ON THE 'WORLD' OF MINKOWSKI 667
The event is the most elementary notion. We shall use it from now on in this work in the sense of a four-dimensional volume of space-time which is small in all four dimensions. We do not posit whether events themselves have structure or not, but it is preferable to assume that they have no space-time structure, which means that the event has no parts which are external to each other in space-time. The order of events is fourfold, as previously shown.
The aggregate or manifold of all point-events is then called the world. The point-events are given by four numbers representing the co-ordinates, three giving the 'space' co-ordinates, and the fourth the 'time' co-ordinate.
The term 'space-time continuum' or 'space-time manifold' is used often and implies that the numbers x, y, z, t, are to vary continuously.
In such a space-time continuum all happenings are structurally the intersections of world-lines, and if we could describe the world-lines of all points of the universe we would have a full account of the universe, 'past' and 'future'. We see that all physics, with the rest of our problems, must then be considered as a chapter of the general structural and semantic study of continuous manifolds of four dimensions.
But we are already acquainted with such theories. For instance, the internal theory of surfaces may be considered as a part of the subject in two and three dimensions. We have seen that different surfaces are characterized by the expression for the line elementor by
that group of transformations which leaves the line element invariant. We know already that in the E as well as riemannian geometries we have similar expressions and characteristic transformations.
If physics is to be considered a branch of the theory of four-dimensionat manifolds, we should naturally look for some such transformations. The manifold represents the world, the generalized* theory of relativity gives the desired answer. Minkowski proposed a postulate, which he calls the postulate of an absolute world, or the world-postulate which asserts the invariance of all the laws of nature in relation to linear transformations, for which the function is invariant.
The reader is already familiar with the expression, which gives
the invariant length in E geometry in two dimensions, and which
gives it in three dimensions. It would be natural to expect that in four dimensions we should have an expression of the type but in this case our expression is.It should be noticed that the above different types of expressions haye different origins. The first two arise in pure geometry, and the last has its roots in physics. The problem was to bring an experimental expression into harmony with a familiar geometrical expression. Minkowski introduced the expression, where * is as usual the square root of minus one,. Then of course becomes
*I use the term 'generalized' to embrace the unified field theory and eventually the quantum theory, although, for our purpose, I utilize only the special and general theory.