In order to arrive at what we want for the theory of relativity, we now have one more generalization to make: we have to substitute the “interval” between events for the distance between points. This takes us to space-time. We have already seen that, in the special theory of relativity, the square of the interval is found by subtracting the square of the distance between the events from the square of the distance that light would travel in the time between them. In the general theory, we do not assume this special form of interval, except at a great distance from matter. Elsewhere, we assume to begin with a general form, like that which Riemann used for distances. Moreover, like Riemann, Einstein only assumes his formula for neighboring events, that is to say, events which have only a small interval between them. What goes beyond these initial assumptions depends upon observation of the actual motion of bodies, in ways which we shall explain in later chapters.
We may now sum up and re-state the process we have been describing. In three dimensions, the position of a point relatively to a fixed point (the “origin”) can be determined by assigning three quantities (“co-ordinates”). For example, the position of a balloon relatively to your house is fixed if you know that you will reach it by going first a given distance due east, then another given distance due north, then a third given distance straight up. When, as in this case, the three co-ordinates are three distances all at right angles to each other, which, taken successively, transport you from the origin to the point in question, the square of the direct distance to the point in question is got by adding up the squares of the three co-ordinates. In all cases, whether in Euclidean or in non-Euclidean spaces, it is got by adding multiples of the squares and products of the co-ordinates according to an assignable rule. The co-ordinates may be any quantities which fix the position of a point, provided that neighboring points must have neighboring quantities for their co-ordinates. In the general theory of relativity, we add a fourth co-ordinate to give the time, and our formula gives “interval” instead of spatial distance; moreover we assume the accuracy of our formula for small distances only. We assume further that, at great distances from matter, the formula approximates more and more closely to the formula for interval which is used in the special theory.
We are now at last in a position to tackle Einstein’s theory of gravitation.
CHAPTER VIII:
EINSTEIN’S LAW OF GRAVITATION
Before tackling Einstein’s new law, it is as well to convince ourselves, on logical grounds, that Newton’s law of gravitation cannot be quite right.
Newton said that between any two particles of matter there is a force which is proportional to the product of their masses and inversely proportional to the square of their distance. That is to say, ignoring for the present the question of mass, if there is a certain attraction when the particles are a mile apart, there will be a quarter as much attraction when they are two miles apart, a ninth as much when they are three miles apart, and so on: the attraction diminishes much faster than the distance increases. Now, of course, Newton, when he spoke of the distance, meant the distance at a given time: He thought there could be no ambiguity about time. But we have seen that this was a mistake. What one observer judges to be the same moment on the earth and the sun, another will judge to be two different moments. “Distance at a given moment” is therefore a subjective conception, which can hardly enter into a cosmic law. Of course, we could make our law unambiguous by saying that we are going to estimate times as they are estimated by Greenwich Observatory. But we can hardly believe that the accidental circumstances of the earth deserve to be taken so seriously. And the estimate of distance, also, will vary for different observers. We cannot, therefore, allow that Newton’s form of the law of gravitation can be quite correct, since it will give different results according to which of many equally legitimate conventions we adopt. This is as absurd as it would be if the question whether one man had murdered another were to depend upon whether they were described by their Christian names or their surnames. It is obvious that physical laws must be the same whether distances are measured in miles or in kilometers, and we are concerned with what is essentially only an extension of the same principle.
Our measurements are conventional to an even greater extent than is admitted by the special theory of relativity. Moreover, every measurement is a physical process carried out with physical material; the result is certainly an experimental datum, but may not be susceptible of the simple interpretation which we ordinarily assign to it. We are, therefore, not going to assume to begin with that we know how to measure anything. We assume that there is a certain physical quantity, called “interval,” which is a relation between two events that are not widely separated; but we do not assume in advance that we know how to measure it, beyond taking it for granted that it is given by some generalization of the theorem of Pythagoras such as we spoke of in the preceding chapter.
We do assume, however, that events have an order, and that this order is four-dimensional. We assume, that is to say, that we know what we mean by saying that a certain event is nearer to another than to a third, so that before making accurate measurements we can speak of the “neighborhood” of an event; and we assume that, in order to assign the position of an event in space-time, four quantities (co-ordinates) are necessary—e.g. in our former case of an explosion on an airship, latitude, longitude, altitude and time. But we assume nothing about the way in which these co-ordinates are assigned, except that neighboring co-ordinates are assigned to neighboring events.
The way in which these numbers, called co-ordinates, are to be assigned is neither wholly arbitrary nor a result of careful measurement—it lies in an intermediate region. While you are making any continuous journey, your co-ordinates must never alter by sudden jumps. In America one finds that the houses between (say) Fourteenth Street and Fifteenth Street are likely to have numbers between 1400 and 1500, while those between Fifteenth Street and Sixteenth Street have numbers between 1500 and 1600, even if the 1400’s were not used up. This would not do for our purposes, because there is a sudden jump when we pass from one block to the next. Or again we might assign the time co-ordinate in the following way: take the time that elapses between two successive births of people called Smith; an event occurring between the births of the 3000th and the 3001st Smith known to history shall have a co-ordinate lying between 3000 and 3001; the fractional part of its co-ordinate shall be the fraction of a year that has elapsed since the birth of the 3000th Smith. (Obviously there could never be as much as a year between two successive additions to the Smith family.) This way of assigning the time co-ordinate is perfectly definite, but it is not admissible for our purposes, because there will be sudden jumps between events just before the birth of a Smith and events just after, so that in a continuous journey your time co-ordinate will not change continuously. It is assumed that, independently of measurement, we know what a continuous journey is. And when your position in space-time changes continuously, each of your four co-ordinates must change continuously. One, two, or three of them may not change at all; but whatever change does occur must be smooth, without sudden jumps. This explains what is not allowable in assigning co-ordinates.
To explain all the changes that are legitimate in your co-ordinates, suppose you take a large piece of soft india-rubber. While it is in an unstretched condition, measure little squares on it, each one-tenth of an inch each way. Put in little tiny pins at the corners of the squares. We can take as two of the co-ordinates of one of these pins the number of pins passed in going to the right from a given pin until we come just below the pin in question, and then the number of pins we pass on the way up to this pin. In the figure, let O be the pin we start from and P the pin to which we are going to assign co-ordinates. P is in the fifth column and the third row, so its co-ordinates in the plane of the india-rubber are to be 5 and 3.