We have seen in the previous chapter that for a pair of events outside each other’s light cones, different observers will not in general agree on which is earlier and which is later. At first sight one might think that this ambiguity could mix up the question of causality. Thus it is a truism that if A is a cause of B, it must occur either earlier than B or at the same time as B. An event that is expected tomorrow cannot be taken as a cause of what is already happening today. For, by the cause of an event B, one means one of the conditions A that, being present and active, leads to the arising of B. Thus a fire acting today may set off some explosives now, but tomorrow’s fire will not set off these explosives today.
It seems clear that if we were allowed to interchange the order of past, present, and future in a completely arbitrary way, the result would be confused, both in physics and in everyday life. Suppose, for example, that one observer sees that the burning of some fuel is followed by the heating of water, and that for another observer, the water was heated first, while the fuel was seen to be burned later. Or suppose that one observer first became hungry and then ate food which satisfied his hunger, while another observer who was satisfied ate food and then became hungry. One could go on to multiply such instances without limit, showing clearly that we could not make sense of the world if we could make arbitrary changes in the time order of events. The question is then to see whether or not the relativistic ambiguity in time order of events outside the light cone of an observer will confuse the problem of cause and effect.
In answering this question we first recall that, as pointed out in Chapter 14, no object, influence, or force, etc., can move or otherwise be transmitted faster than the speed of light c. It is then easy to see that as long as this condition is satisfied, the relativistic ambiguity in time order will not mix up the question of causality. For if one event A is to be a cause of another event B, there must be some kind of physical action of contact between them. (If there is no physical contact at all, then one cannot be a cause of the other.) But if such physical action is not transmitted faster than light, then any two events which are causally connected will, as we have seen in the previous chapter, have a unique and unambiguous time order. In other words, if a cause A is earlier than its effect B for one observer, this relationship will hold for all observers. Therefore, the order of cause and effect will be invariant, so that no confusion will result in this order when different observers consider the same set of events, each in his own frame of reference.
On the other hand, if any influence could go faster than light, then the notion of the order of cause and effect would become completely mixed up. To see this, consider an extreme case, in which one assumes that physical influences could be transmitted with infinite velocities, so that two distant observers could be in simultaneous contact. Let one of these observers, , be represented in Figure 28–1 by the world line OA, while the other, , with the same speed, is taken to have the world line MN. By our hypothesis,
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the observer at the time represented by O could be in immediate physical contact with the other observer at the time represented by M, so that each could, for example, use this contact to signal to the other.
Now, if there were no principle of relativity with the invariance of the speed of light, this assumption would entail no logical self-contradictions. Indeed, it corresponds only to the “common-sense” notion that what we see at a given moment is in immediate contact with us and is all happening at the same time, which we call “now.”
Let us go on, however, to consider the further implications of the principle of relativity, as developed by Einstein, i.e., that the laws of physics have the same form for all observers, and that all observers ascribe the same speed to light. As we have seen earlier, it follows from this that two relatively moving observers do not agree with each other as to which set of events is simultaneous. Thus the observer with world line OE will regard the events on the line OF as simultaneous with O, while (with world line OA) regards events on OB as simultaneous. But according to the principle of relativity all the general laws holding in the frame of also hold in the frame of . Therefore, if it is assumed that can be in contact with an event M
Figure 28–1
that is simultaneous to O in his
observer, , can be in frame, the relatively moving contact with an event S that is simultaneous to O in his frame of reference.
122 The Special Theory of Relativity
Let us now consider the observer with world line MN. At the time corresponding to S his world line would intersect that of an observer , with world line M!N!, moving at the same speed as . Two observers at the same point S, but with different speeds, can evidently be in essentially immediate contact, so that there need be no time lapse (or a negligible one) for a signal to pass from one to the other. Then, according to the principle of relativity, should have the same ability to signal immediately to at O (which is simultaneous with S in his frame) that has to signal to at M. The observer could then signal to at O, and could signal to at M (which is simultaneous to O in the frame of reference of and ). By this cycle of signals S could communicate with M and vice versa. But M is in the past of S. So, in effect, S could communicate with his own past at M, and tell his past self what his future is going to be. But on learning this M could decide to change his actions, so that his future at S would be different from what his later “self” said it was going to be. For example, the past self could do something that would make it impossible for the future one to send the signal. Thus, there would arise a logical self-contradiction.
By a generalization of the above line of argument, it can easily be shown that a similar contradiction would arise if one assumed that physical contact were transmitted with any speed greater than that of light.
We see then that as long as we accept Einstein’s theory of relativity it leads to an absurdity to suppose that there is any action through physical contact capable of constituting the basis of a signal that is transmitted faster than light. In other words, either we have to assume that no physical action faster than light is possible, or else we have to give up Einstein’s form of the principle of relativity. But thus far this form of the principle of relativity has been factually confirmed. Besides, as we have already seen, no physical actions have ever been discovered which are actually transmitted faster than light (e.g., material objects cannot be accelerated to the speed of light, because this would require infinite energy, while no fields are known which propagate influences faster than light). Of course, Einstein’s theory is, like all other theories, in principle capable of being falsified as experiments are extended into broader domains. But as long as we remain in the domain where this theory is valid (and thus far, known experiments are and have been in such a domain), it will not be possible to have physical actions that are transmitted faster than light.
Given an event O, any other event P must, as we have seen, fall into one of a certain set of regions of space-time, which are subject to an invariant distinction (see Figure 27–2). If it is inside or on the light cone, then this event is either in the future of O or in the past of O. Because this distinction is independent of the frame of reference, it can be said to be in a certain sense “absolute” (at least in the domain in which Einstein’s theory holds). The “forward” light cone through O, plus all that is in it, is then called the “absolute future” of O, while the corresponding “backward” light cone, and all that is in it, is called the “absolute past” of O. The region outside the light cone has very aptly been called the “absolute elsewhere” of O. For this region has no direct contact with O at all, and therefore it is essentially “elsewhere.” Thus, if we consider a distant star, we have contact only with what that star was a long time ago. That this star exists “now” is only a likely inference, based on our general knowledge of the properties of stars. But actually
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we do not know that it exists “now.” For example, it may already have exploded. Later we (or other observers) may see this explosion. But whatever happens to this star in what is “now” our absolute elsewhere can have no contact with us “now.” (Later we shall have to be represented by another point on our world line, which is different from the one that represents us now.)
When two events are in each other’s absolute elsewhere, so that they can have no physical contact, it makes no difference whether we say they are before or after each other. Their relative time order has a purely conventional character, in the sense that one can ascribe any such order that is convenient, as long as one applies his conventions in a consistent manner. And as we have seen, observers, moving at different speeds, and correcting for the time , taken by light to reach them from a point at a distance r by the formula , will arrive at different conventions for assigning such events as before, after, and simultaneous with some event taking place in the immediate neighborhood of the observer. But as long as there is no physical contact, which is the basis of the relationship of causal connection of events, it does not matter what we say about which is before and which is after. On the other hand, as we have seen, where such casual contact is possible, the order of events is unambiguous, so that the Lorentz transformation will never lead to confusion as to what is a cause and what is an effect.