# Relativity of Simultaneity

Updated on June 22, 2014

## Relativization of Simultaneity

In the way of thinking we take to be intuitive, there is an absolute truth as to the matter whether two, or more, events are simultaneous with each other. Take any statement p about whether two events a and b are simultaneous: we except that this statement is true or false regardless of what moving frame one is in. If it is true for us, it ought to be true for any one who might be moving relative to our earth-frame as well. In other words, we take it that there is a matter of fact - making the statement true or false - regarding simultaneity (or lack thereof) for any events.

One of the most astounding consequences of relativistic physics is that truth regarding event-simultaneity is not absolute but relative to which moving inertial framework one happens to be on. The statement p affirming simultaneity of events a1, a2, ..., an may be true for framework X but false for framework Y.

No - this does not vindicate rampant relativistic views which claim, self-indulgently, that there are no absolute truths whatsoever. (Communication would be impossible if this were literally true anyway.) Relativistic physics is still deterministic in the old-fashioned way - quantum physics is more interesting in this respect but even at the quantum level we have rigorous deterministic, albeit stochastic, laws. Also, we are saying, notice, that whether p is true or false is relative to the moving frame of the agent and not to such voguish referents as culture or subjective opinions (standard excuses for shirking rigorous thinking.)

The statement p-at-X-at-t, for some framework X and local time measurement t (where p is, as before, the statement about simultaneity of events) is true or false, absolutely. This suggests to me that the type of logic known as Modal Logic is, once again, proven to be the one we need to catch dynamic aspects of the reality we want to talk about. Modal logics (there are many) extend the basic propositional logic we teach at beginner's level in Symbolic Logic. Modal Predicate Logics extend standard Predicate Logic - but they are less well "behaved" in respects that are interesting to the logicians. The modal logic we seem to need here, for tracking p-at-X-at-t, is actually bimodal: truth-values (still only true and false) are relativized to frames and to internal frame-timeflows. There are such modal logics available. The trick is to find what properties you need for the accessibility relationship: accessibility here means all the ordered pairs of worlds <w1, w2> such that w1 "looks" into w2. For the bimodal logic, you will have two such accessibility relations, R1 for moving frames and R2 as the standard "<" ordering relation of the time flow. The properties of the relations determine what logical truths you get - and you don't want any such truths that are undesirable for the subject at hand. If something about this sounds odd - that you get variable logical truths - this is because Modal Logic also requires that there is a variety of modal species - not such logical modals but many others, including physical-theoretical, which we are addressing here. I hope this paragraph whets your appetite for Modal Logic - you will need a basic acquaintance with standard Propositional Logic first.

In spite of the initial shock, it may sound by now that we can live with relativized simultaneity without running into absurdity. There are, however, certain peculiar consequences - and it is to those and related issues that we turn to next.

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## Train and Lightning

In a famous thought experiment, meant to disabuse us of the notion that there are absolute facts or non-relative truths about event-simultaneity, Einstein asks you to think of a moving train racing down a railway along whose banks construction workers are busy. It is stormy and two lightning bolts strike. The construction workers experience the two lightning bolts to be discharged simultaneously. (We are not talking about psychological impressions and we are not supposed to be trusting mere subjective notions about what happened. There is a method for ascertaining simultaneity, which involves use of mirrors positioned so that the competent observer would attest that the two lightning bolts occurred simultaneously. In other words, we take this to be a physical event that is attested to.)

Now consider an observer on the moving train. This person would not have the impression that the two lightning bolts struck simultaneously. (Once again, we are not interested in variable subjective perceptions, which may be influenced psychologically: the verification procedure, involving the mirrors, is assumed as used again.) Indeed, depending on the direction in which the train is moving, the observer on the train would have to attest to the occurrence of one bolt has having occurred before the occurrence of the other lightning bolt. The bolt toward which the train is moving would be considered as having struck before the bolt behind the train, the one from which the train is moving away.

Isn't this obvious? Simultaneity is relative to the moving frame. How could anyone have missed this before? Did we have to wait for Einstein to point all this out?

In the times of classical thinking about the physics of the natural world, we would have considered the earth frame to be privileged - so that it is the one that gives us the truth as to the temporal relationship between the two events. We would count the train observer's perception to be inaccurate and caused precisely by the fact that his or her frame is not stationery relative to the privileged frame (the earth.)

Well, isn't the earth itself a moving frame? Why should it be taken to be privileged? Why should there be any privileged frames anyway? When you start thinking along these lines, you make the step toward the kind of thinking that underpins modern physics. It wasn't philosophical observations like this that brought about acceptance of relativistic physics but there had been developments in thinking that were pointing to this direction. Poincare, a famous scientist, had written along these lines and Mach, after whom the speed of sound unit is named, had made contributions to what is known as the Positivistic mode of thinking. (The word "positivism" has other meanings too.)

The conceit that privileged frames must exist reflected Newtonian thinking which, to work, requires that absolute measurements of spatial and temporal properties are in principle available (even though we can never have access to them!) A background absolute space (which is also ... everywhere) is considered real and substance-like for Newtonian physics; the same is the case for a very quaint substance, moving in one direction only, which is time. The Newtonian picture continues to come across as intuitive. It is still the relativistic physics that paints a peculiar, science-fiction-like, view of the universe. There are certain odd consequences following from the relativization of simultaneity.

## No "Privileged" Explanations

If you read the preceding text, you came across a way of thinking that eschews search for privileged explanations when an array of satisfactory explanations are available. We do have criteria for choosing among different explanations: for instance, Ockham's razor (bidding us to accept the explanation that stipulates fewer entities in its ontology, olther things being equal). Nevertheless, if two explanatory accounts X and Y pass the basic criteria (like Ockham's razor), and can do the job of explaining phenomenon Z equally well, then we should not be allowing ourselves to think that one of them, X or Y, must be the correct explanation (excwpt that we don't know which one it is.) It may be intuitive to think that there is a "correct" explanation. The trademark thinking that we find in the beginning of 20th-century science points rather in the direction of giving this way of thinking up: there may well be more than one explanations which, though different, can be selected indefferently. We say that physical events may be overdertermined. ("Overdetermination" has other meanings, and there is a claim, I cannot confirm or disconfirm, that Freud was the first to use this term. Whatever conclusions Freud may have drawn from overdetermination of psychical events is a different matter...)

Consider the following illustrative story:

We have two plains, A and B, with observers on plain A trying to explain an oddity about the mechanics of apparent movement on plane B. Observers on A can only view projections of the movements of actors on plane B. Plane A is flat. We don't know whether plane B is similarly flat or if, instead, it has some hump or other along it. The peculiarity that is observed is the following: for a segment of B, the moving actors there always appear smaller until they re-emerge, so to speak, to the remaining part of B where they acquire their initial dimensions again. Here are two plausible explanations. The moral of the story is that either explanation will do equally well with the other one. There is no point in discussing which is the "correct" explanation. What do you think about this? Do you agree?

Explanation 1 uses geometry. There may well be a hump over which the movers on plane B have to trek, as they continue moving at their standard speed, and this accounts for why their projections (observed at plane A) shrink when they move over that section. Explanation 2 uses forces instead of geometry. Everything else is the same, according to this explanation, (plane B is flat throughout), but there are certain forces operating over the suspect section of the plane; those forces operate only there and induce shrinking of anything that moves over that segment of B.

Forces or geometry? You might find the geometry explanation more plausible - maybe it is far-fetched to stipulate special forces - but is this the case? We could also have an appropriate combination of 1 and 2 - although, as an explanation, this would run afoul of Ockham's razor.

General Relativity uses spacetime geometry, instead of Newtonian force, to account for universal gravity. So, the above story hits closer to our home than you might have thought at first. We do take it, however, on intutive grounds, that Relativity theory, if it is the "correct" theory, gets it "right" as to how the universe really works. What I have been asking you to contemplate, instead, is the more radical view that we should not bother as to which explanation is the correct one. Either explanation is as good as the other. The reason we have relativistic physics is not because spacetime geometry is somehow a better choice than force-operation (what would be the criteria for deciding which account is "better?")

## Consequences from the Relativization of Simultaneity

According to relativistic physics, whether events are simultaneous with each other or not depends on (is relative to) what moving frame one is located. Events a1, ..., an can be relative for frame X but not so for another inertial frame Y.

The way relativistic physics is set up, each moving inertial frame carries its own light scone with this frame's relative present moment at the intersection of the cone's two segments - the one "behind" which is the relative past of the moving frame, and the one to the "front" which is the relative future of this frame. The movement is over the universal geography of spacetime - which has its own geometry. Events are "there" (like spatially located objects are in classical physics) and each moving frame simply travels over this block universe of events. Not objects but events are the kinds of things that can be determined only if both spatial and temporal characteristics are given: since the universal geography is jointly spatiotemporal and not spatial and temporal separately, the residents of this universal manifold must be determined by both spatial and temporal characteristics: they must be events and not the classical Newtonian objects over which time was supposed to flow evenly and in one direction.

The deeper reason for all this is that the new physics only accepts the velocity of light as a given universal constant to be entered into calculations of relative properties about spatial and temporal characteristics. Accordingly, spatial and temporal properties are treated as flexible - co-varying so as to give us the one and only magnitude we can trust, which is the velocity of light in a vacuum.

Which events are aligned, so to speak, or simultaneous becomes relative to each frame and its given lightcone. The events are lying out there, to put it this way, and are aligned under the moving lightcone that "falls" on them. So, any combination is conceivable in theory. Let's take two moving frames, X and Y, and two outlying events a and b. There is a definable frame of motion for which a and b are simultaneous; there is another one for which a is prior to b; and one in which b is prior to a.

An emerging problem would be that this relativization may upset causal relations, in this way wreaking impermissible havoc with the very foundations of scientific inquiry. Suppose that event a is the cause of event b. Yet there is a definable framework for which b is prior to a - the effect precedes the cause. This would surely spell disaster for the foundations of scienced but, oddly enough, the relativistic theory is saved "by the bell" as it were: whenever this phenomenon arises, it is provable in relativistic theory that the message announcing the effect-cause reversal to the other frame would have to travel faster than the speed of light to make it there; since this is impossible, for relativistic theory, no communication is possible for any case in which the message would overturn cause-effect relations.

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