B Intuition for time dilation in a cesium clock

[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.