is its kinetic energy ?
Compare to classical calculation.
Compare E = mc² to moc² convert to eV ( 1 eV = 1.6 10^-19 J )

The energy required or released in nuclear reactions and decays comes from a change in mass in between the initial and final particles.
In one type of radioactive decay an atom of uranium (m=232.03714 u) decays to an atom of thorium (m=228.02873u) plus an atom
of helium (m= 4.00260 u) where the masses given are in atomic mass units (1 u = 1.6605 10^-27kg)
Calculate the energy released in this decay.


8) relativistic addition of speed

Because no speed can be larger than the speed of light, we can't use the addition of velocities formula anymore
when the speed are large.

 If a space ship is going at V=0.6c (with respect to eArth)  and fire a rocket at u'=0.6c
with respect to the space ship, the speed of the rocket with respect to earth can't be v + u'.
other wise we would get 1.2c and no object can tracel faster than the speed of light.
Instead the correct formula is:  ( v + u' ) / (1 + vu'/c² )

9) ALl the formulas :

source: Cutnell and Johnson, wiley publisher