Einstein's Theories of Relativity and Gravitation / A selection of material from the essays submitted in the competition for the Eugene Higgins prize of $5,000

by J. Malcolm Bird

The World-Frame

This is our conception of a world, if such were possible, entirely free from the influence of energy. We may conceive of it as an amorphous immaterial something containing “point-events” (a point-event being an instant of time at a point in space—a conception, not a definition). These point-events have a fourfold order and definite relation in this Frame, i.e. they can be specified by four variables or coordinates in reference to some base called a reference system, with respect to which they are forward or backward, right or left, above or below, sooner or later. This shows the World-Frame to be four-dimensional. Thus an aggregate of point-events (or an “event,” which implies limited extension in space and limited duration in time)[1] would have what we familiarly describe as length, breadth, height and time. To express these metrical properties most simply we must choose a four-dimensional reference system having a particular form—rectilinear axes (Cartesian coordinates), and a particular motion—uniform and rectilinear, i.e. unaccelerated, and non-rotating with respect to the path of a light ray. We call this an inertial system because Newton’s Law of Inertia holds for such a system alone. This system indicates how observers partition the World-Frame into space and time. It restricts observers to uniform rectilinear motion, and observations to bodies and light-pulses in such motion. Thus gravitational and other forces are discounted, and we obtain World-Frame conditions notwithstanding the fact that observers are in the presence of energy.

Now the separation between point-events which have a definite relation to each other must be absolute. The separation between two points in a plane is defined by the unique distance between them (the straight line joining them). Between point-events the analogue of this unique distance, which we call the “separation-interval” (to indicate its time-like and space-like nature), is also unique. Its unique and absolute character give it great importance as thereby it is the same for all observers regardless of their reference system.

If, in place of the rather cumbersome expression

to indicate the difference between the x-coordinates of two points, we employ the more compact expression dx; if for the benefit of readers who have a little algebra but no analysis we state explicitly that this expression is a single symbol for a single quantity, and has nothing to do with any product of two quantities d and x; and if we extend this notation to all our coordinates: then it is clear from previous essays that the distance S between two points in a plane referred to a rectilinear system OX, OY, is given by the simple equation

. Einstein and Minkowski show that the value for the separation interval

, the analogue of S, referred to an inertial system is given by the equation

which is seen to be a modified extension to four dimensions of the equation for S. We must measure t in the same units as x, y, z. By taking the constant velocity of light (300,000 kilometres per second) as unit velocity, we can measure in length or time indiscriminately.[2]

We will analyse briefly this equation as it epitomizes the Special Theory of Relativity. If the World-Frame had been Euclidean the equation would have been

but this would not satisfy the “transformation equations” which resulted from the Special Theory. These transformation equations arose directly from a reconciliation between two observed facts; (a) the observed agreement of all natural phenomena with the “Restricted Principle of Relativity”—a principle which shows that absolute rectilinear motion cannot be established—(as regards mechanics this was recognized by Newton; the Michelson-Morley and other experiments showed this principle also applied to optical and electro-dynamical phenomena); and (b) the observed disagreement of optical and electro-dynamical phenomena (notably the constancy of light velocity) with the laws of dynamics as given by classical mechanics, e.g., in regard to the compounding of relative velocities. Einstein effected this reconciliation by detecting a flaw in classical mechanics. He showed that by regarding space and time measurements as relative to the observer—not absolute as Newton defined them—there was nothing incompatible between the Principle of Relativity and the laws of dynamics so modified. Newton’s definitions were founded on conception. Einstein’s recognition of the relativity of space and time is based on observation.

Equation (1) shows that the geometry of the World-Frame referred to an inertial system is semi-Euclidean (hyperbolic), and that space and time measurements are relative to the observer’s inertial reference system. The equation shows that the World-Frame has a certain geometrical character which we distinguish as four-dimensional “flatness.” It is everywhere alike (homaloidal). Its flat character is shown by the straight line nature of the separation-interval and of the system to which it is most simply referred.

Thus we have found two absolute features in the World-Frame—(1) Its geometrical character—”flatness”; (2) The separation-interval—which can be expressed in terms of measurable variables called space and time partitions, this partitioning being dependent on the observer’s motion.

We are now in a position to explore the World-Fabric. Already we see that, studied under inertial conditions (free of force), it agrees with the World-Frame.