We have thus a two-fold correction to make in the date of an event when we pass from the terrestrial observer to the traveler. We must first take five-fourths of the time as estimated by the earth dweller, and then subtract three-fourths of the time that it would take light to travel from the event in question to the earth dweller.

Take some event in a distant part of the universe, which becomes visible to the earth dweller and the traveler just as they pass each other. The earth dweller, if he knows how far off the event occurred, can judge how long ago it occurred, since he knows the speed of light. If the event occurred in the direction towards which the traveler is moving, the traveler will infer that it happened twice as long ago as the earth dweller thinks. But if it occurred in the direction from which he has come, he will argue that it happened only half as long ago as the earth dweller thinks. If the traveler moves at a different speed, these proportions will be different.

Suppose now that (as sometimes occurs) two new stars have suddenly flared up, and have just become visible to the traveler and to the earth dweller whom he is passing. Let one of them be in the direction towards which the train is traveling, the other in the direction from which it has come. Suppose that the earth dweller is able, in some way, to estimate the distance of the two stars, and to infer that light takes fifty years to reach him from the one in the direction towards which the traveler is moving, and one hundred years to reach him from the other. He will then argue that the explosion which produced the new star in the forward direction occurred fifty years ago, while the explosion which produced the other new star occurred a hundred years ago. The traveler will exactly reverse these figures: he will infer that the forward explosion occurred a hundred years ago, and the backward one fifty years ago. I assume that both argue correctly on correct physical data. In fact, both are right, unless they imagine that the other must be wrong. It should be noted that both will have the same estimate of the velocity of light, because their estimates of the distances of the two new stars will vary in exactly the same proportion as their estimates of the times since the explosions. Indeed, one of the main motives of this whole theory is to secure that the velocity of light shall be the same for all observers, however they may be moving. This fact, established by experiment, was incompatible with the old theories, and made it absolutely necessary to admit something startling. The theory of relativity is just as little startling as is compatible with the facts. Indeed, after a time, it ceases to seem startling at all.

There is another feature of very great importance in the theory we have been considering, and that is that, although distances and times vary for different observers, we can derive from them the quantity called “interval,” which is the same for all observers. The “interval,” in the special theory of relativity, is obtained as follows: Take the square of the distance between two events, and the square of the distance traveled by light in the time between the two events; subtract the lesser of these from the greater, and the result is defined as the square of the interval between the events. The interval is the same for all observers, and represents a genuine physical relation between the two events, which the time and the distance do not. We have already given a [geometrical construction] for the interval at the end of [Chapter IV]; this gives the same result as the above rule. The interval is “time-like” when the time between the events is longer than light would take to travel from the place of the one to the place of the other; in the contrary case it is “space-like.” When the time between the two events is exactly equal to the time taken by light to travel from one to the other, the interval is zero; the two events are then situated on parts of one light ray, unless no light happens to be passing that way.

When we come to the general theory of relativity, we shall have to generalize the notion of interval. The more deeply we penetrate into the structure of the world, the more important this concept becomes; we are tempted to say that it is the reality of which distances and periods of time are confused representations. The theory of relativity has altered our view of the fundamental structure of the world; that is the source both of its difficulty and of its importance.

The remainder of this chapter may be omitted by readers who have not even the most elementary acquaintance with geometry or algebra. But for the benefit of those whose education has not been entirely neglected, I will add a few explanations of the general formula of which I have hitherto given only particular examples. The general formula in question is the “Lorentz transformation,” which tells, when one body is moving in a given manner relatively to another, how to infer the measures of lengths and times appropriate to the one body from those appropriate to the other. Before giving the algebraical formulæ, I will give a geometrical construction. As before, we will suppose that there are two observers, whom we will call O and O′, one of whom is stationary on the earth while the other is traveling at a uniform speed along a straight railway. At the beginning of the time considered, the two observers were at the same point of the railway, but now they are separated by a certain distance. A flash of lightning strikes a point X on the railway, and O judges that at the moment when the flash takes place the observer in the train has reached the point O′. The problem is: how far will O′ judge that he is from the flash, and how long after the beginning of the journey (when he was at O) will he judge that the flash took place? We are supposed to know O′s estimates, and we want to calculate those of O′.

In the time that, according to O, has elapsed since the beginning of the journey, let OC be the distance that light would have traveled along the railway. Describe a circle about O, with OC as radius, and through O′ draw a perpendicular to the railway, meeting the circle in D. On OD take a point Y such that OY is equal to OX (X is the point of the railway where the lightning strikes). Draw YM perpendicular to the railway, and OS perpendicular to OD. Let YM and OS meet in S. Also let DO′ produced and OS produced meet in R. Through X and C draw perpendiculars to the railway meeting OS produced in Q and Z respectively. Then RQ (as measured by O) is the distance at which O′ will believe himself to be from the flash, not O′X as it would be according to the old view. And whereas O thinks that, in the time from the beginning of the journey to the flash, light would travel a distance OC, O′ thinks that the time elapsed is that required for light to travel the distance SZ (as measured by O). The interval as measured by O is got by subtracting the square on OX from the square on OC; the interval as measured by O′ is got by subtracting the square on RQ from the square on SZ. A little very elementary geometry shows that these are equal.

The algebraical formulæ embodied in the [above construction] are as follows: From the point of view of O, let an event occur at a distance x along the railway, and at a time t after the beginning of the journey (when O′ was at O). From the point of view of O′, let the same event occur at a distance x′ along the railway, and at a time t′ after the beginning of the journey. Let c be the velocity of light, and v the velocity of O′ relative to O. Put

c
β=————
√(c² - v²)