B Intuition for time dilation in a cesium clock

[msumm21]

TL;DR

Wondering if there's an intuitive way to see why cesium clocks slow with velocity, analogous to the common example of a "light clock"

A common way to introduce time dilation is to show the example of a "light clock" which bounces photons back/forth and ticks each time a photon passes a certain point. Wikipedia does it this way, for example. From such a clock, it's easy to see why the constancy of the speed of light would cause the clock to appear to tick slower by exactly the formula given in SR.

I was wondering if there's any way to "see" why a cesium clock (or energy level transitions in atoms) would also slow with movement, due to the constancy of c, analogous to the above clock.

 

[PeroK]

TL;DR Summary: Wondering if there's an intuitive way to see why cesium clocks slow with velocity, analogous to the common example of a "light clock"

A common way to introduce time dilation is to show the example of a "light clock" which bounces photons back/forth and ticks each time a photon passes a certain point. Wikipedia does it this way, for example. From such a clock, it's easy to see why the constancy of the speed of light would cause the clock to appear to tick slower by exactly the formula given in SR.

I was wondering if there's any way to "see" why a cesium clock (or energy level transitions in atoms) would also slow with movement, due to the constancy of c, analogous to the above clock.

Put the cesium clock next to the light clock. They remain synchronized in a reference frame in which they are at rest. Hence, remain synchronized in any frame.

 

[Ibix]

Synchronise the caesium clock with a light clock.

Drive past it at high speed. You know the light clock will tick slowly as measured in your test frame. You know the caesium clock is synchronised with it. Therefore you know the caesium clock is ticking slowly.

 

[Dale]

You could also use Doppler shift.

 

[Vanadium 50]

To turn the argument of @PeroK around, if the two clocks went out of sync, I would be able to use the difference between them to determine my absolute speed. Relativity forbids that.

 

[Orodruin]

While I agree with the light clock argument, it doesn’t seem like what the OP is after. The wording of the OP suggests to me that they are asking for an intuitive description of why the Cs clock slows in the same vein as the light clock argument.

I am also not sure there is a Doppler shift argument accomplishing this.

 

[msumm21]

Put the cesium clock next to the light clock. They remain synchronized in a reference frame in which they are at rest. Hence, remain synchronized in any frame.

I know SR says experimental results are = in all frames and hence the transitions (and everything) must slow like the "light clock", but what I'm looking for is something analogous to the light clock argument, you might call it a more "direct" explanation (if it exists). Edit: might also call this the "why" the Cs clock slows as SR predicts. I know such a "why" question can't always be answered.

While I agree with the light clock argument, it doesn’t seem like what the OP is after. The wording of the OP suggests to me that they are asking for an intuitive description of why the Cs clock slows in the same vein as the light clock argument.

Yes.

 

[robphy]

I would think the answer to the OP is to find a model of the mechanism of the device which can be drawn on a spacetime diagram. That’s what I did in my “Relativity on Rotated Graph Paper” approach. I drew spacetime diagrams of ticking light-clocks.

The advantage of the light-clock is that it is easier to draw than other clocks (for example. a mass on a spring or a pendulum).

 

[PeroK]

what I'm looking for is something analogous to the light clock argument, you might call it a more "direct" explanation (if it exists). Edit: might also call this the "why" the Cs clock slows as SR predicts.

First, you have to explain how a Cesium clock works?

 

[Ibix]

you might call it a more "direct" explanation (if it exists).

A plausibility argument is that it is based on electron transitions in an EM field, and it would be very strange if they didn't obey relativity, what with EM being the driving mystery behind the development of SR.

 

[robphy]

Possibly interesting:

How an atomic clock works, and its use in the global positioning system (GPS)
by engineerguy


So, I’d suggest constructing a spacetime diagram of the essential mechanism.

 

[Dale]

I am also not sure there is a Doppler shift argument accomplishing this.

There certainly is, but you have to start with the experimentally observed relativistic Doppler. Once you have that it is just a bit of algebra to get the dilation of a caesium clock.

 

[Orodruin]

There certainly is, but you have to start with the experimentally observed relativistic Doppler. Once you have that it is just a bit of algebra to get the dilation of a caesium clock.

But that is an additional observation on top of the invariance of the speed of light that the OP wants to start from.

 

[russ_watters]

First, you have to explain how a Cesium clock works?

Why does it matter how a cesium clock works? This interpretation of the question (which I believe to be correct) implies different clocks might be expected to behave differently under Relativity so each different mechanism requires a unique explanation and evidence of how it obeys Relativity. Isn't the whole point of Relativity to avoid that?

 

[PeroK]

Why does it matter how a cesium clock works? This interpretation of the question (which I believe to be correct) implies different clocks might be expected to behave differently under Relativity so each different mechanism requires a unique explanation and evidence of how it obeys Relativity. Isn't the whole point of Relativity to avoid that?

Those would be my thoughts on the subject. But, if the OP wants to explore the specific question, then understanding how a cesium clock works is the first step.

 

[Herman Trivilino]

A common way to introduce time dilation is to show the example of a "light clock" which bounces photons back/forth and ticks each time a photon passes a certain point. Wikipedia does it this way, for example. From such a clock, it's easy to see why the constancy of the speed of light would cause the clock to appear to tick slower by exactly the formula given in SR.

I think you're missing the point. The purpose of examing the light clock is to demonstrate that time dilates. You might be able to do the same with cesium clocks, but then you might ask the same about pendulum clocks, or any other clock. And note that not all clocks are man-made. We use Earth's rotation and its revolution as clocks.

 

[robphy]

From the video I posted, it seems to me that
[from the viewpoint of special relativity] the “cesium clock” is a fundamentally a clock ( a world line partitioned with a equal-intervals of proper-time) that was well-engineered to be self-correcting to approximate an ideal clock over the course of a long time. While the mechanism of the clever engineering between the ticks is difficult to model, the essence is the equal-intervals of the ticks of proper-time on its world line.

For a clock and its mechanism, it may be easier to analyze a mass on a spring, compared to a cesium clock.

---

update: Once one has equal intervals of proper time on an inertial worldline,
one can use methods as I describe in https://www.physicsforums.com/insights/relativity-on-rotated-graph-paper-a-graphical-motivation/ to motivate special relativity.​

---

• Is the goal to understand how the mechanism of such a clock in motion still works in detail? (As opposed to merely comparing it to an adjacent light clock and invoking the relativity-principle?)
• Or is the goal to motivate special relativity from the clock and its mechanism?
  The light-clock has an advantage that its features are in-tune with the radar method of making measurements (because it cleverly encodes them).
• Or something else?

 

[Vanadium 50]

I think we can agree that "I don;t know how a cesium clock works, but I want an intuitive explanation of how these workings change" would be a very odd question to ask, so you are seeing people concluding the actual question must be something else.

 

[vanhees71]

Those would be my thoughts on the subject. But, if the OP wants to explore the specific question, then understanding how a cesium clock works is the first step.

The Cesium clock's working (i.e. the hyperfine transition defining the secondin the SI) is described by QED, which is a relativistic theory, and thus describes time dilation in the way the Oorentz transformation does.

Seen from this fundamental perspective of the "new SI" time dilation indeed is a consequence of the relativistic Doppler effect for the Cs hyperfine-transition radiation.

 

[hutchphd]

TL;DR Summary: Wondering if there's an intuitive way to see why cesium clocks slow with velocity, analogous to the common example of a "light clock"

I was wondering if there's any way to "see" why a cesium clock (or energy level transitions in atoms) would also slow with movement, due to the constancy of c, analogous to the above clock.

While I agree with the light clock argument, it doesn’t seem like what the OP is after.

I agree that this is the request. To which I would give two answers:
1 A clock must agree with all clocks at that location as a matter of practicality. If the readings diverge, one of them cannot be a clock.
2 Understanding the workings of the Cesium atomic clock mechanisms requires Quantum Electrodynamics which obeys all the tenets of General Relativity. Therefore any detailed explanation of the clock must also.
There has to be consistency. If you want a more detailed treatment, I recall somewhere there is a comparison treatment of wristwatch (with a coil spring) done in detail. Perhaps someone (else) can remember where it is published.

 

[FactChecker]

I was wondering if there's any way to "see" why a cesium clock (or energy level transitions in atoms) would also slow with movement, due to the constancy of c, analogous to the above clock.

IMO, you are missing an important point. It is time, itself, that is changing. Anything that changes with time will show those effects with no other rhyme or reason. Many people hated that idea, but eventually the experimental proof was overwhelming and it was undeniable.
I think that if you are looking for non-SR reasons for the slowdown of particular things like Cesium clocks, biological functions, planetary orbits, lifespan of subatomic particles, etc., you will be forever frustrated.

ADDED: In fact, the opposite tends to be true. SR explains some things that nobody could find a non-SR explanation for (including the effects of GR): the strange orbit of Mercury, why electrical current creates magnetic effects, the amount of curvature of light around massive objects, etc.

 

[msumm21]

Thanks all. Some of you mentioned a spring mass system and I can see how time dilation for that would naturally follow from relativistic mass increase, analogous to the light clock using constant c.

There are many posts about time itself changes and therefore everything changes, understood that the theory says everything slows, but what I'm looking for is an alternate, maybe "more direct" explanation for the specific phenomenon. So I think have one now for the light clock and a clock based on a spring/mass system. The Cs clock is overkill I guess, what I'm really after is why spontaneous transitions between energy levels would slow (again, if such an explanation exists). Reading about spontaneous emission now.

There certainly is, but you have to start with the experimentally observed relativistic Doppler. Once you have that it is just a bit of algebra to get the dilation of a caesium clock.

Could you elaborate on this, or do you have a reference? I'm not seeing it at first.

 

[PeroK]

what I'm really after is why spontaneous transitions between energy levels would slow (again, if such an explanation exists).

You may have fundamentally misunderstood the special theory of relativity (SR). What SR does not say is that "when something moves near the speed of light, its mechanical composition and function physically change". Which seems to be what you are assuming.

SR says, in fact, something that is almost the opposite of this: that the laws of physics are the same in all inertial reference frames. That means that all (inertial) motion is relative and there is no such thing as "moving near the speed of light". Only moving near the speed of light relative to something else. An object moving near the speed of light relative to you remains, in its own rest frame, physically and mechanically unchanged. In the same way that you, moving at near the speed of light relative to it, are not physically changed by that relative motion.

SR is, in fact, a theory of spacetime (space and time). The point already made that SR is about "time itself" is very relevant here. You should review your understanding of what SR says at this point, before your confusion becomes deeper.

Note also that "relativistic mass" is a hangover from the early days of SR - although it is still inexplicably fervently promoted in popular science presentations of SR.

I can see how time dilation for that would naturally follow from relativistic mass increase, a

In fact, we have an Insight on why relativistic mass is not a useful concept in SR:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

Our advice is to forget about relativistic mass, generally. When I learned SR (from an undergraduate textbook) relativistic mass was not mentioned - except as a historical footnote.

 

[Herman Trivilino]

Thanks all. Some of you mentioned a spring mass system and I can see how time dilation for that would naturally follow from relativistic mass increase, analogous to the light clock using constant c.

But there is no mass increase. (Relativistic mass is an outdated term, a relic of the way physicists thought about relativity in the early 1900's).

And anyway, the period of such an oscillator is not proportional to $m$, it's proportional to $\sqrt{m}$. So what you think naturally follows doesn't, when you work out the details.

The reason clocks run slow is because time dilates.

You are thinking time dilates because clocks run slow. Your logic is inverted. Clocks are just measuring devices.

 

[Sagittarius A-Star]

Some of you mentioned a spring mass system and I can see how time dilation for that would naturally follow from relativistic mass increase, analogous to the light clock using constant c.

The terminology "relativistic mass" is out-dated. A synonym is $E/c^2$.

So I think have one now for the light clock and a clock based on a spring/mass system.

Here is a calculation of a spring mass system:
https://www.mathpages.com/home/kmath068/kmath068.htm

Here is a calculation of a balance wheel clock:
https://www.physicsforums.com/threa...ausing-it-to-tick-slower.1011025/post-6584697

 

[msumm21]

But there is no mass increase. (Relativistic mass is an outdated term, a relic of the way physicists thought about relativity in the early 1900's).

And anyway, the period of such an oscillator is not proportional to $m$, it's proportional to $\sqrt{m}$. So what you think naturally follows doesn't, when you work out the details.

For the 4-force in SR you need a 1/gamma in the spring constant, the result is gamma^2 under the sqrt, hence the gamma overall, matching the time dilation equation.

 

[FactChecker]

SR says, in fact, something that is almost the opposite of this: that the laws of physics are the same in all inertial reference frames. That means that all (inertial) motion is relative and there is no such thing as "moving near the speed of light". Only moving near the speed of light relative to something else. An object moving near the speed of light relative to you remains, in its own rest frame, physically and mechanically unchanged. In the same way that you, moving at near the speed of light relative to it, are not physically changed by that relative motion.

Excellent. And I would add here that it is only in the other reference frame that the "moving" reference frame's time seems slow. Looking for some non-SR explanation for the slow down would make the symmetric nature of the slow down (each inertial reference frame thinks that the other is running slow) very hard to accept.

 

[Herman Trivilino]

For the 4-force in SR you need a 1/gamma in the spring constant, the result is gamma^2 under the sqrt, hence the gamma overall, matching the time dilation equation.

But I was responding to your claim about mass. You didn't mention the spring constant in that claim.

 

[hutchphd]

The reason clocks run slow is because time dilates.

You are thinking time dilates because clocks run slow. Your logic is inverted. Clocks are just measuring devices.

I dislike the notion of inference. The parameter t in our natural philosophy is what a clock indicates. They are equivalent by our design. Just a quibble that this seems facile to me.

 

[hutchphd]

For the 4-force in SR you need a 1/gamma in the spring constant, the result is gamma^2 under the sqrt, hence the gamma overall, matching the time dilation equation.

So this satisfies you as an explanation for the spring clock? If so, then we are successful (?)

 

[FactChecker]

The reason that one Inertial Reference Frame (IRF) can see another IRF's clocks running slow is surprisingly simple. Suppose that I am observing a moving clock for one of its seconds and the second starts exactly when it passes me. Then it is very far away by the time that its second ends. My reference frame AT THAT SECOND LOCATION records the time. Clearly the elapsed time in my IRF is the difference of the two clocks from the beginning to the end of the moving clock's second. So it all depends on how the clocks in my IRF have been synchronized. That is worth some serious thought.
Question: Why should we think that the clocks in my IRF are synchronized differently from clocks in the moving IRF?
Answer: Because experiments show that both IRFs measure the speed of the same light flash as c.

The slow speed of a moving clock is really about how distant clocks in an IRF are synchronized. That is called the "relativity of simultaneity".

 

[Dale]

Could you elaborate on this, or do you have a reference? I'm not seeing it at first.

Sure, it is just a little algebra. We start with a receiver at rest and an emitter moving away at $v$. The proper frequency emitted is $f_e$ and the received frequency is given by the relativistic Doppler shift $$f_r=f_e \sqrt{\frac{1-v/c}{1+v/c}}$$ Now, we want to obtain the emitted frequency, not in the proper frame, but in the frame of the receiver, which we will denote by $f$. To do that we note that the period at the receiver is $$T_r=\frac{1}{f_r}$$ and in each period the emitter moves a distance $T_r v$ which increases the delay between emitting and receiving by $T_r v/c$. The period at the emitter (in the frame of the receiver) $T$ is therefore $$T=T_r - T_r v/c =(1-v/c)T_r =\frac{1-v/c}{f_e}\sqrt{\frac{1+v/c}{1-v/c}}$$ Finally, we take $f=1/T$ and simplify to get $$f=\frac{f_e}{\sqrt{1-v^2/c^2}}=\gamma f_e$$ which is the usual time dilation factor.

 

[hutchphd]

TL;DR Summary: Wondering if there's an intuitive way to see why cesium clocks slow with velocity, analogous to the common example of a "light clock"

I was wondering if there's any way to "see" why a cesium clock (or energy level transitions in atoms) would also slow with movement, due to the constancy of c, analogous to the above clock.

Please let us know when sufficient light has been directed to clarify your vision. Otherwise no one will know!

 

[msumm21]

lease let us know when sufficient light has been directed to clarify your vision. Otherwise no one will know!

My question boils down to seeing how time dilation manifests in spontaneous transitions between energy levels (i.e. why looking at spontaneous transitions in a moving frame would appear to slow). Some good surrounding info in above posts, the link below might give the answer I'm looking for around page 9, but I've not yet had time to understand it yet.
https://web2.ph.utexas.edu/~vadim/Classes/2022f/FGR.pdf

 

[Ibix]

My question boils down to seeing how time dilation manifests in spontaneous transitions between energy levels (i.e. why looking at spontaneous transitions in a moving frame would appear to slow).

Intuitively, because all the governing theories are relativistic.

link below might give the answer I'm looking for around page 9,

I'm not sure I'd classify nine pages of fairly high density maths as "intuition".

 

[PeroK]

My question boils down to seeing how time dilation manifests in spontaneous transitions between energy levels (i.e. why looking at spontaneous transitions in a moving frame would appear to slow). Some good surrounding info in above posts, the link below might give the answer I'm looking for around page 9, but I've not yet had time to understand it yet.
https://web2.ph.utexas.edu/~vadim/Classes/2022f/FGR.pdf

Time dilation is an elementary consequence of SR, usually derived as part of an introduction to the subject. The material you reference here is advanced particle physics, which has SR (and especially four-vectors) as a key building block.

I can't reconcile your elementary question with this advanced material.

 

[PeterDonis]

My question boils down to seeing how time dilation manifests in spontaneous transitions between energy levels (i.e. why looking at spontaneous transitions in a moving frame would appear to slow).

You aren't going to get an answer to this question by delving into the minutiae of such transitions.

You appear to be thinking of "time dilation" as something that has to be done to the moving object to make it slow down, and has to be done separately to each separate kind of object. That's wrong.

Time dilation is an appearance (and a calculated one at that; what you actually see is the relativistic Doppler shift) due to the way the geometry of spacetime works. It's basically the spacetime equivalent of seeing the angle that an object subtends on your field of view change as the object's orientation relative to you changes in ordinary Euclidean geometry. You haven't done anything to the object itself; you've just changed how you're looking at it.

And since this effect is a geometric property of spacetime, it affects everything and gives the same explanation for how it affects everything; you don't have to concoct a new, separate explanation every time you work with a new type of object or phenomenon.

 

[Dale]

My personal preferred approach:

Start with the spacetime interval in an inertial frame $$ds^2=-c^2 d\tau^2=-c^2 dt^2+dx^2+dy^2+dz^2$$

Then for any coordinate system with one timelike coordinate, $t$, define the time dilation as $$\frac{1}{\gamma} = \frac{d\tau}{dt}$$ which is easy to calculate from the spacetime interval. And since caesium clocks measure proper time $\tau$ the result is obtained.

I think that this is the most intuitive approach since the spacetime interval is so intuitive and it is almost directly obtained from that.

 

[Herman Trivilino]

My question boils down to seeing how time dilation manifests in spontaneous transitions between energy levels (i.e. why looking at spontaneous transitions in a moving frame would appear to slow).

Because time dilates. Therefore all processes must run slow for a moving observer.

 

[msumm21]

Let me try to word this another way.

I'm trying to see how, when viewing a process (transition between energy levels) from a relatively moving frame, the process appears to occur slower. I know you could answer "because time dilates," but I'm looking for a slightly deeper explanation. Again, there seems to be a simple explanation in the case of the "light clock" and frequency of a spring/mass system discussed earlier (by doing the math from a relatively moving frame). So something analogous to that, but for the transition between energy levels.

I'm not sure I'd classify nine pages of fairly high density maths as "intuition".

Oh yeah, that's beyond me at this point, but the concept I saw from that link was that the rate may be proportional to the difference in energy levels, so if the difference in energy levels grows with velocity, then moving things appear to transition between energy levels more slowly. Not certain that is the answer, didn't get to look much yet, but perhaps something simple like that is the answer I'm looking for.

 

[PeterDonis]

I'm trying to see how, when viewing a process (transition between energy levels) from a relatively moving frame, the process appears to occur slower.

The process in question does not involve anything oscillating, so it's not the same kind of process as a light clock or a mass on a spring.

The light emitted during the transition has a particular frequency, which could be viewed as a frequency of oscillation, but then you're just dealing with the Doppler shift of light and correcting for light travel time to compute the emitted frequency as calculated in the frame in which the atom emitting the light is moving.

In other words, the light emitted during a transition between atomic energy levels does not tell you anything about "how fast the transition happened". Atomic clocks that use such transitions are not timing how fast the transitions happen; they are using the frequency of the emitted light.

 

[robphy]

I'm trying to see how, when viewing a process (transition between energy levels) from a relatively moving frame, the process appears to occur slower.

As I mentioned before in #8 , I think you need to articulate a mechanism.
Otherwise, in my opinion, it's a lot of hand-wavy words, possibly with some equations applied...
which probably does not provide an explanation comparable to one explaining a light-clock.

 

[PeterDonis]

As I mentioned before in #8 , I think you need to articulate a mechanism.

First I think the OP needs to be clear about exactly what "process" we are talking about. See my post #41.

 

[hutchphd]

but I'm looking for a slightly deeper explanation

I would argue that you are looking for a shallower explanation than you have been provided. If you make a dress from red cloth, you will have made a red dress......the stitching does not matter.

 

[Dale]

I'm looking for a slightly deeper explanation

See

Start with the spacetime interval in an inertial frame $$ds^2=-c^2 d\tau^2=-c^2 dt^2+dx^2+dy^2+dz^2$$

Then for any coordinate system with one timelike coordinate, $t$, define the time dilation as $$\frac{1}{\gamma} = \frac{d\tau}{dt}$$ which is easy to calculate from the spacetime interval. And since caesium clocks measure proper time $\tau$ the result is obtained.

I can’t see how any answer based on something other than the metric can be deeper. The metric is fundamental.

 

[FactChecker]

Let me try to word this another way.

I'm trying to see how, when viewing a process (transition between energy levels) from a relatively moving frame, the process appears to occur slower. I know you could answer "because time dilates," but I'm looking for a slightly deeper explanation.

I disagree that you are looking for a "deeper" explanation. IMO, you are looking for a superficial, specific, explanation for that one consequence of time dilation. You will not find it. Time dilation is the one and only cause. It would be very strange indeed if the process of transition between energy levels did not slow down when time, itself, dilates. And remember that this effect is only when one IRF is observing the process in another moving IRF. Your alternative, "deeper" explanation would have to be one that disappears within the IRF of the moving cesium clock.
Time dilation is the most profound and "deep" explanation of all, and it affects everything, large or small, in a beautiful way.

 

[PeroK]

Let me try to word this another way.

I'm trying to see how, when viewing a process (transition between energy levels) from a relatively moving frame, the process appears to occur slower. I know you could answer "because time dilates,"

That is the deep explanation. It's call "time dilation"; it's not "clock mechanical malfunction".

but I'm looking for a slightly deeper explanation.

What you are looking for is a superficial one. Because you haven't understood that SR is about spacetime. It's not a theory of clockmaking.

 

[vanhees71]

The process in question does not involve anything oscillating, so it's not the same kind of process as a light clock or a mass on a spring.

The light emitted during the transition has a particular frequency, which could be viewed as a frequency of oscillation, but then you're just dealing with the Doppler shift of light and correcting for light travel time to compute the emitted frequency as calculated in the frame in which the atom emitting the light is moving.

In other words, the light emitted during a transition between atomic energy levels does not tell you anything about "how fast the transition happened". Atomic clocks that use such transitions are not timing how fast the transitions happen; they are using the frequency of the emitted light.

Indeed, and by definition the corresponding frequencies of the emitted light, determined by the difference in the energy levels of the transition, is defined in the rest frame of the atom. That's how, e.g., the Cs standard is used to define the second in the SI.

If now the atom moves relative to a detector (in an inertial frame), the radiation frequency is Doppler shifted, and this immediately leads to "time dilation", as already discussed above. For details on the Doppler effect for light (in vacuum) see

Sect. 4.3 in

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

 

[Ibix]

I'm trying to see how, when viewing a process (transition between energy levels) from a relatively moving frame, the process appears to occur slower.

As others have pointed out, it's not actually the transition rate that you are interested in (although that will decrease by the same factor and you could use it as a very poor clock). The oscillator in the clock is the EM radiation emitted by the transition.

Now here's the fun thing. In any frame except the rest frame of the atom, the emitted frequency depends on the angle at which it's emitted, presumably because the electromagnetic field of the nucleus is not spherically symmetric in those frames. Thus it's not at all clear to me that there is "a" change to the energy levels of the atom. Without going into the maths myself, I would say it isn't even certain that "energy level" is a useful concept in other frames (although others who actually understand relativistic quantum field theory may correct me on that).

It's perfectly fine to want to understand the start-to-finish process of any mechanism. However, that doesn't seem to be what you want. You seem to want a soundbite on "how it works", and I'm not really sure there is such a thing. The whole reason the lightclock gets used in relativity is that it's pretty much the only one that is trivial to analyse. Even the mass on the spring gets messy because the transform of the three-force and three-momentum is not simple.

So the point we're all trying to make is that the description of a Cs clock, even at the level of @Dale's Doppler explanation (which is perfectly fine, but doesn't cover the photon emission at all), is a lot more complex in the moving frame. There probably isn't a soundbite summary of "why it ticks slowly", whereas with the light clock there is nothing more complex than a bit of high-school level geometry, and you can derive the full Lorentz transforms with just two of them.

Understanding that "all clocks must dilate the same" follows directly from the principle of relativity is a much more valuable insight than anything short of a complete QED description of the caesium clock.

 

[Orodruin]

What about this? Create a clock by ensuring that exactly 9,192,631,770/299,792,458 wavelengths of the radiation emitted from the hyperfine Cs transition fit in between two mirrors. Have the emitted light bounce between the two mirrors and count one tick for every 149,896,229 round trips. Call this time ”one second”.

You now have a light clock based on the Cs standard and as a bonus the light clock can be used as a standard ruler of 1 m.