Fig. XVIII
Thus, if a star suddenly becomes visible in the sky, the point-event of that star when it burst into prominence is situated on the surface of our instantaneous light-cone somewhere in the past ([Fig. XVIII]).
In a similar way, the only events we can affect must lie in or on the upper part of the cone. An event which occurs outside the cone can never affect us or be affected by us now. It may, however, affect us ultimately, when, as a result of our light-cone’s rising with us along our world-line, it finally touches the cone’s surface. The event will then become visible, provided it be luminous. Once in the interior of the cone, the event may be perceived in other ways, by sound, etc., but no longer as a result of light transmission in vacuo. We may also state that when two point-events lie on the same time direction, their order of succession will remain the same for all observers; hence, in this case, it will be possible to conceive of a causal relation as existing between them. But if the two events lie on the same space direction, they may be simultaneous, or subsequent and antecedent, or antecedent and subsequent, according to our motion. Events represented by such point-events can never be considered as manifesting any causal relationship one with the other. We see, then, that inasmuch as it is the light-cone which differentiates space from time directions, and events which may be causally connected from those which may not, the irreversibility of time and the problem of causality are linked with the existence of the light-cone.
In a general way, we may say that whereas the classical graph showed an absolute past, present and future for all observers, in the new graph this statement must be modified as follows: If we consider all the observers situated at a definite point at some definite time, that is to say, all observers whose spatio-temporal position is given by the same common point-event, then, regardless of their relative motions, the same instantaneous light-cone holds for all. The apex of the cone is given by the common point-event, and all point-events situated within or on the surface of the cone stand in the instantaneous past or future for all the observers. There exist, therefore, an absolute past and an absolute future even in the theory of relativity. On the other hand, all point-events situated outside the cone will be neither past nor future in any absolute sense, for they may be in the past, the present or the future, according to the motion of the observer.
We may also mention that, thanks to the indefiniteness of the space and time directions, our graph is now dealing with a veritable continuum, the space-time continuum, which cannot be separated in any unique way into space and time.
We should also find that the space-time distance between any two point-events,
and
, for which no absolute expression existed in the classical graph, would now assume an absolute value, the same for all observers. This distance is the Einsteinian interval discovered by Minkowski, which permits us to associate a definite geometry with the four-dimensional continuum of events. The geometry is not Euclidean, but semi-Euclidean. When account is taken on Minkowski’s discoveries, it is seen that a space-time distance, when taken along a line which can be traced inside a light-cone, hence on a world line, is an imaginary magnitude; whereas, when taken along a line lying outside any light-cone, the distance becomes real. However, no great importance need be attributed to time being imaginary and space being real; for we could just as well have conceived of space as imaginary and time as real. All that it is important to note is that there exists a mathematical difference between the various directions in space-time, those which lie inside the cone and are time-like, and those which lie without and are space-like. The former alone can be followed by physical disturbances and material bodies.