Special Theory Of Relativity

Relative Motion

How fast is Car A going relative to Car B?

v = 24 m/s v = 10 m/s

Car A Car B

How fast is Car A going relative to brick wall?

How fast is Car B going relative to brick wall?

How fast is Car B going relative to Car A?

(24 m/s)

(14 m/s)

(10 m/s)

(-14 m/s)

Special Theory Of Relativity

All motion is relative!! The measurement of position and

velocity depends on the what the object is doing and on what the observer is doing! (Frame of Reference!!)

There is no preferred frame of reference. All Frames of reference are equally valid.

Special Theory Of Relativity

Since all motion is relative maybe TIME is “relative” too!!

What occurs in one reference frame, may occur in a different order in another reference frame!!!

Simultaneous in one reference frame is not always simultaneous in another reference frame.

Special Theory Of Relativity

1. All the laws of nature are the same in all uniformly moving reference frames.

2. Speed of light is a constant, independent of the motion of the light source or the observer.

2 postulates of Special Relativity:

Space - Time

Einstein thought these two ideas (Space & Time) as one entity!

Light travels so fast that it does all of it’s moving through space and NONE through time!

You and I are turtles and move so slowly that we do most of our moving through time and very little through space!

Space-Time lineLow speed

High speed

Light spends all of its “time” here moving through space – it experiences no time!

Humans spend all their “time” moving through time – they only experience time!

Time Dilation

The constancy of the Speed of light has some very special ramifications.

One of these is called Time Dilation.

“Moving clocks run slower”

Time DilationWhat is time?

Time is something measured by a clock… but what does a clock measure?

A clock keeps tracks of many repeated motions and indicates the passage of time by the sequential number of these repeated motions!

“Time is how the universe keeps everything from happening at once.”Anonymous

Time Dilation

A simple clock can be made by bouncing a light beam between a light source, mirror, and a detector.

source detector

photon

Mirror

Time Dilation

Why does this happen?

You embark on a 2.0 year journey to the distant stars travelling at 0.99999 C. Upon your return everyone that has been left behind has aged by 50 – 60 years!. You are now “younger that all your friends and your parents!

Imagine that in the distant future being able to ride on a spaceship that can travel close to the speed of light.

Remember these effects only become noticeable near the speed of light!

Time Dilation

Twin Paradox:

Imagine a set of twins. One jumps on board a high speed space ship and is gone from the earth for 2 years ship time.

Upon his return his twin brother greets him, but alas his stay at home twin is an old man!

Is this possible?

Time Dilation

Through Newton’s description of motion we sent men to the moon, he didn’t quite get it right!

As objects approach the speed of light Newton’s equations do not work!

Einstein’s modification to Newton’s equations works for all speeds.

Time Dilation

Relativistic factor:

Where:v = speed of the object

C = speed of light

1 - v2

C2

d = c x t

Time DilationTime Dilation equation:

Where:

to = time as measured by a observer moving with the clock

C = speed of light

1 - v2

C2

t =

to

t = time as measured by a stationary observer

v = speed of the moving observer

Time Dilation

Let us go through some examples on the overhead.

Any real proof to moving clocks run slower?

Length Contraction

One of the weird things about relativity is how it changes time and length measurements. It’s all about perception and reference frames.

Remember from the relativity principle: People travelling on high speed spaceship can say that they are stationary and it is the earth that is moving away from them at a high speed.

From the point of view of those travelling at high speeds they are stationary and space is moving past them.

Length Contraction

Lorentz Contraction:

Moving object contract in the direction of motion.

This is why the Michaelson-Morley experiment. Failed!

v = 0 v = 0.80 C v = 0.999 C v = 1.0 C

?

How a high speed space ship will look to a stationary observer as it reaches different speeds.

Length Contraction

The reason that it takes so little time for high-speed space ships to travel across space is because space has been contracted (shorten) in the direction of motion.

Moving at high speeds actually shortens the distance that has to be travelled!

Distance for a stationary observer …

is only this far for a high speed spaceship.

Length Contraction

Lorentz Contraction:

Where:

v = speed of the object

C = speed of light

1 - v2

C2L = Lo x

L = length of object measured when moving

Lo = rest length of object

Length Contraction

Let us go through some examples.

Addition of Velocities

One more Problem comes up. Let us pretend to fire a missile from a plane.

v = 250 m/s relative to the ground

missile

v = 460 m/s relative to the plane

Ground

How fast is the missile travelling relative to the ground?

Ans: 710 m/s

Addition of Velocities

Now let us try this at very high speeds.

v = 0.75 C relative to some stationary observer

missile

v = 0.60 C relative to the space ship

If we use Newton’s equations:v = 0.75 C + 0.60 C

v = 1.35 C relative to a stationary observer?

Is this possible?

Addition of Velocities

Einstein realized that Newton’s equations no longer worked for high speed objects!

Where:

u = speed of the object relative to the moving ship

C = speed of light

He proposed a new equation:

1 + vuC2

v + uv’ =

v = speed of the ship relative to a stationary observer

v’ = speed of the object relative to a stationary observer

Addition of Velocities

Let us go through some examples on the overhead.

As soon as Einstein developed his ideas about relativity he realized that out of equations came a very special formula.

This equation is not just an artifact. It is the real thing. Energy and mass are related to each other. “Two sides of the same coin”.

ENERGY can be transformed into MASS and MASS can be transformed into ENERGY!

Mass & Energy

Mass & EnergyFor 200 years scientists were wondering how our sun could keep “burning” without running out of fuel!

The answer lies in the conversion of hydrogen into helium and the loss of a little mass each time.

The sun is actually losing 4 million tons of mass every second!

Don’t worry … the sun has a total mass of 2 x 1030 kg … which means it will last for about another 5 billion years!!

Energy released

Less mass

Mass & Energy

Back to relativity… How do make something move?

… by doing work or applying energy to an object

In order to make an object move faster you need to apply more energy. Now here is where it gets s little interesting.

At low speeds most of the energy will go into increasing the object’s speed. However some of the energy that the object receives will be converted to mass. The object will become more massive.

Mass & Energy

At high speeds most of the energy will go into increasing the object’s mass and very little will go into making the object travel faster

As the object becomes more massive, it becomes harder to move faster. Giving it even more energy will only move it a little faster, but will make it even more massive.

In order to make an object travel at the speed of light you would need infinite amount of energy!!

Therefore no object that has mass can ever travel at the speed of light!

Mass & Energy

1 - v2

C2

m =

mo

Where:

mo = rest mass of the object

m = mass of the moving object

Mass Increase:

Mass and Energy

Is there any proof to this mass-energy exchange?

1. Our Sun

2. In particle accelerators, charged particles are accelerated close to the speed of light. These particles momentum are measure to be much greater than their rest mass indicated then to be.

3. Nuclear Bombs produce many 50 Megatons TNT equivalent explosions from only a few kilograms of Hydrogen.

Mass and Energy

How much energy is stored in 1.2 kg of water if it could be converted completely to energy?

E = mc2

E = 1.2 x (3.0 x 108)2

E = 1.1 x 1017 J(Enough energy to power 1000 locomotive engines for 530 years!)

Mass and Energy

To supply the yearly average of 5 x 1011 J of energy per person in Canada we need:

20 tones of Coal

50000 cubic meters of Natural gas

11000 L of Gasoline

0.0056 g of mass conversionor