Special Relativity

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Special Relativity

• An Introduction To High Speed Physics

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What Is Light?
There were two contradicting theories as to the
nature of light
Newton light is corpuscular
Huygens light is a wave
If light is a wave, through what does it
propagate?
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The Aether
Space is permeated by an invisible lumineferous
aether
(light-bearing medium)
Medium through which light can propagate
The Earth must be moving relative to the aether
So light will travel faster or slower, depending
on the orientation
Differences can be determined by experiment
Test for the existence of the aether
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The Michelson-Morley Experiment
Test for the presence of an aether using an
interferometer
A is a half-silvered mirror B/C are mirrors O is
a detector
incoming light
There is a phase difference between the two beams
Light at O should be phase-shifted, but no phase
shift was observed
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Maxwells Predictions
Electric and magnetic fields interact
Accelerating charges produce EM waves
Maxwells equations predict that these waves
propagate through a vacuum at a constant speed
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What Has Gone Wrong?
Consider a train
The resultant velocity of the person is u v
The resultant velocity of the light is c not c v
Is Maxwell wrong? Are Michelson and Morleys
results wrong?
Or is Galileo wrong?
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Einsteins Postulates
We can now state two postulates

• The laws of physics are the same in all inertial
  frames of
• reference

• The speed of light in a vacuum is the same in all
  inertial
• frames of reference

But what is an inertial frame of reference?
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Frames Of Reference
A frame of reference is the coordinate system of
an observer
stationary frame
frame moving at constant velocity v
accelerating frame
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The Galilean Transformation
Consider two inertial frames, S and S
S is moving at velocity v away from S, and vy
vz 0 ms-1
The object has the same y- and z-coordinates in
both frames
The time measured at any instant is the same in
both frames
The x-coordinate is constant in S, but changes
in S
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The Galilean Transformation
At a time t, the x-axis of Sis a distance vt
from the x-axis of S
So the x-coordinate in S is the x-coordinate in
S vt
All other coordinates are unchanged
This transformation between frames can be written
as
Galilean Transformation
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A New Transformation
The Galilean transformation contradicts
Einsteins second postulate
We need to derive a new transformation, with the
properties
We are not assuming that time is absolute
So we need to be careful when referring to time
An event has both space and time coordinates
This can be written as a four-vector (x, y, z, t)
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A Thought Experiment
Alice is on a long train journey, and is rather
bored
She decides to build a clock using her mirror and
a torch
What is the time interval between a pulse leaving
and returning to the torch?
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A Thought Experiment
Bob is standing on platform 9¾ and watches Alice
in the train
What is the time interval Dt in Bobs frame of
reference?
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The Lorentz Factor
The factor g is the Lorentz Factor
For v ltlt c, g(v) 1
As v tends to c, g(v) tends to infinity
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Time Dilation
We have this relationship, but what does it mean?
If a body is travelling slowly w.r.t. an observer
(g 1) time intervals are the same for the body
and the observer
If a body is travelling fast w.r.t. an observer
(g gtgt 1) time intervals appear longer to the
observer
If you are in a spacecraft travelling close to c,
time will pass normally for you, but will speed
up around you
Notice, therefore, that photons do not age
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The Twin Paradox
Once upon a time there were two twins
Ben goes on a journey into space, but Bill stays
on Earth
When Ben returns, which of the twins is oldest?
Bill thinks he will be older, as Ben travelled
very fast away from him
Ben thinks he will be older, as Bill travelled
very fast away from him
Who is right?
Bill is older, because he stayed in the same
inertial frame, but Ben had to accelerate in the
rocket
The viewpoints are not identical
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Length Contraction
How do lengths appear in a different frame?
Similar derivation as for time dilation
There is no change in the directions
perpendicular to travel
In the direction of travel, we can show that
So, for a body travelling with v close to c,
relative to an observer, the body will appear
shorter to the observer, in the direction of v
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How do you measure your velocity?
It is meaningless to have an absolute velocity
Velocity can only be measured relative to another
body
But what about length contraction and time
dilation?
If lengths are shortened, then can you measure
the change?
No the ruler is length-contracted as well
Similarly, clocks slow down, so changes in time
cant be measured
But doesnt relativity define c as a maximum
absolute speed?
Not quite this is a maximum relative speed, as
light has the same speed (c) relative to any frame
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The Lorentz Transformation
We can now derive a relativistic transformation
The y- and z-coordinates will be the same in both
frames, as before
From the Galilean transformation, x x vt
But in frame S, xis length-contracted to g-1x
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The Lorentz Transformation
To get the time transformation is a little
trickier
Notice that the transformations are linear
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The Lorentz Transformation
We can now state the full Lorentz Transformation
This satisfies the conditions for the
transformation
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Can We Go Faster Than Light?
From Newtons Second Law, F ma (constant mass)
So if we provide a continuous force, we can
achieve v gt c
But momentum must be conserved
m0 is the rest mass
The g factor in effect increases the mass, as v
increases
A greater force is needed to provide the same
acceleration
To reach the speed of light, an infinite force
would be required
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Conclusions
We have looked at the theory of Special Relativity
This resolved the conflict between Newtonian
mechanics, and Maxwells equations
It is a special theory, as it doesnt consider
accelerations
This is dealt with in General Relativity
Derivation left as an exercise to the student ?