Let us return for a moment to the moving ball. Four measures, three of distances and one of time, are required in specifying its position with reference to some framework at each point and at each instant. All of these measures can be summed up in one compendious statement—the equations of motion showed how in changing from our room to his accelerated auto we found a new summary, “transformed equations,” which seemed to indicate that the ball had traversed a strong, variable field of force. Is there then in the chaos of observational disagreements anything which is independent of all observers? There is, but it is hidden at the very heart of nature.
Paths Through the World of Four Dimensions
To exhibit this, we must recall a familiar proposition of geometry: the square on the longest side of a right-angled triangle is equal to the sum of the squares on the other two sides. It has long been known that from this alone all the metrical properties of Euclidean space—the space in which for 2,000 years we have imagined we were living—can be deduced. Metrical properties are those depending upon measurement. Now, in the geometry of any space, Euclidean or not, there is a single proposition of a similar sort which tells us how to find the most direct distance between any two points that are very close together. This small distance is expressed in terms of the two sets of distance measurements by which the end-points are located, just as two neighboring positions of our ball were located by two sets of four measurements each. We say by analogy that two consecutive positions of the ball are separated by a small interval of time-space. From the formula for the very small interval of time-space we can calculate mathematically all the metrical properties of the time and space in which measurements for the ball’s motion must be made. So in any geometry mathematical analysis predicts infallibly the truth about all facts depending upon measurements from the simple formula of the interval between neighboring points. Thus, on a sphere the sum of the angles of any triangle formed by arcs of great circles exceeds 180°, and this follows from the formula for the shortest (“geodesic”) distance between neighboring points on the spherical surface.
We saw that it takes four measurements, one for time and three for distances, to fix an elementary event, viz., the position of the centre of our ball at any instant. A system of all possible such sets of four measurements each, constitutes what mathematicians call a four-dimensional space. The study of the four-dimensional time-space geometry, once its shortest-distance proposition is known, reveals all those relations in nature which can be ascertained by measurements, that is, experimentally. We have then to find this indispensable proposition.
Imagine the path taken by a particle moving solely under the influence of gravitation. This being the simplest possible motion of an actual particle in the real world, it is natural to guess that its path will be such that the particle moves from one point of time-space to another by the most direct route. This in fact is verified by forming the equations of the free particle’s motion, which turn out to be precisely those that specify a geodesic (most direct line) joining the two points. On the (two-dimensional) surface of a sphere such a line is the position taken by a string stretched between two points on the surface, and this is the shortest distance on the surface between them. But in the time-space geometry we find a remarkable distinction: the interval between any two points of the path taken is the longest possible, and between any two points there is only one longest path. Translated into ordinary space and time this merely asserts that the time taken between any two points on the natural path is the longest possible.
Recall now that when the line-formula for any kind of space is known all the metrical properties of that space are completely determined, and combine with this what we have just found, namely, the equations of motion of a particle subject only to gravitation are the same equations as those which fix the line-formula for the four-dimensional time-space. Since gravitation alone determines the motion of the particle, and since this motion is completely described by the very equations which fix all the metrical properties of time-space, it follows that the metrical (experimentally determinable) properties of time-space are equivalent to those of gravitation, in the sense that each set of properties implies the other.
The Universe of Space-Time
We have found the thing in nature which is independent of all observers, and it turns out to be the very structure of time-space itself. The motion of the free particle obviously is a thing unconditioned by accidents of observation; the particle under the influence of gravitation alone must go a way of its own. And if some observer in an artificial field of force produced by the acceleration of his reference framework describes the path as knotted, he merely is foisting eccentricities of his own motion upon the direct path of the particle. The conclusion is rational, for we believe that time-space exists independently of any man’s way of perceiving it.