# How to determine which clock runs slower in relativity?

• Max Johnson
In summary, the conversation explores the Twins Paradox thought experiment which questions the different measurements of time between two observers in motion. The concept of relativity of simultaneity is introduced, explaining how the two observers can come to different conclusions about which clock is running slower. The scenario of two planets and a rocket is discussed, where the measurements of time are still affected by relativity of simultaneity, causing the observers to measure a different time between events.
Max Johnson
So I was recently reading Stephen Hawkings' "The Universe in a Nutshell" and came across the famous Twins Paradox thought experiment. My question is, since motion is relative, couldn't we extrapolate that either the observer on Earth is stationary and the rocket is traveling near light-speed (causing the clock on the rocket to run slower) or vice-versa (causing the clock on Earth to run slower)? If this is true, how can both clocks run slower than the other?

I have read vague answers talking about how it has something to do with inertial reference frames, but what if no acceleration is involved? What if the rocket were to fly past Earth and mechanically flip a switch starting both clocks, and then doubling around and again hit the switch to stop both clocks? The clocks would then only be running while it is moving at a constant velocity.

I'm sure I'm overlooking something, I just can't figure out what that is.

Max Johnson said:
I have read vague answers talking about how it has something to do with inertial reference frames, but what if no acceleration is involved?
If no acceleration is involved, the twins can never meet up and compare their clocks.

Max Johnson said:
What if the rocket were to fly past Earth and mechanically flip a switch starting both clocks, and then doubling around and again hit the switch to stop both clocks?
Then you have been accelerating.

Stephanus
Okay, but what if instead there were two planets, stationary with respect to each other, and their clocks were synced. What if the rocket were to fly past the first planet, starting both clocks, and then flew past the second planet which stopped the clocks, while the rocket also relayed its clocks' measurements wirelessly to the second planet. Thus, the rocket is never accelerating, yet its clock's reading can be compared to the second planets' clock (which is synced with the first planets').

Now you run into problems with relativity of simultaneity. The clocks of the planets cannot start "at the same time" in both the rocket frame and the planet frame.

Would you mind explaining this a little more? Why couldn't the clocks on the planets each only measure the time the rocket passes themselves, and then relay the times to each other to measure the time elapsed?

Max Johnson said:
Would you mind explaining this a little more? Why couldn't the clocks on the planets each only measure the time the rocket passes themselves, and then relay the times to each other to measure the time elapsed?
This is how the time would be measured between the events in the planets' rest frame, yes. It will not be the same as the time measured on board the space ship. The time on the spaceship between the events would be shorter.

However, the planets would still be time dilated in the space ship's rest frame. This is because the spaceship would not agree that the clocks are synchronised.

Orodruin said:
This is how the time would be measured between the events in the planets' rest frame, yes. It will not be the same as the time measured on board the space ship. The time on the spaceship between the events would be shorter.

However, the planets would still be time dilated in the space ship's rest frame. This is because the spaceship would not agree that the clocks are synchronised.

So then if the clock on the rocket would still run slower, and no acceleration is involved in this scenario (all data relayed while the rocket is moving), why couldn't we also say the planets are the ones moving and thus they are the ones who measure a slower time (since motion is relative)?

Max Johnson said:
So then if the clock on the rocket would still run slower, and no acceleration is involved in this scenario (all data relayed while the rocket is moving), why couldn't we also say the planets are the ones moving and thus they are the ones who measure a slower time (since motion is relative)?

I just said that this is the case. Fin the rocket frame, the clocks on the planets are moving slow. There is no such thing as absolute motion. This is not a problem because of relativity of simultaneity.

So are you saying that the planets would claim the rocket's clock is slower, while the rocket would claim the planets' clocks are slower? How can both be true?

Max Johnson said:
So are you saying that the planets would claim the rocket's clock is slower, while the rocket would claim the planets' clocks are slower? How can both be true?

Consider exactly what it means to say that clock B is running slower than clock A. We initially synchronize the two clocks so that they both read 12:00 noon. At some later time, A looks at his clock and sees that it reads 1:00; at the same time B's clock reads 12:30 so A concludes that B's clock is running slow by a factor of two.

However, because of the relativity of simultaneity the two events "Clock A reads 1:00" and "Clock B reads 12:30" which happen at the same time according to A do not happen at the same time according to B. According to B, the event "Clock B reads 12:30" happens at the same time as the event "Clock A reads 12:15" and it's A's clock that is running slow by a factor of two.

Because the two observers have different definitions of "at the same time", they can come to different conclusions about what A's clock read at the same time that B's clock read at 12:30, and hence different conclusions about which clock is running slow. They're both right.

Max Johnson said:
So I was recently reading Stephen Hawkings' "The Universe in a Nutshell" and came across the famous Twins Paradox thought experiment. My question is, since motion is relative, couldn't we extrapolate that either the observer on Earth is stationary and the rocket is traveling near light-speed (causing the clock on the rocket to run slower) or vice-versa (causing the clock on Earth to run slower)?

If you haven't yet come across http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html, give it a try. It does a pretty good job of explaining the twin paradox.

The underlying reason is quite analogous to the following situation in classical physics:

Two cars leave the same point with the same speed at the same time, but in different (not opposite, just at an angle) directions. If both decide to measure the velocity component in their own direction of motion, they will both find that they are the ones with the largest velocity component.

In analogy, two observers moving relative to each other in relativity have different notions of which direction in space time is the time direction. What one considers the time direction, the other will consider "mostly in the time direction, but with a part in the space direction as well". This is what the Lorentz transformations describe, e.g., ##t' = \gamma (t - vx/c^2)##.

Why is it stated that a moving clock moves slower. Is it due to the fact that the person on Earth is in fact moving away from the rocket in their perspective?

TheWonderer1 said:
Why is it stated that a moving clock moves slower. Is it due to the fact that the person on Earth is in fact moving away from the rocket in their perspective?
No - that's the Doppler effect. Time dilation is something that occurs as well as that. It turns out - as Orodruin was alluding - that two objects in relative motion have different notions of "now" and "the future". An object in motion has a "future" that is at a slight angle to your "future" (crazy as that may sound - you may wish to google for the "rapidity" which is the angle I am talking about), which means that its clocks tick slower from your perspective. Of course, it can consider itself at rest and you to be moving, so your clocks tick slowly from its perspective. This is not paradoxical because the two perspectives disagree on what "now" means, which means that they aren't measuring the same thing.

Awesome. That's really fascinating. Does this follow for two moving clocks bc they have two different angles of what is going on?

TheWonderer1 said:
Awesome. That's really fascinating. Does this follow for two moving clocks bc they have two different angles of what is going on?

If by "moving" you mean "moving relative to one another"... then yes. Draw a simple spacetime diagram and you'll even see the different angles.

With the right math everything is very simple. An (ideal) clock shows its proper time, which is given by
$$\tau_2-\tau_1=\int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda \sqrt{g_{\mu \nu} \dot{q}^{\mu}(\lambda) \dot{q}^{\nu}(\lambda)},$$
where ##q^{\mu}(\lambda)## is the trajectory of the clock in some coordinates and ##g_{\mu \nu}## the pseudometric tensor components. This is the general-relativistic equation (and thus includes both the effects of motion (time dilation), acceleration, and gravitation).

It's interesting that this is still an issue since there are plenty of experiments proving this postulate about what time a clock measures at pretty high accuracy.

https://en.wikipedia.org/wiki/Time_dilation#Experimental_confirmation

[EDIT: Thanks Nugatory for pointing out my nonsense!]

Presently I'm reading an interesting book on the issue of time and how it is understandable why Einstein received the Nobel prize for the only theory he ever created which has not withstood the development of physics, his naive-photon picture of the photoelectric effect, rather than for his greatest achievement, which without doubt is the discovery of the general theory of relativity. It's just because of a debate about time and what's the "right measure of time" between Einstein and his followers within the physics community and some philosophers, most importantly Henri Bergson:

J. Canales, The Physicist and the Philosopher, Princeton University Press (2015)

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Max Johnson said:
[..] what if no acceleration is involved? What if the rocket were to fly past Earth and mechanically flip a switch starting both clocks, and then doubling around and again hit the switch to stop both clocks? The clocks would then only be running while it is moving at a constant velocity.

I'm sure I'm overlooking something, I just can't figure out what that is.
A telling variant which you almost brought up, is that "doubling around" is replaced by another rocket flying in opposite direction. When passing each other, the second rocket (which is heading towards Earth) synchronizes its clock with the first one. In that way no acceleration is involved. And that changes nothing in the prediction, which was based on the assumption that acceleration forces have no effect.

Max Johnson said:
So are you saying that the planets would claim the rocket's clock is slower, while the rocket would claim the planets' clocks are slower? How can both be true?

Hi Max,

It's important to realize that there is no such thing as a "slower" clock, all standard clocks always run at their normal rate (in their respective rest frames). A clock is only MEASURED to be slower in a reference frame in which it is moving but this is not an intrinsic rate change for any of the clocks involved. You can think of it as a geometric projection effect.

I would suggest trying to familiarize yourself with spacetime diagrams, how to draw and interpret them. I find them to be a very useful tool in analyzing Special Relativity scenarios as they help me visualize what's going on.

en.wikipedia.org/wiki/Minkowski_diagram
www.phys.vt.edu/~takeuchi/relativity/notes/section12.html

If twin A sits there, and twin B moves around in circles at near the speed of light (and somehow isn't turned to jelly), and the two twins compare their watches whenever B swings by A, both twins will agree that B's watch is running slower than A's.

Yes, and precisely this experiment is realized and found a quantitative agreement with SRT to a very high precision. Of course, nobody puts twins in ultra-centrifuges, but one uses the lifetime of atomic states/hyperfine transitions of Li+ ions in the Experiment Storage Ring (ESR) at GSI (at moderate speeds of about 0.33c):

B. Botermann et al., Test of time dilation using stored Li+ ions as clocks at relativistic speed, Phys. Rev. Lett. 113, 120405 (erratum Phys. Rev. Lett. 114, 239902 (2015))
http://dx.doi.org/10.1103/PhysRevLett.113.120405
http://dx.doi.org/10.1103/PhysRevLett.114.239902
http://arxiv.org/abs/1409.7951

This is, of course, only one of a plethora of experiments, testing (S)RT:

https://en.wikipedia.org/wiki/Tests_of_special_relativity
https://en.wikipedia.org/wiki/Tests_of_general_relativity

1977ub said:
If twin A sits there, and twin B moves around in circles at near the speed of light (and somehow isn't turned to jelly), and the two twins compare their watches whenever B swings by A, both twins will agree that B's watch is running slower than A's.
That would be a misleading conclusion which could make one think that motion or acceleration has an intrinsic influence on clock operation causing them to measure time incorrectly. Rather, I think the standard interpretation is that even though the clocks run at the same standard rate they show different time between successive encounters because their separate paths through spacetime have different "lengths" in terms of elapsed proper time, A > B.

But I believe the OP was asking about a symmetric scenario where two inertial observers in relative motion measure mutual time dilation (each measures the other's clock to be "slower").

Vitro said:
That would be a misleading conclusion which could make one think that motion or acceleration has an intrinsic influence on clock operation causing them to measure time incorrectly. Rather, I think the standard interpretation is that even though the clocks run at the same standard rate they show different time between successive encounters because their separate paths through spacetime have different "lengths" in terms of elapsed proper time, A > B.

But I believe the OP was asking about a symmetric scenario where two inertial observers in relative motion measure mutual time dilation (each measures the other's clock to be "slower").

What conclusion do you find misleading? I just think it is an interesting contrast to all the usual "twin paradox straight line paths" examples which sometimes people have trouble with. The circling twin is constantly shifting his IF while the stationary twin remains in a single IF - so yes the circling twin is taking a longer spacetime path. In this case, both twins will agree about which clock is moving slower and who is aging more.

1977ub said:
The circling twin is constantly shifting his IF while the stationary twin remains in a single IF - so yes the circling twin is taking a longer spacetime path.

That is not quite right. The circling twin is indeed taking a "longer" (actually, "shorter" sounds better to me) spacetime path, and this is indeed why less time passes on his path. But this is not because he's "shifting his inertial frame"; it's the other way around. Because he is not following an inertial path through spacetime, there is no inertial frame in which he is more than momentarily at rest.

However, the bit that Vitro was objecting to was:
both twins will agree that B's watch is running slower than A's.
That's not exactly wrong, but it could easily mislead. It would be more accurate to say, as you yourself do a few sentences later, that both clocks are running at the same rate (one second per second) and one of them is measuring a shorter time interval than the other.

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1977ub said:
What conclusion do you find misleading? I just think it is an interesting contrast to all the usual "twin paradox straight line paths" examples which sometimes people have trouble with. The circling twin is constantly shifting his IF while the stationary twin remains in a single IF - so yes the circling twin is taking a longer spacetime path. In this case, both twins will agree about which clock is moving slower and who is aging more.
"Clocks running slower/faster" is very misleading to beginners who readily take that literally leading to discussions like "can we make better clocks which are not affected by motion or gravity" or "if two clocks show different times which one represents the real passage of time, if any" and not realizing there's nothing wrong with the clocks, they don't actually run slower of faster but instead they truly experience more or less elapsed time. The discussion in this very thread is an example of such misunderstanding.

Max Johnson said:
Would you mind explaining this a little more? Why couldn't the clocks on the planets each only measure the time the rocket passes themselves, and then relay the times to each other to measure the time elapsed?
I think I have the answer for this question. Someone in this forum, Janus, has epxlained to me very clear, and I save the link for you.
Here, I requote it.
Stephanus said:
Does clock synchronization for 2 points at rest (or at the same frame) depends to another frame?
Janus said:
Yes.

Consider two clocks at rest with respect to the A frame. They are not running and are set to zero. to synchronize them, you emit a light signal from a point halfway between them that will start each clock when it hits them like this:

The light expands out as a circle, striking both clocks simultaneously and starting both.

Now consider the same situation as viewed from the B frame which is in motion with respect to the A frame. The same light is emitted from the point half way between the two clocks. because the speed of light is invariant. it expands out in a circle from the point of emission. However, now one clock is moving towards the source point and the other away from it and one clock is struck by the light and starts running before the other like this.

Once running both clocks run at the same rate but out of step with each other. These are the same clocks and the same light as in the first animation, they are just being considered from a different frame of reference.

Nugatory said:
That is not quite right. The circling twin is indeed taking a "longer" (actually, "shorter" sounds better to me) spacetime path, and this is indeed why less time passes on his path. But this is not because he's "shifting his inertial frame"; it's the other way around. Because he is not following an inertial path through spacetime, there is no inertial frame in which he is more than momentarily at rest.

He has a momentary co-moving IF that is ever-shifting.

## 1. What is the theory of relativity?

The theory of relativity is a fundamental concept in physics that explains how the laws of physics are the same for all observers in all inertial reference frames, regardless of their relative motion.

## 2. How does time dilation occur in relativity?

Time dilation occurs in relativity when an object is moving at a significant fraction of the speed of light. As the object's velocity increases, time appears to slow down for that object relative to a stationary observer.

## 3. How can we determine which clock runs slower in relativity?

We can determine which clock runs slower in relativity by comparing the elapsed time on two synchronized clocks, one that is stationary and one that is moving at a high velocity. The clock that is moving will appear to run slower due to time dilation.

## 4. Is time dilation the same for all objects in relativity?

No, time dilation is not the same for all objects in relativity. The extent of time dilation depends on the velocity of the object relative to the observer. The faster an object is moving, the greater the time dilation will be.

## 5. What are some real-world applications of time dilation in relativity?

Some real-world applications of time dilation in relativity include the Global Positioning System (GPS), where satellites in orbit experience time dilation due to their high velocities. This must be accounted for in order for the GPS system to function accurately. Time dilation also plays a role in particle accelerators and space travel.

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