Special Theory of Relativity

Special Theory of Relativity
The Special Theory of Relativity
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TABLE OF CONTENTS
q Introduction
q The Constancy of the Speed of Light
The Michelson-Morley Experiment
Einstein "Explains" the Michelson-Morley Experiment
q Exploring the Consequences of Einstein's "Explanation"
q The Parable of the Surveyors
q Spacetime
Spacetime Diagrams
The Dimensions of Spacetime
More Spacetime Diagrams and Some Discussion
The Significance of the Minus Sign
q Further Consequences of Einstein's Explanation
Time Dilation
Length Contraction
Simultaneity
A Little About Language
Relative Speeds
Mass-Energy Equivalence
Tachyons
Superluminal Connections
The "Speed" of Objects
The Lorentz Contraction is Invisible
q The Twin Paradox
q A Favorite Puzzle
q Conclusion
q Author and Copyright
INTRODUCTION
In this document we discuss Einstein's Special Theory of Relativity. The treatment is non-mathematical, except for a brief use of Pythagoras' theorem
about right triangles. We concentrate on the implications of the theory. The document is based on a discussion of the the theory for an upper-year
liberal arts course in Physics without mathematics; in the context of that course the material here takes about 4 or 5 one-hour classes.
Einstein published this theory in 1905. The word special here means that we restrict ourselves to observers in uniform relative motion. This is as
opposed the his General Theory of Relativity of 1916; this theory considers observers in any state of uniform motion including relative acceleration. It
turns out that the general theory is also a theory of gravitation.
Sometimes one hears that the Special Theory of Relativity says that all motion is relative. This is not quite true. Galileo and Newton had a similar
conception. Crucial to Newton's thinking is that there is an absolute space, independent of the things in that space:
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"Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable
dimension or measure of the absolute spaces; which our senses determine by its position to bodies .. because the parts of space cannot be seen, or
distinguished from one another by our senses, there in their stead we use sensible [i.e. perceptible by the sense] measures of them ... but in
philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of
them." -- Principia I, Motte trans.
For Newton, the laws of physics, such as the principle of inertia, are true in any frame of reference either at rest relative to absolute space or in
uniform motion in a straight line relative to absolute space. Such reference frames are called inertial. Notice there is a bit of a circular argument here:
the laws of physics are true in inertial frames, and inertial frames are ones in which the laws of physics are true.
In any case, from the standpoint of any such inertial frame of reference all motion can be described as being relative. If you are standing by the
highway watching a bus go by you at 100 km/hr, then relative to somebody on the bus you are travelling in the opposite direction at 100 km/hr.
This principle, called Galilean relativity, is kept in Einstein's Theory of Relativity.
Many of the consequences of the Special Theory of Relativity are counter-intuitive and violate common sense. Einstein correctly defined common
sense as those prejudices that we acquire at an early age.
THE CONSTANCY OF THE SPEED OF LIGHT
Once we realise that light is some sort of a wave, a natural question is "what is waving?" One answer to this question is that it is the luminiferous
ether. The idea behind this word is that there is an all-pervading homogenous massless substance everywhere in the universe, and it is this ether that is
the medium through which light propagates. Note that this ether could define Newton's absolute space.
A rough analogy is to a sound wave travelling through the air. The air is the medium and oscillations of the molecules of the air are what is "waving."
The speed of sound is about 1193 km/hr with respect to the air, depending on the temperature and pressure. Thus if I am travelling through the air at
1193 km/hr in the same direction as a sound wave, the speed of the wave relative to me will be zero.
The speed of light is measured to be about 1,079,253,000 km/hr, and presumably this is its speed relative to the ether. Presumably the ether is
stationary with respect to the fixed stars. This section investigates these two presumptions.
Galileo attempted to measure the speed of light around 1600. He and a colleague each had a lantern with a shutter, and they went up on neighboring
mountains. Galileo opened the shutter on his lantern and when his colleague saw the light from Galileo's lantern he opened the shutter on this lantern.
The time delay between when Galileo opened the shutter on his lantern and when he saw the answering light from his colleague's lantern would allow
him to calculate the speed of light. This is absolutely correct experimental procedure in principle. However, because of our human reaction times the
lag between when the colleague saw the light from Galileo's lantern until when he could get the shutter of his lantern open is so long that the light
could have circled the globe many many times.
In 1676 Römer successfully measured the speed of light, although his results differed from the accepted value today by about 30%
The Michelson-Morley Experiment
In this sub-section we discuss a famous experiment done in the late nineteenth century by Michelson and Morley. Some knowledge of the fact that
light is a wave and can undergo interference is assumed. A discussion of this occurs in the the first two sections of the document http://www.upscale.
utoronto.ca/GeneralInterest/Harrison/DoubleSlit/DoubleSlit.html.
In is ironic that Michelson himself wrote in 1899, "The more important fundamental laws and facts of physical reality have all been discovered
and they are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly
remote .... Our future discoveries must be looked for in the 6th place of decimals." At this time there were a couple of small clouds on the
horizon. One of those clouds was his own experiment with Morley that we describe in this sub-section. As we shall see, the experiment played a part
in the development of the Special Theory of Relativity, a profound advance.
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Recently some people, especially John Horgan in his book The End of Science (1996), have been making similar claims about how the enterprise of
science is complete. My opinion is that they are no more correct than was Michelson. I certainly hope they are wrong, because if they are correct all
the fun goes out of physics. In fact, as we shall see, I think there are already a couple of clouds on the horizon. One cloud is the failure of our theories
of cosmology to account for recent observations of the universe. The other is the failure of the quark model to produce any truly useful results.
Before we turn to the experiment itself we will consider a "race" between two swimmers.
We have two identical
swimmers, 1 and 2, who each
swim the same distance away
from the raft, to the markers, and
then swim back to the raft. The
"race" ends in a tie.
Now the raft and markers are
being towed to the left. In this
case the race will no longer be a
tie. In fact, it is not too hard to
show that swimmer 2 wins this
race.
A small Flash animation illustrating the above race may be found here.
These notes are intended to be non-mathematical, with the exception of a brief use of Pythagoras theorem about right triangles. However, some people
would like to see a little bit of the math. Thus, a proof that swimmer 2 above wins the race may be found here. Below, a further small amount of math
will appear, but will always be labelled as a Technical note.
One of the difficulties that students experience in learning about the theories of relativity is that it is easy to ask questions of themselves and/or others
that are not well formed. Insisting on complete statements often makes the problems disappear. One common case of sloppy language leading to
poorly formed questions involves the concept of speed. If we say, for example, that the swimmers in the above examples swim at 5 km/hr we have not
made a complete statement; we should say that the swimmers swim at 5 km/hr with respect to the water. If we are stationary with respect to the water
then they swim at 5 km/hr with respect to us. But if we are moving at, say, 5 km/hr with respect to the water in the direction that one of the swimmers
is swimming, that swimmer will be stationary relative to us.
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Now we consider the Michelson interferometer, shown
schematically to the right. The light source is the red star to
the left of the figure. The light from it is incident on a half-
silvered mirror, which is drawn as a blue line; this is a
"crummy" mirror that only reflects one-half of the light
incident on it, transmitting the other half. The two light
beams then go to good mirrors, drawn as green rectangles,
which reflect the light. The reflected light actually follows
the same path as the incident beam, although I have drawn
them slightly offset. When beam 1 returns to the half-silvered
mirror, one half is reflected down; the other half is
transmitted back toward the light source but I haven't
bothered to draw that ray. Similarly, when beam 2 returns to
the half-silvered mirror, one half is transmitted; the other half
is reflected towards the source although I haven't drawn that
ray either. The two combined beams go from the half-
silvered mirror to the detector, which is the yellow object at
the bottom of the figure.
If the distance from the half-silvered mirror to mirror 1 is
equal to the distance to mirror 2, then when the two rays are
re-combined they will have travelled identical distances.
Thus, they will be "in phase" and will constructively interfere
and we will get a strong signal at the detector. If we slowly move mirror 1 to the right, that ray will be travelling a longer total distance than ray 2; at
some point the two rays will be "out of phase" and destructively interfere. Moving mirror 1 a bit further to the right, at some point the two rays will be
"in phase" again, giving constructive interference.
Say we have the interferometer adjusted so we are getting constructive interference at the detector. Then the "race" between the two beams of light is
essentially a tie. This may remind you of the race of the swimmers above.
Except that if we have the apparatus sitting on the earth, we have to remember that the speed of the earth in its orbit around the sun is on the order of
108,000 km/hr relative to the ether, depending on the season and time of day. So the situation is more like the second race above when the raft is
being towed through the water. The interferometer is being "towed" through the ether.
Michelson and Morley did this experiment in the 1880's. The arms of the interferometer were about 1.2 meters long. The apparatus was mounted on a
block of marble floating in a pool of mercury to reduce vibrations. They adjusted the interferometer for constructive interference, and then gently
rotated the interferometer by 90 degrees.
Given the speed of light as 1,079,253,000 km/hr relative to the ether and the speed of the earth equal to some number like 108,000 km/hr relative to
the ether, they calculated that they should easily see the combined beams going through maxima and minima in the interference pattern as they rotated
the apparatus.
Except that when they did the experiment, they got no result. The interference pattern did not change!
It was suggested that maybe the speed of the earth due to its rotation on its axis was cancelling its speed due to its orbit around the sun. So they waited
12 hours and repeated the experiment. Again they got no result.
It was suggested that the Earth's motion in orbit around the Sun cancelled the other motions. So they waited six months and tried the experiment
again. And again they got no result.
It was suggested that maybe the mass of the earth "dragged" the ether along with it. So they hauled the apparatus up on top of a mountain, hoping that
the mountain would be sticking up into the ether that was not being dragged by the earth. And again they got no result.
Thus, this attempt to measure the motion of the earth relative to the ether failed.
Lorentz was among many who were very puzzled by this result. He proposed that when an object was moving relative to the ether, its length along its
direction of motion would be contracted by just the right amount needed to explain the experimental result. If the length of the object when it is at rest
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with respect to the ether is L0, then if is is moving at speed v through the ether its length becomes L given by:
where c is the speed of light relative to the ether. If you chose to look at the brief mathematical supplement above, the structure of this equation may
look familiar to you.
Einstein "Explains" the Michelson-Morley Experiment
When Einstein was 16, in 1895, he asked himself an interesting question:
"If I pursue a beam of light with the velocity c I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest. However,
there seems to be no such thing, whether on the basis of experience or according to [the theory of electricity and magnetism]. From the very beginning
it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws
as for an observer who, relative to the earth, was at rest. For how, otherwise, should the first observer know, i.e.. be able to determine, that he is in a
state of uniform motion?" -- As later written by Einstein in "Autobiographical Notes", in Schilpp, ed., Albert Einstein: Philosopher-Scientist.
He continued to work on this question for 10 years with the mixture of concentration and determination that characterised much of his work. He
published his answer in 1905:
"... light is always propagated in empty space with a definite velocity c which is independent of the state of [relative] motion of the emitting body ....
The introduction of a `luminiferous ether' will be superfluous inasmuch as the view here to be developed will not require an `absolutely stationary
space' provided with special properties." -- Annalen Physik 17 (1905).
Put another way, the speed of light is 1,079,253,000 km/hr with respect to all observers.
As we shall see, this one statement is equivalent to all of the Special Theory of Relativity, and everything else is just a consequence.
Notice that the statement also explains the null result of the Michelson-Morley experiment. However, although the evidence is not certain it seems
quite likely that in 1905 Einstein was unaware of the experiment (cf. Gerald Holton, "Einstein, Michelson and the 'Crucial' Experiment," which has
appeared in Thematic Origins of Scientific Thought, pg. 261. and also in Isis 60, 1969, pg. 133.).
EXPLORING THE CONSEQUENCES OF EINSTEIN'S "EXPLANATION"
Here we will begin to see why Einstein's statement about the constancy of the speed of light leads to all of the strange consequences such as time
dilation, length contraction, etc. But first we should take a few moments to carefully explore just what we mean when we say some event occurred at
some particular place at some particular time
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We imagine a lattice of meter
sticks, such as shown to the right,
and at each intersection we place a
clock. This lattice represents an
inertial frame of reference, and we
imagine that we are at rest relative
to the lattice.
We synchronise the clocks to the
"Reference Clock." To do this
correctly requires taking into
account that if we are standing by
one of the clocks looking at the
Reference Clock, the time that we
see on the Reference is not the
current time, but is the time it was
reading when the light we see left
the clock. Thus we have to account
for the small but finite time it takes
light to travel from the Reference
Clock to us standing beside another
clock. A bit tedious, but fairly
straightforward.

We imagine some event occurs. We define its position by where it happened relative to the lattice of meter sticks and we define the time when it
happened as the time read by the nearest clock.
Of course, in practice nobody ever does this sort of thing.
Usually we don't bother to draw the whole lattice, but rather represent
it by a set of coordinate axes, x and y, and a single clock measuring
time t, as shown to the right. We have also put an observer, whom we
shall name Lou, at rest in his coordinate system.
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Next we imagine that Lou has a light bulb at the "origin" of his
coordinate system. At some time t which we shall call zero he turns on
the light. The light moves away from the light bulb at 1,079,253,000
km/hr as measured by Lou's system of rods and clocks. At some time t
later the light will form a sphere with the light bulb right at the center.
There are two animations of this situation. One is a "simple" animated
gif with a file size of 22k; it may be accessed by clicking here. The
other is a Flash animation with a file size of 16k; it may be accessed
by clicking here.

Now, Lou has a twin sister Sue, whom we shall assume was born at
the same time as Lou (a biological impossibility). Sue has her own
lattice of meter sticks and clocks and she is at rest relative to them.
Just as for Lou, we represent Sue's rods and clocks as shown to the
right.
Sue is an astronaut, and is in her rocket ship which is travelling at one-
half the speed of light to the right relative to Lou. Of course, relative
to Sue, Lou is travelling at half the speed of light to the left.





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Let us imagine that Sue, travelling at half the speed of light relative to
Lou, goes by Lou and he turned on the light bulb just at the moment
that Sue passed by it. Sue will call this time zero as measured by her
clocks.
Relative to Sue, the light bulb is travelling to the left at half the speed
of light. However, because of Einstein's "explanation", the speed of
light relative to her is exactly 1,079,253,000 km/hr. Thus, at some
later time she will measure that the outer edge of the light forms a
perfect sphere with her at the middle.
There are both a animated gif and Flash animation of the above. To access the 18k gif animation click here. To access the 18k Flash animation click
here.
There is also a Flash animation of both Sue and Lou. To access the 22k animation click here.
If we think about the above a moment, it is clear that something weird is going on. Lou claims that the light forms a sphere with the light bulb at the
center. Sue claims the light forms a sphere with her at the center. But except for the moment when the light bulb was first turned on, the light bulb and
Sue are at nowhere near the same place. Evidently the position and time of the outer edge of the sphere as measured by Lou's system of rods and
clocks and as measured by Sue's system of rods and clocks are not as our common sense would predict.
Note that the only assumption we have made here is the constancy of the speed of light. Thus, to avoid this sort of weirdness one must come up with
another explanation of the null result of the Michelson-Morley experiment.
We shall close this section by being slightly mathematical. The only mathematics
that we shall use is Pythagoras' Theorem for right triangles.
This theorem says that for any right triangle such as the one shown to the right:
x2 + y2 = h2
Now, when Lou measures the position of the outer edge of the sphere of light he can use Pythagoras' Theorem to calculate the radius r of the sphere:
x2Lou + y2Lou = r2Lou
But the radius at time t is just the speed of the light, c, times the time:
rLou = c tLou
So:
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x2Lou + y2Lou = (c t)2Lou
Notice that we don't need to label c as being the speed of light relative to Lou, since it is the same number for all observers, including Sue.
Now, Sue measures the position of the outer edge of the sphere of light with her rods and clocks and will conclude that:
x2Sue + y2Sue = (c t)2Sue
I will write the relations for Sue and Lou in a form which will be useful later:
x2Lou + y2Lou - (c t)2Lou = 0 = x2Sue + y2Sue - (c t)2Sue






THE PARABLE OF THE SURVEYORS
In this section we do a diversion: a fairy tale.
Once upon a time there was a kingdom in which all positions were measured relative to the town square of the capitol.
This kingdom had a sort of strange religion that dictated that all North-South distances were to be measured in sacred units of feet; East-West
distances were measured in everyday units of meters.
Despite this religious requirement all positions in the kingdom could be uniquely specified.
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There were two schools or surveying in operation. One, the daytime school, used a compass to determine the direction of North. The other, the
nighttime school, used the North star to determine the direction of North.
As the sophistication of the measuring instruments increased, people began to notice that the daytime and nighttime measurements didn't quite agree.
This is because magnetic North as determined by a compass is not in exactly the same direction as the North star. The figure to the right illustrates,
although the actual difference is much less than in the diagram.
Finally, a young fellow named Albert attended both schools of surveying. He was also an irreligious person so he did not take the religious
requirement of measuring North-South distances in feet seriously. He converted those North-South distances to everyday units by multiplying by k,
the number of meters in a foot. He then discovered that although the daytime and nighttime numbers for the position of a particular place differed
slightly, there was a constant:
E2night + (k N)2night = E2day + (k N)2day
What he is calculating, of course, is the distance squared between the town square and a particular location using Pythagoras' Theorem.
The original source for the above story is E.F. Taylor and J.A. Wheeler, Spacetime Physics (Freeman, 1966), pg. 1.
SPACETIME
In the parable of the surveyors, we converted North-South distances from sacred units of feet to everyday units of meters, and found that for the two
rotated reference frames, the daytime and nighttime frames, there was a constant for the position of a particular place in the kingdom relative to the
town square:
E2night + (k N)2night = E2day + (k N)2day
In the section before that Sue and Lou were observing the same sphere of light expanding outwards and saw that here too there was a constant:
x2Lou + y2Lou - (c t)2Lou = x2Sue + y2Sue - (c t)2Sue
Notice the similarity to the surveyor system. Take time, measured in sacred units of seconds, and convert to everyday units of meters by multiplying
the time by the speed of light. Take the normal position coordinates x and y plus the time coordinate, square them and combine them: the result is the
same number for both Sue and Lou.
Thus we are led to the idea the time is just another coordinate, i.e. that time is the fourth dimension. The fact that there is a minus sign between the
square of the normal spatial coordinates and the square of the time coordinates indicates that there is some difference between space and time, but it is
not a large difference.
Thus, we tend to write spacetime as a single word as a mnemonic to remind us of all this.
Note that the speed of light, c, is now only a conversion factor for units. If we had started out measuring time in everyday units of meters instead of
sacred units of seconds, the speed of light would just be one.
Spacetime Diagrams
The spacetime diagram is a useful visualisation technique.
The time axis is vertical, and of course we have multiplied t by c so we are measuring time in meters, the same as the other coordinates.
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