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In 2019, researches in Japan (M. Takamoto, H. Katori et al.) carried out experiments comparing two transportable ${}^{87}\text{Sr}$ clocks, one ("clock $\mathsf 1$") placed at the ground floor and the other ("clock $\mathsf 2$") placed in the observatory floor of the Tokyo Skytree broadcasting tower, nominally $450~{\rm m}$ above the ground floor. There's an article filed, with a public supplement and with a public article copy available.

The article introduces the symbols $\nu_{\mathsf 1}$ and $\nu_{\mathsf 2}$ to denote "the clock frequency at location $\mathsf 1$" and "the clock frequency at location $\mathsf 2$", respectively; and also referred to as "the clock laser frequencies".

Ramsey spectra (Excitation probability over Detuning) measured in the observatory floor and the ground floor with fit curves are shown (Fig. 2 b,c) with the maximum peak for clock $\mathsf 2$ at the reported Detuning value $\approx 21.18~{\rm Hz}$ and the maximum peak for clock $\mathsf 1$ apparently at Detuning value $0~{\rm Hz}$. (For reference, let's denote these two quantities as $f^{RSmax}_{\mathsf 2 \leftarrow\mathsf 1}$ and $f^{RSmax}_{\mathsf 1 \leftarrow\mathsf 1}$, resp.)

This is surely a quite significant finding, considering that

  • the full width at half maximum of the maximum peak of either Ramsey spectrum is apparently also in the order of $\approx 20~{\rm Hz}$,

  • the CIPM recommended frequency of the relevant unperturbed ${}^1 S_0 - {}^3 P_0$ optical transition of ${}^{87}\text{Sr}$ is given with accuracy better than $1~{\rm Hz}$, as $f({}^{87}\text{Sr}) = 429 \, \, 228 \, \, 004 \, \, 229 \, \, 873.7 (0.5)~{\rm Hz}$, and

  • the experiments involve corrections of about $+2.6~{\rm Hz}$, and systematic uncertainties much less than $\pm 0.1~{\rm Hz}$; as listed in Table S1 of the supplement.

Now, from the outset, in its abstract, the article claims that "A clock at a higher altitude ticks faster than one at a lower altitude, in accordance with Einstein’s theory of general relativity";
and it goes on to refer to "frequency shift $\Delta \nu = \nu_{\mathsf 2} - \nu_{\mathsf 1} \approx 21.18~{\rm Hz}$".
Therefore

My question:

Is it correct to conclude that clock $\mathsf 2$, while located at the observatory floor of the Tokyo Skytree broadcasting tower, ticked significantly faster than clock $\mathsf 1$, while located at the ground floor; in particular that (rounding down to integer ${\rm Hz}$)

  • the clock frequency at location $\mathsf 2$ had the value $\nu_{\mathsf 2} \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 897~{\rm Hz}$ while

  • the clock frequency at location $\mathsf 1$ had the value $\nu_{\mathsf 1} \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 876~{\rm Hz}$ ?

(Or, to consider at least one alternative explicitly:
Is it instead correct to conclude that both clocks ticked as good as equally fast, both, at their respective locations, at approximately $\nu_2 \approx \nu_1 \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 876~{\rm Hz}$; but the reported "frequency shift" value is instead attributable to the (suitably signed) difference between the tick frequency of a clock and the frequency of a receiver in response to ticks of that clock: $$-(\nu_{\mathsf 2} - \nu^{\text{rec}}_{\mathsf 1 \, \leftarrow \, \mathsf 2}) \approx (\nu_{\mathsf 1} - \nu^{\text{rec}}_{\mathsf 2 \, \leftarrow \, \mathsf 1}) \approx 21.18~{\rm Hz}$$ ?)

user12262
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    That's how much time dilation GR predicts, using the standard Schwarzschild metric equation. I don't understand why you say that 21 extra ticks per 429 trillion ticks is significantly faster. – PM 2Ring Jun 30 '22 at 21:14
  • A convenient form of the gravitational time dilation equation is $$\Delta t_0 = \Delta t_{\infty} \sqrt{1 - r_s/r}$$ where $t_{\infty}$ is time measured by the observer at infinity, $t_0$ is time measured by the observer at distance $r$ from the centre of the body, and $r_s$ is the Schwarzschild radius of the body, which is ~8.870056 mm for the Earth. – PM 2Ring Jun 30 '22 at 21:20
  • @PM 2Ring: "why 21 extra ticks per 429 trillion ticks is significantly faster." -- The OP lists four reasons that in the given context the measured value $21~{\rm Hz}$ is significantly different from $0~{\rm Hz}$: $$ \mathsf 1. \qquad 21 > 10 $$ where Fig. 2 (b) and (c) show the peak maximum and the nearby minimum of the Ramsey spectra within $10~{\rm Hz}$. $$ \mathsf 2. \qquad 21 \gg 0.5 $$($0.5~{\rm Hz}$ being the accuracy of the nominal transition freq.) $$ \mathsf 3. \qquad 21 > 2.6 $$, (comparison to the "corrections")$$ \mathsf 2. \qquad 21 \gg 0.1 $$ (comparison to syst. uncert.) – user12262 Jun 30 '22 at 21:48
  • Seems like you are asking if the 21 Hz could be the result of unaccounted experimental error in their setup, which is unknowable with the information we have. If you use the simple approximate formula of the frequency ratio (or time dilation ratio) being $1+gh/c^2$ I believe you get the ~21 Hz value – RC_23 Jun 30 '22 at 22:02
  • @PM 2Ring: p.s. Please note carefully what the OP is asking. It is not put into question that $$\Delta \nu \approx 21~{\rm Hz}$$. But I'm asking for short: Is $\Delta \nu$ defined as $\nu_{\mathsf 2} - \nu_{\mathsf 1}$, as Katori et al. suggest? Or what else?. (Also to consider: How does Katori et al.'s presentation relate to that in WP:Redshift ?) – user12262 Jun 30 '22 at 22:05
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    This may help - Why can't I do this to get infinite energy? – mmesser314 Jun 30 '22 at 23:52
  • @mmesser314: "This may help - {pse/q/178417}" -- It just might, thanks. So: What would therefore be the conclusion? - Were Takamoto, Katori et al. correct to express$$\Delta\nu=\nu_{\mathsf 2}-\nu_{\mathsf 1}$$with $\nu_{\mathsf 2}$ being the frequency of clock $\mathsf 2$ "at its location" in the Tokyo Skytree observation deck, $\nu_{\mathsf 1}$ the frequency of clock $\mathsf 1$ "at its location" at the ground floor, and $\Delta \nu$ measured at $\approx 21~{\rm Hz}$? Or were Takamoto, Katori et al. wrong? Please submit your answer! – user12262 Jul 01 '22 at 05:42
  • I'm still not quite sure what your exact question is. By definition, $\Delta\nu$ is the difference in the clock frequencies, so yes $$\Delta\nu=\nu_{\mathsf 2}-\nu_{\mathsf 1}$$ is correct. (And their value of 21 Hz agrees with my rough calculation). I had a brief look at the article, and I don't claim to fully understand it, but from what I gather, the 2 clocks exchanged frequency data with each other via 4 laser signals. Bear in mind that those laser signals get Doppler shifted travelling up & down the tower, as explained by mmesser314 in their linked answer. – PM 2Ring Jul 01 '22 at 10:45
  • Here are my calculations of g, time dilation, and nu, using Sage. https://sagecell.sagemath.org/?z=eJxdT7tuwzAM3PUVhLtIDuIH_ZKGrP4AtWNRILUFR4hrp4oy5O8r1m6ddpGOd-Qd-QQv8_k-w_P57p0x4O2Hgd6OR2_nib1bf4UDyIzpNvzaHMfWmrHnF2e6A6mCsS4oqFSjsKwkczSgWy4T2WRZVZt9IZheuLpoZIKVCcxpYcoKk0rVgk23fGVQIcosKxGVbIqkCQkXZyfPoyESbOBOhMY8RYghZMXAO0jBiTdc2wauhdgw7OCU4n9GbK6v05-bKcP3S8r103mew56CKIP5nrYkXVOBP8W35WoYejaIjznTLaJDaYrOjYEc0vDma09gxS_EBxhWIO0LHRF1hQ==&lang=sage – PM 2Ring Jul 01 '22 at 12:25
  • @PM 2Ring: "still not quite" -- Added notation may help: $f^{RSmax}{\mathsf 2 \leftarrow\mathsf 1}$ for the freq. shown in Fig 2b and $f^{RSmax}{\mathsf 1 \leftarrow\mathsf 1}$ wrt. Fig. 2c. Clearly: $$f^{RSmax}{\mathsf 2 \leftarrow\mathsf 1}- f^{RSmax}{\mathsf 1 \leftarrow\mathsf 1}=21.18~{\rm Hz}-0~{\rm Hz}.$$ "By def., $\Delta \nu=\nu{\mathsf 2}-\nu_{\mathsf 1}$"_ -- If you so insist, I can oblige by rephrasing: Is $$\nu_{\mathsf 2}-\nu_{\mathsf 1}=f^{RSmax}{\mathsf 2 \leftarrow\mathsf 1}- f^{RSmax}{\mathsf 1 \leftarrow\mathsf 1},$$ as Takamoto, Katori et al. suggest ?? – user12262 Jul 01 '22 at 13:28
  • @PM 2Ring: "Here are my calculations [...]" -- No big deal. The point is: Does $$\nu_{\mathsf 2}-\nu_{\mathsf 1}= 21.18~{\rm Hz},$$ even $$\nu_{\mathsf 2}\approx 429,, 228,, 004,, 229,, 897~{\rm Hz}$$ $$\nu_{\mathsf 2}\approx 429,, 228,, 004,, 229,, 876~{\rm Hz}$$ represent "shift[ing] travelling up & down the tower" (you pointed out above)?? Or does $$\nu_{\mathsf 1}- \nu^{\text{rec}}{\mathsf 2 , \leftarrow , \mathsf 1}\approx 21.18~{\rm Hz}$$ with $$\nu{\mathsf 2}-\nu_{\mathsf 1}\approx 0~{\rm Hz}$$ represent "shift[ing] travelling up & down the tower"? – user12262 Jul 01 '22 at 17:44

1 Answers1

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Is it correct to conclude that clock , while located at the observatory floor of the Tokyo Skytree broadcasting tower, ticked significantly faster than clock , while located at the ground floor

Is it instead correct to conclude that both clocks ticked as good as equally fast, both, at their respective locations … but the reported "frequency shift" value is instead attributable to the (suitably signed) difference between the tick frequency of a clock and the frequency of a receiver in response to ticks of that clock

The distinction you are drawing is a distinction without a difference. It is only a matter of your arbitrary choice of reference frame. The first statement would correspond to a non-inertial reference frame where both clocks are at rest. The second would correspond to an inertial reference frame where both clocks are accelerating.

The choice between the two descriptions is completely arbitrary and makes no real difference. The authors apparently were primarily using the first description, but if you prefer the second then you are free to re-cast their results in such terms. Neither is inherently right or wrong. Either is acceptable and is justified both experimentally and theoretically.

Dale
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  • Dale: "[...] distinction without difference [...] matter of choice of ref. frame." -- Mark an event in which (${}^{87}\text{Sr}$ atoms of) clock $\mathsf 1$ took part: $\varepsilon_{\mathsf 1}^{mark}$. Right away let it count, say, exactly $10^{14}$ ticks; ending in event $\varepsilon_{\mathsf 1}^{done}$. Let $\gamma$ be clock $\mathsf 1$'s worldline between these two events. Do you mean that the value $$\frac{10^{14}}{\int_{\gamma} {\rm d}s}$$ is a matter of choice ? Do you mean that Takamoto, Katori et al.'s $\nu_{\mathsf 1}$ denotes something else ?? Further, on to clock $\mathsf 2$ ... – user12262 Jul 03 '22 at 18:53
  • @user12262 that quantity is invariant. Note, your question was not about this quantity nor even about any other quantity. Your question was about the interpretation of a quantity. The two interpretations are equally valid and simply a matter of the choice of reference frame. Similarly, the quantity in your comment could also be interpreted in multiple ways. E.g. as an accurate measure of time in one frame or a slow measure of time in another frame – Dale Jul 03 '22 at 19:10
  • Dale: "[...] is invariant." -- So we agree that $\nu_{\mathsf 1}$ and $\nu_{\mathsf 2}$ are invariant quantities, defined pretty much as just sketched, matching Takamoto, Katori et al.'s description as "the clock frequency at location $\mathsf 1$, or $\mathsf 2$, resp.", right? _"Your question was about the interpretation [...]" -- My question is whether Takamoto, Katori et al. measured $$\nu_{\mathsf 2} - \nu_{\mathsf 1} \approx 21~{\rm Hz} \ne 0~{\rm Hz},$$ or not; specificly whether instead $$\nu_{\mathsf 2} - \nu_{\mathsf 1} \approx 0~{\rm Hz}.$$ (I might edit my OP) [contd.] – user12262 Jul 03 '22 at 19:55
  • Dale: "the quantity in your comment could also be interpreted in multiple ways. [...]" -- Let's denote as $\varphi$ the worldline of clock $\mathsf 2$ from having taken part in event $\varepsilon_{\mathsf 2}^{mark}$ until having taken part in event $\varepsilon_{\mathsf 2}^{done}$, having counted exactly $10^{14}$ of its ticks. Is the value $$\frac{10^{14}}{\int_{\varphi} {\rm d}s} - \frac{10^{14}}{\int_{\gamma} {\rm d}s}$$ a matter of interpretation ?? Hardly! Due to Takamoto, Katori et al.'s efforts, it's even as good constant (during their experiment). But is it subject to mistakes? – user12262 Jul 03 '22 at 19:58
  • Your question wasn’t about the value itself. It was about how to describe the meaning of the quantity in words. That is the interpretation. Either set of words are a valid description of the quantity in the question. – Dale Jul 03 '22 at 20:35
  • I remind you that comments are not intended for discussion. If you wish to have a discussion then you should post on a discussion-oriented forum like physicsforums.com – Dale Jul 03 '22 at 20:39
  • Dale: "Your q. wasn’t about the value itself." -- It's not meant to doubt the reported value of $\approx 21~{\rm Hz}$, correct. Instead, it's questioning whether the quantity being measured as having had this rep.d value is defined as $\nu_{\mathsf 2}-\nu_{\mathsf 1}$. "It was about how to describe the meaning of the quantity in words." -- At most, in some sense, to ask: Which quantity has been measured instead?. A wrong declaration of quantity being measured is: wrong. "Either set of words [...]" -- Pls. edit your answer explicitly quoting "the first statement" and "the second". – user12262 Jul 03 '22 at 21:13
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    @user12262 I have edited the answer as requested. The point is that neither description is “a wrong declaration of quantity being measured”. Both are valid and the choice between them is arbitrary. – Dale Jul 03 '22 at 21:35
  • Dale: Thanks for your edits (I honestly wasn't quite sure). Now: I still disagree, and I see a conflict to your own description of $\nu_{\mathsf 1}$ and $\nu_{\mathsf 2}$ as invariants. (IMHO, the option remains that Takamoto, Katori et al. perhaps didn't scrupulously think and write of $\nu_{\mathsf 2}$ as "frequency of clock $\mathsf 2$ at its location", but perhaps -- I shudder to type -- meant to imply "as we receive it at ground level", or sth. to this effect.) p.s. https://physics.stackexchange.com/questions/318804/what-exactly-is-meant-by-clock-rate-in-the-theory-of-relativity – user12262 Jul 03 '22 at 21:41