B Intuition for time dilation in a cesium clock

[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.

 

[Sagittarius A-Star]

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 mentioned, this is the Doppler effect on the energy (or frequency) of an emitted light pulse.

Components of the four-momentum of the light-pulse "photon" in the receiver frame ($\vec{n}$ is a unit vector in the travel direction of the light-pulse):
$\mathbf P = \begin{pmatrix} P_t \\ P_x \\ P_y \\ P_z \end{pmatrix} = {E \over c^2}c \begin{pmatrix} 1 \\ \vec{n} \end{pmatrix} = {h\nu \over c} \begin{pmatrix} 1 \\ \vec{n} \end{pmatrix}$

Components of the four-frequency in the receiver frame:
$\mathbf N = \begin{pmatrix} N_t \\ N_x \\ N_y \\ N_z \end{pmatrix} = {c \over h}\mathbf P = \nu \begin{pmatrix} 1 \\ \vec{n} \end{pmatrix}$

Do a Lorentz-transformation of the received time-component of the four-frequency into the (primed) cesium atom-frame:
$N'_t = \gamma (N_t - \beta N_x)$$$\nu_0 =\nu' = \gamma ( \nu - \beta \nu \cos {\varphi_R}) = \nu \gamma (1 - \frac{v}{c} \cos {\varphi_R})$$If you set the angle between the direction of movement of the cesium atom and the dirction of the light pulse in the receiver frame $\varphi_R$ to $90°$, then you get the transverse Doppler effect in the receiver frame.
$$\nu = \nu_0 / \gamma$$.

 

[Herman Trivilino]

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.

Why?

 

[meopemuk]

Somewhat related to this discussion. Decay law of a moving unstable particle was calculated in a number of articles. Few examples:

E. V. Stefanovich, Quantum effects in relativistic decays. Int. J. Theor. Phys. 35 (1996), 2539​

M. Shirokov, Decay law of moving unstable particle. Int. J. Theor. Phys. 43 (2004), 1541​

K. Urbanowski, On the Velocity of Moving Relativistic Unstable Quantum Systems. Adv. High Energy Phys. 2015 (2015), 461987​

The conclusion was that decay becomes slower, but the slowdown is slightly different from the gamma factor predicted by special relativity.
Eugene.

 

[Dale]

Somewhat related to this discussion. Decay law of a moving unstable particle was calculated in a number of articles. Few examples:

E. V. Stefanovich, Quantum effects in relativistic decays. Int. J. Theor. Phys. 35 (1996), 2539​

M. Shirokov, Decay law of moving unstable particle. Int. J. Theor. Phys. 43 (2004), 1541​

K. Urbanowski, On the Velocity of Moving Relativistic Unstable Quantum Systems. Adv. High Energy Phys. 2015 (2015), 461987​

The conclusion was that decay becomes slower, but the slowdown is slightly different from the gamma factor predicted by special relativity.
Eugene.

If they had solid evidence for Lorentz violations this would be published in journals with >90th percentile articles. Not the 18th percentile.

 

[Vanadium 50]

The conclusion was that decay becomes slower, but the slowdown is slightly different from the gamma factor predicted by special relativity.

False.

And shame on you for saying so.

The Shirokov paper, at least in part, discusses non-exponential decays. The inputs equations are wrong, as it happens (but maybe the author is just sloppy) but in any event, of course you will not have a single exponential lifetime, now altered by γ if you didn't start with one.

This is an irrelevant side track involving a low quality paper. If you didn't recoghnize that, you shouldn't have brought it up. If you did, and posted it anyway, you really shouldn't have brought it up.

 

[vanhees71]

One should, however, be aware that indeed the exponential decay low is an approximation (Wigner-Weisskopf approximation of time-dependent perturbation theory).

 

[Vanadium 50]

Sure. And exponential decay assumes independence of events, which is usually* true in nuclear decays but not always so in atomic/chemical transitions. But the key point is that you won't end up with an exponential after the boost if you didn't have one before.

* The exponential is the Fourier transform of the Lorentzian line shape. The a0(980) and f0(980)'s line shape is not Lorentizan, as the lines cross K-Kbar threshold, opening up a new channel and widening the right half of the peak. While both decays are too short to measure lifetimes directly, one would not expect a single exponential. One would, however, expect to see this unusual shape boosted by gamma in another frame.

 

[hutchphd]

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.

As I recall, the frequency of the Cesium clock transition is a hyperfine splitting. This involves nuclear spin magnetic moments coupling to electronic magnetic moments as well as the nuclear electric quadupole moment coupling to the electric field gradient from the electron(s). If you wish to solve the energy level splitting in other than the CM frame, (i.e one moviing at high v relative to it) then good luck. It certainly will not bring forth clarity.
Perhaps lost in this discussion what a fabulous learning aid is the concept of the light clock. As I understand it this also can be attributed to Einstein, and we blithely take it as obvious. I think it a wonderful thought.
As a practical matter, pointed to by @Orodruin, the workings of the Cesium clock are essentially moot, because here in the 21st century it literally defines "time". So the question devolves to whether the light clock is a proper clock, and I contend we all agree that physics requires it.

 

[vanhees71]

Sure. And exponential decay assumes independence of events, which is usually* true in nuclear decays but not always so in atomic/chemical transitions. But the key point is that you won't end up with an exponential after the boost if you didn't have one before.

* The exponential is the Fourier transform of the Lorentzian line shape. The a0(980) and f0(980)'s line shape is not Lorentizan, as the lines cross K-Kbar threshold, opening up a new channel and widening the right half of the peak. While both decays are too short to measure lifetimes directly, one would not expect a single exponential. One would, however, expect to see this unusual shape boosted by gamma in another frame.

Exactly, and the precise line shape can never be Lorentzian, i.e., you cannot have a constant width. It's only a good approximation for the spectral function if the width is pretty small (compared to the (pole) mass).

Of course, all this has nothing to do with kinematical effects of special relativity. The relevant transition matrix elements are Poincare covariant anyway. Of course, the lifetime (the inverse width of the spectral function), is time-dilated in a frame, where the particle moves. What's listed in the particle data booklet is of course the "proper lifetime", i.e., the mean life time of the unstable particle in its rest frame.