Space-Time

The A B C of Relativity by Bertrand Russells, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. V. SPACE-TIME

V. SPACE-TIME

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.

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.

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This book is part of the public domain. Bertrand Williams (2004). THE A B C OF RELATIVITY. Urbana, Illinois: Project Gutenberg. Retrieved October 2022, from https://www.gutenberg.org/files/67104/67104-h/67104-h.htm

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