HPS 0410 Einstein for Everyone

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Philosophical Significance of the Special Theory of Relativity
What does it all mean?

Morals About Time

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

This chapter continues the discussion of the last chapters on morals that we might try to draw from special relativity. This chapter collects morals pertaining to time.

The following chapter will describe another moral pertaining to time drawn from special relativity: the thesis of the conventionality of simultaneity. Several decades ago, this thesis engendered such an energetic discussion that our presentation here is long enough to require its own chapter.

For a discussion of philosophical morals that can be drawn from relativity theory concerning space and time, see my paper, "What can we Learn about the Ontology of Space and Time from the Theory of Relativity," on philsci-archive. Beware. The discussion is at a more advanced level than presumed in this class, so it is only for the adventurous.

7. Time is the fourth dimension

With the transition to relativity theory, we no longer conduct our physics in a three-dimensional space; we now employ the four-dimensional spacetime introduced by Minkowski.

This slogan "time is the fourth dimension" is a mischievous slogan, used, as far as I can tell, to intimidate novices. They are supposed to be awed by the apparent profundity of the claim while at the same time never being able quite to grasp its content at the insightful depth apparently accessible to the mischief making sloganeer. If you meet such a sloganeer, you should ask "what precisely do you mean?" Keep in mind the confusion favored by sloganeers sketched below and insist on a precise answer!

There is no interesting content to the claim. The problem lies in the vagueness of the statement of the thesis. There are two readings possible for it and neither yields results of importance.

In a trivial and true reading, we allow that space and time taken together form a manifold of four dimensions. What that just means is that four numbers are needed to locate an event in spacetime. Three of them are the usual spatial coordinates and the last is a time coordinate. That is true and was always true in classical physics as well. There is nothing of novel interest in this reading beyond the usual banalities about how things change with time. The idea that this sort of spatial representation of time is possible is as old as a pocket book calendar in which the passage of time is represented by a sequence of boxes or list of dates.

almanac almanac

There is a profound but false version of the slogan. What if time were a fourth dimension just like the three dimensions of space? That would be extraordinary. It mean that we could move about in the time dimension just as we move about in the space dimension. But time is not just like space in relativity theory. The theory keeps the timelike direction in spacetime quite distinct from the spacelike; the light cone structure does this quite effectively. So relativity theory contradicts this profound reading.

Underlying the profound reading is a simple fallacy. We note that in a spacetime formulation of relativity theory, time is usefully represented spatially in a diagram. So we can infer time must be like space in some aspects or this device would fail. It does not follow that time is like space in all aspects. Analogously, we can represent the spectrum of colors spatially with color wheels and rainbows. That does not mean that colors are spatial. Red is not the fifth dimension of space.

There is an interesting entanglement of space and time in relativity theory captured in the relativity of simultaneity. But the slogan of time as the fourth dimension is a defective and misleading way of expressing it.

Einstein rarely took the trouble to correct misreadings of his work. There were too many of them! However, this misreading of "time is the fourth dimension" as the moral of relativity theory was one that he did pause to correct. In 1946, when he wrote his Autobiographical Notes, he offered the following correcting. It says in more restrained terms what I have said above.

"First a remark concerning the relation of the theory to "four-dimensional space." It is a wide-spread error that the special theory of relativity is supposed to have, to a certain extent, first discovered, or at any rate, newly introduced, the four-dimensionality of the physical continuum. This, of course, is not the case. Classical mechanics, too, is based on the four-dimensional continuum of space and time. But in the four- dimensional continuum of classical physics the subspaces with constant time value have an absolute reality, independent of the choice of the reference system. Because of this [fact], the four-dimensional continuum falls naturally into a three-dimensional and a one-dimensional (time), so that the four-dimensional point of view does not force itself upon one as necessary. The special theory of relativity, on the other hand, creates a formal dependence between the way in which the spatial co-ordinates, on the one hand, and the temporal co-ordinates, on the other, have to enter into the natural laws."

8. Change is illusion

The relativity of simultaneity establishes that the future is as determinate as the past and present.


This moral is intended to negate a common sense idea we have about the future. It is the idea that the future is unresolved, whereas the present and past are or have happened and so are fixed. The notion is captured well enough by comparing the outcome of the last presidential election with the next. The outcome of the last election is known and fixed; it is a part of the determinate past. The outcome of the next election is open; it is a part of the indeterminate future.

We popularly imagine that the moment of the now advances through history converting the indeterminate possibilities of the future into the fixed actualities of the present and the determinate facts of the past. That there was a rainstorm on February 1 last is a determinate fact. Whether there will be one on February 1 next year in open. It may also be sunny or snowy. Which it is will become a determinate fact when our present has advanced to February 1 next year. determinateness of the weather


pen writing
This notion of determinateness and indeterminateness will figure prominently in what follows. That is awkward since the notions are not supplied as a theoretical term in special relativity. In the theory, we simply have a spacetime of events. Which are past, present or future depends on where we locate the present moment. Future events are later than present events; past events are earlier than present events. The theory does not tell us where to locate the present and has no physical property that corresponds to determinateness or indeterminateness. The notions are supplied by us externally.

What is determinateness? Perhaps the best we can say is this:

If we conceive of past and present events as fixed and future events as open, determinateness is that property of past and present events that distinguishes them from future events with regard to fixity and openness.

It is sometimes thought that merely employing a four dimensional spacetime in physics is already enough to overturn the idea that the future is indeterminate. For in a spacetime diagram, we see past, present and future laid out as equally real. This argument is flawed. It depends essentially on confusing the reality of a picture of a thing with the reality of the thing. My diary has equally real squares in it for yesterday and tomorrow. We would not infer from that, that yesterday and tomorrow are equally real (or squares).

The argument from spacetime is also less relevant in the present context since spacetime can also be used with classical physics. So whatever moral we might get from it is equally available from classical physics. Putnam, Rietjdk and others have tried to use what is distinctive about the Minkowski spacetime of relativity theory, the relativity of simultaneity, to get a stronger result about the determinateness of the future. They combine the way the relativity of simultaneity tangles up future and past with two observers in relative motion to get the result.

In brief, their argument goes as follows. Consider some possible event in our future: will there be a blizzard next February 1? We can always find a position and motion for a possible observer who would in our present, judge next February 1 to be in his present. For that observer, whether or not there is a blizzard here on February 1 is a present fact--it is determinate. Since that is true now of that observer, should we not also assume that the blizzard (or otherwise) of next February 1 is determinate?

The figure shows the spacetime diagram that goes with the argument.


The argument is:

Earth observer:
Event "Spaceship now" is simultaneous with respect to event "Earth now."
Therefore Event "Spaceship now" is determinate with respect to event "Earth now."

Spaceship observer:
Event "Earth later" is simultaneous with respect to event "Spaceship now"
Therefore event "Earth later" is determinate with respect to event "Spaceship now."

Event "Earth later" is determinate with respect to event "Earth now."

There are two weaknesses in the argument.

First, we are asked to accept that simultaneity and determinateness go hand in hand. That is, we must accept that:

  "Spaceship now" is simultaneous with respect to the event "Earth now."

entails that

  "Spaceship now" is determinate with respect to the event "Earth now."

I see no good reason to accept this. The notion of determinateness itself is sufficiently unclear as to leave me uncertain of its connection to simultaneity.

Second, it is not clear that determinateness is transitive. Transitivity is the property that allows us to chain together judgments of determinateness as is done in the little argument above.

For example, the "greater than" relation ">" on numbers is transitive. If A>B and B>C, then A>C. Transitivity can fail. It fails for the "is a friend of" relation. We may have A is a friend of B; and B is a friend of C. It does not follow that A is a friend of C.

Whether transitivity is admissible for the determinateness relation depends on what "determinate" means. That meaning remains sufficiently vague for a definite conclusion to be elusive.

However if the determinateness relation coincides the simultaneity relation, then its transitivity is dubious. For the simultaneity relation, as it appears in the argument, is not transitive. Simultaneity judgments from different observers cannot be chained together. We cannot infer that the events "Earth later" and "Earth now" are simultaneous. Why should it be different with determinateness?

How serious are the weaknesses? In my view, they are very serious. To resolve them, we need to find some independent basis upon which to judge the properties carried by determinateness, so that we can decide if determinateness coincides with simultaneity and is transitive. Yet, as the earlier discussion showed, determinateness is a notion we supply from outside relativity theory and in a way that its properties are left vague.

The one property that seemed secure is:

Future events are indeterminate with respect to past and present events.

If we conjoin this as an additional premise to the above argument, we end up concluding a contradiction:

Future event "Earth later" is AND is not determinate with respect to present event "Earth now."

If our premises lead us to a contradiction, we know that at least one of them is false and should be rejected. (This is a reductio ad absurdum.) Which should it be? There are three choices:

Simultaneity coincides with determinateness.

Determinateness is transitive.

Future events are indeterminate with respect to past events.

Given our slender grasp on determinateness, I see no reason to protect either of the first two from rejection. They entered in the first place as unsubstantiated suppositions.

People who favor the determinateness of the future, however, are likely to want to keep the first two and reject the third. Absent an independent characterization of determinateness, that rejection looks like circular reasoning. The prior commitment to the determinateness of the future is merely being reasserted.


9. Causal Theory of Time (H. Reichenbach):

Einstein defined simultaneity in terms of a light signaling operation. We can generalize his procedure to define the nature of time in terms of signaling with any causal process. To say that "an event P is earlier than an event Q" simply means that it would be possible for some causal process to pass from P to Q.

Reichenbach here attempted to solve an old problem in philosophy, rather nicely expressed in a lament by Augustine:

"What, then, is time?
If no one asks me, I know:
if I wish to explain it to one that asketh,
I know not."

This traditional problem is already captured in the dictionary game. You want to know what time is? Look up the definition of time in the dictionary. And then look up the definition of the definition and soon enough you are back at time, in a closed circuit. There seems no, simple, non-circular way to finish the defining sentence "Time is..."

Definition of time
Definition of duration

Reichenbach Reichenbach's causal theory of time aims to solve this problem. It will complete the "Time is..." sentence with talk of causes. To be more precise, it looks at the time order of events, the notions of earlier and later. Just what does it mean to say that two events are separated in time? Reichenbach's answer is in terms of causal connectibility.

Event P is earlier than event Q just means that event P could causally affect event Q

by, for example, the transmission of a light or signal from P to Q.

The inspiration for this approach is Einstein's 1905 treatment of simultaneity. In Einstein's special theory of relativity, two events A and A' are simultaneous if they are hit by light signals emitted at the same moment from their spatial midpoint; or if light signals they emit arrive at their spatial midpoint at the same time. The figure shows the propagation of light signals. Events A and A' are simultaneous; as are B and B'; and C and C'. Einstein turned this result into a definition. Two events are defined as simultaneous if they could be hit by such light signals. A variant form of this definition was the centerpiece of the first section of Einstein's paper.

The later version of the definition shown here is better suited to Reichenbach's project of reconstructing time from causal relations. The original 1905 version (described in the next chapter) has one light signal traveling from place A to place B and back to A. Einstein's definition required us to identify the event at place A midway in time between the departure and return of the light signal. That event is defined as simultaneous with the reflection event at place B. Identifying that event midway beween departure and return presumes that we have a well-functioning clock at place A, so that it already presumes a notion of temporal passage prior to the operations with light signals.

Reichenbach extended this thinking to all the time relations between events, being before and being after. It is a truth that P is earlier than Q just if a causal signal could pass from P to Q. Reichenbach now proposed that this truth be a definition.

There is something important and right about the approach. We cannot allow notions like time to become too distant from the physical processes of the world. Special relativity has reminded us that our notions of time must respond to those processes and the physical theories that govern them. Time is deeply entangled with causation. We will see just how much more profound that entanglement is when we deal with the spacetimes of general relativity.

However, in my view, Reichenbach's approach goes too far. We do not just see the entanglement of space and time in his theory. We see the reduction of time order to causal order. Causation becomes the fundamental idea and time order is derived from it. The difficulty is that we end up with a primitive notion, causation, that we seem to understand less well than the thing we started with, time order. So now we must ask "what is causation?" We will have a harder time answering. Theories of the nature of causation remain diverse and controversial. (For my diatribe on causation see "Causation as Folk Science.") Time remains far less problematic; our theories of time are some of the best developed of all physics. A theory that reduces the less problematic to the more problematic seems to me to be most problematic.

My Picks

Among the candidate morals of this and the last two chapters, everyone will find their own favorites, although it can be quite hard to make the selection. For what it is worth, here are my picks. They have actually mostly been embedded in the earlier critical discussion.

Common sense tracks the latest science. That is, common sense lags behind our latest science, which is very slowly incorporated into that nebulous "what everyone knows." Doesn't everyone now know that matter is made of atoms; or that the air is part oxygen and that oxygen is the bit that matters for our survival? Yet all this was once the most advanced science. The process seems to be continuing with special relativity. Many people somehow know that "nothing goes faster than light" but they are not sure where is comes from. The moral is not solely derived from special relativity, but special relativity does supply a nice instance of it.

Mature theories are very stable in the domains for which they were devised. They are fragile elsewhere. This is what I think should be learned from the long history of fragility of scientific theories, with the advent of special relativity an excellent example. While theories do not retain unqualified validity when we move to new domains, the mature theories remain essentially unaltered in their original domains. We need relativity theory for motions close to the speed light, yet we still use ordinary Newtonian theory for motions at ordinary speed. That does not seem likely to change.

Beware of theories or parts of theories that are designed to escape experimental or observational test. This is the part the verificationists got right. There is something very fishy about theoretical entities with properties so perfectly contrived that we cannot ever put them to observational or theoretical test. We should treat them with the highest suspicion. Asking for the means of verification or falsification is a good test if one is suspicious. Finding clear conditions for verification or falsification is an assurance that a healthy connection between the theory and experience is possible.

Be ready to abandon concepts that hide empirical content. This is the part that the operationists got right. One cannot develop conceptual schemes without making presumptions about the world, yet those very presumptions can be contradicted by emerging science, making acceptance or even formulation of appropriate new theories difficult. A related concern is that some concepts may have no real basis in experience at all (e.g. ether state of rest!). Asking for an operational definition of the concept is a healthy but not final test. If it admits an operational definition, then at least we know it has a connection to possible experience.

Infer to common causes. When you have the choice, the better explanation is the one that posits fewer coincidences and that is the one you should infer to.

Space isolates us causally. The novel results about space and time itself provide some of the most interesting results of special relativity. If we try to look beyond the theory and still have outcomes that pertain to space and time, I think the most important is simply the idea of upper limit of speed of light to causal interactions. That tells us that we are quite powerfully causally isolated from other parts of the universe. Nearby galaxies are already millions of light years away. That means that just sending a signal from our galaxy to another will require eons of time. Conversely, something happening there now will not affect us for the corresponding eons.
If one wishes to press further, special relativity has revealed a relatedness of space and time that we did not formerly suspect. It is hard to know how best to express this entanglement. I think the best way is still our familiar relativity of simultaneity.

What you should know

Copyright John D. Norton. February 2001; October 2002; July 2006; February 2, 13, September, 23, 2008; February 1, 2010; February 11, 2013. February 9, 2017; October 1, 2019. Section renumbering, October 20, 2019. September 22, 2020.