VCE Physics
Unit 3
Topic 3
Special Relativity
The World at the
Speed of Light.
Einstein’s
Contribution.
Unit Outline
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.
Galilean Relativity
Galileo Galilei
1564 - 1642
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:
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.
There is no absolute inertial frame of
reference: all velocity measurements
are relative to the frame of reference.
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.
FRAMES OF REFERENCE
Frames 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.
Galilean Motion
VTRAIN = 25.0 ms-1
VBALL = 5.0 ms-1
In Galileo’s world, the idea of relative
motion is clearly understood.
This can be shown with a simple
example.
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.
The boy in the carriage sees the ball
travel away from him at 5.0 ms-1
But, an observer standing beside the track,
sees the ball moving to the right at 30.0 ms-1.
Stationary observer
Remember, according to
Galileo, there is no
absolute inertial frame of
reference: all velocity
measurements are relative
to the frame of reference.
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.
Isaac Newton
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).
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
Isaac Newton, aged 26
commonly written as F = ma
These Laws explained Galilean relativity
and using Newton's laws, physicists in the
18th and 19th century were able to predict
Law 3
the motions of the planets, moons, comets,
cannon balls, etc.
In classical Newtonian mechanics, time was universal
(Action Reaction Law)
For every action there is an
equal and opposite reaction.
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 Clouds
Lord Kelvin
In 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.
The Revolution
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.
Mercury
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.
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.
Maxwell’s Contribution
• Throughout 1700’s and
Charles Coulomb
1800’s, many individual laws
about electricity and
magnetism had been
discovered, such as
Coulomb’s law of
electrostatic force.
1 q1q2
F
 2
40 d
One interesting consequence of
Maxwell’s unification is that you
can calculate the velocity of
electromagnetic waves based on
properties of capacitors and
inductors.
c
1
 0 0
c = Speed of EM Waves
μ0 = Permeability of free
space
ε0 = Susceptibility of free
space
In Maxwell’s own words:
This velocity is so nearly that of
James Clerk Maxwell (1831James Maxwell light, that it seems we have
strong reasons to conclude that
1879) had, by 1855, unified
light itself (including radiant heat,
some laws and finally by 1873
and other radiations if any) is an
had found that all of these laws
electromagnetic disturbance in
could be summarised by four
the form of waves propagated
partial differential equations.
through the electromagnetic field
A triumph of unification!
according to electromagnetic
(Which of course is the holy
laws.
grail of Physics)
th
19
•
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).
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 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.
The Speed of Light
In 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.
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.
The result of the Michelson-Morley Speed = c
Speed = c
Ether Speed = v experiment was that the speed of
Ether Speed = v
the Earth through the ether (or the
speed of the ether wind) was zero.
Speed = c + v
Expected 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!
Speed = c
Actual result
Michelson Morley in Detail
The experiment was set up using a “monochromatic”
(single colour) light source split into two beams.
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.
ether
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.
Michelson Morley in Detail 2
NO CHANGE IN THE PATTERN
The travel times for the two
beams were compared in a very COULD EVER BE DETECTED
WHEN THE EQUIPMENT
sensitive manner.
0
If the travel times were different TURNED THROUGH 90
Double
the two beams, when combined,
Slit
would have produced an
Incident
Interference
“interference pattern”.
Light
Pattern
This is the same as the pattern
produced when a
Michelson and Morley repeated their
monochromatic beam of light is
experiment many times up until 1929, but
allowed to pass through two
always with the same results and
narrow slits.
conclusions.
Michelson won the Nobel Prize in Physics in 1907. Probably
the only prize ever awarded for a failed experiment.
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
It took nearly 20 years to develop the
theory to match this experimental result.
Newton versus Maxwell
Under Galileo and Newton, the speed
of light would vary depending the
inertial frame of reference.
The Speed of
Light (c) is
3 x 108 ms-1
No, its c - 1000
No, its c - 100
1000 ms-1
No, its c - 10
100 ms-1
10 ms-1
Whilst under Maxwell, the speed of
light is constant no matter what
the inertial frame of reference.
I agree it’s c
I agree it’s c
c = 3 x 108 ms-1
1000 ms-1
I agree it’s c
100 ms-1
10 ms-1
Einstein’s
Insight
It 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.
Fig
1
Fig
2
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!
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.
Introducing Relativity
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.
Special 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)
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.
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
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 History
An 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
In 1894, as a 15 year old, he was expelled from
Catholic College for disruptive behaviour.
Einsteinaged 14
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 Professor
Einstein and 1st wife
Mileva
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.
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.
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
The first of these consequences is known as “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.
Time Dilation
In order to demonstrate this change in the rate at which
time passes, let us produce a simple clock.
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.
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.
Mirror
Mirror
“1 Tick”
Photon
Earth
Earth
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.
Hence, if the photon is to strike the mirrors, the space
observer requires it to travel on a diagonal path, as shown.
Earth
Time Dilation
Clock as seen by Earth
bound observer
Short Distance
Clock as seen by Space observer
Long Distance
Speed = c
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
Speed = c
Earth
Earth
Earth
The space observer thus concludes the
Earth clock runs slow compared to his
clock.
This same argument holds true for the
earth bound observer, who would see
the space observer’s clock running
slow.
and this gets bigger
this must also get
bigger
If this has to stay the same
Thus, MOVING CLOCKS RUN SLOW.
Time Dilation
The mathematical
representation of Time Dilation
is shown in the formula:
t = γto
where γ =
•
•
•
•
•
•
So
1
t=
to
2
1 - v2
c
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.
2
1 - v2
γ is called the “Lorentz Factor”
c
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.
My Clock
If v = c, the term v2/c2 = 1 and the square root
term becomes zero. Dividing a number by
zero equals infinity.
My observation
Thus, when v = c the time interval becomes
of the moving
infinite. In other words, time stops passing.
clock
The
Twins
Paradox
Twins, 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
The 2nd consequence of light
having a constant speed is
Length Contraction.
An observer sees two set
squares, one stationary in his
inertial frame, the other in an
inertial frame moving near
the speed of light.
How does the speed
difference affect the apparent
size of the set square ?
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.
y
v=0
v = 0.8c
x
Remember: Speed = distance
time
to keep
this the
same
with this
having
become
less
This must
also be
less
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)
Length Contraction
Land available on Mars and its
free !
The Martians send an
advertising rocket to fly past
Earth.
v = 0.1c
What is the best speed for the
rocket so stationary Earthlings
can read the sign ?
v = 0.86c
v = 0.90c
v = 0.98c
Length Contraction
The mathematical representation of Length
Contraction is shown in the formula.
L = Lo/γ
2
v
1- 2 )
c
So, L = Lo(
Where:
The formula has a number of consequences:
L = Length of moving
1. If the v << c, the square root term approaches
object as measured
by stationary
1 and the length is unaffected, ie. L = Lo
observer.
2. As v approaches c, v2/c2 approaches 1 and the
Lo = Length of stationary
square root term approaches 0. Thus, the
object measured by
length approaches 0 ie. L = 0.
stationary observer.
So, a photon of light travelling at c from the Sun to
(“Proper Length”)
the Earth makes the journey in no time and travels
v = speed of moving
no distance !!!!!!
object.
c = Speed of Light.
The moving observer’s view
of the length contracted
world
The stationary observer’s
view of the length
contracted Superman
Mass Dilation
The 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 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.
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.
Mass – Newton v Einstein
Newtonian physics gives good
results at speeds less than 10% of
the speed of light.
Einstein’s relativity deals with faster
speeds.
The mass of an object does not
change with speed, it changes
only if we cut off or add a piece to
the object .
As an object moves faster its mass
increases. (As measured by a
stationary observer).
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.
Mass approaches infinity as speed
approaches c. To reach c would
require infinite force.
Kinetic Energy = ½mv2, since
mass does not change an
increase in KE means an increase
in speed.
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.
Apparent Mass
Mass - How Fast, How Heavy ?
The mass of an object at rest is called its rest mass (m0)
At low velocities the increase in mass is small.
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).
As speed increases, apparent mass increases rapidly.
m0
Mathematically:
So, m =
2
m = γmo
1 - v2
c
6m0
where:
m = Apparent Mass of the object
m0 = Rest Mass of the object
4m0
v = speed of object.
c = Speed of Light.
2m0
1. When v << c, the square root term
approaches 1, and m = m0
m0
2. As v approaches c, the square root
term approaches 0, and m approaches
infinity.
0
20
40
60
80
100 There is insufficient energy in the
% of Speed of Light
universe to accelerate even the smallest
Speed of object as seen by a
particle up to the speed of light !!!!!!!!!!
stationary observer
Energy & Mass
Increasing the speed of a mass
requires energy.
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.
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.
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
m=
2
1 - v2
c
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
Rest Energy
If 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 Joules
This 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 Mass
As 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
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
As an object’s speed increases
more and more of the energy
goes into increasing mass and
less and less into increasing
velocity.
The Speed of Light. A Limit ?
t
t0
v2
1 2
c
v2
L  L0 ( 1  2 )
c
m0
m
v2
1 2
c
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
v =0.75c
v’=0.75c
v” = 0.96c
Imagine that you are standing between two space-ships
moving away from you.
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.
However, in special relativity, the velocities are
added together as
In classical Newtonian
v” = v’ + v
mechanics, two different
1 + v.v’
velocities and are added
c2
together by the formula
This formula is called the
v” = v’ + v
relativistic addition of velocities.
where v” is the sum of the
Note that if v’ = c and/or v = c,
two velocities.
then v” = c, and for small
velocities v, v’ << c, then the
classical formula is regained.
Special Relativity
Experimental Proofs
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.
Experimental
Proofs
Time Dilation and
Length Contraction
Mass Dilation
Radioactive particles
from outer space called MU-MESONS
bombard the earth at speeds close to c.
Their behaviour is perfectly predicted by the formulae
Protons when accelerated in particle
accelerators show mass increase exactly as
predicted by the formula.
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.
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.)
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:
f = f0 c – v
c+v
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.
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 Relativity
Conclusion
• 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