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.

 

[PeroK]

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]

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]

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’

 

[vanhees71]

Also see my SRT review article here:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[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:

https://www.physicsforums.com/threa...multaneity-easier-to-see-with-a-train.468826/

 

[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]

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]

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:
https://www.physicsforums.com/threads/synchronize-clocks-in-relative-motion.930548/

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.

 

[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!

 

[Orodruin]

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]

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]

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]

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]

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

 

[Orodruin]

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.

To be honest, you cannot learn relativity without understanding relativity of simultaneity. If you are having problems, just glazing over simultaneity when it is mentioned is probably your big hurdle to learning relativity. Understanding what simultaneity means and that it is relative is crucial in understanding what is going on in relativity. As stated in my signature, the reason for the large majority of misconceptions about relativity that you will find out there are based on people glossing over the relativity of simultaneity and implicitly assuming that "simultaneous" has a meaning without reference to a specific frame.

 

[vanhees71]

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

Hm, I hope you like my approach:

https://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

 

[Erland]

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.

I think you are completely wrong. In my opinion, when learning relativity, one should learn relativity of simultaneity BEFORE one learns time dilation and length contraction. Indeed, one should learn relativity of simultaneity as an immediate consequence of the two postulates of SR. The simplest example is the one with a light pulse in the middle of a train, which reaches the front and back of the train simultaneously according the observer on the train, but not according to the observer on the ground. Very simple to understand, in my opinion. This is how Iearned it in my teens, and this made it easy for me to understand how BOTH observers see time slowing down and lengths contract for the other observer.

Einsteins 1905 example might seem simple, but, as has been pointed out, it becomes complex when one clock changes its intertial reference frame several times, and with it the opinion about which events are simultaneous and not.

 

[vanhees71]

Time dilation is among the most simple of all these effects. You just have to define proper time of a moving clock (idealized to a point-like body). That's all. It's not more complicated than understanding that in usual Euclidean geometry you can travel different distances between two points, depending on the path you take (whether I move, e.g., from Frankfurt to New York directly or via Sydney, Australia it makes quite some difference in my travel distance ;-)).

 

[Orodruin]

It's not more complicated than understanding that in usual Euclidean geometry you can travel different distances between two points, depending on the path you take

Technically that is differential ageing, not time dilation. Time dilation is more related to parametrising both curves by the $x$-coordinate and comparing the path length up to a given value of that parameter.

 

[Bartolomeo]

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.

That goes straight from Einstein 1905 paper. He doesn't mention that the observer is non inertial. According to Einstein, from the point of view of a moving observer a clock at rest „is ticking faster“. Since rotating observer can never be „at rest“, he always measures that a clock in the center is ticking faster. But, Einstein's calculation works perfectly well for inertial observer also.

Here it is:

https://www.fourmilab.ch/etexts/einstein/specrel/www/ - § 7. Theory of Doppler's Principle and of Aberration

From the equation for $\omega'$ it follows that if an observer is moving with velocity $v$ relatively to an infinitely distant source of light of frequency $\nu$, in such a way that the connecting line “source-observer” makes the angle $\varphi$ with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency of the light perceived by the observer is given by the equation

$$\nu= \nu' \frac {(1-\cos\varphi \cdot v/c)}{\sqrt {1-v^2/c^2}}$$

We see that, in contrast with the customary view, when $v=-c, \nu'=\infty$

It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.

So, in Transverse condition ray of light from the source moves at right angle to direction of motion of the observer. If observer rotates, it always comes at right angle to direction of its motion. Rotating observer always moves at tangential to wavefront. In classical case there is no Doppler effect. So, $\cos \pi/2 = 0$ and purely Transverse effect is seen

$$\nu= \frac {\nu'}{\sqrt {1-v^2/c^2}}$$

According to celebrated Einstein's 1905 paper, both rotating and the inertial observer (who momentarily coincides with the rotating one) will see, that clock in the center of the circumference is ticking faster.

 

[vanhees71]

Technically that is differential ageing, not time dilation. Time dilation is more related to parametrising both curves by the $x$-coordinate and comparing the path length up to a given value of that parameter.

Proper time is parametrization invariant as well as Poincare invariant. That's why it makes life so much easier than all these discussions with Lorentz transformations and Minkowski diagrams, although one should of course treat them in the introductory lecture on SR to some extent.

 

[Orodruin]

Proper time is parametrization invariant as well as Poincare invariant. That's why it makes life so much easier than all these discussions with Lorentz transformations and Minkowski diagrams, although one should of course treat them in the introductory lecture on SR to some extent.

Indeed, but time-dilation as usually presented is comparing a coordinate time to proper time. This requires a time coordinate and therefore a frame dependent simultaneity convention.

 

[PeroK]

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.

Trying to learn relativity without confronting the issue of simultaneity is like building a house without a key foundation. One day your perceived understanding will collapse.

For example, this thread started with your post, including:

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.

It's clear, therefore, that your knowledge of SR is not yet on a solid foundation. In my view, you do need to confront the simultaneity issue.

As for its being convoluted, all you need is a light source in the middle of a vehicle. In the vehicle's frame light from the source hits both ends of the vehicle simultaneously. Yet, in a frame where the vehicle is moving the light reaches the rear of the vehicle first. Thus, simultaneity is frame dependent. It's that simple.

 

[Janus]

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.

Assume you have two rows of clocks as shown in the following animation. You have arranged things such that, even though the top row is moving relative to the bottom row at some fraction of the speed of light, the top row of clocks have had their distances and tick rates adjusted so that they line up with the lower row of clocks and tick at the same rate at seen by the lower row, so you get this( in this GIF the clocks run from 12:00 to 2:00 and then reset):

Now consider the same scenario, but as seen from the view upper row, where the lower row would be seen as moving from right to left, you end up with this:

The bottom row of clocks are spaced closer together, run slower and are not synced to each other. However, whenever a clock in the upper row and one in the lower row pass each other, they still read exactly the same time, just as they do in the first animation.

In the first animation, we had to adjust the distance between the top row of clocks to counteract length contraction(notice how in the second animation, done from the upper row's rest frame, the clocks are further apart), the tick rate to counteract Time dilation, the relative clock reading to counteract the Relativity of Simultaneity( in its own rest frame, the upper row of clocks are not in sync.). We had to do this in order to get that nice arrangement of evenly spaced clocks all ticking together as seen by the bottom row. So for example, if the distance between the lower row clocks is 0.5 light hr, and the relative speed between the rows is 0.5c, Then the upper row of clocks is length contracted by a factor of 0.866 as measured by the lower row, and for them to appear to be 0.5 light hr apart, they must be ~0.577 light hr apart as measured in their own frame. The same happens for tick rate and the simultaneity between the individual clocks.

when you switch to the top row, you measure the proper distances, tick rates, etc for that frame, but now the lower row is time dilated, length contracted, etc. But is all works together to ensure that no matter which row you are watching from, when an upper clock passes a lower clock, they both read the same time (12:00, 1:00, 2:00, etc)