Special Relativity

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Lecture 4 Special Relativity Space and Time are
connected
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What is a theory?
A set of statements or principles devised to
explain a group of facts or phenomena, especially
one that has been repeatedly tested or is widely
accepted and can be used to make predictions
about natural phenomena. American Heritage
Dictionary
The Theory of Special Relativity
ROBUST must be refutable!
In physics, the term theory is generally used for
a mathematical framework which is capable of
producing experimental predictions for a given
category of physical systems. Wikipedia
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What is a theory of relativity?
A theory of relativity is both a description of
the space-time fabric that underlies physics and
a constraint on the mathematical forms that the
theories or laws of physics can assume within
that fabric.
The Theory of Special Relativity
A theory of relativity answers the following
questions

• Question 1 What is an acceptable space-time
  structure
• coordinate system (or reference frame) in
  which
• to express the laws of physics?

Question 2 Of the acceptable space-time
coordinate systems, what sets of them are
equivalent in the sense that the laws of physics
are the same covariant in all equivalent
coordinate systems.
Question 3 What quantities have the same value
are invariant when measured in any of a set of
equivalent frames?
Question 4 What are the rules transformation
laws - by which noninvariant quantities can be
transformed from their values in any one frame to
any equivalent frame?
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Why special?

• Real theories of relativity
• Newtonian or Galilean Relativity
• Special Relativity
• General Relativity

The Theory of Special Relativity
Einsteins Postulate 1 the same laws of
electrodynamics and optics physics will be
valid for all frames of reference for which the
equations of mechanics hold good.
Einsteins Postulate 2 light is always
propagated in empty space with a definite
velocity c which is independent of the state of
motion of the emitting or observing body.
c 186,000 miles per second
Einsteins Postulate 3 The Dynamical
Postulate Relativistic momentum and energy are
always conserved in a closed system.
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What is a frame of reference?
Frames of Reference
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Frames of Reference
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vc
Relative velocity according to Newton
Velocity of car in S frame v Velocity of car
in S frame
70 mph 60 mph 10 mph
Only approximately true!
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v
Relative velocity according to Einstein
Velocity of rocket does not equal velocity of
shuttle velocity of rocket with respect to the
shuttle
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Relative velocity of light
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Lorentz Contraction Moving objects are
shorter in the direction of their motion than
when they are at rest.
Time Dilation Moving clocks tick more slowly
than when they are at rest.
Special Relativity Space Time Phenomena
BUT What does the length of a moving object,
or the tick of a moving clock mean?
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• Definition of the length of a moving object
• The distance between the simultaneous positions
  of its end points. Or...
• The time it takes to go past a given point
  multiplied by its velocity.

How do we measure the length of a moving object?

• How do you define events as being simultaneous?
• Einsteins procedure
• First, put clocks everywhere in space and
  synchronize them!

In my relativity theory I set up a clock at
every point in space, but in reality I find it
difficult to provide even one clock in my room.
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• Einsteins procedure
• First, put clocks everywhere in space and
  synchronize them!

How do we measure the length of a moving object?
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• Einsteins procedure
• First, put clocks everywhere in space and
  synchronize them!

Wait a minute. S wants to synchronize his clocks
too.
How do we measure the length of a moving object?
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• Einsteins procedure
• First, put clocks everywhere in space and
  synchronize them!
• Next, we make a light clock.

In the clocks rest frame
How do we measure the length of a moving object?
In the S frame
Dilated time (different but synchronized clocks)
?
Proper time (same clock)
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• Einsteins procedure
• First, put clocks everywhere in space and
  synchronize them!
• Next, we make a light clock.

In the clocks rest frame
How do we measure the length of a moving object?
In the S frame
A little math. Dont vorry about it.
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• Einsteins procedure
• First, put clocks everywhere in space and
  synchronize them!
• Next, we make a light clock.
• Finally, we measure the length of time it takes
  the object
• to pass some particular clock.

How do we measure the length of a moving object?
We measure the time it takes the object to pass
by clock B.
L
Event 1 Leading edge is at B. Trailing edge is
at A.
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• Einsteins procedure
• First, put clocks everywhere in space and
  synchronize them!
• Next, we make a light clock.
• Finally, we measure the length of time it takes
  the object
• to pass some particular clock.

How do we measure the length of a moving object?
We measure the time it takes the object to pass
by clock B.
L
L
Event 1 Leading edge is at B. Event 2
Trailing edge is at B.
L
Time interval between events
time dilation ?
?
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Time Dilation Consider a clock (or any
physical system) that experiences a time
difference of ?t between events. If that clock
(or system) travels at a speed v with respect to
an observer, the observer will measure, using his
own synchronized clocks, a dilated time
difference, ?t, given by the formula
Time Dilation and Lorentz Contraction
Lorentz Contraction Consider an object of
length L as measured when it is at rest. If it
moves with speed v in the direction of that
length, then its length, L, as measured by a
stationery observer, using her own synchronized
clocks, will be contracted according to the
formula
Note When v c
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The strange case of the cosmic ray muons
Simple physical example of Time Dilation and
Lorentz Contraction
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Twins in Space e-mail each other once a year
Twins in Space!
SPACE TIME get twisted
Tess sends 20 messages and all are received
Tom sends 52 messages and all are received
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Space-Space Twisting
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The Twin Experiment
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Relativity in your pocket GPS (Global
Positioning Systems) require nine major
relativistic corrections!
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Einsteins Postulate 3 The Dynamical
Postulate Relativistic momentum and energy are
always conserved in a closed system.
Consequence of all 3 postulates
The Third Postulate of Special Relativity
What does it mean?
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What is energy?
Energy is the ability to do work.
What is work?
Work is done when a force is exerted through a
distance.
How much work is done?
If F is the force and d is the distance, Work F
x d.
What happens to the energy?
It is always conserved but often changes from one
form to another.
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What forms of energy are there?
There is energy of motion, called kinetic energy.
How much kinetic energy does an object have?
If an object of mass m has velocity v, it has
kinetic energy equal to ½mv2.
What other forms of energy are there?
There is heat energy, which is really the kinetic
energy of atoms and molecules all together in a
substance.
Are there other forms of energy?
Yes. All other forms are called potential energy
because it is stored up and waiting to be turned
into kinetic energy or other forms of potential
energy.
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What are some examples of potential energy?
There is gravitational potential energy. If you
hold an object up, it has gravitational potential
energy. If you drop it, the force of gravity acts
to convert the gravitational potential energy to
kinetic energy.
Potential energy seems to be associated with
forces. Is there electrical potential energy?
Yes, and magnetic potential energy,
too. Electrical charges move under the action of
the forces each exerts on the other and magnets
do the same.
How about elastic forces, like in rubber bands or
springs?
They are really electrical forces between atoms
and molecules. Stretched springs have potential
energy.
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Are all forms of energy kinetic or potential?
According to classical physics, yes. But I showed
that mass is also a form of energy.
What kind of energy is mass?
It is sort of a potential energy, but not due to
force fields like gravitational or electrical
energy are.
What do you mean?
An object of mass m that is not moving and not in
a force field still has an energy equal to mc2,
where c is the speed of light. If it is moving
with speed v, its total energy is
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What are the units of energy?
Energy units English Units the
foot-pound Metric Units the joule
1 joule (J) 1 kilowatt-hour 1 electron volt
(eV) 1.6 X 10-19 J Electron mass 0.511 X 106
eV/c2
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Conversion between Mass and Energy
Nucleus
Rest Energy
Neutron
939.6 MeV
Hydrogen-2
1875.6 MeV
Creating Energy from Mass
Hydrogen-3
2809.4 MeV
Helium-4
3728.5 MeV
Example Fusion of Hydrogen-2 and Hydrogen-3


?

17.6 MeV
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Conversion between Mass and Energy
Beam Particle
Rest Energy
Pion
0.139 GeV
Creating Mass from Energy
Proton
0.938 GeV
Example pion production in proton-proton
scattering
target proton
beam proton
Beam protons kinetic energy must be at least
15/14 times the pions mass energy