# I Time dilation again, Einstein or Resnick?

#### exmarine

I continue to see references to time dilation that I don’t understand. Maybe the bluntest one is in a textbook by Resnick. Introduction to Special Relativity, p. 93: “Indeed we find that the phenomena are reciprocal. That is, just as A’s clock seems to B to run slow, so does B’s clock seem to run slow to A;”

Here is Einstein’s much more plausible version from his 1905 paper, in my opinion anyway.

“From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by (up to magnitudes of fourth and higher orders), t being the time occupied in the journey from A to B.”

Notice that he carefully gives the initial conditions; (1) the clocks were synchronized, (2) in a common stationary reference frame; (3) which clock was set in relative motion, clock A; and finally which clock lost some time, clock A. I cannot read that paragraph and think the effects are reciprocal. I do not think that an observer riding along with clock A could claim that it was actually clock B that was running slow and losing some time.

Resnick’s version does not seem logically possible, that somehow both clocks could “seem to be running slow” relative to each other. Any help understanding this would be appreciated.

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#### PeroK

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I continue to see references to time dilation that I don’t understand. Maybe the bluntest one is in a textbook by Resnick. Introduction to Special Relativity, p. 93: “Indeed we find that the phenomena are reciprocal. That is, just as A’s clock seems to B to run slow, so does B’s clock seem to run slow to A;”

Resnick’s version does not seem logically possible, that somehow both clocks could “seem to be running slow” relative to each other. Any help understanding this would be appreciated.
Resnick is, of course, correct. Time dilation is reciprocal. How far have you got with studying SR? Time dilation of a clock moving in your reference frame would normally be one of the first things covered.

#### Orodruin

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The ingredient you are clearly missing is relativity of simultaneity. This is crucial in order to understand why time dilation is reciprocal.

Also, see my PF Insight on time dilation and the twin paradox.

#### Dale

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I cannot read that paragraph and think the effects are reciprocal. I do not think that an observer riding along with clock A could claim that it was actually clock B that was running slow and losing some time
Both Einstein and Resnick are correct. They do not contradict each other. What do you think would happen in Einstein’s scenario if the same arrangement was made in frame K’

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#### Erland

Both Einstein and Resnick are right, of course. Notice that Resnick's and Einsteins scenario's are somewhat different. In Einstein's scenario, one clock is first at rest relative to the other, then it moves towards the other clock. So the two clocks do not move with constant velocity relative to each other. This is somewhat unfortunate from a pedagogical point of view, it would have been better to say that both clocks all the time move with a constant velocity relative to each other, which, as I understand it, is Resnick's scenario.

As has been pointed out, it is the relativity of simultaneity which is the crux of the matter, and using that, the paradox dissolves.
An easy way to see this is to consider a train, which contains one of the clocks, while the other clock is at rest relative to a platform which the train passes. In addition to the nice links in earlier posts, I would like to refer to an old post of my own, where I develop this scenario:

#### Ibix

Imagine driving at 30mph along a straight road. Another car drives at 30mph along another straight road that makes an angle $\theta$ with yours. The other car will fall behind because it's speed in your direction is only $30\cos\theta$. Of course, the other driver will say it's you who is falling behind, and for the exact same reason. There's no logical problem with both of you saying the other falls behind - it's just that you have slightly different definitions of going forwards.

This is very closely analogous to the difference of opinion over time in relativity. You have different notions of what "not moving in space, only time" means, so disagreeing over clock rates is like observing that mile markers on the other road don't measure miles on your road. The relevant factor is the hyperbolic cosine of the velocity rather than the trigonometric cosine of the angle, though, because spacetime is Minkowski, not Euclidean.

#### vanhees71

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It is, however, very easy since you can formulate everything in a manifest covariant way, and the time an ideal clock shows is its proper time, which is a Lorentz scalar. If you have a clock moving on a trajectory $x^{\mu}(\lambda)$ (where $\lambda$ is an arbitrary parameter for the world line) then the time difference between two points (given by the values $\lambda_1$ and $\lambda_2$ of the worldline parameter) on its trajectory it will show, is given by
$$\Delta \tau=\int_{\lambda_1}^{\lambda_2} \mathrm{d} \lambda \sqrt{\eta_{\mu \nu} \dot{x}^{\mu}(\lambda) \dot{x}^{\nu}(\lambda)},$$
where the dot denotes the derivative with respect to the world-line parameter $\lambda$. Note that this is both Lorentz invariant (i.e., it doesn't matter in which inertial reference frame you define the space-time components $x^{\mu}$ of the trajectory) and parameterization invariant (i.e., it doesn't depend on the used parameter $\lambda$ of the world line).

It's much easier to use covariant or even invariant quantities to understand a problem than struggling with cumbersome explicit transformations from one frame of reference to another!

#### exmarine

Wow, lots of good answers already. Thanks! I need to study them and then respond later.

But I can answer Dale in #4 now: If by "K' viewpoint", you mean that A is still the clock set in relative motion, then I get the exact same answer. A matrix (Lorentz Transform) and its inverse is not a big deal, even with possible origin offsets.

If by "same arrangement" you mean that clock B is the one set in relative motion, then I would get that clock B was running slower and losing time.

Maybe this is related to an earlier question I posted some time ago, and got no serious responses: Is it possible to synchronize two clocks already in relative motion? If so, exactly how? And I am referring to their running rates, not some trivial possible origin offsets. If I remember correctly, Einstein was always careful to specify that the clocks were "good" or some such word, meaning to me at least, that at some time in the past they had occupied a common reference frame and had their running rates synchronized...

Thanks.

#### jbriggs444

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Is it possible to synchronize two clocks already in relative motion?
No. Each will be time dilated as viewed from a frame of reference in which the other is at rest.

The muons in above example are accelerating (exposed to g force). So their dilation is due to time dilation as described by GR.
The acceleration of gravity is negligible for this scenario. SR applies. GR is irrelevant.

#### Ibix

be this is related to an earlier question I posted some time ago, and got no serious responses: Is it possible to synchronize two clocks already in relative motion? If so, exactly how? And I am referring to their running rates, not some trivial possible origin offsets.
What was not serious about my answer? Apart from the spelling of "pass"?

#### exmarine

What was not serious about my answer? Apart from the spelling of "pass"?
Bad choice of words on my part, sorry. What I was looking for was HOW operationally to do it if it was possible. For example, how could you even tell what the other guy’s running rate was on the fly so to speak? You would have to know that in order to set one running 1/gamma slower like you said. Which one would you set slower, or would it matter? Would you get into an infinite recursion if not careful? Then which one would you accelerate in order to set the relative velocity back to zero? Would it matter? According to the answers in this thread, it sounds like it wouldn’t. Maybe modern Doppler radar could be used?

Thanks.

PS. Actually wouldn’t one set his own clock to run gamma faster than the other’s apparent rate??

#### Ibix

The thread exmarine and I are referring to:
What I was looking for was HOW operationally to do it if it was possible.
First you need to work out what you mean by "synchronised". Einstein defines clocks at rest as synchronised if both agree how behind the other appears to be (i.e. if we both look at our the other's clock when our own reads 12.00 and see the other's reading 11.59) and if that relationship is maintained. He never even defines "synchronised" for clocks in relative motion.

What I assumed you meant was this. I fill spacetime with a set of mutually at rest clocks, then let you move through this field. You adjust your clock's rate so that it always shows the same time as the nearest of my clocks. That way, your clock is showing my frame's time. This is pretty much what the GPS clocks do, although for other reasons.

You are correct that you want your clock to tick fast, and that's what I said (a shorter pendulum to tick once every $1/\gamma$ seconds).

I'm sure there are other ways to define synchronisation between moving clocks. I'm not aware of any obvious standard for what it would mean - but I haven't thought about it particularly hard or looked for one.

#### Erland

About synchronization of clocks, at least the following holds:

Given two clocks moving with constant nonzero velocity relative to each other. Each clock is assumed to be at rest relative to some inertial frame. Then there is no inertial observer moving with constant velocity relative to each one of the clocks for which the two clocks can be set to show the same time for more than an instant.

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#### Ibix

About synchronization of clocks, at least the following holds:

Given two clocks moving with constant nonzero velocity relative to each other. Each clock is assumed to be at rest relative to some inertial frame. Then there is no inertial observer moving with constant velocity relative to each one of the clocks for which the two clocks can be set to show the same time for more than an instant.
I don't think that's true - there's a frame where the clocks have equal and opposite velocities, for instance. And anyway, we were considering adjusting one clock's tick rate to match another frame. The clock is broken as far as Einstein synchronisation is concerned, but it's a simple linear adjustment to the rate.

#### Erland

I don't think that's true - there's a frame where the clocks have equal and opposite velocities, for instance.
OK, you are right. So, for the conclusion to hold, we need to add that relative to the observer, the clocks should have unequal speeds.

#### exmarine

Imagine driving at 30mph along a straight road. Another car drives at 30mph along another straight road that makes an angle $\theta$ with yours. The other car will fall behind because it's speed in your direction is only $30\cos\theta$. Of course, the other driver will say it's you who is falling behind, and for the exact same reason. There's no logical problem with both of you saying the other falls behind - it's just that you have slightly different definitions of going forwards.

This is very closely analogous to the difference of opinion over time in relativity. You have different notions of what "not moving in space, only time" means, so disagreeing over clock rates is like observing that mile markers on the other road don't measure miles on your road. The relevant factor is the hyperbolic cosine of the velocity rather than the trigonometric cosine of the angle, though, because spacetime is Minkowski, not Euclidean.
That cosine trip analogy is really nice. Thanks!

So it is DURING the trip that A thinks clock B is running slower than his own? Then when he reaches B (and stops I assume) he realizes it is his own clock that was really running slower? But I really need to see the math for that. I’ll show you what I think I understand about the math. I don’t see how to get your result.

I don’t feel like messing with the equation editor on here – can I attach a file? I’ll type it up in Word, make a pdf, and see if I can attach it.

Thanks!

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#### Orodruin

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So it is DURING the trip that A thinks clock B is running slower than his own? Then when he reaches B (and stops I assume) he realizes it is his own clock that was really running slower?
The ”running slower” part really is a red herring as it depends on how you decide to compare clocks that are not colocated due to the relativity of simultaneity. Again, I suggest reading the PF Insight I linked to in #3. It is essentially making the same comparison to Euclidean space as presented by Ibix.

#### PeroK

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So it is DURING the trip that A thinks clock B is running slower than his own? Then when he reaches B (and stops I assume) he realizes it is his own clock that was really running slower? But I really need to see the math for that. I’ll show you what I think I understand about the math. I don’t see how to get your result.
During the inertial phase of the trip, both clocks are time dilated relative to the other. In the same way that both are moving relative to the other and neither is in absolute motion. Neither clock is "really" running faster than the other. It's all relative.

Consider the following:

In some inertial reference frame both A and B are moving to the left at some speed ($c/2$, say). A accelerates instantaneously to the right until it is not moving in the original frame. A and B now converge with A at rest and B still moving to the left. When they meet A accelerates to the left so that they are both moving at their original speed.

Now, if A and B's clocks are observed from the original reference frame (in which A was at rest for the journey and B was moving at $c/2$), whose clock will appear to run slower during the experiment?

#### PeroK

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Here is Einstein’s much more plausible version from his 1905 paper, in my opinion anyway.

“From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by (up to magnitudes of fourth and higher orders), t being the time occupied in the journey from A to B.”
I would not discourage you from reading what Einstein had to say, but you need to be careful. In this case, you could get the impression (as I think you have) that time dilation is not symmetrical. But, in fact, it takes more than time dilation to explain what Einstein describes. To be precise, this is actually a case of "differential aging". There are, in fact, three factors involved:

Time dilation
Relativity of simultaneity
That A twice changes its inertial reference frame

There is nothing wrong with what Einstein says, of course. But, this experiment he describes is perhaps not the best place to start understanding time dilation.

#### exmarine

During the inertial phase of the trip, both clocks are time dilated relative to the other. In the same way that both are moving relative to the other and neither is in absolute motion. Neither clock is "really" running faster than the other. It's all relative.

Consider the following:

In some inertial reference frame both A and B are moving to the left at some speed ($c/2$, say). A accelerates instantaneously to the right until it is not moving in the original frame. A and B now converge with A at rest and B still moving to the left. When they meet A accelerates to the left so that they are both moving at their original speed.

Now, if A and B's clocks are observed from the original reference frame (in which A was at rest for the journey and B was moving at $c/2$), whose clock will appear to run slower during the experiment?
Perfect! Now I see it. Thanks to everyone.

#### Orodruin

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There is nothing wrong with what Einstein says, of course. But, this experiment he describes is perhaps not the best place to start understanding time dilation.
I think there is an important point to be made here. Although Einstein was not wrong, the theory was just in development. For a pedagogical presentation, a student will be much better off with a modern textbook. If one so desires, it is possible to go back and read the original papers later for an understanding of how the theory developed. Much has happened in how we view relativity since Einstein’s seminal papers.

#### Mister T

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So it is DURING the trip that A thinks clock B is running slower than his own?
Whether he thinks it or not is not relevant. It's a fact that they each will observe the others' clocks as running slow. But to make those observations they'll each need at least one other clock not co-located with their first. You need to understand that issues arise with synchronizing those clocks, as each can't do it in a way that the other will agree is valid.

Then when he reaches B (and stops I assume) he realizes it is his own clock that was really running slower?
He doesn't need to stop, but it's important to understand that when they reunite to compare clocks, the difference in elapsed time is a difference in two proper times whereas the comparisons in the above discussion each involve only one proper time.

I don’t feel like messing with the equation editor on here – can I attach a file?
Responders can't quote from it!

#### 1977ub

If you have one traveler moving in a circle around the other, you have removed the constant inertial frame from the one traveler, and the result is less paradoxical.

The observer in the middle will find that the circling traveler has a slower clock, and the circling traveler will simply agree: the non-circling observer has a clock which is ticking faster than his own.

#### exmarine

You guys must be professors if you are concerned about how to teach this stuff. Then you might be interested in my opinion about learning it. Forget about teaching “simultaneity” for beginners! I’ve been studying physics when possible for the last 8 years since I retired, have acquired about a dozen textbooks, been through all of Susskind’s lectures on Special, General, Cosmology, and my eyes still glaze over when anyone mentions that. It is far too convoluted and tedious for beginners – long trains, bolts of lightning at various places and times, etc., etc. Instead, I highly recommend this clock example from Einstein’s paper. It is elegantly simple. The entire scenario can be accurately stated in one sentence and easily visualized. Just be sure to also show the view in the other reference frame like you guys showed me! The straight line from the empirical evidence for the speed of light in all inertial reference frames, the derivation of the Lorentz Transform from that (Resnick’s elementary book is pretty good on that), and then to the world-line plots and a little math for Einstein’s clock experiment seems like the simplest way to teach it. And the hyperbolic or non-Pythagorean nature of the LT is important to emphasize for beginners.

Thanks

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