Missives from a fly bottle
barang dot sg
Updated 6 August 2020
 
About this site

A philosophy site by Mark in Malaysia.

My real interest is the mind-body problem and I’m gradually trying to figure out what I want to say about this.

Meanwhile, here’s some other stuff that I’ve found to be more tractable over the years, hard as they are in themselves.

Wittgenstein’s grave in Cambridge receives more than a hundred visitors every year, not that there’s a contest.
Rememberably, I used to teach logic and philosophy in a local university, but I now enjoy messing with this site and trying to figure stuff out. There is too much of it. Apart from mind and body, there is probability, and colour, and causation, and time, and existence, and money, and people, and power, and counterfactual conditionals, and why the packet nasi lemak nowadays got no timun. Why?

I discovered philosophy when I was twenty-one and never looked back. Duck finally found water and I cannot even begin to describe what the subject did for me. If you’re looking for a good introduction to the subject, this one did something for me.



I also had many reliable teachers at all levels to whom I am very grateful. I hope that some people will find this site useful in return.
Time slows down when you’re moving

Now if all moving clocks run slower, if no way of measuring time gives anything but a slower rate, we shall just have to say, in a certain sense, that time itself appears to be slower in a space ship.
– Richard Feynman, Lectures on Physics

Time slows down when you’re moving. So goes a well-known slogan of the special theory of relativity. Most people have heard of this and, for some people, it is the one thing that they know about the theory of relativity. What is less well-known, however, is how misleading the slogan can be. In what follows, I would like to unpick the slogan, explain what it means, and highlight a particular way in which it can mislead – an instructive way, hopefully. But I’ll begin with two simpler ways in which the slogan can mislead because this will help to introduce the issue. A rudimentary grasp of special relativity is presupposed in what follows.

The slogan is, ‘Time slows down when you’re moving,’ and the first way it can mislead is this. The phrase ‘when you’re moving’ may foster the wrong impression that motion is an intrinsic, or absolute, property that a body might possess, whereas a main point of relativity is that motion should only ever be understood in a relative sense.—You never move simpliciter but only relative to some other body, e.g., me. This is meant to be understood even though the slogan does not mention it.

A second, related misunderstanding is that the averred slowing down of time is something that will be apparent or otherwise manifest to the individual who is ‘moving.’ For example, if you are whizzing about the galaxy in your spaceship, then the misunderstanding is that you yourself will observe events on board your ship to unfold at a slower pace than normal, or in slow motion, as I will sometimes say. But this is not what the slogan means. Rather, it is only an “outside observer” like me—someone relative to whom you are moving—who will observe events in your ship to unfold in slow motion.

Thus, if I, here on Earth, could somehow observe the goings-on in your galactic ship, then I would observe everything on board to be transpiring at a slower rate than corresponding events on Earth. This includes things like the beating of one’s heart and the ageing of the cells in one’s body: I would observe every such process at your end to be occurring in slow motion. For short, if you are moving relative to me, then I will find that your clock ticks slower than mine, where a ticking clock is meant to be representative of any event or process that unfolds in time.

In contrast, as far as you are concerned, everything aboard your ship behaves as normal. You certainly don’t observe anything in your ship to occur in slow motion and you find your on-board clock to be ticking at its normal rate. But your clock does slow down for me. So the averred slowing down of time is something that manifests itself to me rather than to you.

These two simple ways in which the slogan can mislead are relatively well-known and I mention them just to get started.

The slogan again is, ‘Time slows down when you’re moving.’ A third way in which this can mislead is a little more involved and it’s really this that I’d like to talk about here. It concerns the use of the phrase ‘time slows down’ rather than (say) ‘clock slows down.’ Recall that the essential meaning of the slogan is this: if you are moving relative to me, then I will find that your clock slows down. But notice that the slogan does not say, ‘Clock slows down when you’re moving.’ It says instead, ‘Time slows down when you’re moving.’ And my question is whether this is really the same thing. Put another way, if I find, quite generally, that events in your frame of reference are unfolding in slow motion, can I correctly describe this by saying that I find that your time has slowed down?

Some people think so. In his book, The Wonderful World of Relativity, Andrew Steane, Professor of Physics at Oxford University, writes:
You see the idea: whatever processes the astronaut has going on in the rocket, including his own heart-beat … the space-station friend must observe them to be going in ‘slow motion.’ It applies to chemical reactions, to biological phenomena, to radioactive decay, to anything. But when all processes are slowed down like this, we may as well say that time itself has been slowed down: that is the summary, and the phenomenon is called ‘time dilation.’
The Wonderful World of Relativity, Oxford University Press (2011), p. 104.
Many others agree. Here’s a short video entitled, ‘Relativity Paradox,’ featuring Michael Merrifield, Professor of Astronomy at Nottingham University. Early on in the video, Merrifield says:
You see something going past you, something that’s travelling relative to you … the clocks in that reference frame seem to be going slow relative to your reference frame, so that time, you know, the clocks tick more slowly, and everything goes more slowly, biological processes go more slowly, radioactive decay goes more slowly, and so on. So, really, time really is slowed down when something is moving relative to you. (0:21-0:38)

Again, the phrase ‘time slows down’ is used to label a situation in which events are unfolding in slow motion, i.e., a situation in which a clock slows down, as I am putting it for brevity.

Here’s another video entitled, ‘Length Contraction and Time Dilation,’ from Henry Reich, producer of the popular youtube channel minutephysics. Speaking of time dilation, Reich says:
Suppose I have a clock with me that ticks every two seconds. But if you’re moving at a third the speed of light to my left, then from your perspective the time coordinates at which my clock now ticks are slightly further apart. According to you it takes about 2.12 seconds between each tick instead of two seconds. Time literally is running slower for me relative to you … (0:39-0:57)

It probably needs no saying, but these are well-qualified and knowledgeable people saying these things.

A final example is from astrophysicist Valerio Faraoni’s textbook, Special Relativity:
The formula for time dilation [Δt = γΔτ] is often interpreted by saying that time “runs slower” for a moving clock than for a clock at rest, or that a moving clock “runs slower” than a clock at rest by a factor γ.
Special Relativity, Springer (2013), p. 20.
Having shown these examples, I should emphasize that not everyone freely exchanges the phrases ‘clock slows down’ and ‘time slows down’ in this way. More cautious expositors say simply that a moving clock is observed to run slower without bringing the concept of time into it at all. Here’s an example from the website, Einstein Online:
time dilation
In special relativity: From the point of view of an observer (more precisely: an inertial observer), a moving clock goes slower than an identically built clock at rest. All other processes moving alongside the clock (for instance: everything happening aboard a rocket speeding by) are slowed down in an identical fashion.
The talk here is solely of a clock and other processes slowing down. There is no mention of time itself slowing down. You can also find many other accounts in the literature that are cautious in the same way. So the ready equation of ‘clock slows down’ with ‘time slows down,’ while common, is hardly universal.

My question was whether ‘time slows down’ can really be equated with ‘clock slows down’ in the manner just described. If I find that your clock has slowed down, do I thereby find that your time has slowed down?

To properly answer this, we need to clarify what it means for someone to find that your time has “slowed down” (or to find that your time is “running slow,” which I take to mean the same thing). It is not obvious what this is supposed to mean, nor do the authors above in general explain it. Notice, in contrast, that it is relatively clear what it means for someone to find that your clock has slowed down, i.e., to find that events in your frame of reference are “unfolding in slow motion.” But finding that your time has slowed down is not obviously the same thing. What does it actually mean?

There is, in fact, a common and established practice of speaking of time slowing down whenever events around us are found to be unfolding in slow motion. The authors above may mean, consciously or otherwise, to be tapping into this practice, so it is worth examining some familiar instances of it to see how the practice arises.

Consider the following cases.

Many beauty products are designed to slow down the ageing process (e.g., retard the formation of wrinkles) and people often speak within this cosmetic context of slowing time down, or even bringing it to a stop altogether. For example, the face cream shown on the right bears the prominent caption: Slow down time.

In a different kind of case, accident or trauma victims often report that the harrowing events that befell them happened in slow motion. Such cases too are typically then spoken of in terms of time slowing down. Thus, ‘Why do accidents and emergencies make time slow down so radically?,’ asks the following article from www.psychologytoday.com.

There is also the common science-fiction trope in which the protagonist is portrayed as having the ability to slow time down or bring it to a stop altogether, e.g., by pressing a button on a special device. In the limiting case, where time is stopped, the protagonist finds his or her surroundings to be “frozen.” Here’s an entertaining sample of this genre:


In all of these cases, there is an accepted connection between surrounding events unfolding in slow motion and time slowing down. (In the limit, surrounding events “freeze” and time is said to stop.) But why? What is the basis of this connection?

We can begin to see the answer by considering the inverted case where surrounding events occur in fast motion. This case is simpler to grasp and may easily be ‘re-inverted’ later. So suppose—science-fiction-wise—that you can “speed up” what’s happening around you by flipping a certain switch. It may help to imagine yourself ensconced within a booth of some sort within which events unfold as normal; it’s just events outside that speed up when you flip the switch. Then it seems to me that you would have a way of travelling into the future. Tucked safely within the booth, you flip the switch and everything outside unfolds faster than normal—a hundred times faster, say. People outside walk and talk a hundred times faster, their biological processes speed up a hundredfold, day and night zip by speedily, and so on. If you unflip the switch and exit the booth after a day (say) by your watch, you would find that a hundred days have elapsed outside: you would effectively have travelled into the future.
 
Note. One might dispute that this is a case of travelling into the future: is it not rather a case of making events around you speed up? This is a delicate question that would take too long to examine here; my own opinion is that the distinction in question may be without a difference. But, in any case, even if the above is not time travel into the future, it effectively is, and this will suffice to explain the linguistic practice that we are in the midst of accounting for.

So if you find surrounding events to have sped up, you may take yourself to be travelling into the future; or, at least, travelling through time faster than your surrounding fellows. You are whisked along the time line faster than them, as it were, and so, compared to them, time passes faster (or otherwise speeds up) for you. This conclusion—that time speeds up for you—is exactly what we should expect, actually. If you are listening to a boring speech and wish that time would speed up, i.e., pass faster, then you are wishing for the speech itself to be sped up and over with more quickly. So our bottom line is this: if events in your surroundings speed up, then time speeds up for you.
 
Note. What if only some events in your surroundings speed up?—e.g., just the events in your neighbour’s house? This case presents no significant complication. Here, time speeds up for you when compared to your neighbour and the goings-on in his house, but not when compared to the rest of your surroundings. So you travel into the future relative only to what is going on in your neighbour’s house. This circumscribed form of time travel may seem strange but that is all. It is exactly as if your ensconcing booth were that much larger and contained everything except your neighbour and his house.

In a similar way, if you find surrounding events to have slowed down, then you are travelling through time slower than your surrounding fellows. Compared to them, time passes slower (or otherwise slows down) for you. This result is simply the inverse of the previous one but it may be harder to appreciate because it involves a form of time travel—viz., travelling through time slower than your fellows—that has no familiar name. To see what I mean, consider that if you enter a time machine at noon and exit an hour later by your watch, then you would have ‘travelled into the past’ if you exit and find that it is earlier than noon outside. And you would have ‘travelled into the future’ if you exit and find that it is later than 1 pm outside. But what if you exit and find that it is between noon and 1 pm outside? This would be the case of travelling through time slower than your fellows. It lacks a familiar name and so we have a poor handle on it. Nevertheless, if you are faced with a fast-approaching deadline—“the clock is ticking”—and wish that time would slow down, i.e., pass more slowly, then this form of time travel would clearly be useful. You are wishing, in essence, for the clock to tick slower; or, more generally, for events in your surroundings to unfold more slowly. And so the bottom line in this case is the ‘inverse’ result already stated: if events in your surroundings slow down, then time slows down for you.

At this point, it should be reasonably clear why people speak of time slowing down whenever they find events in their surroundings to be unfolding in slow motion. So far as I can see, this common and established practice is both sensible and sound. Unfortunately, it turns out now that, if our relativity expositors above mean to be tapping into this practice, then they have misconstrued it badly. Consider again our initial example from Andrew Steane’s, The Wonderful World of Relativity:
… whatever processes the astronaut has going on in the rocket, including his own heart-beat … the space-station friend must observe them to be going in ‘slow motion.’ … But when all processes are slowed down like this, we may as well say that time itself has been slowed down …
It should be clear what this passage means. In finding that all processes have slowed down in the astronaut’s rocket, the space-station friend may say that time has slowed down for the astronaut. (He may regard the astronaut as moving through time slower than himself.) That is the intended meaning. But notice that, according to the common practice just examined, this gets things the wrong way around. Imagine that you are the space-station friend, observing the astronaut. The astronaut is moving relative to you and so, as before, you find his or her behaviour and biological processes to be occurring in slow motion. According to the common practice, you should therefore say that time has slowed down for you when compared to the astronaut. (We saw this above: if events in your surroundings slow down, then time slows down for you.) But our relativity expositors would have you say the opposite. According to them, you should say that time has slowed down for the astronaut when compared to you. So their way of talking is at odds with common practice.

We can also see this in a different way—perhaps more directly—by considering what it would be like to observe someone who was travelling into the future. Recall the booth that we imagined above, into which one could enter and flip a switch that would make events outside “speed up” a hundredfold, thereby enabling one to travel into the future. If the booth contained a window, then a man inside would see events outside unfold a hundred times faster, as he travelled into the future. But what would a woman outside see if she peeked inside and observed the time-travelling man? The answer should be clear: she would see everything within the booth unfolding a hundred times slower; happening in slow motion. To see this, imagine that the man enters the booth at noon and travels into the future for just one minute (by his clock), so that, upon exiting the booth, he finds that a hundred minutes have elapsed outside. If the woman observes the man all this while through the window, she would see his clock tick off just one minute in a hundred minutes of her own time. So she would see his clock ticking a hundred times slower; indeed she would see the man himself moving about in slow motion. Quite generally then, someone who is moving about in slow motion in this way is someone who is travelling into the future relative to oneself, i.e., travelling faster through time than oneself. Notice again that this is the opposite of what our relativity expositors above would say. By their lights, someone who is moving about in slow motion in this way is moving slower through time than oneself.

The slogan we were examining was, ‘Time slows down when you’re moving,’ and I was questioning the use of the word ‘time’ therein. What is agreed on all sides is that, if someone is moving relative to you, then you will observe events in their frame of reference to unfold in slow motion. But it does not follow—at least not on the common way of talking examined above—that their time has slowed down. On the contrary, their sluggish comportment would indicate that they were travelling into the future relative to you, i.e., moving faster through time than you. So time speeds up when you’re moving, not slows down.
 
Note. In a way, we should already know this because time-travel into the future is a well-known relativistic consequence of motion through space: someone who is moving relative to you is travelling into the future relative to you. But time passes faster, not slower, for someone who is travelling into the future.

All of this leads me to believe that our relativity expositors may be using the phrase ‘time slows down’ in some idiosyncratic way that is not tied to the common linguistic practice just examined. I wonder, in particular, if they are using it merely as a sexy way of saying ‘clock slows down’—and no more. So if you are moving past me and I find that your clock has slowed down, i.e., that events in your frame of reference are happening in slow motion, they would feel free to re-word this and say that I find that your time has slowed down. That’s just how they are using the phrase ‘time slows down.’ If this were true, then they might be accused, at most, of using the phrase in a misleading way, given that there is an established practice of using it in a different way. This may strike you as a minor transgression since the theory of relativity has not itself been misrepresented in any substantial way. After all, if you are moving relative to me, then I will find your clock to slow down, and what does it matter if someone casually and idiosyncratically says instead that I have found your time to slow down? Anyone can see that the reference to ‘time’ is incidental and that the essential meaning is that I find events in your frame of reference to be happening in slow motion. No harm done, really.

I would readily agree to this except for one thing. There is some evidence that people have been substantially misled by this casual and idiosyncratic equation of ‘time slows down’ with ‘clock slows down.’ Let’s turn at once to the case that I have in mind because it was this that got me started in the first place. A discussion of this case will occupy the rest of this essay.

In the relativity literature, especially the popular literature, you will repeatedly find touted a certain intriguing metaphysical idea that is offered by its proponents as an intuitive way of understanding why time slows down when you’re moving, as the slogan goes. They claim to have a simple metaphysical idea that enables one to understand, at a gut level, why this happens. Briefly put, the idea is that there is a trade-off between one’s speed through space and one’s speed through time, so that the faster one moves through space, the slower one must move through time. The idea is quite intriguing and worth studying in itself. Let’s see how it goes.

One person closely associated with the idea is Brian Greene, Professor of Mathematics and Physics at Columbia University. He has articulated it in numerous places, two of which I will mention. One is the following 4-minute video for the World Science Festival entitled, ‘Why Does Time Slow for a Clock in Motion?’ In the video, Greene says:
Is there a mental mnemonic, some sort of intuitive way of thinking about why it is that time slows down when an object is in motion?
And there is … (0:43-0:53)
He then proceeds to sketch the main idea pretty clearly and, if the idea is new to you, it will help to watch the video before we turn to the second source below.


A second place where Greene provides this explanation is a web essay called, ‘Special Relativity in a Nutshell.’ I quote the relevant passages here, which flesh out the same idea a little more:
To get a fuller sense of what Einstein found, imagine that Bart has a skateboard with a maximum speed of 65 miles per hour. If he heads due north at top speed—reading, whistling, yawning, and occasionally glancing at the road—and then merges onto a highway pointing in a northeasterly direction, his speed in the northward direction will be less than 65 miles per hour. The reason is clear. Initially, all his speed was devoted to northward motion, but when he shifted direction some of that speed was diverted into eastward motion, leaving a little less for heading north. …

Special relativity declares a similar law for all motion: The combined speed of any object’s motion through space and its motion through time is always precisely equal to the speed of light. At first, you may instinctively recoil from this statement since we are all used to the idea that nothing but light can travel at light speed. But that familiar idea refers solely to motion through space. We are now talking about something related, yet richer: an object’s combined motion through space and time. …

[So] when you look at something like a parked car, which from your viewpoint is stationary—not moving through space, that is—all of its motion is through time. The car, its driver, the street, you, your clothes are all moving through time in perfect synch: second followed by second, ticking away uniformly.

But if the car speeds away, some of its motion through time is diverted into motion through space. And just as Bart’s speed in the northward direction slowed down when he diverted some of his northward motion into eastward motion, the speed of the car through time slows down when it diverts some of its motion through time into motion through space. This means that the car’s progress through time slows down, and therefore time elapses more slowly for the moving car and its driver than it elapses for you and everything else that remains stationary. …

Moreover, the maximum speed through space is reached when all light-speed motion through time is fully diverted into light-speed motion through space—one way of understanding why it is impossible to go through space at greater than light speed. Light, which always travels at light speed through space, is special in that it always achieves such total diversion.

And just as driving due east leaves no motion for traveling north, moving at light speed through space leaves no motion for traveling through time! Time stops when traveling at the speed of light through space. A watch worn by a particle of light would not tick at all. Light realizes the dreams of Ponce de León and the cosmetics industry: It doesn’t age.
The broad meaning of these passages should be clear and I will also show below that the idea is mathematically sound. Meanwhile, I have one passing observation. The passages are meant to constitute a simple and intuitive explanation of why time slows down when you’re moving. I noted above that what really needs explaining, at bottom, is why your clock slows down when you’re moving. So there is a question here about what Greene takes the relation between ‘time slows down’ and ‘clock slows down’ to be. This question will matter soon but we can leave it unanswered for now.

For the moment, let us consider an alternative rendering of essentially the same metaphysical idea to see how a different author expresses it. The following account is by Lewis Carroll Epstein, an American physicist and physics popularizer, and it is the earliest account that I know of, of the idea in question. It is also the most detailed, forming just one part, in Epstein’s hands, of a larger metaphysical treatment that we cannot examine here, unfortunately. In his fascinating book, Relativity Visualized, Epstein writes:
To understand the Special Theory of Relativity at the gut level, a good myth must be invented, and here it is. …

Why can’t you travel faster than light? THE REASON YOU CAN’T GO FASTER THAN THE SPEED OF LIGHT IS THAT YOU CAN’T GO SLOWER. THERE IS ONLY ONE SPEED. EVERYTHING, INCLUDING YOU, IS ALWAYS MOVING AT THE SPEED OF LIGHT. How can you be moving if you are at rest in a chair? You are moving through time.

Why are the clocks moving through space perceived to run slower and slower as they travel faster and faster? Because a clock properly runs through time, not space. If you compel it to run through space, it is able to do so only by diverting some of the speed it should use for traveling through time. As it travels through space faster and faster, it diverts more and more speed. How much can it possibly divert? The clock can divert ALL of its speed. Then it is going through space as fast as it possibly can, but there is nothing left for traveling through time. The clock stops ticking. It stops aging.

All this can be depicted in a diagram, which is essentially the cosmic speedometer diagram. Nothing can ever be done to alter the speed of anything. Only its direction of motion through spacetime can be altered. At rest a thing is perceived to speed through time from O to A; but set in motion to the right, R, or to the left, L, its velocity is tilted right or left. In the extreme, the velocity is tilted all the way to C, in which case all the perceived motion is through space and none through time. The diagram is easily calibrated. If the distance from O to A is 1 year, then the distance from O to C must be 1 light year – the distance light travels in 1 year.
Relativity Visualized, Insight Press (1981), pp. 78-81.
This is essentially the explanation given by Greene above and you can also find numerous other advocates of the same idea, especially on the wild wild web. I have not found the idea expressed in any standard text on the subject, however—e.g., you won’t find it in Taylor & Wheeler’s well-known text, Spacetime Physics. Possibly for this reason, some people seem to regard it as a bit of an unorthodox or fringe idea; a useful mnemonic or calculation heuristic perhaps, but one devoid of genuine metaphysical significance. Be that as it may, I have not seen the idea carefully assessed anywhere and have had to evaluate it for myself. What follows are some things that struck me as I turned it over in my mind. As I hope is clear, its key features are these:

1. All objects move in spacetime at the speed of light.
2. An object’s speed in spacetime can never be altered; only the direction in which it moves in spacetime can be altered.
3. Depending on which direction it moves in spacetime, some of an object’s speed will be parcelled out in the space direction; the rest in the time direction. (If all of its speed is parcelled out in the space direction, then it has no speed in the time direction, and vice versa.)

This account is professed to explain, among other things, why you cannot travel through space faster than the speed of light, why time slows down when you move through space, and why time stops when you move through space at the speed of light. This much should also be clear, given everything so far said.

Before assessing the idea, let me first show that it is mathematically sound. In particular, it should be able to deliver the following well-known result of special relativity: if you are moving past me at some fraction β of the speed of light, where 0 < β < 1, then I will find your clock to be running √(1 – β2) times as fast as mine, where this latter number is also a fraction. For example, if you are moving past me at ⅘ of the speed of light, then I will find your clock to be running ⅗ times as fast as mine. In this example, β = ⅘, but the result holds quite generally. I will take this result for granted here because this essay is long enough as it is. It falls right out of the ‘time dilation formula,’ which may be found in any standard account of the subject.

To see that this accords with the metaphysical idea in question, consider the diagram on the right, which is what both Epstein and Greene have in mind. The grey quarter-circle has a radius of 1, which number we may take to represent the speed of light. An object travels through spacetime at the speed of light in some arbitrary direction, indicated by the diagonal arrow, such that its speed through space is β, as shown, where 0 < β < 1. By Pythagoras’ theorem, its speed through time must therefore be √(1 – β2). In comparison, my own speed through time will be 1, or the speed of light, since I may regard myself as standing still in space. And so the object’s speed through time is indeed √(1 – β2) times as fast as mine.

Mathematically then, the idea is quite correct. But what can we say about the intriguing metaphysics that underlies it? This is the real question before us. Let me summarize everything that has been said so far before providing an assessment.

If you are moving relative to me, then I will find events in your frame of reference to happen in slow motion. This is the phenomenon captured in the popular slogan, ‘Time slows down when you’re moving.’ It’s a rather baffling and unexpected phenomenon, which is why the metaphysical idea touted by theorists like Greene and Epstein is so intriguing: it promises us an intuitive way of understanding why the phenomenon happens; why time slows down when you’re moving, as the slogan puts it. I have emphasized a number of times that the phenomenon is essentially that your clock slows down when you’re moving—i.e., that events in your frame of reference happen in slow motion—but people tend to echo the popular slogan and say that your time slows down when you’re moving, without pausing to consider whether this is really the same thing.

Indeed, we saw above that, on a common and established way of talking about time, it is not the same thing at all. On this common way of talking, the fact that your clock is slowing would mean that you are travelling into the future relative to me—i.e., moving faster through time relative to me, and not slower. In other words, as your clock slows down, your time actually speeds up. By these lights, time speeds up when you’re moving, and not slows down. Anyone who continues to brandish the slogan ‘Time slows down when you’re moving’ in the face of this common and established way of talking needs therefore to account for their contravention of it.

I suggested above that our relativity expositors may do this by claiming to be using the phrase ‘time slows down’ in a loose and casual way; as merely a sexy way of saying ‘clock slows down,’ and nothing more. On this casual way of talking, time does slow down when you’re moving, simply because your clock slows down when you’re moving. But this would just be a manner of speaking; a way of talking that may be tolerated in an informal setting. So far as I can see, this is also the only plausible way in which the slogan ‘Time slows down when you’re moving’ may be justified, given that it contravenes the common and established way of talking. (That is, short of challenging that way of talking altogether, for which see below.) One essentially has to say that the phrase ‘time slows down,’ as used in the slogan, is nothing more than a colloquial way of saying ‘clock slows down.’

Having said that, notice now that this way of justifying one’s use of the slogan ‘Time slows down when you’re moving’ is not available to theorists like Greene and Epstein because of the nature of their intriguing “explanation” for why the phenomenon occurs. Recall that their explanation was essentially this:
Time slows down when you move through space because the faster you move through space, the slower you move through time. This trade-off is necessary to ensure that your overall speed through spacetime remains constant.
A centerpiece of this explanation is the idea of moving slower through time as a counterweight to moving faster through space. Indeed, this idea is so central to the explanation that Greene and Epstein cannot reasonably be supposed to be employing the expression ‘move slower through time’ (or the equivalent ‘time slows down’) in a loose or casual fashion, e.g., as just a colloquial way of saying ‘clock slows down.’ Rather, these expressions are obviously meant to be temporal analogues of their spatial counterparts, e.g., ‘move slower through time’ is meant to be analogous to ‘move slower through space,’ and so on. This means that, while the idea of time slowing down figures prominently in the explanation, the idea of your clock slowing down is not, at this stage, coming into it at all. We saw above that some of our relativity expositors may reasonably claim to be using ‘time slows down’ as just short for ‘clock slows down’ whenever they brandish the slogan ‘Time slows down when you’re moving.’ Greene and Epstein, however, cannot do so. And so the “escape route” of the previous paragraph is not available to them, whereupon they seem bound for a head-on collision with the common and established way of talking about time. When Greene and Epstein say, ‘Time slows down when you’re moving,’ what they mean by these words is precisely what the common way of talking denies.

Another way to see this is to notice that, at some point in Greene and Epstein’s explanation—most likely after having demonstrated that your time slows down when you’re moving—the connection has to be made to the phenomenon of events in your frame of reference happening in slow motion, i.e., the phenomenon of your clock slowing down. This, after all, is what ultimately needs to be explained. We can thus infer that Greene and Epstein would uphold the following equivalence: your time slows down if and only if your clock slows down. But this equivalence is something that the common linguistic practice would again deny. As we saw, on the common way of talking, your time speeds up if and only if your clock slows down. So the common way of talking is again contravened.

Having committed themselves this far, it seems to me that the only recourse left open to Greene and Epstein is to challenge the common and established way of talking about time – to deny its correctness. For if that way of talking were correct, then time does not slow down when you’re moving, and Greene and Epstein would be left trying to “explain” a non-existent phenomenon. Unfortunately, I don’t think that the prospects for challenging the common way of talking are all that good. In the next and final section, I’ll explain why and draw an appropriate conclusion.


We discussed the common and established way of talking about time in §3 and I concluded at the time that it was both sensible and sound. I still stand by that conclusion, but having just seen its acute consequences for theorists like Greene and Epstein, we should probably say a little more. Rehashing the entire discussion of §3 would be rather pointless so I will shore it up instead from a different angle. There is a rather important objection to the common way of talking about time that I failed to consider previously although it deserves to be examined. Now seems like a good time to do so. The objection stems from a certain “intuition” that one might have about time that directly contradicts the common way of talking. Since the intuition is quite natural to have, and it is fairly clear that both Greene and Epstein have it, disarming it will go some way towards vindicating the common way of talking about time. Let me explain what the intuition is before trying to disarm it.

We saw above that, on the common way of talking, if I find events in your frame of reference to be occurring in slow motion, then I should regard myself to be moving slower through time than you. But many people find this counterintuitive. Surely, in such a case, I should regard you to be moving slower through time than me? For example, if your clock ticks off just one hour in the time that mine ticks off six, then am I not the speedy one through time and you the laggard? This is the intuition in question. (You may have it.) Greene seems to have it, judging by his words in the video shown previously:
Einstein showed that if you watch me as I’m walking, you will say that time elapses more slowly on my watch than it does on your watch. Time slows for an object in motion. (0:07-0:21)
Indeed, all of our previous relativity expositors may have had the intuition too, since they tend to say similar things. But, as we have seen, they have some room to maneouvre and cannot quite be pinned down on this in the way that Greene and Epstein can.

In the extreme case, and somewhat more vividly, if I find my surroundings and everyone in them to have “frozen,” then, on the common way of talking, I should say that time has stopped for me. We saw this previously, but one might again find this counterintuitive. Surely time has stopped for the frozen people, rather than for me? After all, they’re not moving, not aging, their biological processes have stopped: if time has stopped for anyone, it would be for them, and not for me? Epstein appears to share this intuition, in a passage already quoted:
… The clock can divert ALL of its speed. Then it is going through space as fast as it possibly can, but there is nothing left for traveling through time. The clock stops ticking. It stops aging.
These intuitions are both natural and tempting. Indeed, when you first study the theory of relativity and learn of how everything slows down in a moving person’s frame of reference, how their clocks slow down, their biology retards, and so on, you never think to question the (oft-made) ensuing assertion that time slows down for someone who is moving. It sounds like such a correct moral to draw; the natural upshot. It is only when you consider the common way of talking about time—the kinds of things we saw in §3—that these intuitions can come to seem suspect. I believe in the end that the common way of talking is correct and that the intuitions are misplaced, but I do understand how compelling they can be, having grappled with them myself. It is important to disarm them, so here goes.

Consider the case where I find events in your frame of reference to be unfolding slower than corresponding events in mine—e.g., six times slower, as in our first example above. So, among other things, your clock ticks six times slower than mine. In this case, as we saw, it can be quite “intuitive” for me to regard you as moving six times slower through time than me. As before:
If your clock ticks off just one hour in the time that mine ticks off six, then am I not the speedy one through time and you the laggard?
Why would this “intuition” be wrong?

From what I can see, the problem is that the intuition rests on a tempting but ultimately incoherent analogy with a certain spatial case that is superficially similar. To see the spatial case, suppose that you move just one mile through space in the time that I move six:


Obviously then, you move six times slower through space than me, as the diagram makes clear. But the mistake now is to think that our case above is just the “temporal analogue” of this spatial case. In our case, you move just one hour through time in the time that I move six. This analogous diagram then springs to mind, in which you move six times slower through time than me:


If you don’t think about it too much, a diagram like this can be compelling. It may well be the source of the “intuition” that you are moving slower through time than me. Unfortunately, the diagram is incoherent because it contains contradictory depictions. On the one hand, it depicts you at an earlier point in time than me, just as the previous diagram depicts you at an earlier point in space. On the other hand, it locates you at the same point in time as me because the supposition was that you move one hour through time at the same time as I move six.

This contradiction does not infect the original spatial diagram because that diagram coherently locates us at different places at the same time, as the following dotted lines make explicit:


In contrast, the “temporally analogous” diagram is incoherently trying to locate us at different times at the same time:


So the diagram is essentially incoherent and the “intuition” that rests upon it just as confused. The diagram and intuition look sensible at first sight but they do not stand up to scrutiny. Trying to depict our case in a handy diagram is not actually a bad idea in itself, but a better attempt would be this:


In this diagram, you are depicted, without contradiction, as moving one hour through time at the same time as I move through six. The contradiction is removed by “stretching” your one hour to match my six: this also captures the relativistic phenomenon of time dilation. Notice that this diagram elicits a very different “intuition,” viz., that it takes you just one hour to reach a point in time that I need six hours to get to. This implies, of course, that you are moving six times faster through time than me, matching the common way of talking, rather than six times slower, which was the confused intuition.

Once we see how the confused intuition rests on a mistaken spatial analogy, the common way of talking about time can come to seem more natural. This is true also of the extreme case in which I find my surroundings and everyone in them to be “frozen.” The confused intuition here would be that time has stopped for the frozen people, i.e., that their speed through time is zero. The sounder intuition is that they take no time at all to reach a point in time that I need a finite amount of time, no matter how large, to get to. This implies that the frozen people are moving infinitely faster through time than me. And so, compared to them, I am standing still in time, which means that it is my speed through time that is zero. This again matches the common way of talking.

On their own, these informal diagrams and remarks have merely circumstantial value, but, in conjunction with the considerations adduced in §3, they do persuade me to endorse the common and established way of talking about time. (There is a reason why that way of talking is common and established.) And so I conclude that the intriguing metaphysical picture touted by Greene, Epstein, and like-minded theorists, does not quite survive scrutiny.—It clashes head-on with the common way of talking about time and comes off worse. My overall opinion is that theorists like Greene and Epstein may have been misled by the loose and casual slogan, ‘Time slows down when you’re moving,’ to look for an explanation for this alleged phenomenon, when, strictly speaking, it is only your clock (and not your time) that slows down when you’re moving.

And so the well-known slogan, ‘Time slows down when you’re moving,’ is not quite as innocent as it might first appear to be. Arguably, the slogan is outright wrong; at best, it can be seriously misleading. Time doesn’t slow down when you’re moving, except in a manner of speaking. Your clock slows down when you’re moving, but your time actually speeds up.

March 2020