and
between P and Q are different. But for both, the distance from P direct to Q is the same. In each case the right triangle tells us that:
Imagine an observer so dominated by his coordinate system that he knows no way of relating P with Q save by their horizontal and vertical separation. His whole scheme of things would be shattered by the suggestion that other observers on other reference frames find different horizontal and vertical components. We have to show him the line PQ. We have to convince him that this length is the absolute property enjoyed by his pair of points; that horizontals and verticals are merely relations between the points and the observer, result of the observer’s having analyzed the distance PQ into two components; that different observers effect this decomposition differently; that this seems not to make sense to him only because of his erroneous concept of a fundamental difference between verticals and horizontals.
The Four-Dimensional World of Events
We too have created a distinction in our minds corresponding to no sufficient reality. Our minds seize on time as inherently separable from space. We see the world made up of things in a continuum of three space dimensions; to make this dead world live there runs through it a one-dimensional time continuum, imposed from without, unrelated.
But did you ever observe anything suggesting the presence of time in the absence of space, or vice versa? No; these vessels of the universe always occur together. Association of the space dimensions into a manifold from which time is excluded is purely a phenomenon of the mind. The space continuum cannot begin to exist until the time dimension is supplied, nor can time exist without a place to exist in.
The external world that we observe is composed, not of points, but of events. If a point lacks position in time it does not exist; give it this position and it becomes an event. This world of events is four-dimensional—which means nothing more terrifying than that you must make four measures to locate an event. It does not mean, at all, that you must visualize four mutually perpendicular lines in your accustomed three-space or in a four-space analogous to it. If this world of four dimensions seems to lack reality you will be able to exhibit no better reality for your old ideas. Time belongs, without question; and not as an afterthought, but as part of the world of events.