Let us now tie together the different threads in SR. Any actual event in an observer’s history, or better, in the history of any physical entity (I am not allotting any special status to consciousness or mentality, namely, to an observer taken literally), is an interval. What is simultaneous with this event, according to our causal conception of simultaneity, is what can influence and be influenced by the physical entity during this interval, what can then interact with it. This is determined independently of frame of reference, and is therefore absolute.
The set of events that stand in this double causal relation to an event e is determined as follows. The past light cone that has as its apex the last moment of e consists of all events that can, as a whole, influence e or some part of e; the future light cone that has as its apex the first moment of e consists of all events that can, as a whole, be influenced by e or by some part of e; their intersection thus consists of all events that can, as a whole, both influence some parts of e and be influenced by some parts of e. (As a whole, in the sense that each part of such an event can both influence some parts of e and be influenced by some parts of e.) This intersection is a double cone, called the Alexandroff intervalFootnote 8 and illustrated as follows:
Any event inside the Alexandroff interval of an event can both be influenced by some earlier parts of the given event and influence some of its later parts. Any event outside the Alexandroff interval either cannot influence the event, or cannot be influenced by it, or both. Parts of an event outside the Alexandroff interval of a given event may be able to influence or be influenced by the given event, but not the event as a whole. The Alexandroff interval of a given event thus consists of the events simultaneous with that event
Although geometrically, the Alexandroff interval is a double cone, to describe it as such might be somewhat misleading: from an ordinary human perspective, it looks much more like a cylinder. If we measure distances in metres and time in seconds, then the opening angle of the Alexandroff double cone is 179.9999996 degrees; if we measure them in kilometres and hours, it is 179.99999990: a straight angle for all standard intents and purposes.
Accordingly, from this perspective, any actual event has a substantial set of distant events simultaneous with it. If we go back to the grandmother, say in Europe, talking with her grandchildren in distant Australia, then her present, designated by her use of ‘now’ in the question ‘What are you doing now, children?’ and standing for a vague interval of a few minutes, coincides with that of her distant grandchildren. Special Relativity doesn’t force us to change anything in this use of our concept of distant simultaneity. And this would be true for the great majority of standard ascriptions of simultaneous occurrence to distant actual events. Thus, frame-independent distant simultaneity between events exists in Special Relativity, and it applies to most common ascriptions of simultaneity.
Some distinctions from common ways of thinking of simultaneity should, however, be noted. My definition of simultaneity by means of causal concepts was intended to capture, or at least approximate, our concept of simultaneity, and not our judgments of simultaneity. It was not meant to yield a concept that coincides with all our pre-theoretical judgments of simultaneity. Before we learn about the finiteness of the speed of light, we take the night sky to show us the stars as they are at the moment we see them; but once we learn about that, as we had done long ago with Römer, before SR was developed, we judge what we see to belong to the past. Our concept of simultaneity was not affected by this discovery, although our applications of it were.—All the same, concept and judgments are not independent: we explain many of our concepts and learn them by means of particular applications, so the correctness of a substantial part of these applications is presupposed by the meaningfulness of the concept. Accordingly, ordinary applications of a concept must as a rule be preserved by its analysis, if it is to be meaningful or coherent. As can be seen, the causal definition above does that.
The Alexandroff interval conception of simultaneity, although it follows from our ordinary criteria of judging one event simultaneous with another, is in another respect an extension of our simultaneity concept to new circumstances. In pre-theoretical applications of our concept of simultaneity, we never considered, say, cases in which the event e2 judged simultaneous with event e1 is sufficiently remote from e1 so that the temporal extension or ‘height’ of e1′s Alexandroff double cone at e2 has shrunk to a small fraction of its maximal height, at e1. In this respect this conception is an optional, albeit natural way of extending our simultaneity concept to new circumstances. It does, however, incorporate our pre-relativistic criteria and ratify the application of this concept in ordinary circumstances.
Extension-wise, the concept of simultaneity as applied in SR is different from the one that applies in the world of Newtonian mechanics. Simultaneity in SR is limited in space: if an event lasts time t, then according to the Alexandroff interval conception, events simultaneous with it exist only up to a distance of ct/2 from its location. In the world of Newtonian mechanics, by contrast, since there is no upper limit on the speed of propagation of causal influence, any event has at any distance from it events simultaneous with it.
However, this aspect of the Newtonian concept of simultaneity is not part of our pre-theoretical one but is a result of its application within the world as described by Newtonian mechanics (which is not our world). We introduce, explain and apply our concepts of at the same time, earlier and later not by means of a description of the temporal relations between ourselves and distant galaxies but when talking about mundane events. Pace Stein, our ordinary concept of at the same time does not involve any ‘“intuitions” of something like “cosmic simultaneity”, or a “cosmic present”’ (, p. 162). The Newtonian theoretical extrapolation of our spatiotemporal concepts might be the simplest one, but it is still an extrapolation, and one that in fact proved to be wrong. Accordingly, although Special Relativity forces us to abandon some aspects of our concept of simultaneity as applied in the world of Newtonian mechanics, this does not mean that it modifies the concept or makes it inapplicable in our world.
Special Relativity did have some unforeseen consequences for the applicability of our concept of simultaneity. The fact that when we look beyond the moon, talk of simultaneity doesn’t always make sense wasn’t anticipated (the Alexandroff interval has finite dimensions, and therefore the simultaneous has spatial boundaries). Equally unanticipated was the fact that events lasting for a few nanoseconds can’t be said to be exactly simultaneous with similar events occurring just a few miles away (as we converge towards a point-event, the Alexandroff interval converges to zero in all dimensions). But these unexpected facts about inapplicability to the very distant or the extremely brief do not mean that there is anything problematic in the ordinary, pre-theoretical applications of the concept.
Another unexpected result, related to those just mentioned, is that if one event e1 lasts less time than another event e, then it is possible that e1 happens while e does, although no part of e is simultaneous with e1. This happens if the Alexandroff interval of e contains e1, while e is outside the Alexandroff interval of e1. For instance, when we watch (= e) an almost instantaneous nuclear reaction in a bubble chamber (= e1), the reaction occurs while we are watching it, although no interval in our history can be said to be simultaneous with it. In causal terms this means that although the scientists during the experiment can both influence and be influenced by the nuclear reaction, no stage in the scientists’ history could, as a whole, both influence and be influenced by the reaction. This result shows the limitations of the extrapolation of our concept of simultaneity to new kinds of circumstance, and therefore, although unanticipated, it does not reveal any problem with the ordinary applications of the concept of distant simultaneity. I also think that this result captures the actual way we came to describe the temporal relations between such events.
Relativistic space–time is usually divided into three or four regions relative to a given event: a time-like region, which can be further divided into a future region and a past region; a light-like surface, which I shall here count as belonging to the time-like regions; and a space-like region, containing all events which are neither in the future nor in the past of the given event. This trifold classification—past, future, and space-like—is justified relative to point-events. However, if we consider actual events, which always have a duration, then relative to each actual event space–time should be divided into four regions. To the three regions mentioned above we should add a present region, constituted by the Alexandroff interval. Moreover, given, first, that actual events always have vague boundaries, and secondly, that as mentioned above, a more realistic diagram of lines representing the propagation of light should have them roughly horizontal, the vague boundaries between the past and present regions, as well as those between the present and future ones, should merge in the vicinity of the event. The space-like region should start only some distance from the event, this distance depending on the degree of vagueness of the event’s beginning and end. The resulting diagram should therefore have only three regions near an event: past, present and future:
This diagram of space–time without idealisation to point-events clearly shows that from an ordinary perspective, Special Relativity affects the applicability of our concept of simultaneity when very large distances relative to the event duration are concerned, but that otherwise there is no problem in the classification of less distant events as simultaneous. Closer to home, we have only past, present and future