Uncovering the Mystery of Time Dilation in Mechanical Watches

[GZADC]

I am currently taking an undergraduate modern physics course that offers a brief overview of relativity. Let us consider a scenario where we have one clock on Earth in a rest frame, and one clock traveling in space at a constant speed v. we have some sort of mechanism for viewing the traveling clock in real time. Relativity tells us that the two clocks will display different times. Now, if we bring both clocks to rest and compare them side by side, experimental evidence tells us they will still display different times. I can't fathom how once they are both in the same frame, two standard mechanical watches would display different times. It makes since for an atomic clock to display different times, but I don't quite understand how a regular mechanical watch would be affected by the velocity differences.

In other words, I am asking what is the underlying process that causes the hands on a mechanical watch to move at different speeds.

 

[WannabeNewton]

Unfortunately relativity doesn't provide an underlying fundamental process for time dilation as applied to any and all mechanical clocks. The usual reasoning is as follows: since time dilation can be explicitly (and easily) derived for light clocks using the basic postulates of special relativity, the principle of relativity implies that time dilation must also affect any and all ideal isochronous clocks.

 

[PeterDonis]

In other words, I am asking what is the underlying process that causes the hands on a mechanical watch to move at different speeds.

There isn't one. Or rather, trying to think of it as a process that "changes the speed of the watch hands" is thinking of it backwards.

Here's a better way to think of it: the watch is traveling along some curve in spacetime--the usual name for that curve is "worldline". Along that curve, as far as the watch is concerned, *all* time-dependent processes go "at the same rate". (The usual way of expressing this in relativity is that "proper time" is an invariant along any given worldline, and the rate of all time-dependent processes along that worldline is determined by proper time along that worldline.) For example, if you were moving along with the watch, you would see it ticking away normally, and you would not be able to tell anything about your or the watch's state of motion just by watching its ticking.

Now consider two watches that travel along different worldlines in spacetime--that is, they are in different states of motion--but the worldlines begin and end at the same event (i.e., the watches start out co-located, then separate, then come back together again). In general, the "length" of the two worldlines between those two events--i.e., the amount of proper time elapsed along each worldline--will be different, simply as a matter of geometry: two different curves between the same pair of points can have different lengths. And since proper time along each worldline determines the rate of all time-dependent processes along that worldline, including the ticking of watches, we expect the two watches in this case to show different elapsed times, in general, when they come back together. Again, this is purely a matter of the geometry of spacetime: there's no mysterious "process" at work that makes the watches "go at different rates"; it's just ordinary proper time along different curves.

Now, *if* you happen to be following a particular curve in spacetime (say, the one being followed by one watch in the above example), the proper time along that curve will seem "normal" to you, and proper time along other curves will seem "dilated" because it's going at a different rate, simply because of the geometry of spacetime. (Strictly speaking, the curve you are following must be inertial, and spacetime must be flat, for the proper time along all other curves between the same two points to seem dilated compared to yours.) So "time dilation" is really a side effect of spacetime geometry; it's not something separate that has to have a separate cause.

 

[GZADC]

There isn't one. Or rather, trying to think of it as a process that "changes the speed of the watch hands" is thinking of it backwards.

Here's a better way to think of it: the watch is traveling along some curve in spacetime--the usual name for that curve is "worldline". Along that curve, as far as the watch is concerned, *all* time-dependent processes go "at the same rate". (The usual way of expressing this in relativity is that "proper time" is an invariant along any given worldline, and the rate of all time-dependent processes along that worldline is determined by proper time along that worldline.) For example, if you were moving along with the watch, you would see it ticking away normally, and you would not be able to tell anything about your or the watch's state of motion just by watching its ticking.

Now consider two watches that travel along different worldlines in spacetime--that is, they are in different states of motion--but the worldlines begin and end at the same event (i.e., the watches start out co-located, then separate, then come back together again). In general, the "length" of the two worldlines between those two events--i.e., the amount of proper time elapsed along each worldline--will be different, simply as a matter of geometry: two different curves between the same pair of points can have different lengths. And since proper time along each worldline determines the rate of all time-dependent processes along that worldline, including the ticking of watches, we expect the two watches in this case to show different elapsed times, in general, when they come back together. Again, this is purely a matter of the geometry of spacetime: there's no mysterious "process" at work that makes the watches "go at different rates"; it's just ordinary proper time along different curves.

Now, *if* you happen to be following a particular curve in spacetime (say, the one being followed by one watch in the above example), the proper time along that curve will seem "normal" to you, and proper time along other curves will seem "dilated" because it's going at a different rate, simply because of the geometry of spacetime. (Strictly speaking, the curve you are following must be inertial, and spacetime must be flat, for the proper time along all other curves between the same two points to seem dilated compared to yours.) So "time dilation" is really a side effect of spacetime geometry; it's not something separate that has to have a separate cause.

Thank you very much for this explanation, really makes more sense now.

 

[Chestermiller]

I am currently taking an undergraduate modern physics course that offers a brief overview of relativity. Let us consider a scenario where we have one clock on Earth in a rest frame, and one clock traveling in space at a constant speed v. we have some sort of mechanism for viewing the traveling clock in real time. Relativity tells us that the two clocks will display different times. Now, if we bring both clocks to rest and compare them side by side, experimental evidence tells us they will still display different times. I can't fathom how once they are both in the same frame, two standard mechanical watches would display different times. It makes since for an atomic clock to display different times, but I don't quite understand how a regular mechanical watch would be affected by the velocity differences.

In other words, I am asking what is the underlying process that causes the hands on a mechanical watch to move at different speeds.

The clocks are not running at different speeds, they are running at the same speed. But, the clock that was traveling in space for a while and then turned around, and then came back to rest at Earth has traveled into the future of the clock that remained on earth. It is just as if it took a shortcut through space-time. This is the closest analogy I can think of.

Chet

 

[PeterDonis]

the clock that was traveling in space for a while and then turned around, and then came back to rest at Earth has traveled into the future of the clock that remained on earth.

I don't think this is a good way to describe what's happening, because when the traveling clock gets back, the stay-at-home clock is right there; the traveling clock is in the present of the stay-at-home clock at that point, not its future.

It is just as if it took a shortcut through space-time.

This, OTOH, I think is a good short way of saying what I was saying; the traveling clock takes a different path through spacetime, and different paths can have different lengths. The only twist in spacetime is that the traveling path is *shorter* instead of longer, because of the different signature of the metric.

 

[WannabeNewton]

The clocks are not running at different speeds, they are running at the same speed.

That's certainly not true in light of the OP because as it stands your statement is conflating (the lack of) clock synchrony with isochronism. What you're referring to is the latter, which regards the uniformity in tick rate of a single ideal clock, and not the former which regards the relative tick rates of two ideal clocks in relative motion, which is what the OP is referring to. Two ideal clocks can be isochronous but they won't be synchronous if in relative motion.

 

[Chestermiller]

I don't think this is a good way to describe what's happening, because when the traveling clock gets back, the stay-at-home clock is right there; the traveling clock is in the present of the stay-at-home clock at that point, not its future.

That's one perspective. But, as an astronaut who has returned to Earth after being gone one year (according to both my mechanical clock and my own biological clock), if everyone I knew has aged 10 years, technology has advanced 10 years, 3 presidents have been elected, babies have been born and are now 10 years old, and all the newspapers, calendars, and clocks on Earth now say 10 years later, call it what you want, but I call it having traveled into Earth's future.

Chet

 

[PAllen]

That's one perspective. But, as an astronaut who has returned to Earth after being gone one year (according to both my mechanical clock and my own biological clock), if everyone I knew has aged 10 years, technology has advanced 10 years, 3 presidents have been elected, babies have been born and are now 10 years old, and all the newspapers, calendars, and clocks on Earth now say 10 years later, call it what you want, but I call it having traveled into Earth's future.

Chet

I'd argue that a more precise way of saying this is that since separating (now back together), the astronaut has 'traveled' one year into their future while the Earth has 'traveled' 10 years into its future. The Earth certainly doesn't think it is in its 'future', it thinks it is in its present.

 

[Yashbhatt]

That's one perspective. But, as an astronaut who has returned to Earth after being gone one year (according to both my mechanical clock and my own biological clock), if everyone I knew has aged 10 years, technology has advanced 10 years, 3 presidents have been elected, babies have been born and are now 10 years old, and all the newspapers, calendars, and clocks on Earth now say 10 years later, call it what you want, but I call it having traveled into Earth's future.

Chet

I agree with PAllen. The astronaut thinks that he has traveled into the Earth's future because he compares the time elapsed on Earth with the time he "thinks" has elapsed.

 

[PhilDSP]

The difference in clock rates should occur purely on the basis of an increase in momentum between the clock in the rest frame and the clock traveling at v, shouldn't it?.

The clock in the Earth rest frame contains molecules that are exchanging photons almost entirely with molecules at rest with respect to it. It gains no relativistic momentum. However the clock flying by at v exchanges a very large amount of photons of origin in molecules from the Earth. It gains relativistic momentum with respect to objects in the Earth rest frame.

The same could be said of the transference of energy from EM fields generated by objects on the Earth. The increase of momentum is one means of energy storage. But it also affects the rate of movement of the clock's mechanism in relative terms because of the apparent extra "weight" of the clock's components. This is taking a semi-classical approach to relativistic effects.

 

[PAllen]

The difference in clock rates should occur purely on the basis of an increase in momentum between the clock in the rest frame and the clock traveling at v, shouldn't it?.

The clock in the Earth rest frame contains molecules that are exchanging photons almost entirely with molecules at rest with respect to it. It gains no relativistic momentum. However the clock flying by at v exchanges a very large amount of photons of origin in molecules from the Earth. It gains relativistic momentum with respect to objects in the Earth rest frame.

The same could be said of the transference of energy from EM fields generated by objects on the Earth. The increase of momentum is one means of energy storage. But it also affects the rate of movement of the clock's mechanism in relative terms because of the apparent extra "weight" of the clock's components. This is taking a semi-classical approach to relativistic effects.

It is possible to take such an approach. All it does is say that in any inertial frame, physical processes run slower for a body moving in that frame. If you insist on such an interpretation, then you obviously don't talk about 'going into the future'. Instead, you say Earth aged 10 years and the rocket 1 year because it was moving (fast) in the inertial frame.

However, it doesn't change that the explanation is very different in other inertial frames (e.g. one for the outgoing rocket trip). It also doesn't generalize well to GR.

What also doesn't generalize to GR is any notion of a frame variant but 'objective, global, simultaneity'.

 

[Chestermiller]

I'd argue that a more precise way of saying this is that since separating (now back together), the astronaut has 'traveled' one year into their future while the Earth has 'traveled' 10 years into its future. The Earth certainly doesn't think it is in its 'future', it thinks it is in its present.

Fair enough.

Chet