Special Relativity

Option H: Relativity (22 hours) H1: Introduction to Relativity

Option H: Relativity (22 hours)

H1: Introduction to Relativity

© Kari Eloranta2013

Jyväskylän Lyseon lukio

International Baccalaureate

January 24, 2013

© Kari Eloranta 2013 Option H: Relativity (22 hours) H1: Introduction to Relativity


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 Einstein’s 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 Einstein’s 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.



H2 Concepts and postulates of special relativity

H2.1 Inertial Frame of Reference

H.2.1 Inertial Frame of Reference

According to the Newton’s 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.

Newton’s first law is valid only in reference frames that are either at rest ormoving with constant velocity.

Inertial Frame of ReferenceAn inertial reference frame is a reference frame in which Newton’s First Law isvalid.

Inertial reference frames cannot be in accelerated motion. They are in uniformmotion with respect to other inertial frames of reference.



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 attime t = t ′= 0, and S ′ moves along the x-axis of S 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 − v t (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 toeach other, such that the velocity is well below the speed of light c.




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.




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 timedilation, length contraction and mass-energy equivalence.



H3 Relativistic Kinematics

H3.4 Lorentz factor

H3.4 Lorentz factor

Lorentz factorThe expression

γ=

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 than0.1c , the Lorentz factor is γ≈ 1. As the speed v approaches the speed of lightc , the Lorentz factor approaches ∞.



H3 Relativistic Kinematics

H3.4 Lorentz factor

H3.4 Asymptotic Behaviour of Lorentz factor

0

1

2

3

4

5

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

γ

The asymptotic behaviour of the Lorentz factor. As the speed v approaches thespeed of light c (v → c), the Lorentz factor γ approaches infinity (γ→∞).



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 = 2.998×108ms−1 the speed of

light 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.





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 =

1√

1− v2

c2

−1

m0c2= γm0c2

−m0c2= (γ−1)m0c2 (6)

where v is the speed of the object, c = 2.998×108ms−1 the speed of light in

vacuum, m0 the rest mass of the object, and γ the Lorentz factor.

As the speed of the object v approaches the speed of light c, the kinetic energyEkin approaches infinity ∞.





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 = γm0c2=

1√

1− v2

c2

m0c2 (7)

where m0 is the rest mass of the object, γ Lorentz factor, v the speed of theobject, and c = 2.998×10

8ms−1 the speed of light in vacuum.

As the speed of the object approaches to speed of light, the total energyapproaches infinity.





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 beconserved 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 Momemtum ~p

The relativistic linear momentum is

p = γm0u =

1√

1− u2

c2

m0u (8)

where γ is the Lorentz factor, m0 rest mass of the object, u speed of the object,and c = 2.998×10

8ms−1 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.


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Special Relativity

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