The special theory of relativity arose as a way of accounting for the
facts of electromagnetism. We have here a somewhat curious history. In
the eighteenth and early nineteenth centuries the theory of electricity
was wholly dominated by the Newtonian analogy. Two electric charges
attract each other if they are of different kinds, one positive and
one negative, but repel each other if they are of the same kind; in
each case, the force varies as the inverse square of the distance,
as in the case of gravitation. This force was conceived as an action
at a distance, until Faraday, by a number of remarkable experiments,
demonstrated the effect of the intervening medium. Faraday was no
mathematician; Clerk Maxwell first gave a mathematical form to the
results suggested by Faraday’s experiments. Moreover Clerk Maxwell gave
grounds for thinking that light is an electromagnetic phenomenon,
consisting of electromagnetic waves. The medium for the transmission of
electromagnetic effects could therefore be taken to be the ether, which
had long been assumed for the transmission of light. The correctness
of Maxwell’s theory of light was proved by the experiments of Hertz in
manufacturing electromagnetic waves; these experiments afforded the
basis for wireless telegraphy. So far, we have a record of triumphant
progress, in which theory and experiment alternately assume the leading
role. At the time of Hertz’s experiments, the ether seemed securely
established, and in just as strong a position as any other scientific
hypothesis not capable of direct verification. But a new set of facts
began to be discovered, and gradually the whole picture was changed.
The movement which culminated with Hertz was a movement for making
everything continuous. The ether was continuous, the waves in it were
continuous, and it was hoped that matter would be found to consist
of some continuous structure in the ether. Then came the discovery
of the electron, a small finite unit of negative electricity, and
the proton, a small finite unit of positive electricity. The most
modern view is that electricity is never found except in the form of
electrons and protons; all electrons have the same amount of negative
electricity, and all protons have an exactly equal and opposite amount
of positive electricity. It appeared that an electric current, which
had been thought of as a continuous phenomenon, consists of electrons
traveling one way and positive ions traveling the other way; it is no
more strictly continuous than the stream of people going up and down
an escalator. Then came the discovery of quanta, which seems to show
a fundamental discontinuity in all such natural processes as can be
measured with sufficient precision. Thus physics has had to digest new
facts and face new problems.
But the problems raised by the electron and the quantum are not those
that the theory of relativity can solve, at any rate at present; as
yet, it throws no light upon the discontinuities which exist in nature.
The problems solved by the special theory of relativity are typified by
the Michelson-Morley experiment. Assuming the correctness of Maxwell’s
theory of electromagnetism, there should have been certain discoverable
effects of motion through the ether; in fact, there were none. Then
there was the observed fact that a body in very rapid motion appears
to increase its mass; the increase is in the ratio of =OP= to =MP=
in the figure in the preceding chapter. Facts of this sort gradually
accumulated, until it became imperative to find some theory which would
account for them all.
Maxwell’s theory reduced itself to certain equations, known as
“Maxwell’s equations.” Through all the revolutions which physics has
undergone in the last fifty years, these equations have remained
standing; indeed they have continually grown in importance as well as
in certainty—for Maxwell’s arguments in their favor were so shaky that
the correctness of his results must almost be ascribed to intuition.
Now these equations were, of course, obtained from experiments in
terrestrial laboratories, but there was a tacit assumption that the
motion of the earth through the ether could be ignored. In certain
cases, such as the Michelson-Morley experiment, this ought not to have
been possible without measurable error; but it turned out to be always
possible. Physicists were faced with the odd difficulty that Maxwell’s
equations were more accurate than they should be. A very similar
difficulty was explained by Galileo at the very beginning of modern
physics. Most people think that if you let a weight drop it will fall
vertically. But if you try the experiment in the cabin of a moving
ship, the weight falls, in relation to the cabin, just as if the ship
were at rest; for instance, if it starts from the middle of the ceiling
it will drop onto the middle of the floor. That is to say, from the
point of view of an observer on the shore it does not fall vertically,
since it shares the motion of the ship. So long as the ship’s motion
is steady, everything goes on inside the ship as if the ship were not
moving. Galileo explained how this happens, to the great indignation
of the disciples of Aristotle. In orthodox physics, which is derived
from Galileo, a uniform motion in a straight line has no discoverable
effects. This was, in its day, as astonishing a form of relativity
as that of Einstein is to us. Einstein, in the special theory of
relativity, set to work to show how electromagnetic phenomena could be
unaffected by uniform motion through the ether if there be an ether.
This was a more difficult problem, which could not be solved by merely
adhering to the principles of Galileo.
The really difficult effort required for solving this problem was in
regard to time. It was necessary to introduce the notion of “proper”
time which we have already considered, and to abandon the old belief in
one universal time. The quantitative laws of electromagnetic phenomena
are expressed in Maxwell’s equations, and these equations are found
to be true for any observer, however he may be moving.[3] It is a
straight-forward mathematical problem to find out what differences
there must be between the measures applied by one observer and the
measures applied by another, if, in spite of their relative motion,
they are to find the same equations verified. The answer is contained
in the “Lorentz transformation,” found as a formula by Lorentz, but
interpreted and made intelligible by Einstein.
The Lorentz transformation tells us what estimate of distances and
periods of time will be made by an observer whose relative motion is
known, when we are given those of another observer. We may suppose that
you are in a train on a railway which travels due east. You have been
traveling for a time which, by the clocks at the station from which
you started, is _t_. At a distance _x_ from your starting point, as
measured by the people on the line, an event occurs at this moment—say
the line is struck by lightning. You have been traveling all the time
with a uniform velocity _v_. The question is: How far from you will you
judge that this event has taken place, and how long after you started
will it be by your watch, assuming that your watch is correct from the
point of view of an observer on the train?
[3] So long as he has no considerable acceleration. The treatment of
acceleration belongs to the _general_ theory of relativity.
Our solution of this problem has to satisfy certain conditions. It has
to bring out the result that the velocity of light is the same for all
observers, however they may be moving. And it has to make physical
phenomena—in particular, those of electromagnetism—obey the same
laws for different observers, however they may find their measures of
distances and times affected by their motion. And it has to make all
such effects on measurement reciprocal. That is to say, if you are in
a train and your motion affects your estimate of distances outside the
train, there must be an exactly similar change in the estimate which
people outside the train make of distances inside it. These conditions
are sufficient to determine the solution of the problem, but the
method of obtaining the solution cannot be explained without more
mathematics than is possible in the present work.
Before dealing with the matter in general terms, let us take an
example. Let us suppose that you are in a train on a long straight
railway, and that you are traveling at three-fifths of the velocity
of light. Suppose that you measure the length of your train, and find
that it is a hundred yards. Suppose that the people who catch a glimpse
of you as you pass succeed, by skilful scientific methods, in taking
observations which enable them to calculate the length of your train.
If they do their work correctly, they will find that it is eighty
yards long. Everything in the train will seem to them shorter in the
direction of the train than it does to you. Dinner plates, which you
see as ordinary circular plates, will look to the outsider as if they
were oval: they will seem only four-fifths as broad in the direction
in which the train is moving as in the direction of the breadth of the
train. And all this is reciprocal. Suppose you see out of the window a
man carrying a fishing rod which, by his measurement, is fifteen feet
long. If he is holding it upright, you will see it as he does; so you
will if he is holding it horizontally at right angles to the railway.
But if he is pointing it along the railway, it will seem to you to
be only twelve feet long. All lengths in the direction of motion are
diminished by twenty per cent, both for those who look into the train
from outside and for those who look out of the train from inside.
But the effects in regard to time are even more strange. This matter
has been explained with almost ideal lucidity by Eddington in _Space,
Time and Gravitation_. He supposes an aviator traveling, relatively to
the earth, at a speed of 161,000 miles a second, and he says:
“If we observed the aviator carefully we should infer that he was
unusually slow in his movements; and events in the conveyance moving
with him would be similarly retarded—as though time had forgotten to
go on. His cigar lasts twice as long as one of ours. I said ‘infer’
deliberately; we should _see_ a still more extravagant slowing down
of time; but that is easily explained, because the aviator is rapidly
increasing his distance from us and the light impressions take longer
and longer to reach us. The more moderate retardation referred to
remains after we have allowed for the time of transmission of light.
But here again reciprocity comes in, because in the aviator’s opinion
it is we who are traveling at 161,000 miles a second past him; and when
he has made all allowances, he finds that it is we who are sluggish.
Our cigar lasts twice as long as his.”
What a situation for envy! Each man thinks that the other’s cigar lasts
twice as long as his own. It may, however, be some consolation to
reflect that the other man’s visits to the dentist also last twice as
long.
This question of time is rather intricate, owing to the fact that
events which one man judges to be simultaneous another considers to be
separated by a lapse of time. In order to try to make clear how time
is affected, I shall revert to our railway train traveling due east at
a rate three-fifths of that of light. For the sake of illustration, I
assume that the earth is large and flat, instead of small and round.
If we take events which happen at a fixed point on the earth, and ask
ourselves how long after the beginning of the journey they will seem to
be to the traveler, the answer is that there will be that retardation
that Eddington speaks of, which means in this case that what seems an
hour in the life of the stationary person is judged to be an hour and a
quarter by the man who observes him from the train. Reciprocally, what
seems an hour in the life of the person in the train is judged by the
man observing him from outside to be an hour and a quarter. Each makes
periods of time observed in the life of the other a quarter as long
again as they are to the person who lives through them. The proportion
is the same in regard to times as in regard to lengths.
But when, instead of comparing events at the same place on the earth,
we compare events at widely separated places, the results are still
more odd. Let us now take all the events along the railway which, from
the point of view of a person who is stationary on the earth, happen
at a given instant, say the instant when the observer in the train
passes the stationary person. Of these events, those which occur at
points towards which the train is moving will seem to the traveler to
have already happened, while those which occur at points behind the
train will, for him, be still in the future. When I say that events
in the forward direction will seem to have already happened, I am
saying something not strictly accurate, because he will not yet have
seen them; but when he does see them, he will, after allowing for the
velocity of light, come to the conclusion that they must have happened
before the moment in question. An event which happens in the forward
direction along the railway, and which the stationary observer judges
to be now (or rather, will judge to have been now when he comes to know
of it), if it occurs at a distance along the line which light could
travel in a second, will be judged by the traveler to have occurred
three-quarters of a second ago. If it occurs at a distance from the two
observers which the man on the earth judges that light could travel
in a year, the traveler will judge (when he comes to know of it) that
it occurred nine months earlier than the moment when he passed the
earth dweller. And generally, he will ante-date events in the forward
direction along the railway by three-quarters of the time that it would
take light to travel from them to the man on the earth whom he is just
passing, and who holds that these events are happening now—or rather,
will hold that they happened now when the light from them reaches him.
Events happening on the railway behind the train will be post-dated by
an exactly equal amount.
We have thus a two-fold correction to make in the date of an event when
we pass from the terrestrial observer to the traveler. We must first
take five-fourths of the time as estimated by the earth dweller, and
then subtract three-fourths of the time that it would take light to
travel from the event in question to the earth dweller.
Take some event in a distant part of the universe, which becomes
visible to the earth dweller and the traveler just as they pass each
other. The earth dweller, if he knows how far off the event occurred,
can judge how long ago it occurred, since he knows the speed of light.
If the event occurred in the direction towards which the traveler is
moving, the traveler will infer that it happened twice as long ago as
the earth dweller thinks. But if it occurred in the direction from
which he has come, he will argue that it happened only half as long
ago as the earth dweller thinks. If the traveler moves at a different
speed, these proportions will be different.
Suppose now that (as sometimes occurs) two new stars have suddenly
flared up, and have just become visible to the traveler and to the
earth dweller whom he is passing. Let one of them be in the direction
towards which the train is traveling, the other in the direction from
which it has come. Suppose that the earth dweller is able, in some way,
to estimate the distance of the two stars, and to infer that light
takes fifty years to reach him from the one in the direction towards
which the traveler is moving, and one hundred years to reach him from
the other. He will then argue that the explosion which produced the
new star in the forward direction occurred fifty years ago, while the
explosion which produced the other new star occurred a hundred years
ago. The traveler will exactly reverse these figures: he will infer
that the forward explosion occurred a hundred years ago, and the
backward one fifty years ago. I assume that both argue correctly on
correct physical data. In fact, both are right, unless they imagine
that the other must be wrong. It should be noted that both will have
the same estimate of the velocity of light, because their estimates
of the distances of the two new stars will vary in exactly the same
proportion as their estimates of the times since the explosions.
Indeed, one of the main motives of this whole theory is to secure that
the velocity of light shall be the same for all observers, however they
may be moving. This fact, established by experiment, was incompatible
with the old theories, and made it absolutely necessary to admit
something startling. The theory of relativity is just as little
startling as is compatible with the facts. Indeed, after a time, it
ceases to seem startling at all.
There is another feature of very great importance in the theory we
have been considering, and that is that, although distances and times
vary for different observers, we can derive from them the quantity
called “interval,” which is the same for all observers. The “interval,”
in the special theory of relativity, is obtained as follows: Take
the square of the distance between two events, and the square of the
distance traveled by light in the time between the two events; subtract
the lesser of these from the greater, and the result is defined as
the square of the interval between the events. The interval is the
same for all observers, and represents a genuine physical relation
between the two events, which the time and the distance do not. We
have already given a geometrical construction for the interval at the
end of Chapter IV; this gives the same result as the above rule. The
interval is “time-like” when the time between the events is longer than
light would take to travel from the place of the one to the place
of the other; in the contrary case it is “space-like.” When the time
between the two events is exactly equal to the time taken by light to
travel from one to the other, the interval is zero; the two events are
then situated on parts of one light ray, unless no light happens to be
passing that way.
When we come to the general theory of relativity, we shall have to
generalize the notion of interval. The more deeply we penetrate into
the structure of the world, the more important this concept becomes;
we are tempted to say that it is the reality of which distances and
periods of time are confused representations. The theory of relativity
has altered our view of the fundamental structure of the world; that is
the source both of its difficulty and of its importance.
The remainder of this chapter may be omitted by readers who have not
even the most elementary acquaintance with geometry or algebra. But for
the benefit of those whose education has not been _entirely_ neglected,
I will add a few explanations of the general formula of which I have
hitherto given only particular examples. The general formula in
question is the “Lorentz transformation,” which tells, when one body
is moving in a given manner relatively to another, how to infer the
measures of lengths and times appropriate to the one body from those
appropriate to the other. Before giving the algebraical formulæ, I
will give a geometrical construction. As before, we will suppose that
there are two observers, whom we will call =O= and =O=′, one of whom is
stationary on the earth while the other is traveling at a uniform speed
along a straight railway. At the beginning of the time considered, the
two observers were at the same point of the railway, but now they are
separated by a certain distance. A flash of lightning strikes a point
=X= on the railway, and =O= judges that at the moment when the flash
takes place the observer in the train has reached the point =O=′. The
problem is: how far will =O=′ judge that he is from the flash, and
how long after the beginning of the journey (when he was at =O=) will
he judge that the flash took place? We are supposed to know =O=′s
estimates, and we want to calculate those of =O=′.
[Illustration]
In the time that, according to =O=, has elapsed since the beginning of
the journey, let =OC= be the distance that light would have traveled
along the railway. Describe a circle about =O=, with =OC= as radius,
and through =O′= draw a perpendicular to the railway, meeting the
circle in =D=. On =OD= take a point =Y= such that =OY= is equal to =OX=
(=X= is the point of the railway where the lightning strikes). Draw
=YM= perpendicular to the railway, and =OS= perpendicular to =OD=. Let
=YM= and =OS= meet in =S=. Also let =DO′= produced and =OS= produced
meet in =R=. Through =X= and =C= draw perpendiculars to the railway
meeting =OS= produced in =Q= and =Z= respectively. Then =RQ= (as
measured by =O=) is the distance at which =O′= will believe himself to
be from the flash, not =O′X= as it would be according to the old view.
And whereas =O= thinks that, in the time from the beginning of the
journey to the flash, light would travel a distance =OC=, =O′= thinks
that the time elapsed is that required for light to travel the distance
=SZ= (as measured by =O=). The interval as measured by =O= is got by
subtracting the square on =OX= from the square on =OC=; the interval
as measured by =O′= is got by subtracting the square on =RQ= from the
square on =SZ=. A little very elementary geometry shows that these are
equal.
The algebraical formulæ embodied in the above construction are as
follows: From the point of view of =O=, let an event occur at a
distance _x_ along the railway, and at a time _t_ after the beginning
of the journey (when =O′= was at =O=). From the point of view of =O′=,
let the same event occur at a distance _x′_ along the railway, and at a
time _t′_ after the beginning of the journey. Let _c_ be the velocity
of light, and _v_ the velocity of =O′= relative to =O=. Put
_c_
β = ————————————
√(_c_² - _v_²)
Then
_x′_ = β(_x_ - _vt_)
( _vx_ )
_t′_ = β(_t_ - —————)
( _c²_ )
This is the Lorentz transformation, from which everything in this
chapter can be deduced.