Relativity
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.