Everybody who has ever heard of relativity knows the phrase
“space-time,” and knows that the correct thing is to use this phrase
when formerly we should have said “space _and_ time.” But very few
people who are not mathematicians have any clear idea of what is meant
by this change of phraseology. Before dealing further with the special
theory of relativity, I want to try to convey to the reader what is
involved in the new phrase “space-time,” because that is, from a
philosophical and imaginative point of view, perhaps the most important
of all the novelties that Einstein has introduced.
Suppose you wish to say where and when some event has occurred—say
an explosion on an airship—you will have to mention four quantities,
say the latitude and longitude, the height above the ground, and the
time. According to the traditional view, the first three of these
give the position in space, while the fourth gives the position in
time. The three quantities that give the position in space may be
assigned in all sorts of ways. You might, for instance, take the
plane of the equator, the plane of the meridian of Greenwich, and the
plane of the ninetieth meridian, and say how far the airship was from
each of these planes; these three distances would be what are called
“Cartesian co-ordinates,” after Descartes. You might take any other
three planes all at right angles to each other, and you would still
have Cartesian co-ordinates. Or you might take the distance from London
to a point vertically below the airship, the direction of this distance
(northeast, west-southwest, or whatever it might be), and the height of
the airship above the ground. There are an infinite number of such ways
of fixing the position in space, all equally legitimate; the choice
between them is merely one of convenience.
When people said that space had three dimensions, they meant just this:
that three quantities were necessary in order to specify the position
of a point in space, but that the method of assigning these quantities
was wholly arbitrary.
With regard to time, the matter was thought to be quite different. The
only arbitrary elements in the reckoning of time were the unit, and
the point of time from which the reckoning started. One could reckon
in Greenwich time, or in Paris time, or in New York time; that made a
difference as to the point of departure. One could reckon in seconds,
minutes, hours, days, or years; that was a difference of unit. Both
these were obvious and trivial matters. There was nothing corresponding
to the liberty of choice as to the method of fixing position in space.
And, in particular, it was thought that the method of fixing position
in space and the method of fixing position in time could be made wholly
independent of each other. For these reasons, people regarded time and
space as quite distinct.
The theory of relativity has changed this. There are now a number of
different ways of fixing position in time, which do not differ merely
as to the unit and the starting point. Indeed, as we have seen, if one
event is simultaneous with another in one reckoning, it will precede
it in another, and follow it in a third. Moreover, the space and time
reckonings are no longer independent of each other. If you alter the
way of reckoning position in space, you may also alter the time
interval between two events. If you alter the way of reckoning time,
you may also alter the distance in space between two events. Thus space
and time are no longer independent, any more than the three dimensions
of space are. We still need four quantities to determine the position
of an event, but we cannot, as before, divide off one of the four as
quite independent of the other three.
It is not quite true to say that there is no longer any distinction
between time and space. As we have seen, there are time-like intervals
and space-like intervals. But the distinction is of a different sort
from that which was formerly assumed. There is no longer a universal
time which can be applied without ambiguity to any part of the
universe; there are only the various “proper” times of the various
bodies in the universe, which agree approximately for two bodies which
are not in rapid relative motion, but never agree exactly except for
two bodies which are at rest relatively to each other.
The picture of the world which is required for this new state of
affairs is as follows: Suppose an event =E= occurs to me, and
simultaneously a flash of light goes out from me in all directions.
Anything that happens to any body after the light from the flash has
reached it is definitely after the event =E= in any system of reckoning
time. Any event anywhere which I could have seen before the event =E=
occurred to me is definitely before the event =E= in any system of
reckoning time. But any event which happened in the intervening time
is not definitely either before or after the event =E=. To make the
matter definite: suppose I could observe a person in Sirius, and he
could observe me. Anything which he does, and which I see before the
event =E= occurs to me, is definitely before =E=; anything he does
after he has seen the event =E= is definitely after =E=. But anything
that he does before he sees the event =E=, but so that I see it after
the event =E= has happened, is not definitely before or after =E=.
Since light takes many years to travel from Sirius to the earth, this
gives a period of twice as many years in Sirius which may be called
“contemporary” with =E=, since these years are not definitely before or
after =E=.
Dr. A. A. Robb, in his _Theory of Time and Space_, suggests a point
of view which may or may not be philosophically fundamental, but is
at any rate a help in understanding the state of affairs we have
been describing. He maintains that one event can only be said to be
definitely _before_ another if it can influence that other in some
way. Now influences spread from a center at varying rates. Newspapers
exercise an influence emanating from London at an average rate of about
twenty miles an hour—rather more for long distances. Anything a man
does because of what he reads in the newspaper is clearly subsequent
to the printing of the newspaper. Sounds travel much faster: it would
be possible to arrange a series of loud speakers along the main roads,
and have newspapers shouted from each to the next. But telegraphing is
quicker, and wireless telegraphy travels with the velocity of light,
so that nothing quicker can ever be hoped for. Now what a man does in
consequence of receiving a wireless message he does _after_ the message
was sent; the meaning here is quite independent of conventions as to
the measurement of time. But anything that he does while the message
is on its way cannot be influenced by the sending of the message, and
cannot influence the sender until some little time after he sent the
message. That is to say, if two bodies are widely separated, neither
can influence the other except after a certain lapse of time; what
happens before that time has elapsed cannot affect the distant body.
Suppose, for instance, that some notable event happens on the sun:
there is a period of sixteen minutes on the earth during which no event
on the earth can have influenced or been influenced by the said notable
event on the sun. This gives a substantial ground for regarding that
period of sixteen minutes on the earth as neither before nor after the
event on the sun.
The paradoxes of the special theory of relativity are only paradoxes
because we are unaccustomed to the point of view, and in the habit
of taking things for granted when we have no right to do so. This is
especially true as regards the measurement of lengths. In daily life,
our way of measuring lengths is to apply a foot rule or some other
measure. At the moment when the foot rule is applied, it is at rest
relatively to the body which is being measured. Consequently the length
that we arrive at by measurement is the “proper” length, that is to
say, the length as estimated by an observer who shares the motion of
the body. We never, in ordinary life, have to tackle the problem of
measuring a body which is in continual motion. And even if we did, the
velocities of visible bodies on the earth are so small relatively to
the earth that the anomalies dealt with by the theory of relativity
would not appear. But in astronomy, or in the investigation of atomic
structure, we are faced with problems which cannot be tackled in this
way. Not being Joshua, we cannot make the sun stand still while we
measure it; if we are to estimate its size, we must do so while it is
in motion relatively to us. And similarly if you want to estimate the
size of an electron, you have to do so while it is in rapid motion,
because it never stands still for a moment. This is the sort of problem
with which the theory of relativity is concerned. Measurement with a
foot rule, when it is possible, gives always the same result, because
it gives the “proper” length of a body. But when this method is not
possible, we find that curious things happen, particularly if the
body to be measured is moving very fast relatively to the observer. A
figure like the one at the end of the previous chapter will help us to
understand the state of affairs.
[Illustration]
Let us suppose that the body on which we wish to measure lengths
is moving relatively to ourselves, and that in one second it moves
the distance =OM=. Let us draw a circle round =O= whose radius is
the distance that light travels in a second. Through =M= draw =MP=
perpendicular to =OM=, meeting the circle in =P=. Thus =OP= is the
distance that light travels in a second. The ratio of =OP= to =OM=
is the ratio of the velocity of light to the velocity of the body.
The ratio of =OP= to =MP= is the ratio in which apparent lengths are
altered by the motion. That is to say, if the observer judges that
two points in the line of motion on the moving body are at a distance
from each other represented by =MP=, a person moving with the body
would judge that they were at a distance represented (on the same
scale) by =OP=. Distances on the moving body at right angles to the
line of motion are not affected by the motion. The whole thing is
reciprocal; that is to say, if an observer moving with the body were to
measure lengths on the previous observer’s body, they would be altered
in just the same proportion. When two bodies are moving relatively
to each other, lengths on either appear shorter to the other than
to themselves. This is the Fitzgerald contraction, which was first
invented to account for the result of the Michelson-Morley experiment.
But it now emerges naturally from the fact that the two observers do
not make the same judgment of simultaneity.
The way in which simultaneity comes in is this: We say that two points
on a body are a foot apart when we can _simultaneously_ apply one end
of a foot rule to the one and the other end to the other. If, now, two
people disagree about simultaneity, and the body is in motion, they
will obviously get different results from their measurements. Thus the
trouble about time is at the bottom of the trouble about distance.
The ratio of =OP= to =MP= is the essential thing in all these matters.
Times and lengths and masses are all altered in this proportion when
the body concerned is in motion relatively to the observer. It will
be seen that, if =OM= is very much smaller than =OP=, that is to say,
if the body is moving very much more slowly than light, =MP= and =OP=
are very nearly equal, so that the alterations produced by the motion
are very small. But if =OM= is nearly as large as =OP=, that is to
say, if the body is moving nearly as fast as light, =MP= becomes very
small compared to =OP=, and the effects become very great. The apparent
increase of mass in swiftly moving particles had been observed,
and the right formula had been found, before Einstein invented his
special theory of relativity. In fact, Lorentz had arrived at the
formulæ called the “Lorentz transformation,” which embody the whole
mathematical essence of the special theory of relativity. But it was
Einstein who showed that the whole thing was what we ought to have
expected, and not a set of makeshift devices to account for surprising
experimental results. Nevertheless, it must not be forgotten that
experimental results were the original motive of the whole theory,
and have remained the ground for undertaking the tremendous logical
reconstruction involved in Einstein’s theories.
We may now recapitulate the reasons which have made it necessary to
substitute “space-time” for space and time. The old separation of
space and time rested upon the belief that there was no ambiguity in
saying that two events in distant places happened at the same time;
consequently it was thought that we could describe the topography of
the universe at a given instant in purely spatial terms. But now that
simultaneity has become relative to a particular observer, this is
no longer possible. What is, for one observer, a description of the
state of the world at a given instant, is, for another observer, a
series of events at various different times, whose relations are not
merely spatial but also temporal. For the same reason, we are concerned
with _events_, rather than with _bodies_. In the old theory, it was
possible to consider a number of bodies all at the same instant, and
since the time was the same for all of them it could be ignored. But
now we cannot do that if we are to obtain an objective account of
physical occurrences. We must mention the date at which a body is to be
considered, and thus we arrive at an “event,” that is to say, something
which happens at a given time. When we know the time and place of an
event in one observer’s system of reckoning, we can calculate its time
and place according to another observer. But we must know the time as
well as the place, because we can no longer ask what is its place for
the new observer at the “same” time as for the old observer. There is
no such thing as the “same” time for different observers, unless they
are at rest relatively to each other. We need four measurements to
fix a position, and four measurements fix the position of an event in
space-time, not merely of a body in space. Three measurements are not
enough to fix any position. That is the essence of what is meant by the
substitution of space-time for space and time.