NOTE: This page is a continuation of the notes and
worksheets for topic 9.2 Space. Two separate pages were
used for this topic because of the large volume of material in the topic.
This will keep download time within acceptable limits.
9.2 Space Continued
INTRODUCTION
TO RELATIVITY
Relativity is the study of the relative motions
of objects. Einstein’s
Theory of Relativity is one of the greatest intellectual
achievements of the 20^{th} Century.
Special Relativity, developed by Einstein in 1905,
deals with systems that are moving at constant velocity (no
acceleration) with respect to each other.
General Relativity proposed in 1916 deals with systems
that are accelerating with respect to each other.
Before commencing our study of Relativity some preliminary
definitions are necessary.
Reference
Frames:
A reference frame can be considered to be a
set of axes with respect to which distance measurements can be made. A set of recording clocks can be considered to be embedded in
the frame to specify time.
An
inertial reference frame is defined as one in which Newton’s First
Law (his law of inertia) is valid.
In other words, an inertial reference frame is one that is
not accelerating.
A noninertial reference frame is one that
is accelerating.
An
Event:
A physical event can be considered to be
something that happens independently of the reference frame used to
describe it – eg lightning flashes.
An event can be characterized in a Cartesian reference frame
by stating its coordinates x, y, z and t.
Brief History of Relativity Before Einstein
The phenomenon of motion has been studied for thousands
of years. To the ancient Greek philosopher Aristotle
it was obvious that objects would assume a preferred state
of rest unless some external force propelled them.
He also believed in the concepts of Absolute Space
and Absolute Time – that is that both space and time
exist in their own right, independently of each other and
of other material things (Refs 1, 2 & 3). Thus, to
Aristotle it was possible to assign absolute values of position
and time to events. Aristotle’s work was held in such
high regard that it remained basically unchallenged until
the end of the sixteenth century, when Galileo showed
that it was incorrect.
The view that motion must be relative – that is,
it involves displacements of objects relative to some reference
system – had its beginnings with Galileo. Galileo’s
experiments and “thought experiments” led him to state what
is now called the Principle of Galilean Relativity: the
laws of mechanics are the same for a body at rest and a
body moving at constant velocity.
Using Galileo’s measurements as a starting point Isaac
Newton developed his Laws of Motion and his Law of Universal
Gravitation. Newton showed that it is only possible
to determine the relative velocity of one reference frame
with respect to another and not the absolute velocity of
either frame. So, as far as mechanics is concerned,
no preferred or absolute reference frame exists.
The Principle of Newtonian Relativity may be stated
as: the laws of mechanics must be the same in all inertial
reference frames.
Thus, due to Galileo and Newton, the concept of Absolute
Space became redundant since there could be no absolute
reference frame with respect to which mechanical measurements
could be made. However, Galileo and Newton retained
the concept of Absolute Time, or the ability to establish
that two events that happened at different locations occurred
at the same time (1). In other words, if an observer
in one reference frame observed two events at different
locations as occurring simultaneously, then all observers
in all reference frames would agree that the events were
simultaneous.
The Newtonian concept of the structure of space and time
remained unchallenged until the development of the electromagnetic
theory in the nineteenth century, principally by Michael
Faraday and James Clerk Maxwell. Maxwell showed that
electromagnetic waves in a vacuum ought to propagate at
a speed of c = 3 x 10^{8} m/s, the speed of
light (1). To 19^{th} Century physicists this
presented a problem. If EM waves were supposed to
propagate at this fixed speed c, what was this speed measured
relative to? How could you measure it relative to
a vacuum? Newton had done away with the idea of an
absolute reference frame (2).
Quite apart from the relativity problem, it seemed inconceivable
to 19^{th} Century physicists that light and other
EM waves, in contrast to all other kinds of waves, could
propagate without a medium. It seemed to be a logical
step to postulate such a medium, called the aether (or
ether), even though it was necessary to assume unusual
properties for it, such as zero density and perfect transparency,
to account for its undetectability. This aether
was assumed to fill all space and to be the medium with
respect to which EM waves propagate with the speed c.
It followed, using Newtonian relativity, that an observer
moving through the aether with velocity u would measure
a velocity for a light beam of (c + u) (5).
So theoretically, if the aether exists, an observer on
earth should be able to measure changes in the velocity
of light due to the earth’s motion through the aether.
The MichelsonMorley experiment attempted to do just this.
[Top]
The Aether Model for
the Transmission of Light
Before moving onto the MichelsonMorley experiment,
we pause to examine in more detail the features of
the aether model for the transmission of light.
When 19^{th} Century physicists chose the aether
as the medium for the propagation of EM waves they were
merely borrowing and adapting an existing concept.
The fact that certain physical events propagate themselves through
astronomic space led long ago to the hypothesis that space is not
empty but is filled with an extremely fine substance, the Aether,
which is the carrier or medium of these phenomena. Indeed
the aether was proposed as the carrier of light
in Rene Descartes’ Dioptrics, which in 1638
became the first published scientific work on optics (4).
In this work, Descartes proposed that the aether was allpervasive
and made objects visible by transmitting a pressure from the object
to the observer’s eye.
Robert Hooke in 1667
developed pressure wave theories that allowed for the
propagation of light (6). In these theories, luminous objects set up vibrations
that were transmitted through the aether like sound waves
through air.
The Dutchman Christiaan
Huygens published a full theory on
the wave nature of light in 1690. According to Huygens,
light was an irregular series of shock waves that proceeded
with great velocity through a continuous medium – the
luminiferous aether. This
aether consisted of minute elastic particles uniformly
compressed together. The movement of light through
the aether was not an actual transfer of these particles
but rather a compression wave moving through the particles.
It was thought that the aether particles were not packed
in rows but were irregular in their orientation so that
a disturbance at one particle would radiate out from it
in all directions
In 1817 the French engineer Augustine Fresnel
and the English scientist Thomas Young independently
deduced that light was a transverse wave motion.
This required a rethink of the nature of the aether, which
until this time had been considered by most scientists
to be a thin fluid of some kind. Transverse
waves can only travel through solid media (or along the
surface of fluids). Clearly, the aether had to
be a solid. The solid also had to be very rigid
to allow for the high velocity at which light travelled
(4).
Clearly, this posed a problem, since such a solid would
offer great resistance to the motion of the planets and
yet no such resistance had been noted by astronomers.
In 1845 George Stokes attempted to solve the dilemma
by proposing that the aether acted like pitch or wax
which is rigid for rapidly changing forces but is fluid
under the action of forces applied over long periods of
time. The forces that occur in light vibrations
change extremely quickly (600 x 10^{12 }times
per second) compared with the relatively slow processes
that occur in planetary motions. Thus, the aether
may function for light as an elastic solid but give way
completely to the motions of the planets (4).
In 1865 the great Scottish physicist James Clerk Maxwell
published his theory of electromagnetism, which summarised
the basic properties of electricity and magnetism in four
equations. Maxwell also deduced that light waves
are electromagnetic waves and that all electromagnetic
waves travelled at 3 x 10^{8} m/s relative to
the aether. The aether was now called the electromagnetic
aether rather than the luminiferous aether (4) and
became a kind of absolute reference frame for electromagnetic
phenomena.
Exercise:
Outline the features of the aether model for the
transmission of light. (Note: This is Syllabus point
9.2.4 Column 1 Dot Point 1.)
Answer:
For several hundred years the aether was believed to be the medium
that acted as a carrier of light waves. The aether was
allpervasive, permeating all matter as evidenced by the
transmission of light through transparent materials.
Originally, the aether was believed to be a very thin, zero density,
transparent fluid. Young and Fresnel showed that light was a
transverse wave which implied that the aether must be solid and very
rigid to transmit the high velocity of light. Stokes (1845)
proposed that the aether acted like wax  very rigid for
rapidly changing forces (like high velocity light travel) but very
fluid for long continued forces (like the movement of the
planets). Maxwell (1865) used this aether as the absolute
reference frame in which the speed of all EM waves is 3 x 10^{8}
m/s.
[Top]
The MichelsonMorley Experiment
In 1887 Albert Michelson and Edward Morley
of the USA carried out a very careful experiment at
the Case School of Applied Science in Cleveland.
The aim of the experiment was to measure the motion
of the earth relative to the aether and thereby demonstrate
that the aether existed. Their method involved
using the phenomenon of the interference of light to
detect small changes in the speed of light due to the
earth’s motion through the aether (5).
The whole apparatus is mounted on a solid stone block
for stability and is floated in a bath of mercury so
that it could be rotated smoothly about a central axis
(5). The earth, together
with the apparatus is assumed to be travelling through
the aether with a uniform velocity u of about 30 km/s.
This is equivalent to the earth at rest with the aether
streaming past it at a velocity –u.
Now in the experiment a beam of light from the source
S is split into two beams by a halfsilvered mirror
K as shown. One half of the beam travels from
K to M_{1} and is then reflected back to K,
while the other half is reflected from K to M_{2}
and then reflected from M_{2} back to K.
At K part of the beam from M_{1} is reflected
to the observer O and part of the beam from M_{2}
is transmitted to O.
Although the mirrors M_{1} and M_{2}
are the same distance from K, it is virtually impossible
to have the distances travelled by each beam exactly
equal, since the wavelength of light is so small compared
with the dimensions of the apparatus. Thus, the
two beams would arrive at O slightly out of phase and
would produce an interference pattern at O.
There is also a difference in the time taken by each
beam to traverse the apparatus and arrive at O, since
one beam travels across the aether stream direction
while the other travels parallel and then antiparallel
to the aether stream direction (see the note below).
This difference in time taken for each beam to arrive
at O would also introduce a phase difference and would
thus influence the interference pattern.
Now if the apparatus were to be rotated through 90^{o},
the phase difference due to the path difference of each
beam would not change. However, as the direction
of the light beams varied with the direction of flow
of the aether, their relative velocities would alter
and thus the difference in time required for each beam
to reach O would alter. This would result in
a change in the interference pattern as the apparatus
was rotated.
The MichelsonMorley apparatus was capable of detecting
a phase change of as little as 1/100 of a fringe.
The expected phase change was 4/10 of a fringe.
However, no such change was observed.
Thus, the result of the MichelsonMorley experiment
was that no motion of the earth relative to the aether
was detected. Since the experiment failed
in its objective, the result is called a null
result. The experiment has since been repeated
many times and the same null result has always been
obtained.
NOTE: This time difference
mentioned above comes about from classical vector work.
After the original beam is split at K the half transmitted
to M_{1} travels with velocity (c + u)
relative to the “stationary” earth, as it is travelling
in the direction of “flow” of the aether. When
it is reflected from M_{1} it travels towards
K with a velocity relative to the earth of (c – u)
against the motion of the aether stream. Thus,
the time taken for the total journey of this beam from
K to M_{1} and back again is:
However, the other beam travels with velocity Ö (c^{2} – u^{2}) towards
M_{2} and then with the same speed in the opposite
direction away from M_{2} after reflection.
Thus, the time for the total journey of the beam from
K to M_{2} and back again is:
Clearly, t_{1} and t_{2} are different.
[Top]
The Role of the MichelsonMorley
Experiment
The
MichelsonMorley experiment is an excellent example of a critical
experiment in science. The
fact that no motion of the earth relative
to the aether was detected suggested quite strongly that
the aether hypothesis was incorrect and that no aether (absolute)
reference frame existed for electromagnetic phenomena.
This opened the way for a whole new way of thinking
that was to be proposed by Albert Einstein in his Theory of Special
Relativity.
It
is worth noting that the null result of the MichelsonMorley
experiment was such a blow to the aether hypothesis in particular
and to theoretical physics in general that the experiment was
repeated by many scientists over more than 50 years. A null result has always been obtained.
Extra
detail  beyond what is required by the Syllabus: The
aether hypothesis had become so entrenched in 19^{th}
Century Physics thinking that many scientists ignored the
significance of the null result and instead, looked for alternative
hypotheses to explain the null result:
u
The FitzgeraldLorentz Contraction Hypothesis
– in which all bodies are contracted in the direction of motion
relative to the stationary aether by a factor of Ö
1 – (v^{2 }/ c^{2}).
This was contradicted when the arms of the MichelsonMorley
interferometer were made unequal in the KennedyThorndyke
experiment.
u
The Aether Drag Hypothesis – in which the
aether was believed to be dragged along by all bodies of finite
mass. This too was
contradicted both on astronomical grounds (see the Bradley
Aberration – ref. 5) and by experiment (see Fizeau’s experiment
– ref. 5).
u
Attempts were also made to modify electromagnetic
theory itself. Emission
theories suggested that the velocity of light is c relative to
the original source and that this velocity is independent of the
state of motion of the medium transmitting the light.
This automatically explains the null result.
It was found though that all such emission theories could be
directly contradicted by experiment.
Eventually,
physicists like Lorentz (1899), Larmor (1900) and Poincare (1905)
showed that the changes needed to make the aether hypothesis
consistent with the null result of the MichelsonMorley experiment
implied that the aether (absolute) reference frame was impossible.
The aether ceased to exist as a real substance (4).
[Top]
Principle of Relativity
A relativity principle is a statement of what the
invariant quantities are between different reference
frames. It says that for such quantities the
reference frames are equivalent to one another, no
one having an absolute or privileged status relative
to the others. So, for example, Newton’s relativity
principle tells us that all inertial reference frames
are equivalent with respect to the laws of mechanics.
As we have seen, for quite a while in the 19^{th}
Century it looked as if there was a preferred or absolute
reference frame (the aether) as far as the laws of
electromagnetism were concerned. However, in
1904 Henri Poincare proposed his Principle of Relativity:
“The laws of physics are the same for a fixed observer
as for an observer who has a uniform motion of translation
relative to him”. Note that this principle applies
to mechanics as well as electromagnetism. Although
his principle acknowledged the futility in continued
use of the aether as an absolute reference frame,
Poincare did not fully grasp the implications.
Poincare still accepted the Newtonian concept of absolute
time. Einstein abandoned it.
[Top]
Einstein’s
Theory of Special Relativity
In 1905, Albert Einstein published his famous paper
entitled: “On the Electrodynamics of Moving Bodies”,
in which he proposed his two postulates of relativity
and from these derived his Special Relativity
Theory.
Einstein’s postulates are:
1. The Principle of Relativity – All the laws of
physics are the same in all inertial reference frames
– no preferred inertial frame exists.
2. The Principle of the Constancy of the Speed of Light
– The speed of light in free space has the same value
c, in all inertial frames, regardless of the velocity
of the observer or the velocity of the source emitting
the light.
The significance of the
first postulate is that it extends Newtonian Relativity
to all the laws of physics not just mechanics.
It implies that all motion is relative – no absolute
reference frame exists. The significance of
the second postulate is that it denies the existence
of the aether and asserts that light moves at speed
c relative to all inertial observers. It also
predicts the null result of the MichelsonMorley experiment,
as the speed of light along both arms of the interferometer
will be c.
Perhaps the greatest
significance of the second postulate, however, is
that it forces us to rethink our understanding of
space and time. In Newtonian Relativity,
if a pulse of light were sent from one place to another,
different observers would agree on the time that the
journey took (since time is absolute), but would not
always agree on how far the light travelled (since
space is not absolute). Since the speed of light
is just the distance travelled divided by the time
taken, different observers would measure different
speeds for light. In Special Relativity,
however, all observers must agree on how fast light
travels. They still do not agree on the
distance the light has travelled, so they must therefore
now also disagree over the time it has taken.
In other words, Special Relativity put an end to
the idea of absolute time (2).
Clearly, since c must
remain constant, both space and time must be relative
quantities.
[Top]
Simultaneity
Let us consider a “thought experiment” (Gedanken)
to illustrate that time is relative. Imagine
two observers O and O’ standing at the midpoints of
their respective trains (reference frames) T and T’.
T’ is moving at a constant speed v with respect
to T. Just at the instant when the two observers
O and O’ are directly opposite each other, two lightning
flashes (events) occur simultaneously in the T frame,
as shown below. The question is, will these
two events appear simultaneous in the T’ frame?
From our T reference frame, it is clear that observer
O’ in the T’ frame moves to the right during the time
the light is travelling to O’ from A’ and B’.
At the instant that O receives the light from A and
B, the light from B’ has already passed O’, whereas
the light from A’ has not yet reached O’. O’
will thus observe the light coming from B’ before
receiving the light from A’. Since the speed
of light along both paths O’A’ and O’B’ is c (according
to the second postulate), O’ must conclude that the
event at B’ occurred before the event at A’.
The two events are not simultaneous for O’, even though
they are for O.
Thus, we can conclude that two
events that are simultaneous to one observer are not
necessarily simultaneous to a second observer.
Moreover, since there is no preferred reference frame,
either description is equally valid. It follows
that simultaneity is not an absolute concept, but
depends on the reference frame of the observer.
[Top]
Length Contraction
When measuring the length of an object it is necessary
to be able to determine the exact position of the
ends of the object simultaneously. If, however,
observers in different reference frames may disagree
on the simultaneity of two events, they may also disagree
about the length of objects.
In fact, using Special Relativity theory, it is
possible to show mathematically and to demonstrate
experimentally that the length of a moving rod
appears to contract in the direction of motion relative
to a “stationary” observer. This is described
by the LorentzFitzgerald Contraction Equation:
where l is the moving length, l_{0} is the rest length (or proper
length) and v is the velocity of the rod relative to
the stationary observer. Note that this contraction
takes place in the direction of motion only.
So, for example, an observer on earth watching a rectangular
spacecraft move past the earth in the horizontal plane
would observe the horizontal length of the craft to
be contracted but the vertical width of the craft
to remain the same as seen by the observer on the
rocket. (Note that this is an over simplification.
Three dimensional objects travelling at relativistic
speeds relative to a given reference frame will appear
to be distorted in other ways as well, to an observer
at rest in that frame. This is outside the scope
of this course.)
[Top]
Time Dilation
Let us consider another thought experiment.
Imagine a “light clock”, as shown below. Time
is measured by light bouncing between two mirrors.
This clock ticks once for one complete up and down
motion of the light.
The light clock is placed in a rocket that travels
to the right at a constant speed v with respect to a stationary observer
on earth. When viewed by an observer travelling
with the clock, the light follows the path shown in
(a) above. To the stationary observer on earth,
who sees the clock moving past at a constant speed,
the path appears as in (b) above.
From (a), the time taken for light to make one complete
trip up and down, t_{0}, is
t_{0} = 2.L / c
 (1)
Remember that this represents one tick or one second
on the light clock as seen by the observer moving
with the clock. From (b), the distance the light
moves between A and B is c.t_{AB}, and the distance moved
by the whole clock in time t_{AB} is v.
t_{AB}.
So, by Pythagoras’ Theorem:
(c.t_{AB} )^{2} = (v.t_{AB}
)^{2} + L^{2}
and therefore:
t_{AB} ^{2} = L^{2}
/ (c^{2} – v^{2})
which can then be rearranged (divide throughout
RHS by c^{2} and take the square root) to
give:
t_{AB} = (L/c)
/ Ö 1 – (v^{2}/c^{2})
and thus, the total time taken by the light for
one complete up and down motion is:
t_{ABC} = (2L/c)
/ Ö 1 – (v^{2}/c^{2})
But from (1) above:
t_{0} = 2.L / c
And so we have:
Clearly, the time interval corresponding
to one tick of the light clock is larger for the observer
on earth than for the observer on the rocket, since
the denominator on the RHS of the above equation is
always less than 1.
The above equation may be interpreted as meaning
that the time interval t for an event to occur, measured by an
observer moving with respect to a clock is longer
than the time interval t_{0} for the same event, measured
by an observer at rest with respect to the clock.
An alternative way of stating this is that clocks
moving relative to an observer are measured by that
observer to run more slowly than clocks at rest with
respect to that observer. That is, time in
a moving reference frame appears to go slower relative
to a “stationary” observer. This result is called
time dilation.
The time interval t_{0} is referred to as the proper
time. t_{0} is always the time for an
event as measured by the observer in the moving
reference frame (Ref 5, pp.6364).
An example is probably a good idea
at this stage. Consider a rocket travelling
with a speed of 0.9c relative to the earth.
If an observer on the rocket records a time for a
particular event as 1 second on his clock, what time
interval would be recorded by the earth observer?
From our time dilation equation we have:
t = 1 / Ö 1 – [(0.9c)^{2}/c^{2}]
t = 2.29 s
So, to an observer on earth, the time taken for
the event is 2.29s. The earth observer sees
that the rocket clock has slowed down. It
is essential that you understand that this is not
an illusion. It makes no sense to ask which
of these times is the “real” time. Since no
preferred reference frame exists both times are as
real as each other. They are the real times
seen for the event by the respective observers.
Time dilation tells us that a moving clock runs
slower than a clock at rest by a factor of 1/Ö 1 – (v^{2}/c^{2}).
This result, however, can be generalised beyond clocks
to include all physical, biological and chemical processes.
The Theory of Relativity predicts that all such processes
occurring in a moving frame will slow down relative
to a stationary clock.
Try this java demonstration (page down when you
get there and click on the Time Dilation applet under
the Relativity section heading):
http://www.walterfendt.de/ph14e/
(Do you have the Walter Fendt site in your Java
Exception Site List? If not see instructions
here.
Otherwise app will not work.)
[Top]
Experimental
Evidence for Time Dilation (Not
Examinable)
The validity of time dilation has been confirmed
experimentally many times. One of these experiments
involves the study of the behaviour of particles called
muons, which are produced by collisions in the earth’s
upper atmosphere. When measured in their own
rest frame they have a lifetime of 2.2 ms. Their speed can reach as high
as 0.99c, which would enable them to travel about
650 m before decaying. Clearly, this distance
is not sufficient to allow the muons to reach the
surface of the earth and yet muons are found in plentiful
supply even in mine shafts beneath the earth’s surface.
The explanation is provided by time dilation.
The lifetime of muons with a speed of 0.99c is
dilated to about 16 ms in the earth’s reference frame.
This much time allows the muons to travel close to
5 km in the earth’s reference frame – sufficient to
reach the ground. (Remember though, if you could
think of a muon carrying a clock along with it, then
this clock would record the normal muon life span
of 2.2 ms.
2.2 ms of
movingmuon time is equivalent to 16 ms
of stationary earth observer's time.)
[Top]
The Twin
Paradox (Not Examinable)
The Twin Paradox is another example of a thought
experiment in relativity. Consider two twins.
Twin A takes a trip in a rocket ship at constant speed
v relative to the earth to a distant point in space
and then returns, again at the constant speed v.
Twin B remains on earth the whole time. According
to Twin B, the travelling twin will have aged less,
since his clock would have been running slowly relative
to Twin B’s clock and would therefore have recorded
less time than Twin B’s clock. However, since
no preferred reference frame exists, Twin A would
say that it is he who is at rest and that the earth
twin travels away from him and then returns.
Hence, Twin A will predict that time will pass more
slowly on earth, and hence the earth twin will be
the younger one when they are reunited. Since
they both cannot be right, we have a paradox.
To resolve the paradox we need to realise that it
arises because we assume that the twins’ situations
are symmetrical and interchangeable. On closer
examination we find that this assumption is not correct.
The results of Special Relativity can only be applied
by observers in inertial reference frames. Since
the earth is considered an inertial reference frame,
the prediction of Twin B should be reliable.
Twin A is only in an inertial frame whilst travelling
at constant velocity v. During the intervals
when the rocket ship accelerates, to speed up or slow
down, the reference frame of Twin A is noninertial.
The predictions of the travelling twin based on Special
Relativity during these acceleration periods will
be incorrect. General Relativity can be used
to treat the periods of accelerated motion.
When this is done, it is found that the travelling
twin is indeed the younger one.
Note that the only way to tell whose clock has actually
been running slowly is to bring both clocks back together,
at rest on earth. It is then found that it
is the observer who goes on the round trip whose clock
has actually slowed down relative to the clock of
the observer who stayed at home.
[Top]
Relativity
and Space Travel
Time dilation and length contraction have raised
considerable interest in regard to space travel.
Consider the following thought experiment. Imagine
that adventurous Toni goes on an excursion to Alpha
Centauri in a space ship at 0.9c. Her friend
Candy
stays behind on earth. Candy knows that
a–Centauri is 4.3 light years away and
so calculates the time for the trip as 4.8 years.
Allowing for a brief stop over when Toni gets there (shopping, cappuccino & cake etc),
Candy expects that Toni will be
back in about 10 years.
Travelling at 0.9c, Toni measures the distance
between earth and a–Centauri
to be contracted to 1.87 light
years and thus calculates the time for the trip as
2.1 years. Thus, she expects to be back on earth
in a little over 4 years.
Clearly, this 2.1 years of rocket time must be equivalent
to 4.8 years of earth time, since both observers must
observe the laws of physics to be the same.
(Note: We are ignoring the brief periods of acceleration
required by Toni.) This equivalence can be
checked using the time dilation equation.
When Toni arrives back on earth she finds that
she has indeed aged a little over four years, whilst
poor Candy is nearly 10 years older than when she
left. (Perhaps the rare a–Centaurian
wolfhound that Toni has
bought for Candy will soothe the upset.)
Seriously, though, the closer v gets to c, the closer
the distance to a–Centauri
and the time required to get
there get to zero as seen by Toni. Obviously,
the minimum time for the journey as seen by Candy is 4.3 years. So, if
Toni travels the distance
in 1 s, then 1 s of her time is equivalent to 4.3
years of Candy's (earth) time. If Toni travelled for 1 hour at this
very high speed, (3600 x 4.3) years or 15480 years
would elapse on earth. If Toni travelled for
a whole year on the rocket at this high speed, 135
million years would pass on earth.
While time dilation and length contraction overcome
one of the great difficulties of space travel, problems
obviously remain in producing such high speeds.
[Top]
Mass Dilation
and the MassEnergy Relationship
Another aspect of the Special Relativity theory
is that the mass of a moving object is greater than
when it is stationary. In fact, the higher the
velocity of the object, the more massive it becomes.
This is called Mass Dilation and is represented
mathematically as:
where m = relativistic mass of particle, m_{0} = rest mass of particle,
v is the velocity of the particle relative
to a stationary observer and c = speed of light.
The interesting question is of course, from where
does this extra mass come?
Using relativistic mechanics it can be shown that
the kinetic energy of a moving body makes a contribution
to the mass of the body. It turns out that
mass = energy/c^{2} or in a more recognizable
form:
E = mc^{2}
which is Einstein’s famous equation. Einstein
originally derived this equation by using the idea
that radiation exerts a pressure on an absorbing body.
This equation states the equivalence of mass
and energy. It establishes that energy can be
converted into mass and vice versa. For example,
when a particle and its antiparticle collide, all
the mass is converted into energy. Mass is converted
into energy in nuclear fission. When a body
gives off energy E in the form of radiation, its mass
decreases by an amount equal to E/c^{2}.
In Special Relativity, the Law of Conservation of
Energy and the Law of Conservation of Mass have been
replaced by the Law of Conservation of MassEnergy.
Note that the fact that mass increases as a body
gains velocity effectively limits all manmade objects
to travel at speeds lower than the speed of light.
The closer a body gets to the speed of light, the
more massive it becomes. The more massive it
becomes, the more energy that has to be used to give
it the same acceleration. To accelerate the
body up to the speed of light would require an infinite
amount of energy. Clearly, this places
a limit on both the speed that can be attained by
a spacecraft and therefore the time it takes to travel
from one point in space to another.
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Verification
of E = mc^{2} (Not Examinable)
Einstein's famous massenergy relationship E = mc^{2}
has been verified experimentally on many occasions. The
results of the latest experimental verification were reported in the
4th March 2006 edition of New Scientist Magazine. Two groups
of scientists studied gammaray emission from radioactive sulfur and
silicon atoms. One group at MIT used a very highprecision
massmeasuring apparatus called a Penning Trap to measure the mass
of particles before and after gammaray emission. The other
group at the US National Institute of Standards and Technology (NIST)
used a highprecision spectrometer to measure the wavelength of each
emitted gammaray, and thus determine their energy. When each
group was completely confident in the accuracy of their results,
they faxed them to each other. Imagine the tension as they
awaited each other's results. In the end the groups showed
that E does indeed equal mc^{2}
to better than 5 parts in 10 million.
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Four Dimensional
Space Time (Not Examinable)
The Theory of Special Relativity shows that space
and time are not independent of one another but are
intimately connected. The theory shows that
objects moving at very high speeds relative to a stationary
observer appear contracted in the direction of motion
and have clocks that appear to run slow. It
seems that some space (length) has been exchanged
for some time. Although the length of the object
is shorter as seen by the stationary observer, each
of the object’s seconds is of longer duration than
each of the stationary observer’s seconds. The
two effects of time dilation and length contraction
balance each other, with space being exchanged for
time. Thus, in relativity it makes very good
sense to speak of spacetime rather than space and
time as separate entities.
Such considerations have led to the concept of
fourdimensional spacetime. In this fourdimensional
spacetime, space and time can intermix with part
of one being exchanged for part of the other as the
reference frame is changed. In 1908, Hermann
Minkowski provided a mathematical treatment of
Special Relativity in which he developed relativistic
kinematics as a fourdimensional geometry (4).
He assigned three spatial coordinates and one time
coordinate to each event. Thus, in Special
Relativity, an event in spacetime may be represented
on the Cartesian plane, by quoting its x, y, z and
t coordinates. The first three coordinates tell
where the event occurred; the fourth coordinate tells
when the event occurred. (See ref. 4 for a good
introduction to Minkowski SpaceTime diagrams.)
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A New
Standard of Length
Length is one of the fundamental quantities
in Physics because its definition does not depend
on other physical quantities. The SI unit of
length, the metre was originally defined as one tenmillionth
of the distance from the equator to the geographic
North Pole (6). The first truly international
standard of length was a bar of platinumiridium alloy
called the standard metre and kept in Paris.
The bar was supported mechanically in a prescribed
way and kept in an airtight cabinet at 0^{o}
C. The distance between two fine lines engraved
on gold plugs near the ends of the bar was defined
to be one metre (7).
In 1961 an atomic standard of length was adopted
by international agreement. The metre was defined
to be 1 650 763.73 times the wavelength of the orangered
light from the isotope krypton86. This standard
had many advantages over the original – increased
precision in length measurements, greater accessibility
and greater invariability to list a few (7).
In 1983 the metre was redefined
in terms of the speed of light in a vacuum.
The metre is now defined as the distance light travels
in a vacuum in 1/299792458 of a second as measured
by a cesium clock (2 & 6). Since the speed
of light is constant and we can measure time more
accurately than length, this standard provides increased
precision over previous standards. The reason
for that particular fraction (1/299792458) is that
the standard then corresponds to the historical definition
of the metre – the length on the bar in Paris.
So, our current standard
of length is actually defined in terms of time in
contrast to the original standard metre, which was
defined directly in terms of length (distance).
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BIBLIOGRAPHY
1.
Hawking, S.W. & Israel, W. (1979).
General Relativity – An Einstein Centenary Survey,
Cambridge, Cambridge University Press
2.
Hawking, S.W. (1988). A
Brief History of Time, London, Bantam Press
3.
Whitrow, G.J. (1984). The Natural Philosophy of Time,
Oxford, Oxford University Press
4.
Born, M. (1965). Einstein’s Theory of Relativity,
New York, Dover Publications Inc.
5.
Resnick, R. (1968). Introduction To Special Relativity,
New York, Wiley
6.
Bunn, D.J. (1990). Physics
for a Modern World, Sydney, The
Jacaranda Press
7.
Halliday, D. & Resnick, R. (1966). Physics Parts I
& II Combined Edition, New York, Wiley
[Top]
SPACE TOPIC PROBLEMS II
1
Describe an experiment that you could perform
in a reference frame to determine whether or not the
frame was noninertial.
2
A spacecraft is travelling at 0.99c.
An astronaut inside the craft records a time of
1 hour for a certain event to occur.
How long would an observer stationary relative
to the craft record for this event?
(7.09 h)
3
A missile travelling at 9/10 the speed of light
has a rest length of 10 m.
Calculate the length of the moving missile as
measured by a stationary observer directly under the
flight path of the missile.
(4.36 m)
4
An electron with a rest mass of 9.11 x 10^{31}
kg is travelling at 0.999c.
Determine the relativistic mass of the
electron. (2.04
x 10^{29} kg)
5
A particular radioactive isotope loses 5 x 10^{2}
J of energy. Calculate
its resultant loss of mass.
(5.6 x 10^{15} kg)
6
The radius of our galaxy is 3 x 10^{20}
m, or about 3 x 10^{4} light years.
(a)
Can a person, in principle, travel from the
centre to the edge of our galaxy in a normal lifetime?
Explain using either time dilation or length
contraction arguments.
(b)
Determine the constant velocity that would be
needed to make the journey in 30 years (proper time).
(299999850 m/s or 0.9999995c)
(Hint  What is the shortest time that a stationary
earth observer could possibly measure for such a
trip?)
7
A new EFT (extremely fast train) is travelling
along the tracks at the speed of light relative to the
earth’s surface.
A passenger is walking towards the front of the
train at 5 m/s relative to the floor of the train.
Clearly, relative to the earth’s surface, the
passenger is moving faster than the speed of light.
Analyse this situation from the point of view
of Special Relativity.
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