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CHAPTER 27

EINSTEIN’S Relativity

LENGTH CONTRACTION

TIME DILATION

SIMULTANEITY

RELATAVISTIC MASS

GENERAL RELATIVITY

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The transformation equations before

Einstein �� = � − �� �� = �

� =

�� = �

These are the Galilean Equations that allow

observers to compare observations in two

different frames moving relative to each

other with constant velocity.

From these equations we get the addition

of velocities transformation equation.

� = − �

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Consider the two speeds, each measured in

a different reference frame:

A policeman on the side of the road

measures your speed with radar:

Another policeman driving towards you at

speed � measures your speed: �

The two speeds are related by this equation

� = − �

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Consider that you are going 80 mph and the

moving policeman is going 20 mph (towards

you). Take the direction the policeman is

going as the � direction.

Use � = − �

Then � = 20 �ℎ = −80 �ℎ

What does the moving policeman measure

for your speed?

� = �−80 �ℎ − 20 �ℎ� = 100 �ℎ

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Einstein’s postulates for the Special Theory

of Relativity:

1. Fundamental laws of physics are identical

for any two observers in uniform relative

motion.

2. The speed of light is independent of the

motion of the light source or observer.

These postulates cannot be satisfied using

the Galilean Equations.

However Einstein found that the following

equations worked.

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�� = ��� − ��� �� = �

� =

�� = � �� −�����

� = 1�1 −����

These are the Lorentz Transformation

Equations.

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LENGTH CONTRACTION

Consider an observer on a railroad car with

a rod of length L0. (L0 is called proper length

because it is at rest relative to the observer.)

He measures the length of the rod to be

��� −��� =�

Use the Lorentz equations to get

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��� = ���� − ����

And

��� = ���� − ����

Then putting these in the equation

For proper length.

� = ���� − ���� − ���� − ���� � = �!��� −��� − ���� − ���"

If the observer on the ground measures the

far end and near end of the rod at the same

time

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�� = ��

Then

� = ���� −��� = ��

Or

� = #$% and � > 1

So the observer on the ground with the rod

moving past in the x direction measures the

rod to be shorter than what is measured by

the observer at rest relative to the rod and

on the car.

Length Contraction is a prediction of the

Lorentz Equations.

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TIME DILATION

Again we will find time dilation a different

way than the book.

Observer on railroad car moving in direction

with firecrackers and another observer on

ground.

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The firecracker at �� explodes and then

some time later the one at �� explodes.

We will consider the time between the two

events.

The time interval between the two events

as measured by the observer on the ground

will be

∆� = �� −��

The transformation equation for time is

�� = � �� −�����

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Which for the other observer is

� = � (�� +����� *

Therefore we get for �� and ��

�� = � (��� +������ *

And

�� = � (��� +������ *

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Put these into the time interval equation

∆� = � (��� +������ * − � (��� +������ *

Or ∆� = � +���� − ���� + ��� ���� −����,

Or ∆� = � +∆�� + ��� ���� −����,

The time interval is different for the two

observers.

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For simplicity consider the two firecrackers

at the same place

��� =��� =��

Then

∆� = � +∆�� + ��� ��� −���,

Or ∆� = �∆��

And � > 1

The time interval for the observer on the

ground with the events moving relative to

her is longer.

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MOVING CLOCKS RUN SLOWLY

If an observer is moving relative to a clock

the interval between ticks will be longer.

This is Time Dilation

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Einstein / Lorentz Transformation Equations

1. Satisfy Einstein’s Second Postulate

2. Predict Length contraction

3. Predict Time Dilation

But are these correct?

Theories are not worth much if they are not

supported by experiment!

Must have an experiment to prove!!!

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Muon “lifetime” Experiment

Muons are particles created high in our

atmosphere.

They rain down continuously at high

velocity at approximately

� = 2.994�100 /2

If at rest they only exist for approximately

3 = 2�104526�7892

The experiment is to set up a detector at

the top of a mountain and stop the muons

in the detector and measure how long they

exist.

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Then ask how far down would they have

traveled if they had not been stopped in the

detector.

distance = (speed) x (time they exist)

9 = �2.994�100 /2��2�10452� = 600

If the mountain is 2000m tall, in the valley

at the bottom of the mountain should

find very few if any muons.

Move the apparatus to the bottom of the

mountain and measure the number of the

same type muons that reach sea level.

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The experiment showed that almost as

many reached sea level as passed through

the atmosphere at the level of the top of

the mountain.

Why?

When moving relative to us the observer

they exist not for

3 = 2�104526�7892

But for

3 = �� = 3;1 −<2.994�1003�100 >�

3 = 32�10452

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Thus distance traveled will be

9 = �2.994�100 /2��32�10452� 9 = 10,000

Well below sea level, thus almost all muons

should reach sea level.

Confirming Time Dilation

What about Length Contraction?

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