Section seven. The apparent incompatibility of the law of propagation of light with the
principle of relativity. There is hardly a simpler law in physics than that according
to which light is propagated in empty space. Every child a school knows or
believes he knows that this propagation takes place in straight lines with a velocity c
equals three hundred thousand kilometers per second at all events. We know with great
exactness that this velocity is the same for all colors, because if this were
not the case, the minimum of emission would not be observed simultaneously for different
colors during the eclipse of a fixed star by its dark neighbor. By means
of similar considerations based on observations of double stars, the Dutch astronomer to Zetter
was also able to show that the velocity of propagation of light cannot depend on
the velocity of motion of the body emitting the light. The assumption that this
velocity of propagation is dependent on the direction in space is in itself improbable.
In short, let us assume that the simple law of the constancy of the
velocity of light see in vacuum is justifiably believed by the child at school,
who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the
greatest intellectual difficulties. Let us consider how these difficulties arise. Of course,
we must refer the process of the propagation of light, and indeed every other
process, to a rigid reference body coordinate system. As such a system,
let us again choose our embankment. We shall imagine the air above it to
have been removed if a ray of light be sent along the embankment. We
see from the above that the tip of the ray will be transmitted with the
velocity C relative to the embankment. Now, let us suppose that our railway
carriage is again traveling along the railway lines with the velocity V, and that
its direction is the same as that of the ray of light, but its
velocity, of course much less. Let us inquire about the velocity of propagation
of the ray of light relative to the carriage. It is obvious that we
can here apply the consideration of the previous section. Since the ray of light
plays the part of the man walking along relatively to the carriage, the velocity
capital W of the man relative to the embankment is here replaced by the velocity
of light relative to the embankment. W is the required velocity of light with
respect to the carriage, and we have W equals c fee. The velocity
of propagation of a ray of light relative to the carriage thus comes out smaller
than C. But this result comes into conflict with the principle of relativity set
forth in section five. For, like every other general law of nature,
the law of the transmission of light in vacuo must, according to the principle
of relativity, be the same for the railway carriage as reference body as when
the rails are the body of reference. But from our above consideration, this
would appear to be impossible. If every ray of light is propagated relative to
the embankment with the velocity C, then for this reason it would appear that
another law of propagation of light must necessarily hold with respect to the carriage,
a result contradictory to the principle of relativity. In view of this dilemma,
there appears to be nothing else for it than to abandon either the principle of
relativity or the simple law of the propagation of light invacuo. Those of you
who have carefully followed the preceding discussion are almost sure to expect that we should
retain the principle of relativity, which appeals so convincingly to the intellect because it
is so natural and simple. The law of the propagation of light invacuo would
then have to be replaced by a more complicated law conformable to the principle of
relativity. The development of theoretical physics shows, however, that we cannot pursue
this course. The epoch making theoretical investigations of H. A. Lorentz on
the electrodynamical and optical phenomena connected with moving bodies show that experience in this domain
leads conclusively to a theory of electromagnetic phenomena, of which the law of the
constancy of the velocity of light in vacuo is a necessary consequence. Prominent theoretical
physicists were therefore more inclined to reject the principle of relativity, in spite of
the fact that no empirical data had been found which were contradictory to this principle.
At this juncture, the theory of relativity entered the arena. As a
result of an analysis of the physical conceptions of time and space, it became
evident that in reality there is not the least incompatibility between the principle of relativity
and the law of propagation of light, and that by systematically holding fast to
both these laws, a logically rigid theory could be arrived at. This theory
has been called the special theory of relativity, to distinguish it from the extended
theory with which we shall deal later. In the following pages. We shall
present the fundamental ideas of the special theory of relativity. End of section seven.
Section eight on the idea of time in physics. Lightning has struck the
rails on our railway embankment at two places A and B, far distant from
each other. I make the additional assertion that these two lightning flashes occurred simultaneously.
If I ask you whether there is sense in this statement, you will
answer my question with a decided yes. But if I now approach you with
the request to explain to me the sense of the statement more precisely, you
find, after some consideration that the answer to this question is not so easy
as it appears at first sight. After some time, perhaps the following answer
would occur to you. The significance of the statement is clear in itself and
needs no further explanation. Of course, it would require some consideration if I
were to be commissioned to determine by observations whether, in the actual case the
two events took place simultaneously or not. I cannot be satisfied with this answer
for the following reason. Supposing that, as a result of ingenious considerations,
an able meteorologist were to discover that the lightning must always strike the places A
and B simultaneously, then we should be faced with the task of testing whether
or not this theoretical result is in accordance with the ready We encounter the same
difficulty with all physical statements in which the conception simultaneous plays apart. The concept
does not exist for the physicist until he has the possibility of discovering whether or
not it is fulfilled in an actual case. We thus require a definition of
simultaneity such that this definition supplies us with the method by means of which,
in the present case he can decide by experiment whether or not both the lightning
strokes occurred simultaneously. As long as this requirement is not satisfied, I allow
myself to be deceived as a physicist, and of course the same applies if
I am not a physicist. When I imagine that I am able to attach
a meaning to the statement of simultaneity, I would ask the reader not to
proceed farther until he is fully convinced on this point. After thinking the matter
over for some time, you then offer the following suggestion with which to test
simultaneity by measuring along the rails. The connecting line AB should be measured up
and an observer placed at the midpoint M of the distance AB. This observer
should be supplied with an arrangement e g. Two mirrors inclined at ninety degrees,
which allows him visually to observe both places A and B at the same
time. If the observer perceives the two flashes of lightning at the same time,
then they are simultaneous. I am very pleased with the suggestion, but
for all that I cannot regard the matter as quite settled because I feel constrained
to raise the following objection. Your definition would certainly be right if only I
knew that the light by means of which the observer at M perceives the lightning
flashes travels along the length A to M with the same velocity as along the
length B to M. But an examination of this supposition would only be possible
if we already had at our disposal the means of measuring time. It would
thus appear as though we were moving here in a logical circle. After further
consideration, you cast a somewhat disdainful glance at me, and rightly so,
and you declare, I maintain my previous definition. Nevertheless, because in reality
it assumes absolutely nothing about light, there is only one demand to be made
of the definition of simultaneity, namely that in every real case, it must
supply us with an empirical decision as to whether or not the conception that has
to be defined is fulfilled. That my definition satisfies this demand is indisputable.
The light requires the same time to traverse the path A to M as for
the path B to M. Is, in reality, neither a supposition nor
a hypothesis about the physical nature of light, but a stipulation which I can
make of my own free will in order to arrive at a definition of simultaneity.
It is clear that this definition can be used to give an exact meaning
not only to two events, but to as many events as we care to
choose, and independently of the positions of the scenes of the events with respect
to the body of reference here the railway embankment footnote. We suppose further that
when three events A, B, and C occur in different places in such
a manner that A is simultaneous with B, and B is simultaneous with C
simultaneous in the sense of the above definition, then the criterion for the simultaneity
of the pair of events A C is also satisfied. This assumption is a
physical hypothesis about the propagation of light. It must certainly be fulfilled if we
are to maintain the law of the constancy of the velocity of light in vacuo
end a footnote. We are thus led also to a definition of time in
physics. For this purpose, we suppose that clocks of identical construction are placed
at the points A, B, and C of the railway line coordinate system,
and that they are set in such a manner that the positions of their
pointers are simultaneously in the sense the same. Under these conditions, we understand
by the time of an event the reading position of the hands of that one
of these clocks which is in the immediate vicinity in space of the event.
In this manner, a time value is associated with every event which is essentially
capable of observation. This stipulation contains a further physical hypothesis, the validity of
which will hardly be doubted without empirical evidence. To the contrary, it has
been assumed that all these clocks go at the same rate if they are of
identical construction. Stated more exactly, when two clocks arranged at rest in different
places of a reference body are set in such a manner that a particular position
of the pointers of the one clock is simultaneous in the above sense with the
same position of the pointers of the other clock, then identical settings are always
simultaneous in the sense of the above definition. End of Section eight. Section
nine the relativity of simultaneity. Up to now our considerations have been referred to
a particular body of reference, which we have styled a railway embankment. We
suppose a very long train traveling along the rails with the constant velocity V and
in the direction indicated in Figure one. People traveling in this train will with
advantage view the train as a rigid reference body coordinate system. They regard all
events in reference to the train, Then every event which takes place along the
line also takes place at a particular point of the train. Also, the
definition of simultaneity can be given relative to the train in exactly the same way
as with respect to the embankment. As a natural consequence, however, the
following question arises. Are two events e g. The two strokes of lightning
A and B, which are simultaneous with reference to the railway embankment, also
simultaneous relatively to the train. We shall show directly that the answer must be
in the negative. When we say that the lightning strokes A and B are
simultaneous with respect to the embankment, we mean the rays of light emitted at
the places A and B where the lightning occurs meet each other at the midpoint
M of the length A to B of the embankment. But the events A
and B also correspond to positions A and B on the train. Let M
prime be the midpoint of the distance A to B on the traveling train.
Just when the flashes as judged from the embankment of lightning occur. This point
M prime naturally coincides with the point M, but it moves toward the right
in the diagram with the velocity V of the train. If an observer sitting
at the position M prime in the train did not possess this velocity, then
he would remain permanently at M, and the light rays emitted by the flashes
of lightning A and B would reach him simultaneously, i e. They would
meet just where he is situated now. In reality, considered with reference to
the railway embankment, he is hastening towards the beam of light coming from B,
whilst he is riding on ahead of the beam of light coming from A.
Hence, the observer will see the beam of light emitted from B earlier
than he will see that emitted from A. Observers who take the railway train
as the reference body must therefore come to the conclusion that the lightning flash B
took place earlier than the lightning flash A. We thus arrive at an important
result. Events which are simultaneous with reference to the embankment are not simultaneous with
respect to the train, and vice versa. Relativity of simultaneity every reference body
coordinate system has its own particular time. Unless we are told the reference body
to which the statement of time refers, there is no meaning in a statement
of the time of an event. Now, before the advent of the theory
of relativity, it had always tacitly been assumed in physics that the statement of
time had an absolute significance, ie that it is independent of the state of
motion of the body of reference. But we have just seen that this assumption
is incompatible with the most natural definition of simultaneity. If we've discard this assumption,
then the conflict between the law of the propagation of light invacuo and the
principle of relativity developed in section seven disappears. We were led to that conflict
by the considerations of section six, which are now no longer tenable. In
that section, we concluded that the man in the carriage who traverses the distance
w per second relative to the carriage, traverses the same distance also with respect
to the embankment in each second of time. But according to the foregoing considerations,
the time required by a particular occurrence with respect to the carriage must not
be considered equal to the duration of the same occurrence as judged from the embankment
as reference body. Hence, it cannot be contended that the man in walking
travels the distance W relative to the railway line in a time which is equal
to one second as judged from the embankment. Moreover, the considerations of Section
six are based on yet a second assumption, which, in the light of
a strict consideration appears to be arbitrary, although it was always pacitly made even
before the introduction of the theory of relativity. End of Section nine