This crude analogy haltingly conveys what is meant by curved time.
It is an idea which is implicit in the Theory of Relativity. This
theory has profoundly modified many of our basic conceptions about
the universe in which we are immersed. It is outside the province of
this book and beyond the power of its author even so much as to
sketch the main outlines of this theory, but certain of its
conclusions are indispensable, since they baldly set forth our
dilemma in regard to the measurement of space and time. We can
measure neither except relatively, because they must be measured one
by the other, and no matter how they vary, these variations always
compensate one another, leaving us in the same state of ignorance
that we were in before.
Suppose that two intelligent beings, one on Mars, let us say, and
the other on the earth, should attempt to establish _the same moment
of time_, by the interchange of light signals, or by any other
method which the most rigorous science could devise. Assume that
they have for this purpose two identically similar and mechanically
perfect chronometers, and that every difficulty of manipulation were
successfully overcome. Their experiment could end only in failure,
and the measure of this failure neither one, in his own place, could
possibly know. If, after the experiment, the Martian, chronometer in
hand, could be instantly and miraculously transported to the earth,
and the two settings compared, they would be found to be different:
how different, we do not know.
The reason for the failure of any such experiment anywhere conducted
can best be made plain by a crude paraphrase of a classic
proposition from Relativity. Suppose it is required to determine the
same moment of time at two different places on the earth's surface,
as must be attempted in finding their difference in longitude. Take
the Observatory at Greenwich for one place, and the observatory at
Washington for the other. At the moment the sun is on the meridian
of Greenwich, the exact time of crossing is noted and cabled to
Washington. The chronometer at Washington is set accordingly, and
the time checked back to Greenwich. This message arrives two seconds,
say, after the original message was sent. Washington is at once
notified of this double transmission interval. On the assumption
that HALF of it represents the time the message took to travel from
east to west, and the other half the time from west to east again,
the Washington chronometer is set one second ahead of the signalled
time, to compensate for its part of the loss. When the sun has
reached the meridian of Washington, the whole process is repeated,
and again as before, half of the time the message has taken to cross
and recross the Atlantic is added to the Greenwich record of noon at
Washington. The number of hours, minutes, seconds, and fractions of
a second between these two corrected records represents the
difference in solar time between the two places, and incidentally
the same moment of time has been established for both--at least, so
it would appear.
But is it established? That each message took an equal time to
travel each way is pure assumption, and happens to be a false one.
The accuracy of the result is vitiated by a condition of things to
which the Relativists have called attention. Our determination might
be defended if Washington and Greenwich could be assumed to remain
at rest during the experiments, and some argument might even be made
in its favor if we could secure any cosmic assurance that the
resultant motion of the earth should be the same when Greenwich
signalled its noon to Washington and Washington its noon to Greenwich.
Our present discussion is merely illustrative, or diagrammatic; so
we will neglect the velocity of the earth in its orbit round the sun,
some forty times greater than that of a cannon ball, and the more
uncertain and more vertiginous speed of the whole solar system
towards its unknown goal. Let us consider only the rotation of the
earth on its axis, the tide-speed of day and night. To fix our idea,
this may be taken, in our latitudes, at eighteen thousand miles per
day, or perhaps half the speed of a Mauser rifle bullet.
So fast, then, will Washington have been moving to meet the message
from Greenwich. So fast will Greenwich have been retreating from
Washington's message.
Now the ultimate effect of motion on the time-determination cannot
be calculated along any such simple lines as these. Indeed, it
cannot be exactly calculated at all, for we have not all the data.
But there is certainly _some_ effect. Suppose one rows four miles up
a river against a current of two miles per hour, at a rowing speed
of four miles per hour. This will take two hours, plainly. The
return trip with the river's gift of two miles per hour will
evidently require but forty minutes. _Two hours and forty minutes_
for the round trip, then, of eight miles.
Now then, to row eight miles in still water, according to our
supposition, would have required but _two hours_. But, some one
objects, the current must help the return trip as much as it
hindered the outgoing! Ah, here is the snare that catches
rough-and-ready common sense! How long would the double journey have
taken _if the river current had been faster than our rowing speed_?
How shall we schedule our trip if we cannot learn the correct speed,
_or if it varies from minute to minute_?
These explanations are necessarily symbolistic rather than
demonstrative, but any one who will seriously follow out these lines
of thought, or, still better, study the attitude of the hard-headed
modern physicist towards our classical geometry and mechanics,
cannot fail to realize how conventional, artificial--even
phantasmal--are the limitations set by the primitive idea of flat
space and straight time.
The inferences which we may draw from our hypothetical experiment
are plain. The settings of the two chronometers would be defective,
they would not show the same time, but each of them would mark the
_local_ time, proper to its own place. There would be no means of
detecting the amount of error, since the messages were transmitted
by a medium involved with them in their transportation. If only
local time can be established, the possibility of a warped
time-plane--the curvature of time--is directly opened up. Doubtless
it is true that on so relatively minute a scale as is offered by the
earth, any deviation from perfect flatness of the time-plane would
be so inconsiderable and imperceptible as to make it scientifically
negligible; but this by no means follows when we consider our
relation to other worlds and other systems.
A similar condition holds with regard to space-distortion. The
Theory of Relativity enforces the conclusion that from the
standpoint of our conventions in regard to these matters, all bodies
involved in transportation undergo a contraction in the direction of
that transportation, while their dimensions perpendicular to the
transportation remain invariable. This contraction is the same for
all bodies. For bodies of low velocity, like the earth, this
distortion would be almost immeasurably slight; but great or little,
no measuring instruments on the body transporting would ever
disclose it, for a measure would undergo the same contraction as the
thing measured.