Until the advent of the special theory of relativity, no one had
thought that there could be any ambiguity in the statement that
two events in different places happened at the same time. It might
be admitted that, if the places were very far apart, there might
be difficulty in finding out for certain whether the events were
simultaneous, but every one thought the meaning of the question
perfectly definite. It turned out, however, that this was a mistake.
Two events in distant places may appear simultaneous to one observer
who has taken all due precautions to insure accuracy (and, in
particular, has allowed for the velocity of light), while another
equally careful observer may judge that the first event preceded
the second, and still another may judge that the second preceded
the first. This would happen if the three observers were all moving
rapidly relatively to each other. It would not be the case that one
of them would be right and the other two wrong: they would all be
equally right. The time order of events is in part dependent upon the
observer; it is not always and altogether an intrinsic relation between
the events themselves. Einstein has shown, not only that this view
accounts for the phenomena, but also that it is the one which ought
to have resulted from careful reasoning based upon the old data. In
actual fact, however, no one noticed the logical basis of the theory
of relativity until the odd results of experiment had given a jog to
people’s reasoning powers.
How should we naturally decide whether two events in different places
were simultaneous? One would naturally say: they are simultaneous
if they are seen simultaneously by a person who is exactly half-way
between them. (There is no difficulty about the simultaneity of two
events in the _same_ place, such, for example, as seeing a light
and hearing a noise.) Suppose two flashes of lightning fall in two
different places, say Greenwich Observatory and Kew Observatory.
Suppose that St. Paul’s is half-way between them, and that the flashes
appear simultaneous to an observer on the dome of St. Paul’s. In that
case, a man at Kew will see the Kew flash first, and a man at Greenwich
will see the Greenwich flash first, because of the time taken by
light to travel over the intervening distance. But all three, if they
are ideally accurate observers, will judge that the two flashes were
simultaneous, because they will make the necessary allowance for the
time of transmission of the light. (I am assuming a degree of accuracy
far beyond human powers.) Thus, so far as observers on the earth are
concerned, the definition of simultaneity will work well enough, so
long as we are dealing with events on the surface of the earth. It
gives results which are consistent with each other, and can be used for
terrestrial physics in all problems in which we can ignore the fact
that the earth moves.
But our definition is no longer so satisfactory when we have two sets
of observers in rapid motion relatively to each other. Suppose we see
what would happen if we substitute sound for light, and defined two
occurrences as simultaneous when they are heard simultaneously by a
man half-way between them. This alters nothing in the principle, but
makes the matter easier owing to the much slower velocity of sound.
Let us suppose that on a foggy night two men belonging to a gang of
brigands shoot the guard and engine driver of a train. The guard is at
the end of the train; the brigands are on the line, and shoot their
victims at close quarters. An old gentleman who is exactly in the
middle of the train hears the two shots simultaneously. You would say,
therefore, that the two shots were simultaneous. But a station master
who is exactly half-way between the two brigands hears the shot which
kills the guard first. An Australian millionaire uncle of the guard
and the engine driver (who are cousins) has left his whole fortune to
the guard, or, should he die first, to the engine driver. Vast sums
are involved in the question of which died first. The case goes to the
House of Lords, and the lawyers on both sides, having been educated at
Oxford, are agreed that either the old gentleman or the station master
must have been mistaken. In fact, both may perfectly well be right. The
train travels away from the shot at the guard, and towards the shot at
the engine driver; therefore the noise of the shot at the guard has
farther to go before reaching the old gentleman than the shot at the
engine driver has. Therefore if the old gentleman is right in saying
that he heard the two reports simultaneously, the station master must
be right in saying that he heard the shot at the guard first.
We, who live on the earth, would naturally, in such a case, prefer
the view of simultaneity obtained from a person at rest on the earth
to the view of a person traveling in a train. But in theoretical
physics no such parochial prejudices are permissible. A physicist on a
comet, if there were one, would have just as good a right to his view
of simultaneity as an earthly physicist has to his, but the results
would differ, in just the same sort of way as in our illustration of
the train and the shots. The train is not any more “really” in motion
than the earth; there is no “really” about it. You might imagine a
rabbit and a hippopotamus arguing as to whether man is “really” a large
animal; each would think his own point of view the natural one, and
the other a pure flight of fancy. There is just as little substance
in an argument as to whether the earth or the train is “really” in
motion. And, therefore, when we are defining simultaneity between
distant events, we have no right to pick and choose among different
bodies to be used in defining the point half-way between the events.
All bodies have an equal right to be chosen. But if, for one body, the
two events are simultaneous according to the definition, there will
be other bodies for which the first precedes the second, and still
others for which the second precedes the first. We cannot therefore
say unambiguously that two events in distant places are simultaneous.
Such a statement only acquires a definite meaning in relation to a
definite observer. It belongs to the subjective part of our observation
of physical phenomena, not to the objective part which is to enter into
physical laws.
This question of time in different places is perhaps, for the
imagination, the most difficult aspect of the theory of relativity. We
are accustomed to the idea that everything can be dated. Historians
make use of the fact that there was an eclipse of the sun visible in
China on August 29 in the year 776 B. C.[1] No doubt astronomers could
tell the exact hour and minute when the eclipse began to be total at
any given spot in North China. And it seems obvious that we can speak
of the positions of the planets at a given instant. The Newtonian
theory enables us to calculate the distance between the earth and (say)
Jupiter at a given time by the Greenwich clocks; this enables us to
know how long light takes at that time to travel from Jupiter to the
earth—say half an hour; this enables us to infer that half an hour ago
Jupiter was where we see it now. All this seems obvious. But in fact it
only works in practice because the relative velocities of the planets
are very small compared with the velocity of light. When we judge that
an event on the earth and an event on Jupiter have happened at the
same time—for example, that Jupiter eclipsed one of his moons when
the Greenwich clocks showed twelve midnight—a person moving rapidly
relatively to the earth would judge differently, assuming that both he
and we had made the proper allowance for the velocity of light. And
naturally the disagreement about simultaneity involves a disagreement
about periods of time. If we judged that two events on Jupiter were
separated by twenty-four hours, another person might judge that they
were separated by a longer time, if he were moving rapidly relatively
to Jupiter and the earth.
[1] A contemporary Chinese ode, after giving the day of the year
correctly, proceeds:
“For the moon to be eclipsed
Is but an ordinary matter.
Now that the sun has been eclipsed,
How bad it is.”
The universal cosmic time which used to be taken for granted is thus no
longer admissible. For each body, there is a definite time order for
the events in its neighborhood; this may be called the “proper” time
for that body. Our own experience is governed by the proper time for
our own body. As we all remain very nearly stationary on the earth,
the proper times of different human beings agree, and can be lumped
together as terrestrial time. But this is only the time appropriate to
_large_ bodies on the earth. For Beta-particles in laboratories, quite
different times would be wanted; it is because we insist upon using
our own time that these particles seem to increase in mass with rapid
motion. From their own point of view, their mass remains constant,
and it is we who suddenly grow thin or corpulent. The history of a
physicist as observed by a Beta-particle would resemble Gulliver’s
travels.
The question now arises: what really is measured by a clock? When we
speak of a clock in the theory of relativity, we do not mean only
clocks made by human hands: we mean anything which goes through some
regular periodic performance. The earth is a clock, because it rotates
once in every twenty-three hours and fifty-six minutes. An atom is a
clock, because the electrons go round the nucleus a certain number of
times in a second; its properties as a clock are exhibited to us in
its spectrum, which is due to light waves of various frequencies. The
world is full of periodic occurrences, and fundamental mechanisms,
such as atoms, show an extraordinary similarity in different parts of
the universe. Any one of these periodic occurrences may be used for
measuring time; the only advantage of humanly manufactured clocks is
that they are specially easy to observe. One question is: If cosmic
time is abandoned, what is really measured by a clock in the wide sense
that we have just given to the term?
Each clock gives a correct measure of its own “proper” time, which,
as we shall see presently, is an important physical quantity. But it
does not give an accurate measure of any physical quantity connected
with events on bodies that are moving rapidly in relation to it. It
gives one datum towards the discovery of a physical quantity connected
with such events, but another datum is required, and this has to be
derived from measurement of distances in space. Distances in space,
like periods of time, are in general not objective physical facts, but
partly dependent upon the observer. How this comes about must now be
explained.
First of all, we have to think of the distance between two events, not
between two bodies. This follows at once from what we have found as
regards time. If two bodies are moving relatively to each other—and
this is really always the case—the distance between them will be
continually changing, so that we can only speak of the distance
between them at a given time. If you are in a train traveling towards
Edinburgh, we can speak of your distance from Edinburgh at a given
time. But, as we said, different observers will judge differently as
to what is the “same” time for an event in the train and an event in
Edinburgh. This makes the measurement of distances relative, in just
the same way as the measurement of times has been found to be relative.
We commonly think that there are two separate kinds of interval between
two events, an interval in space and an interval in time: between your
departure from London and your arrival in Edinburgh, there are 400
miles and ten hours. We have already seen that another observer will
judge the time differently; it is even more obvious that he will judge
the distance differently. An observer in the sun will think the motion
of the train quite trivial, and will judge that you have traveled the
distance traveled by the earth in its orbit and its diurnal rotation.
On the other hand, a flea in the railway carriage will judge that you
have not moved at all in space, but have afforded him a period of
pleasure which he will measure by his “proper” time, not by Greenwich
Observatory. It cannot be said that you or the sun dweller or the
flea are mistaken: each is equally justified, and is only wrong if he
ascribes an objective validity to his subjective measures. The distance
in space between two events is, therefore, not in itself a physical
fact. But, as we shall see, there is a physical fact which can be
inferred from the distance in time together with the distance in space.
This is what is called the “interval” in space-time.
Taking any two events in the universe, there are two different
possibilities as to the relation between them. It may be physically
possible for a body to travel so as to be present at both events, or it
may not. This depends upon the fact that no body can travel as fast as
light. Suppose, for example, that it were possible to send out a flash
of light from the earth and have it reflected back from the moon. The
time between the sending of the flash and the return of the reflection
would be about two and a half seconds. No body could travel so fast
as to be present on the earth during any part of those two and a half
seconds and also present on the moon at the moment of the arrival of
the flash, because in order to do so the body would have to travel
faster than light. But theoretically a body could be present on the
earth at any time before or after those two and a half seconds and also
present on the moon at the time when the flash arrived. When it is
physically impossible for a body to travel so as to be present at both
events, we shall say that the interval[2] between the two events is
“space-like”; when it is physically possible for a body to be present
at both events, we shall say that the interval between the two events
is “time-like.” When the interval is “space-like,” it is possible for
a body to move in such a way that an observer on the body will judge
the two events to be simultaneous. In that case, the “interval” between
the two events is what such an observer will judge to be the distance
in space between them. When the interval is “time-like,” a body can
be present at both events; in that case, the “interval” between the
two events is what an observer on the body will judge to be the time
between them, that is to say, it is his “proper” time between the two
events. There is a limiting case between the two, when the two events
are parts of one light flash—or, as we might say, when the one event
is the seeing of the other. In that case, the interval between the two
events is zero.
[2] I shall define “interval” in a moment.
There are thus three cases. (1) It may be possible for a ray of light
to be present at both events; this happens whenever one of them is the
seeing of the other. In this case the interval between the two events
is zero. (2) It may happen that no body can travel from one event to
the other, because in order to do so it would have to travel faster
than light. In that case, it is always physically possible for a body
to travel in such a way that an observer on the body would judge the
two events to be simultaneous. The interval is what he would judge to
be the distance in space between the two events. Such an interval is
called “space-like.” (3) It may be physically possible for a body to
travel so as to be present at both events; in that case, the interval
between them is what an observer on such a body will judge to be the
time between them. Such an interval is called “time-like.”
The interval between two events is a physical fact about them, not
dependent upon the particular circumstances of the observer.
There are two forms of the theory of relativity, the special and the
general. The former is in general only approximate, but is exact at
great distances from gravitating matter. When the special theory can
be applied, the interval can be calculated when we know the distance
in space and the distance in time between the two events, estimated by
any observer. If the distance in space is greater than the distance
that light would have traveled in the time, the separation is
space-like. Then the following construction gives the interval between
the two events: Draw a line =AB= as long as the distance that light
would travel in the time; round =A= describe a circle whose radius is
the distance in space between the two events; through =B= draw =BC=
perpendicular to =AB=, meeting the circle in =C=. Then =BC= is the
length of the interval between the two events.
[Illustration]
When the distance is time-like, use the same figure, but let =AC= be
now the distance that light would travel in the time, while =AB= is the
distance in space between the two events. The interval between them is
now the time that light would take to travel the distance =BC=.
Although =AB= and =AC= are different for different observers, =BC= is
the same length for all observers, subject to corrections made by the
general theory. It represents the one interval in “space-time” which
replaces the two intervals in space and time of the older physics. So
far, this notion of interval may appear somewhat mysterious, but as we
proceed it will grow less so, and its reason in the nature of things
will gradually emerge.