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Transcript of Relativity 20130121
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Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
Kari Eloranta
2013Jyvskyln Lyseon lukio
International Baccalaureate
January 24, 2013
Kari Eloranta 2013 Option H: Relativity (22 hours)
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Option H: Relativity (22 hours)
H1 Introduction to relativity
H1.1Frames of Reference
H.1.1 Frame of Reference
All physical measurements are made with respect to a frame of reference.
In Einsteins Special Theory of Relativity, we attach time to all points in acoordinate system.
Frame of Reference in Special RelativityA frame of reference is a coordinate system with respect to measurements aremade. In the Einsteins Special Theory of Relativity, the coordinate system consistsof position coordinates (x,y,z), and time t attached to each coordinate point.
You can think of the coordinate system as an infinite information collectionsystem with synchronised clocks that can record the position and time of anyevent in the system.
Kari Eloranta 2013 Option H: Relativity (22 hours)
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Option H: Relativity (22 hours)
H2 Concepts and postulates of special relativity
H2.1 Inertial Frame of Reference
H.2.1 Inertial Frame of Reference
According to the Newtons first law of motion, an object at rest remains at rest,
and an object in uniform motion continues the motion at constant speed along astraight line unless acted upon an unbalanced force.
Newtons first law is valid only in reference frames that are either at rest or
moving with constant velocity.
Inertial Frame of ReferenceAn inertial reference frame is a reference frame in which Newtons First Law is
valid.Inertial reference frames cannot be in accelerated motion. They are in uniformmotion with respect to other inertial frames of reference.
Kari Eloranta 2013 Option H: Relativity (22 hours)
O i H R l i i (22 h )
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Option H: Relativity (22 hours)
H1 Introduction to relativity
H1.2 Galilean Transformation
H.1.2 Galilean Transformation
Galilean TransformationAssume two inertial frames of reference S and S such that their origins coincide at
time t= t
= 0, and S
moves along the x-axis ofS by speed v. If coordinates(x,y,z) and (x,y,z) specify a position in S and S, respectively, the relationbetween the coordinates is
x
=
x
vt (1)y=y (2)
z= z (3)
This is known as Galilean Transformation.Galilean Transformation is based on the classical addition of velocities. It is validin the inertial frames of reference that move by the velocity v with respect to
each other, such that the velocity is well below the speed of light c.
Kari Eloranta 2013 Option H: Relativity (22 hours)
O ti H R l ti it (22 h )
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Option H: Relativity (22 hours)
H2 Concepts and postulates of special relativity
H2.2 Postulates of Special Relativity
Principle of Relativity
Principle of RelativityThe laws of physics are the same in all inertial frames of reference.
The first postulate is a rational assumption. After all, anything else would notmake sense.
The value of velocity depends on the selected frame of reference. However,acceleration is independent of the choice of a reference frame.
As a result, a physicist will measure the same force independently of his linearuniform motion.
If a frame of reference is in accelerated motion, the measurement of forcechanges.
Coriolis force, for example, is a consequence of a laboratory on Earth not beingan inertial frame of reference.
Kari Eloranta 2013 Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
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Option H: Relativity (22 hours)
H2 Concepts and postulates of special relativity
H2.2 Postulates of Special Relativity
Constancy of the Speed of Light
Constancy of the Speed of LightThe speed of light in vacuum is the same for all observers.
The second postulate is totally against common sense.Nowadays, it is an experimentally confirmed scientific fact.
The consequences of the second postulate lead to amazing things such as time
dilation, length contraction and mass-energy equivalence.
Kari Eloranta 2013 Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
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Option H: Relativity (22 hours)
H3 Relativistic Kinematics
H3.4 Lorentz factor
H3.4 Lorentz factor
Lorentz factorThe expression
=1
1 v2
c2
(4)
is called the Lorentz factor.
From the expression we see that when the speed v of the object is less than
0.1c, the Lorentz factor is 1. As the speed v approaches the speed of lightc, the Lorentz factor approaches .
Kari Eloranta 2013 Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
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Option H: Relativity (22 hours)
H3 Relativistic Kinematics
H3.4 Lorentz factor
H3.4 Asymptotic Behaviour of Lorentz factor
01
2
3
45
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v/c
The asymptotic behaviour of the Lorentz factor. As the speed v approaches thespeed of light c (v c), the Lorentz factor approaches infinity ().
Kari Eloranta 2013 Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
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Option H: Relativity (22 hours)
H4 Some Consequences of Special Relativity
H4.3 Relativistic Mass and Energy
H4 Mass Energy Equivalence
Albert Einstein proposed 1905 that mass is just one form of energy.
Mass Energy EquivalenceThe total energy of an object at rest is
Erest=m0c2 (5)
where m0 is the rest mass of the object, and c= 2
.99810
8
m s1
the speed oflight in vacuum.
Mass and energy are different manifestations of the same thing.
In natural processes, a small amount of mass can be converted into largeamount of energy, and vice versa.
Mass is an invariant quantity: it does not change as the speed of the objectincreases.
Kari Eloranta 2013 Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
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p y ( )
H4 Some Consequences of Special Relativity
H4.3 Relativistic Mass and Energy
H4 Relativistic Kinetic Energy
A moving object has kinetic energy.
At relativistic speeds, the inertia of a moving object increases.
Relativistic Kinetic EnergyThe relativistic kinetic energy of an object is
Ekin=
11 v
2
c2
1
m0c2= m0c2m0c2= (1)m0c2 (6)
where v is the speed of the object, c= 2.998108
m s1
the speed of light invacuum, m0 the rest mass of the object, and the Lorentz factor.
As the speed of the object v approaches the speed of light c, the kinetic energyEkin approaches infinity .
Kari Eloranta 2013 Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
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p y ( )
H4 Some Consequences of Special Relativity
H4.3 Relativistic Mass and Energy
H4 Total Relativistic Energy
Total energy of an object is its rest energy plus kinetic energy.
Relativistic Total EnergyThe relativistic total energy of an object is
Etotal=
Erest+
Ekin=m0c
2=
1
1 v2
c2m0c
2
(7)
where m0 is the rest mass of the object, Lorentz factor, v the speed of theobject, and c= 2.998108m s1 the speed of light in vacuum.
As the speed of the object approaches to speed of light, the total energyapproaches infinity.
Kari Eloranta 2013 Option H: Relativity (22 hours)
Option H: Relativity (22 hours)
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H4 Some Consequences of Special Relativity
H4.3 Relativistic Mass and Energy
H6 Relativistic Momentum and Energy
Linear momentum of an isolated system is conserved.
If linear momentum is conserved in one inertial frame of reference, it has to be
conserved in other frames of reference as well.From the assumption above it follows that, at relativistic speeds, the expressionof linear momentum has to be modified.
Relativistic MomemtumpThe relativistic linear momentum is
p= m0u=1
1 u2
c2
m0u (8)
where is the Lorentz factor, m0 rest mass of the object, u speed of the object,and c= 2.998108m s1 the speed of light in vacuum.
From the expression we see that a constant net force acting on an object causesa decreasing acceleration at relativistic speeds.
Kari Eloranta 2013 Option H: Relativity (22 hours)