Transcript Relativity - PAC

Relativity
Lesson 1
1.
2.
3.
4.
5.
6.
Galilean Transformations (one frame moving relative to
another) Michelson Morley experiment– ether.
Speed of light constant
Simultaneity.
Inertial Observers. Frames of Reference.
2 Postulates of Special relativity – c is constant and all
laws are the same for all inertial observers.
Light Clocks and how they work; Pythagorean
treatment
Click for good background site
IB Physics – Relativity
Inertial frames of reference
• An inertial frame of reference moves at a
constant velocity.
• It is not accelerated
y
An Event
0
9
Lightning strikes at
x = 60 m and t = 3 s
3
6
x
x=0
x = 10
x = 20
x = 30
x = 40
x = 50
x = 60
x = 70
Tsokos, 2005, p553
IB Physics – Relativity
Absolute Rest
• There is no such thing as absolute rest.
• In an inertial frame of reference there
are no experiments you can do which
prove you are moving. (Unless you look
outside!)
• Newton’s Laws consider zero and
constant velocity to be identical
IB Physics – Relativity
Galilean Transformations
Consider a stationary frame and a frame moving
at velocity v in the x direction.
x  x  vt
t  t
u  u  v
y  y
z  z
Relatively
easy?
IB Physics – Relativity
Use of the Galilean Transforms
y
y
origins coincide when clocks at
both origins show zero
velocity, v with respect to ground
0
9
6
9
x
3
0
3
y
6
y
x
x  x  vt
vt
the train has moved
away; when the clocks
show 3 s, lightning
strikes at x = 60 m
Calculate xand t  if v = 15ms-1
0
0
9
9
3
6
x
3
6
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x
Tsokos, 2005, p553
Calculating relative velocity
y
y
x  ut
v
u
v
x
vt
Tsokos, 2005, p553
x
An object rolling on the floor of the ‘moving’ frame appears
to move faster as far as the ground observer is concerned
A ball rolls on the floor of the train at 2ms-1 (with respect to the
floor). The train moves with respect to the ground at (a) 12 ms-1 to
the right and (b) 12 ms-1 to the left. Find the velocity of the ball
relative to the ground.
IB Physics – Relativity
Nature of Light
Oscillating magnetic and electric fields at right angles to
each other.
Maxwell’s equations predict that the speed of light is
independent of the velocity of the light source.
c
1
 0 0
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Conflict with Galilean relativity.
V
C
Observer in the train measures the speed of light = C
Observer on the ground measures the speed of light as C + V
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The Michelson-Morley Experiment
Read in Kirk p149
Explain how the null result from this
experiment shows that the speed of light is
unaffected by the motion of the Earth.
Link to Michelson-Morley powerpoint
Link to Michelson-Morley explanation
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Modified equations
x 
x  vt
2
2
v
1 2
c
v
t  t 1  2
c
uv
u 
 uv 
1  2 
 c 
IB Physics – Relativity
Postulates of Special Relativity
• The laws of physics are the same for all
inertial observers.
• The speed of light in a vacuum is the
same constant for all inertial observers
Animations showing time dilation and
length contraction.
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Postulates of special relativity video
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The laws of Physics are the same
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Electromagnetism and relativity
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Speed of light is constant video
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Maxwells equations lead to constant c
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Simultaneity
• Two events which happen together are
simultaneous.
• If two events are simultaneous in one frame
of reference they may not be in another.
• See train example in Kirk p143
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Simultaneity video
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Light clocks
S
x  0
x  0
B
B
B
A
A
A
vD t
x=0
S
x = vD t
B
L
A
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Time dilation video
IB Physics – Relativity
Relativity
Lesson 2
1.
2.
3.
4.
5.
6.
7.
Time dilation
Lorentz Factor
Proper time
Lorentz contraction
Proper length
Twin paradox and symmetric situations
Muon decay; evidence for time dilation
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Time dilation
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Time dilation
Dt 
Dt 
2
v
1 2
c
Think about what this means?
????
IB Physics – Relativity
Proof of formula
The proof of the time dilation formula is a standard
requirement in the exam.
Carefully work through the proof using Pythagoras’
Theorem.
Make sure you understand each step.
Hints
The “prime” notation refers to measurements in the ‘moving’ frame
The speed of light is the same for all observers.
IB Physics – Relativity
The Lorentz factor

1
Lorentz Factor
8
2
v
1 2
c
7
6
5
4
3
2
1
Work out  below
0
0
0.2
0.4
0.6
0.8
1
1.2
v/c
v
0.1c
0.2c
0.3c
0.4c
0.5c
0.6c
0.7c
0.8c

For what values of v is  significant ?
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0.9c
c
Example of time dilation
Dt  Dt 
If the train passengers measure a time interval of Dt1 = 6 s
and the train moves at a speed v = 0.80c, calculate the
length of the same time interval measured by a stationary
observer outside the train standing on the ground
IB Physics – Relativity
Atomic clocks prove time dilation!
IB Physics – Relativity
Proper time
A proper time interval is the time separating
two events that take place at the same
point in space
observed time interval =  x proper time interval
Note that the proper time interval is the shortest
possible time separating two events
IB Physics – Relativity
Examples of proper time
1.
The time interval between the ticks of a clock carried on a fast rocket is half
of what observers on Earth record. How fast is the rocket moving with
respect to the Earth? What are the two events here?
2.
A rocket moves past an observer in a laboratory with speed = 0.85c. The
lab observer measures that a radioactive sample of mass 50 mg (which is at
rest in the lab) has a half life of 2.0 min. What half-life do the rocket
observers measure? Again, what are the two events?
3.
In the year 2010, a group of astronauts embark on a journey toward
Betelgeuse in a spacecraft moving at v = 0.75c with respect to the Earth.
Three years after departure from the Earth (as measured by the
astronaut’s clocks) one of the astronauts announces that she has given
birth to a baby girl. The other astronauts immediately send a radio signal
to Earth announcing the birth. When is the good news received on Earth
(according to the Earth Clocks)?
Tsokos, 2005, p562
In each case first suggest in which frame
the proper time is directly measured
IB Physics – Relativity
Length Contraction
Another consequence of the invariance of the speed of light
is that the distance between two points in space contracts
according to an observer moving relative to the two points.
The contraction is in the same direction as the relative
motion.
Measured by
L
Measured by observer
in a moving frame with
respect to the object
L0

observer who
is stationary
with respect to
the object
A paradox on length contraction
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Proper Length
The proper length of an object is the length recorded
in a frame where the object is at rest
Any observers moving relative to the object measure a
shorter length (Lorentz contraction);
length 
proper length

IB Physics – Relativity
Examples
1.
An unstable particle has a life time of 4.0 x 10-8 s in its own rest frame. If it
is moving at 98% of the speed of light calculate;
a)
Its life time in the lab frame
b)
The length traveled in both frames.
Kirk, 2003, p145
2.
Electrons of speed v = 0.96c move down the 3 km long SLAC linear
accelerator.
a)
How long does take according to lab observers?
b)
How long does it take according to an observer moving along with the
electrons?
c)
What is the speed of the accelerator in the rest frame of the electrons?
Tsokos, 2005, p566
IB Physics – Relativity
The twin paradox
Read about this in Kirk
p 146.
•The paradox arises
because both twins
view a symmetrical
situation. Explain
why?
•Explain why it is not a
paradox.
Link to twin paradox
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The muon experiment
This offers direct experimental evidence of time dilation
Key points
Muons have an average lifetime of 2.2 x 10-6 s in their
own rest frame.
They are created 10 km up in the atmosphere with
velocities as large as 0.99c.
Show that without special relativity muons are unlikely
to be detected on Earth.
Muon decay explanation
IB Physics – Relativity
Relativity
Lesson 3
1.
2.
3.
4.
5.
Velocity addition
Rest mass and relativistic mass
Use of E = mc2 as total energy
Acceleration of electrons by a p.d.
Use of MeVc-2 and MeVc-1
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Galilean Velocity addition
S
S
u
v
u
train stationary
u  u  v
ground stationary
Easy!
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What if
u  u  v
u would be bigger than c
….and my special
theory of relativity does
not allow this
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u approaches c ?
Relativistic velocity formulae
Just a small correction needed
u  v
u
u v
1 2
c
prove the inverse
formulae from this;
Notice the similarity to Galileo's
formulae with a small correction
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uv
u 
uv
1 2
c
Now try this
An electron has a speed of 2.00 x 108 ms-1 relative to a
rocket, which itself moves at a speed of 1.00 x 108 ms-1 with
respect to the ground. What is the speed of the electron with
respect to the ground.
Answer; u = 2.45 x 108 ms-1
Tsokos, 2005, p567
1.
The trick is to clearly sort out your reference frames.
s
s
u   2108
v  1108
rocket
ground
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And this; a bit harder
2. Two rockets move away from each other with speeds of of 0.9c to right and
0.8c to the left with respect to the ground. What is the relative speed of
each rocket as measured from the other. Answer ± 0.988c
Tsokos, 2005, p567
0.8c
0.9c
A
B
Again the trick is to be very careful with
your frames of reference
IB Physics – Relativity
Learn this solution
s
s
u  0.8c
v  0.9c
rocket A
ground
rocket B
u is then the velocity of A with respect to B.
Write down then check the velocity of B with respect to A
IB Physics – Relativity
Apply the same approach for the question in Kirk on p147
By now you should have realised that at low
velocities the relativistic formulae approximate to the
Galilean equations.
Check this out.
Show also that the relativistic formulae do not allow
objects to have velocities greater than c
............. but F = ma allows velocities > c so we might
have to look at mass again!!!
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Galileo and Newton allow v > c
v
constant acceleration
c
decreasing acceleration as
speed approaches c
but I don’t allow
this
t
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Inertial mass
Defined as m = F/a
For a constant force Einstein predicted that as v approaches c the
acceleration will decrease to zero. This ensures that speed can
never exceed the speed of light
This means the inertial mass must be approaching infinite.
IB Physics – Relativity
Rest Mass
The rest mass, m0, of an object is the mass as measured in a
frame where the object is at rest
If mass is measured in a frame moving with respect to the
object;
m =  m0
Lorentz Factor
m/m0
8
7
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
v/c
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Try these
1.
Find the speed of a particle whose relativistic mass is double its rest mass.
Answer = 0.866c
2.
A constant force, F, is applied to a particle at rest. Find the acceleration in
terms of the rest mass, m0. Find the acceleration again if the same force is
applied to the same particle when it moves at a speed 0.8c.
3.
Find the rest energy of an electron and its total energy when it moves at a
speed equal to 0.8c. Give values in MeV.
Tsokos, 2005, p571
IB Physics – Relativity
Mass - Energy
The energy needed to create a particle at rest is called the rest
energy given by;
E0 = m0c2
If this particle accelerates it gains kinetic energy and the mass also
will increase so total energy;
E = mc2
or
E = m0c2
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Proper energy; mc2
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Kinetic energy (T) and momentum (p)
The formulae; Ek = ½ mv2 and p = mv no longer work at
high speeds.
Total energy of a particle;
E = Ek + m0c2
So total energy is the kinetic energy plus the rest mass
energy
IB Physics – Relativity
Now try this
An electron is accelerated through a p.d. of 1.0 x 106V.
Calculate its velocity; (use the data book to find the electron
rest mass and charge). Answer = 0.94c
Hint; First calculate the energy gained then find the rest mass
Kirk, 2003, p148
energy. This will give you a value for .
***??**
IB Physics – Relativity
Relativistic formulae
At high speeds you must use the following formulae;
p = m0v
for momentum
E = Ek + m0c2 for kinetic energy
E2 = p2c2 + m02c4 for total energy
E = m0c2
the derivations of these are not required.
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Useful units
The rest mass of an electron = 9.11 x 10-31 Kg
This is equivalent to energy E0 = m0c2
= 9.11 x 10-31 x 9 x 1016 = 8.199 x 10-14 J
This is more conveniently quoted in MeV.
so E0 = 8.199 x 10-14 / 1.6 x 10-19 = 0.512 MeV
Working backwards we know E0 /c2 = m0
So mass can be measured in
MeVc-2
 electron rest mass = 0.512 MeVc-2
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Similarly
KeVc-2 and
GeVc-2 etc.
may be used
Similarly MeVc-1 may be used for momentum
and problems are much
easier to solve.
IB Physics – Relativity
So try these using the new units
1.
Find the momentum of a pion (rest mass 135 MeVc-2) whose speed is
0.80c. Answer 180 MeVc-1
2.
Find the speed of a muon (rest mass = 105 MeVc-2) whose momentum is
228 MeVc-1. Answer = 0.91c
3.
Find the kinetic energy of an electron (rest mass 0.511 MeVc-2) whose
momentum is 1.5MeVc-1. Answer = 1.07 MeV
Tsokos, 2005, p581
Work through the example in Kirk on p.150
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Relativity
Lesson 4
General Relativity (without the maths)
1. The equivalence principle
2. Spacetime
3. Gravitational redshift
4. Black holes
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The Nature of Mass
Gravitational Mass
mg = W/g
These are
completely
different
Inertial Mass
mi = F/a
I do not
think so!
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Einstein’s Happiest Thought
Sitting in a chair in the Patent office at Berne (in
1907), a sudden thought occurred to me.
" If a person falls freely he will not feel his own
weight”.
I was startled and this simple thought made a deep
impression on me. It impelled me towards a theory
of gravitation. It was the happiest thought in my life.
I realised that ........for an observer falling freely from the roof of
a house there exists – at least in his immediate surroundings – no
gravitational field.
The observer therefore has the right to interpret his state as at
rest or in uniform motion...................
McEvoy & Zarate, 1995, p32
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The equivalence principal
a
1. An object inside an
accelerating rocket in outer
space will “fall”.
2. An object in a stationary
rocket inside a gravitational
field will fall
Planet
So both situations are equivalent
Path of “falling” objects
IB Physics – Relativity
and similarly
1. An object inside a rocket in
outer space moving at constant
velocity is weightless.
v
2. An object inside a rocket
accelerating due to a
gravitational field feels
weightless
?????
Again both
situations are
equivalent
Equivalence animations
Planet
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Einstein’s elevator
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The path of light
v
y1
x
y
x
Inertial observer
outside rocket
Observer inside rocket
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The bending of light
a
y1
y2
x
y
y2
x
Inertial observer
outside rocket
Observer inside rocket
IB Physics – Relativity
But according to the equivalence principle
So gravity bends light
towards the planet
a
Is equivalent to
x
y
y2
y
y2
x
Planet
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Space time
Einstein viewed space time like a
rubber sheet extending into the x,
y, z and t dimensions
Flat space time.
Objects move in a “straight line”
Space time is “curved” by mass.
An object moves along the path of least resistance.
This means they take the shortest path between two
points in curved space time.
McEvoy & Zarate, 1995, p32
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General relativity
An object is “just”
captured by the
depression.
Model of a planet in orbit
Model of a meteorite crashing into the Earth.
McEvoy & Zarate, 1995, p32
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Space-time movie
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So what’s so great about general relativity?
Matter tells space how to curve and
then space tells matter how to move
........
The beauty of this simple model is....
we don’t need forces.
“objects move in a
straight line in
curved space-time”
IB Physics – Relativity
More on space-time
Events can be given an x, y, z and t coordinate in space-time
to describe where and when they occurred.
Think?
ct
a
b
c
1.
Why is the time axis multiplied by c?
2.
What is the gradient of a line on this
axis?
3.
What is the velocity of lines a, b and
c?
4.
What is wrong with d?
d
45o
space-time diagram
x
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Gradient of a space-time diagram
cDt
grad 
Dx
c
 grad 
v
or
1
v

grad c
now try the questions on the previous slide
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Even light bends in space time
General relativity predicts that the path of light is deviated by
curved space-time.
“There was to be a total eclipse of the
Sun on 29 May 1919, smack in the
middle of a bright field of stars in the
cluster Hyades.”........Arthur Eddington
led an expedition to the island of
Principe off the coast of Africa to
photograph the eclipse.
Eddington found that the position of
the stars appeared different from
pictures taken at a different time. He
concluded that the light had curved
around the sun by the exact amount
that Einstein had predicted.
hyperphysics.com, 2006
McEvoy & Zarate, 1995, p32
IB Physics – Relativity
Gravitational lensing
Light from objects (e.g.quasars) which are very far away can
be bent round massive galaxies to produce multiple images;
the galaxy behaves like a lens.
Kirk, 2003, p155
Researchers at Caltech have
used the gravitational
lensing afforded by the
Abell 2218 cluster of
galaxies to detect the most
distant galaxy known (Feb,
15th 2004) through imaging
with the Hubble Telescope.
Spot the multiple images
IB Physics – Relativity
Wikiepedia.org, 2006
Black Holes
When a star uses up its nuclear fuel it collapses.
If the remnant mass of the star is greater than 3 x
the sun’s mass there is no mechanism to stop it
collapsing to a singularity. The curvature of
space-time near a singularity is so extreme that
even light cannot escape.
2GM
v
r
2GM
c
r
Escape velocity
For a photon
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Schwarzchild Radius; RSch
At a distance, RSch, from a
singularity the escape velocity
is the speed of light.
2GM
c
r
 RSch
2GM
 2
c
At a distance less than RSch from
a singularity;
escape velocity > C
 Light cannot escape
RSch is also called the event horizon. Everything trapped within the
event horizon is not observable in our universe.
IB Physics – Relativity
Try this
1. Calculate the Schwarzschild radius for a star of one solar
mass; (M = 2 x 1030 Kg)
Tsokos, 2005, p591
IB Physics – Relativity
Time in gravitational fields
General relativity predicts that time runs slower in
places where the gravitational field strength is stronger.
For example; the Earth’s field weakens as you go
further away from the Earth’s surface.
1
g 2
r
This means that time runs
more slowly at the
ground floor than at the
top floor of a building.
IB Physics – Relativity
Gravitational Red Shift
Gravitational time dilation, or clocks running slower in
strong gravitational fields, leads us directly to the prediction
that the wavelength of a beam of light leaving the Earth’s
surface will increase with height.
c  f
but Period
or
c

f
1
T
f
IB Physics – Relativity
  cT
Red-shift Explained
Large
wavelength
As the light gains height so
time runs more quickly because
the gravitational field weakens.
  cT
So as time speeds up,
the period increases and
hence the wavelength
also increases
Short
wavelength
Planet
IB Physics – Relativity
Calculating frequency shifts.
Consider a photon of frequency f0 leaving the surface of the Earth. It gains
potential energy given by mgDh and the frequency is reduced to f.
E  mc  hf 0
2
Photons do not have
mass but have an
effective mass given by;
hf 0
m 2
c
Do not get h the
Planck's constant
confused with Dh
the height gain.
Conserving energy we get;
hf 0  hf  mgDh
hf 0 gDh
 hf 0  hf 
c2
f 0  f gDh

 2
f0
c
Df g Dh

 2
f0
c
IB Physics – Relativity
Blue shift
Similarly, a beam of light directed from outer space towards
the Earth is blue-shifted i.e. the wavelength decreases as the
gravitational intensity increases. The same formula is used to
calculate red-shift or blue-shift.
Examples to try
1. A photon of energy 14.4 KeV is emitted from the top of a 30 m tall tower
toward the ground. What shift of frequency is expected at the base of the tower?
Ans; Df = 1.16 x 104 Hz
Tsokos, 2005, p589
2. A UFO travels at such a speed to remain above one point on the Earth at a
height of 200 Km above the Earth’s surface. A radio signal of frequency
110MHz is sent to the UFO. (i) What is the frequency received by the UFO?
(ii) If the signal was reflected back to Earth, what would be the observed
frequency of the return signal? Explain your answer. Ans; (i) f = 1.1 x 108 Hz
(shift is v. small). (ii) return signal is the same frequency as the emitted signal
Kirk, 2003, p153
IB Physics – Relativity
Experimental support
In 1960 a famous experiment called
the Pound-Rebka experiment was
carried out at Harvard University to
verify gravitational blue/red-shift. The
frequencies of gamma ray photons
were measured at the bottom and top
of the Jefferson Physical Laboratory
tower. Very small frequency shifts
were detected as predicted by the
theory.
Atomic clocks are very sensitive and
precise. Atomic clocks sent to high
altitudes in rockets have been shown
to run faster than a similar clocks left
on Earth
IB Physics – Relativity
An alternative view of black holes
The gravitational intensity near black holes is very strong and approaches
infinite at the event horizon. Time effectively stops and any object falling
into a black hole will appear to stop at the event horizon according to a far
observer. Light emitted from a black hole will therefore be infinitely redshifted; hence no light is emitted.
ct
This photon is trapped in the black hole.
This photon will escape
Space-time diagram of the
formation of a black hole
Rs
x
IB Physics – Relativity
Beyond relativity!
IB Physics – Relativity
Bibliography
Kirk, T; Physics for the IB diploma, OUP, UK, 2003
McEvoy, J.P. & Zarate, O; Stephen Hawking for beginners,
Icon Books, U.K., 1995.
Tsokos, K.A.; IB Physics, Cambridge University press, U.K.,
2005.
hyperphysics.com, 2006
http://en.wikipedia.org/wiki/Gravitational_lensing, Mar 06
©Neil Hodgson
Sha Tin College
IB Physics – Relativity
Videology
1. Postulates of special relativity
2. Speed of light is constant
3. Simultaneity
4. Time dilation
5. Atomic clocks prove time dilation
IB Physics – Relativity