Special Relativity

The World at the Speed of Light.

Einstein’s Contribution.

VCE PhysicsUnit 3

Topic 3

To achieve the outcome the student should demonstrate the knowledge and skills to: describe Maxwell’s prediction that the speed of light depends only on the electrical and magnetic properties of the medium it is passing through and not on the speed of the source or the speed of the medium;contrast Maxwell’s prediction with the principles of Galilean relativity (no absolute frame of reference, all velocity measurements are relative to the frame of reference);interpret the results of the Michelson Morley experiment in terms of the postulates of Einstein’s special theory of relativity;-the laws of physics are the same in all inertial frames of reference-the speed of light has a constant value for all observers;compare Einstein’s postulates and the postulates of the Newtonian model;use simple thought experiments to show that -the elapse of time occurs at different rates depending on the motion of the observer relative to the event;-spatial measurements are different when measured in different frames of reference;explain the concepts of proper time and proper length as quantities that are measured in the frame of reference in which the objects are at rest;explain movement at speeds approaching the speed of light in terms of the postulates of Einstein’s special theory of relativity;model mathematically time dilation, length contraction and mass increase with respectively the equations t = toγ, L = Lo/γ, m = moγ where γ = 1/(1-v2/c2)1/2 explain the relation between the relativistic mass of a body and the energy equivalent according to Einstein’s equation E = mc2

explain the equivalence of work done to increased mass energy according to Einstein’s equation E = mc2

compare special relativistic and non relativistic values for time, length and mass for a range of situations.

Unit Outline

Galilean Relativity

FRAMES OF REFERENCEFrames of reference can be of 2

types:1. Inertial Frames. These are

systems (or groups of objects) which are either at rest or moving with constant velocity.

2. Non Inertial Frames. These are systems which are accelerating.

One of the earliest of the great minds to ponder motion, both on Earth and in the heavens, was Galileo Galilei.He developed the principle of Galilean Relativity. This is best shown with a simple example:

Galileo Galilei1564 - 1642

Generalizing these observations Galileo postulated his relativity hypothesis: any two observers in inertial frames of reference with respect to one another will obtain the same results for all mechanical experiments.

Imagine an observer in a house by the sea shore and another in the windowless hull of a ship. Neither will be able to determine that the ship is moving at constant velocity by comparing the results of experiments done inside the house or on the ship. In order to determine motion these observers must look at each other.

There is no absolute inertial frame of reference: all velocity measurements are relative to the frame of reference.

Galilean Motion In Galileo’s world, the idea of relative motion is clearly understood.

This can be shown with a simple example.

But, an observer standing beside the track, sees the ball moving to the right at 30.0 ms-1.

The boy in the carriage sees the ball travel away from him at 5.0 ms-1

A train carriage is travelling to the right at a constant velocity of 25.0 ms-1.A boy standing in the carriage throws a ball to the right at a constant velocity of 5.0 ms-1.

So what is the ball’s “correct” speed ? 5.0 ms-1 or 30.0 ms-1?

BOTH answers are CORRECT. There is no single “correct” answer. The speed of an object depends on where the observer is when the speed was measured.

VBALL = 5.0 ms-1

Stationary observer

VTRAIN = 25.0 ms-1

Remember, according to Galileo, there is no absolute inertial frame of reference: all velocity measurements are relative to the frame of reference.

Isaac Newton

Law 1 (The Law of Inertia)A body will remain at rest, or in a state of uniform motion, unless acted upon by a net

external force.Law 2 The acceleration of a body is

directly proportional to net force applied and inversely

proportional to its mass. Mathematically, a = F/m more commonly written as F = ma

Law 3 (Action Reaction Law)For every action there is an

equal and opposite reaction.

These Laws explained Galilean relativity and using Newton's laws, physicists in the 18th and 19th century were able to predict the motions of the planets, moons, comets, cannon balls, etc.

Isaac Newton, aged 26

The next great mind to influence mankind’s understanding of the operation of the universe was Isaac Newton (1642 – 1727), when he developed his 3 laws, first mentioned in his 1687 book Philosophiae naturalis principia mathematica (or just Principia).

In classical Newtonian mechanics, time was universal and absolute.

The Clouds Gather For more than two centuries after its inception (in about the 1680’s), the Newtonian view of the world ruled supreme, to the point that scientists developed an almost blind faith in this theory. And for good reason: there were very few problems which could not be accounted for using this approach.

Newton aged 38

Nonetheless, by the end of the 19th century, new experimental evidence, difficult to explain using the Newtonian theory, began to accumulate, and the novel theories required to explain this data would soon replace Newtonian physics.

19th Century CloudsIn 1884 Lord Kelvin (of temperature scale fame) in a lecture delivered in Baltimore, Maryland, mentioned the presence of “Nineteenth Century Clouds'' over the physics of the time, referring to certain problems that had resisted explanation using the Newtonian approach.

Among the problems of the time were: a) Light had been recognized as a wave, but the properties

(and the very existence!) of the medium that conveys light appeared inconsistent.

b) The equations describing electricity and magnetism were inconsistent with Newton's description of space and time.

c) The orbit of Mercury, which could be predicted very accurately using Newton's equations, presented a small but disturbingly unexplained discrepancy between the observations and the calculations.

d) Materials at very low temperatures do not behave according to the predictions of Newtonian physics.

e) Newtonian physics predicted that an oven at a stable constant temperature has infinite energy.

Lord Kelvin

The Revolution

The first two problems concerning the nature of light and electricity and magnetism required the introduction of the Special Theory of Relativity. The third item concerning Mercury’s orbit required the introduction of the General Theory of Relativity. The last two items low temperature materials and infinite energy ovens can be understood only through the introduction of a completely new mechanics: quantum mechanics.

The first quarter of the 20th century witnessed the creation of the revolutionary theories which explained these phenomena. They also completely changed the way we understand Nature.

Albert Einstein

The new theories that superseded Newton's had the virtue of explaining everything Newtonian mechanics did (with even greater accuracy) while extending our understanding to an even wider range of phenomena.

Mercury

Maxwell’s Contribution • Throughout 1700’s and 1800’s, many individual laws about electricity and magnetism had been discovered, such as Coulomb’s law ofelectrostatic force.

One interesting consequence of Maxwell’s unification is that you can calculate the velocity of electromagnetic waves based onproperties of capacitors and inductors.

221

04

1

d

qqF

James Maxwell

00

1

c

James Clerk Maxwell (1831-1879) had, by 1855, unified some laws and finally by 1873 had found that all of these laws could be summarised by four partial differential equations. A triumph of unification!(Which of course is the holy grail of Physics)

In Maxwell’s own words:This velocity is so nearly that of light, that it seems we have strong reasons to conclude that light itself (including radiant heat, and other radiations if any) is anelectromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

Charles Coulomb

c = Speed of EM Wavesμ0 = Permeability of free spaceε0 = Susceptibility of free space

19th Century Physics• Around this time, Physicists were trying to find a way to measure the

ABSOLUTE VELOCITY of an object relative to some fixed point which was

COMPLETELY AT REST.

• But what, in our universe, is completely at rest ?

Certainly not the Earth, which as well as spinning

on its axis at 500 ms-1 (1800 kmh-1), travels around

the sun at 30 kms-1 (108,000 kmh-1).

The sun, of course, is in orbit around the centre

of our galaxy at 250 kms-1 (900,000 kmh-1).

And our galaxy is in some kind of orbit amongst

the other galaxies (velocity unknown).

SO MUCH FOR USING THE EARTH AS A

STATIONARY LABORATORY.

The Ether By the 1880’s scientists knew that waves transferred energy from one place to another and their movement depended upon them travelling through a MEDIUM (water waves in water, sound waves in air and other materials).

The ether was a hypothetical medium in which it was believed that electromagnetic waves (visible light, infrared radiation, ultraviolet radiation, radio waves, X-rays), would propagate.

This led them to believe that ALL waves required a medium for travel, and so to development of the concept of the luminiferous ether, (or aether) which was the name given to the medium through which light supposedly travelled from the sun to earth.

The Speed of LightIn 1887, Albert A. Michelson and Edward W. Morley working at the Case School in Cleveland, Ohio, tried to measure the speed of the ether, (or more precisely the speed of the Earth through the ether).

They expected to find the speed of light (symbol, c) differed depending on its direction with respect to the “ether wind”. This result would accord with Galilean relativity.

Ether Speed = vThe result of the Michelson-Morley experiment was that the speed of the Earth through the ether (or the speed of the ether wind) was zero.

Speed = c

Speed = c + v

Expected result

Ether Speed = v

Speed = c

Speed = c

Actual result

Therefore, they also showed that there is no need for any ether at all, and it appeared that the speed of light (in a vacuum) was independent of the velocity of the observer!

Michelson explained his experiment to his children this way: two swimmers race; one struggles upstream and back while the other swims the same distance across and back. The second swimmer will always win, if there is any current in the river.

Michelson Morley in Detail

The interference fringes produced by the two

reflected beams were observed in the telescope. It

was found that these fringes did not shift when the

table was rotated. That is, the time required to travel

one leg of the interferometer never varied with the

time required to travel its normal counterpart. They

NEVER got a changing interference pattern.

Using an interferometer floating on a pool of mercury, they tried to determine the existence of an ether wind by observing interference patterns between the two light beams. One beam travelling with the "ether wind" as the earth orbited the sun, and the other at 90º to the ether wind.

The experiment was set up using a “monochromatic” (single colour) light source split into two beams.

ether

The result proved to be an extremely perplexing and frustrating to the physicists of the day who firmly believed in the ether theory.The result proved, beyond doubt, that the speed of light is CONSTANT, no matter how fast an observer was travelling when measuring it.In other words, it led to the death of the ether concept and, more importantly, the death of Galilean Relativity

NO CHANGE IN THE PATTERN COULD EVER BE DETECTED WHEN THE EQUIPMENT TURNED THROUGH 900

It took nearly 20 years to develop the theory to match this experimental result.

Michelson Morley in Detail 2

Michelson and Morley repeated their experiment many times up until 1929, but always with the same results and conclusions.

Michelson won the Nobel Prize in Physics in 1907. Probably the only prize ever awarded for a failed experiment.

Incident Light

Double Slit

Interference Pattern

The travel times for the two beams were compared in a very sensitive manner.If the travel times were different the two beams, when combined, would have produced an “interference pattern”. This is the same as the pattern produced when a monochromatic beam of light is allowed to pass through two narrow slits.

Newton versus MaxwellUnder Galileo and Newton, the speed of light would vary depending the inertial frame of reference.

Whilst under Maxwell, the speed of light is constant no matter what the inertial frame of reference.

The Speed of Light (c) is

3 x 108 ms-1

10 ms-1

No, its c - 10

100 ms-1

1000 ms-1

No, its c - 1000

c = 3 x 108 ms-1

10 ms-1

I agree it’s c

I agree it’s c

100 ms-1

1000 ms-1

I agree it’s c

No, its c - 100

Einstein’s InsightIt was Einstein who finally found an answer to the seemingly unbelievable result – that the speed of light in inertial frames of reference is always the same.The answer was to change the understanding of the term simultaneity. The figures to the left, seen from two different

inertial frames, help clarify the concept of simultaneity:Fig 1:In the inertial frame of the wagon, the lamps are switched on simultaneously and the two light impulses reach the girl at the same time.

Fig 2:In the inertial frame of the observer outside the wagon, it seems that the left lamp is switched on first, although for the girl in the wagon the lamps are switched on simultaneously.

Two physical events that occur simultaneously in one inertial frame are only simultaneous in any other inertial frame if they occur at the same time and at the same place.

This means:TIME IS RELATIVE!

Fig 1

Fig 2

Introducing RelativitySpecial Relativity deals with large velocity differences between frames of reference (Inertial Frames).General Relativity deals with large acceleration differences between frames of reference (Non inertial Frames)

Einstein developed the theory of Special Relativity in 1905 and the more comprehensive and far more complex theory of General Relativity about 10 years later.

Speeds of objects

Inertial Frames Non inertial Frames

Very much less than c Newton’s Laws

Newton’s Laws Plus Fake Forces

Close to c Special Relativity General Relativity

At low speeds, Newton’s laws are adequate to explain motion.But the relativity theories need to be applied to objects travelling at or near c, the speed of light.

Special Relativity• The theory of Special Relativity was

developed by Einstein in 1905 when, as a 26 year old, he was working as a clerk in the Swiss Government Patents Office.

• Basically the theory states:1. The laws of physics are identical for all

observers, provided they are moving at constant velocity with respect to one another, i.e., they are all in inertial frames of reference.

2. The SPEED OF LIGHT is CONSTANT. This is true no matter how fast the observer is travelling relative to the source of light.This theory was completely at odds with the classical physics of Aristotle, Galileo and Newton.

Einstein – The Patent Clerk

Einstein’s Early HistoryAn only child, Albert Einstein was born in

Ulm, Germany, on the 14th of March 1879.

His parents - Herman, an electrical engineer,

and Pauline, were worried their son may be

retarded, as he did not speak his first words

until after his 3rd birthday.Earliest known picture of Einstein - as a 3 year old

Einstein- aged 14

In 1894, as a 15 year old, he was expelled from Catholic College for disruptive behaviour.

In 1896, he managed to talk himself into a place at the Swiss Federal Polytechnic Academy in Zurich, graduating in 1900 (at age 21), as a secondary school teacher of Maths and Physics.

At age 23 he married his university sweetheart Mileva Maric

From Student to ProfessorEinstein and 1st wife

Mileva

In 1908, he submitted a further paper to the University of Bern leading to an offer of employment as a lecturer.In 1909 he received an offer of an associate professorship in physics at the University of Zurich.He jumped into various university professorships throughout German speaking Europe, finally landing Europe’s most prestigious post as physics professor at Kaiser-Wilhelm Gesellschaft in Berlin.

Einstein did not take up a teaching position immediately, but in 1902 obtained a position as a Patent’s Clerk at the Swiss Patents Office in Bern where he worked until 1909.During his time there he completed an astonishing number of papers on theoretical physics, mostly completed in his spare time.He submitted one of his papers to the University of Zurich for which he obtained his PhD degree in 1905.

Special Relativity• After studying the results of the Michelson -

Morley experiments, Einstein proposed the following:

• THE SPEED OF LIGHT IS ALWAYS THE SAME, REGARDLESS OF WHO MEASURES IT AND HOW FAST THEY ARE GOING RELATIVE TO THE LIGHT SOURCE.

• From this simple statement a number of startling consequences arise:

Time Dilation

• It requires that, depending on the motion of an observer, time must pass at different rates.

• Two observers, one stationary, the other moving near the speed of light, observe the same event.

• In order for each to get the same speed for the event, each must see it occur during different time intervals.

• The faster the observer travels the slower the rate at which his time appears to pass to stationary observer.

The first of these consequences is known as “Time Dilation”

Time Dilation

A photon of light (travelling at c) is bouncing backwards and forwards between two parallel mirrors. One back and forth motion of the photon represents one tick of the clock.

Hence, if the photon is to strike the mirrors,, the space observer requires it to travel on a diagonal path, as shown.

In order to demonstrate this change in the rate at which time passes, let us produce a simple clock.

“1 Tick”

In the time, we (standing on Earth), see the photon bounce back and forth once, the space observer sees the Earth move a little way along its orbit path.

Mirror

Mirror

Earth Earth Earth

Photon

An observer, stationary in space with respect to the Sun, sees the Earth (with its attached “clock”), go zooming past on its orbit around the sun.

Time Dilation

Short Distance Speed = c

Earth Earth Earth

Long Distance

Speed = c

Thus, MOVING CLOCKS RUN SLOW.

This same argument holds true for the earth bound observer, who would see the space observer’s clock running slow.

The space observer thus concludes the Earth clock runs slow compared to his clock.

Clock as seen by Earth bound observer

Clock as seen by Space observer

Since the photon MUST travel at the Speed of Light, c, the only logical outcome for the space observer is to conclude that the photon on Earth takes a LONGER TIME to cover the APPARENTLY LONGER DISTANCE it needs to travel.

Remember, Speed = distance time

and this gets bigger

this must also get bigger

If this has to stay the same

Time Dilation

• The formula has a number of consequences:• If v << c, the term v2/c2 approaches zero and

the square root term approaches 1. • Thus t = to and no change in time (the rate at

which time passes) is observed. • As v approaches c (say v = 0.9c), the

stationary observer sees the moving observer’s clock tick over only 0.4 sec for every 1 second on his own clock.

• If v = c, the term v2/c2 = 1 and the square root term becomes zero. Dividing a number by zero equals infinity.

• Thus, when v = c the time interval becomes infinite. In other words, time stops passing.

where: t = moving observer’s time

as measured by the stationary observer.

to = time measured by stationary observer’s clock. (“proper time”) v = speed of moving

observer. c = Speed of Light.

The mathematical representation of Time Dilation is shown in the formula:

t = γto

My Clock

My observation of the moving clock

where 1

1 -c2

v2γ =

to

1 -c2

v2t =So

γ is called the “Lorentz Factor”

The Twins ParadoxTwins, Adam and Eve, are thinking how they will age if one of them goes on a space journey, travelling at say 0.866c. Will Eve be younger, older, or remain the same age as her brother if she does a round trip of some years duration ?Assume that Adam and Eve’s clocks are synchronized before Eve leaves. At 0.866c, Adam will “see” Eve’s time pass at exactly half the rate his time passes.So when Eve returns, she will have aged by 1 year for every 2 years Adam has aged. Thus, Eve is younger than Adam. However, can you turn the discussion around and say that Eve has been at rest in her space-ship while Adam has been on a "space journey" with planet Earth? In that case, Adam must be younger than Eve at the reunion!Adam is at rest all the time on Earth, i.e., he is in the same inertial frame all the time, but Eve is not - she will have felt forces when her space-ship accelerates and retards, and Adam will not feel such forces. So the argument is not an interchangeable one. The travelling twin is the younger upon their reunion. P.S. Eve's space-ship has to consume fuel, which means that it costs to keep yourself young!

Length Contraction

An observer sees two set squares, one stationary in his inertial frame,

v = 0 v = 0.8c

y

x

Remember the stationary observer sees the moving “frames” clock running slow.To get the same value for c in each frame, he must measure the length of the set square (in the direction of travel) to be shorter than his own stationary ruler.

Remember: Speed = distance time

with this having become

less

This must also be

lessto keep this the same

IMPORTANT NOTE: The length contraction only occurs in the direction of travel (x direction) and measurements at right angles to that direction are unaffected ! (no contraction in the y direction)

How does the speed difference affect the apparent size of the set square ?

the other in an inertial frame moving near the speed of light.

The 2nd consequence of light having a constant speed is Length Contraction.

Length ContractionLand available on Mars and its free !The Martians send an advertising rocket to fly past Earth.

What is the best speed for the rocket so stationary Earthlings can read the sign ?

v = 0.1c

v = 0.86c

v = 0.90c

v = 0.98c

Length Contraction

The formula has a number of consequences:

1. If the v << c, the square root term approaches 1 and the length is unaffected, ie. L = Lo

2. As v approaches c, v2/c2 approaches 1 and the square root term approaches 0. Thus, the length approaches 0 ie. L = 0.

The mathematical representation of Length Contraction is shown in the formula. L = Lo/γ

Lo( 1 -c2

v2

So, L = )

Where: L = Length of moving

object as measured by stationary

observer. Lo = Length of stationary

object measured by stationary observer. (“Proper Length”) v = speed of moving

object. c = Speed of Light.

So, a photon of light travelling at c from the Sun to the Earth makes the journey in no time and travels no distance !!!!!!

The moving observer’s view of the length contracted world

The stationary observer’s view of the length contracted Superman

Mass DilationThe third effect of the invariance of the speed of light is mass dilation. As the speed of an object increases so too does its mass !!!!!!

Under Newton, mass is an absolute quantity for each object and it is conserved, never changing for each object.This invariance of mass is the basis of Newton’s 2nd Law (F = ma), and our own every day experience seems to verify that mass is absolute.

Under Einstein mass is whatever we measure it to be. We must use an operational definition for mass. He showed that the mass of an object depends on how fast the object is moving relative to a stationary observer.

Mass – Newton v EinsteinNewtonian physics gives good results at speeds less than 10% of the speed of light.

The mass of an object does not change with speed, it changes only if we cut off or add a piece to the object .

F = ma means that to accelerate a mass requires a force, by supplying sufficient force you can make an object go as fast as you like.

Kinetic Energy = ½mv2, since mass does not change an increase in KE means an increase in speed.

Einstein’s relativity deals with faster speeds.

As an object moves faster its mass increases. (As measured by a stationary observer).

Mass approaches infinity as speed approaches c. To reach c would require infinite force.

Since mass changes with speed, a change in K.E. must involve both a change in speed and a change in mass.At speeds close to c most of the change occurs to the mass.

Mass - How Fast, How Heavy ?

0 20 40 60 80 100

m0

2m0

4m0

6m0

% of Speed of LightSpeed of object as seen by a

stationary observer

Ap

par

ent

Mas

s

An object travelling at 20% of the speed of light (60,000 kms-1) has an apparent mass only 2% greater than its rest mass (m0).

The mass of an object at rest is called its rest mass (m0)

As speed increases, apparent mass increases rapidly.

m0

1 -c2

v2So, m =

where: m = Apparent Mass of the object m0 = Rest Mass of the object v = speed of object. c = Speed of Light.

1. When v << c, the square root term approaches 1, and m = m0

2. As v approaches c, the square root term approaches 0, and m approaches infinity.

Mathematically: m = γmo

There is insufficient energy in the

universe to accelerate even the smallest

particle up to the speed of light !!!!!!!!!!

At low velocities the increase in mass is small.

Energy & Mass

The fact that feeding energy into a body increases its mass suggests that the rest mass m0 of a body, multiplied by c2, can be considered as a quantity of energy.

This EM radiation can be detected and its energy measured. It turns out to be 2m0c2 where m0 is the mass of the electron (and the positron). So each particle must have possessed so called “rest energy” of m0c2

The truth of this is best seen in interactions between elementary particles. For example, if a positron and an electron collide at low speed (so there is very little kinetic energy) they both disappear in a flash of electromagnetic radiation.

Increasing the speed of a mass requires energy.

Einstein recognised the fundamental importance of the interchangeability of mass and energy which is summarised in his famous equation:

E = mc2

where m is the Apparent Mass. Remember,

m0

1 -c2

v2m =

Rest EnergyIf an object is at rest it possesses “rest mass energy” or more simply “rest energy”Einstein’s equation is then written as:

E = m0c2

Where E = Energy (joules) m0 = Rest Mass (kg)

c = 3 x 108 ms-1

How much energy does 1 kg of mass, at rest, represent ?E = m0c2

= (1)(3 x 108)2 = 9 x 1016 JoulesThis represents the average annual output of a medium sized Power Station

A Hiroshima sized atomic bomb releases about 1014 Joules, (100,000 billion joules).How much mass has been converted ?

E = m0c2

Thus m0 = (1014)/(3 x 108)2 = 1.1 x 10-3 kg = 1.1 g

As can be seen a tiny mass converts to a huge amount of energy

Moving MassAs a mass begins to move it possesses BOTH rest mass energy AND energy of motion (Kinetic Energy).

Expressing Einstein’s equation as: E = mc2

Includes both rest mass and kinetic energy

The Kinetic Energy of a fast moving particle can be calculated from:

K.E. = mc2 – m0c2

The relativistic energy of a particle can also be expressed in terms of its momentum (p) in the expression:

E = mc2 = p2c2 + m02c4

As an object’s speed increases more and more of the energy goes into increasing mass and less and less into increasing velocity.

This is essentially defining the kinetic energy of an object as the excess of the object’s energy over its rest mass energy. For low velocities this expression approaches the non-relativistic kinetic energy expression.

For v/c << 1, KE = mc2 – m0c2 ≈ ½ m0v2

The Speed of Light. A Limit ?

2

2

0

2

2

0

2

2

0

1

)1(

1

cv

mm

c

vLL

cv

tt

These equations together are called “The Lorentz Transforms”.Each Lorentz Transform has a limiting factor.

If v > c, then:• t becomes negative, and time runs

backward !!!!!! (the bullet hits you BEFORE it is fired from the gun).

• L becomes negative, and an object has a length less than zero!!!!!,

• m becomes negative and objects have a mass less than zero!!!!!

• Thus, c (the speed of light) is the limiting factor.

• Speeds greater than c are not possible.

Relativistic Speed Addition

One space-ship moves to the left with a speed of 0.75 c (relative to you) and the other one moves to the right also with a speed of 0.75 c (again relative to you). At what speed will each space-ship see the other moving away? 0.75 c + 0.75 c = 1.5 c?No, their relative speed will be 0.96 c (according to the relativistic addition of velocities), and it cannot, of course, be faster than the speed of light c.

This formula is called the relativistic addition of velocities.Note that if v’ = c and/or v = c, then v” = c, and for small velocities v, v’ << c, then the classical formula is regained.

Imagine that you are standing between two space-ships moving away from you.

In classical Newtonian mechanics, two different velocities and are added together by the formula v” = v’ + vwhere v” is the sum of the two velocities.

v” = v’ + v 1 + v.v’

c2

However, in special relativity, the velocities are added together as

v =0.75c v’=0.75c

v” = 0.96c

Special RelativityExperimental Proofs

R adioactive partic lesfrom outer space called MU -MESO N S

bom bard the earth at speeds close to c.Their behaviour is perfectly predicted by the form ulae

Tim e D ila tion andLength C ontraction

Protons when accelerated in partic leaccelerators show m ass increase exactly as

predicted by the form ula.

Mass D ila tion

Experim enta lP roofs

Experimental proof of the for each of the areas of Time Dilation, Length Contraction and Mass Dilation are available on Earth. These are shown below.

Relativistic Doppler Effect The Doppler Effect:Motion towards or away from a source will cause a change in the observed frequency f (or wavelength λ) as compared to the emitted frequency. All wave phenomena (e.g., water, sound, and light) behave in this way.

Suppose a source emits light of frequency f (or wavelength λ, remember that c = fλ). Then, an observer moving with a speed v away from the source, will observe the frequency:

If you are driving towards a red traffic light (λ0 = 650 nm) at a speed of approximately v = 0.17 c, the traffic light will actually appear to be green (λ = 550 nm)! (0.17 c is approximately 5.0 x107 ms-1.)

This formula is called the relativistic Doppler formula. Note that f < f0 for all 0 < v < c, i.e., the frequency which the observer sees, is smaller than the "original" frequency in the inertial frame of the source.

f = f0 c – v c + v

Observers moving away from the source will see a redshift in the frequency of the light, since light with lower frequencies are "more red" and light with higher frequencies are "more blue." While observers moving towards the source will see a corresponding blueshift.

Special RelativityConclusion

• I leave the last word to Einstein himself who, when asked to describe Special Relativity in laymen's terms, said:

• “Put your hand on a hot stove for a minute, and it seems like an hour.

• Sit with a pretty girl for an hour and it seems like a minute.

• That’s relativity”

THE END

Ollie Leitl 2005