                  9.2 Space Continued    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.

# 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 20th 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 non-inertial 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 108 m/s, the speed of light (1).  To 19th 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 19th 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 Michelson-Morley experiment attempted to do just this.

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### The Aether Model for the Transmission of Light

Before moving onto the Michelson-Morley experiment, we pause to examine in more detail the features of the aether model for the transmission of light.  When 19th 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 all-pervasive 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 1012 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 108 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 all-pervasive, 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 108 m/s.

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### The Michelson-Morley 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 half-silvered mirror K as shown.  One half of the beam travels from K to M1 and is then reflected back to K, while the other half is reflected from K to M2 and then reflected from M2 back to K.  At K part of the beam from M1 is reflected to the observer O and part of the beam from M2 is transmitted to O.

Although the mirrors M1 and M2 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 anti-parallel 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 90o, 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 Michelson-Morley 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 Michelson-Morley 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 M1 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 M1 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 M1 and back again is: However, the other beam travels with velocity Ö (c2 – u2) towards M2 and then with the same speed in the opposite direction away from M2 after reflection. Thus, the time for the total journey of the beam from K to M2 and back again is: Clearly, t1 and t2 are different.

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### The Role of the Michelson-Morley Experiment

The Michelson-Morley 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 Michelson-Morley 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 19th 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 Fitzgerald-Lorentz Contraction Hypothesis – in which all bodies are contracted in the direction of motion relative to the stationary aether by a factor of Ö 1 – (v2 / c2).  This was contradicted when the arms of the Michelson-Morley interferometer were made unequal in the Kennedy-Thorndyke 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 Michelson-Morley experiment implied that the aether (absolute) reference frame was impossible.  The aether ceased to exist as a real substance (4).

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### 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 19th 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.

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### 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 Michelson-Morley 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 re-think 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.

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### 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.

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### 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 Lorentz-Fitzgerald Contraction Equation: where l is the moving length, l0 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.)

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### 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, t0, is

t0 = 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.tAB, and the distance moved by the whole clock in time tAB is v. tAB.

So, by Pythagoras’ Theorem:

(c.tAB )2 = (v.tAB )2  +  L2

and therefore:               tAB 2  =  L2 / (c2 – v2)

which can then be re-arranged (divide throughout RHS by c2 and take the square root) to give:

tAB   =  (L/c)  /  Ö 1 – (v2/c2)

and thus, the total time taken by the light for one complete up and down motion is:

tABC   =  (2L/c)  /  Ö 1 – (v2/c2)

But from (1) above:      t0 = 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 t0 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 t0 is referred to as the proper time.  t0 is always the time for an event as measured by the observer in the moving reference frame (Ref 5, pp.63-64).

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/c2]

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 – (v2/c2).  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.walter-fendt.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.)

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### 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 moving-muon time is equivalent to 16 ms of stationary earth observer's time.)

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### 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 re-united.  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 non-inertial.  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.

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### 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.

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### Mass Dilation and the Mass-Energy 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, m0 = 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/c2 or in a more recognizable form:

E = mc2

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/c2.

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 Mass-Energy.

Note that the fact that mass increases as a body gains velocity effectively limits all man-made 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 = mc2(Not Examinable)

Einstein's famous mass-energy relationship E = mc2 has been verified experimentally on many occasions.  For example, in one verification, two groups of scientists studied gamma-ray emission from radioactive sulfur and silicon atoms.  One group at MIT used a very high-precision mass-measuring apparatus called a Penning Trap to measure the mass of particles before and after gamma-ray emission.  The other group at the US National Institute of Standards and Technology (NIST) used a high-precision spectrometer to measure the wavelength of each emitted gamma-ray, 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 mc2 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 space-time rather than space and time as separate entities.

Such considerations have led to the concept of four-dimensional space-time.  In this four-dimensional space-time, 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 four-dimensional geometry (4).  He assigned three spatial coordinates and one time coordinate to each event.  Thus, in Special Relativity, an event in space-time 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 Space-Time 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 ten-millionth of the distance from the equator to the geographic North Pole (6).  The first truly international standard of length was a bar of platinum-iridium 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 0o 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 orange-red light from the isotope krypton-86.  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 re-defined 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

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SPACE TOPIC PROBLEMS II

1        Describe an experiment that you could perform in a reference frame to determine whether or not the frame was non-inertial.

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 102 J of energy.  Calculate its resultant loss of mass.  (5.6 x 10-15 kg)

6        The radius of our galaxy is 3 x 1020 m, or about 3 x 104 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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