# How to determine which clock runs slower in relativity?

Tags:
1. Jul 11, 2015

### 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.

2. Jul 11, 2015

### Orodruin

Staff Emeritus
If no acceleration is involved, the twins can never meet up and compare their clocks.

Then you have been accelerating.

3. Jul 11, 2015

### Max Johnson

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').

4. Jul 11, 2015

### Orodruin

Staff Emeritus
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.

5. Jul 11, 2015

### Max Johnson

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?

6. Jul 11, 2015

### Orodruin

Staff Emeritus
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 space ship 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 space ship would not agree that the clocks are synchronised.

7. Jul 11, 2015

### Max Johnson

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)?

8. Jul 11, 2015

### Orodruin

Staff Emeritus
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.

9. Jul 11, 2015

### Max Johnson

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?

10. Jul 11, 2015

### Staff: Mentor

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.

11. Jul 11, 2015

### Staff: Mentor

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.

12. Jul 11, 2015

### Orodruin

Staff Emeritus
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., $t' = \gamma (t - vx/c^2)$.

13. Jul 11, 2015

### TheWonderer1

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?

14. Jul 11, 2015

### Ibix

No - that's the Doppler effect. Time dilation is something that occurs aswell 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.

15. Jul 11, 2015

### TheWonderer1

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

16. Jul 11, 2015

### Staff: Mentor

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.

17. Jul 13, 2015

### vanhees71

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)

Last edited: Jul 13, 2015
18. Jul 13, 2015

### harrylin

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.

19. Jul 14, 2015

### Vitro

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

20. Jul 16, 2015

### 1977ub

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.