# Did Chou, Hume, Rosenband, Wineland (2010) account for redshift when analysing their experiment with clocks at varying heights?

In widely reported experiments by C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland ("Optical Clocks and Relativity" (2010)) they

[...] first compared the frequencies of [their] two $${\rm Al}^+$$ clocks [namely one $$\text{Mg-Al}$$ optical clock, and one $$\text{Be-Al}$$ optical clock] at the original height difference of $$\Delta h = h(\text{Mg-Al}) - h(\text{Be-Al}) = -17~{\rm cm}$$ [...] [... and then] elevated the optical table on which the $$\text{Mg-Al}$$ clock was mounted [...] by $$33~{\rm cm}$$, and compared the frequencies again.

The results of the comparison (fractional difference in frequency between two $${\rm Al}^+$$ clocks at different heights) for 13 separate data sets are shown in Fig. 3 (B). Chou et al. claim to have found

[...] a fractional frequency change of $$(4.1 \pm 1.6) \times 10^{-17}$$.

For interpretation of this result they suggest:

For small height changes on the surface of the Earth, a clock that is higher by [...] $$\Delta h$$ runs faster by $$\frac{\delta f}{f_0} = \frac{g \, \Delta h}{c^2}$$ [... which] corresponds to a clock [fractional frequency] shift of about $$1.1 \times 10^{-16}$$ per meter of change in height.

However, there is no explicit mentioning of redshift or of blueshift ...
... i.e. the prescription that a constant height difference $$\Delta h$$ (as opposed to just spatial separation) is attributed to a signal source $$S$$ and a receiver $$R$$ which are held rigidly to each other if (discrete) signal indications $$\mathcal S$$ stated by signal source $$S$$ at rate $$\nu_S^{\mathcal S}$$ are accompanied by corresponding reception indications stated by the receiver $$R$$ at rate

$$\nu_R^{(\circledR \, \mathcal S)} = \nu_S^{\mathcal S} \, \, \text{Exp} \! \left[ \frac{\| \mathbf a_S + \mathbf a_R \| \, \Delta h}{2 \, c^2} \right],$$

provided the conditions of the equivalence principle are satisfied, i.e. the accelerations of signal source $$S$$ and receiver $$R$$ have (as good as) the same direction:

$$\frac{ \| \mathbf a_S + \mathbf a_R \|}{ \| \mathbf a_S \| + \| \mathbf a_R \| } \approx 1,$$

and

$$\| \mathbf a_R \| \approx \| \mathbf a_S \| \, \, \text{Exp} \! \left[ \frac{\| \mathbf a_S + \mathbf a_R \| \, \Delta h}{2 \, c^2} \right].$$

Therefore my question:

When determining the values of fractional difference in frequency between their two clocks, as presented in Fig. 3 (B), did Chou, Hume, Rosenband, and Wineland account for redshift or blueshift, corresponding to the reported height differences and the accelerations of their clocks (as they well should have)?

In other words, assuming that both clocks were held at (as good as) equal accelerations $$\mathbf g$$:

Can it be expected from the results of Chou, Hume, Rosenband, and Wineland, that, if a receiver $$R$$ is being held at a height of $$33~{\rm cm}$$ above an $${\rm Al}^+$$ clock $$S$$, on the surface of the Earth, then $$R$$ finds its reception rate of $$S$$'s clock ticks $$\mathcal S$$ roughly as

$$\nu_R^{(\circledR \, \mathcal S)} \approx \nu_S^{\mathcal S} \, \, \text{Exp} \! \left[ \frac{-1.1 \times 10^{-16}}{3} \right] \approx \nu_R^{(\circledR \, \mathcal A)} \, \text{Exp} \! \left[ \frac{-1.1 \times 10^{-16}}{3} \right] \times (1 - (4.1 \pm 1.6) \times 10^{-17}),$$

where $$\mathcal A$$ denotes the set of clock tick indications of an $${\rm Al}^+$$ clock being held exactly as high as receiver $$R$$?

• I'm not entirely sure what your question is asking. Gravitational time dilation and redshift when going up a gravitational field are different ways of describing the exact same effect. You don't "subtract out the redshift", because once you do there's nothing left. Commented Oct 4, 2021 at 22:15
• @knzhou: "Grav. time dilation and redshift when going up a grav. field are different ways of describing the exact same effect." -- Maybe. But: Chou/Wineland write of "a clock changing its frequency (tick rate) due to having been moved vertically. (In comparison to another one being held still.) That's surely substantially different from "a receiver changing its reception frequency of signals due to the signal source changing height". "You don't "subtract out the redshift", because once you do there's nothing left." -- Amusingly, "accounting for redshift" doubles the result value. Commented Oct 4, 2021 at 22:27
• I think you're just increasing the number of ways to say the same thing. You can think of gravitational time dilation as just a Doppler shift. And you can think of that Doppler shift as a shift of the receiver, or a shift of the source. All 3 of these are equivalent ways of describing the same thing, in the same sense that a time dilation in one frame can become a length contraction effect in another. If you counted "all of them" you would not get the right answer, you would get 3 times the right answer. Commented Oct 4, 2021 at 22:38
• @knzhou: "I think you're just increasing the number of ways to say the same thing." -- I agree that there's only one "physical (geometric-kinematical) thing" to be expressed. But I suggest that there's a correct way to express this one "thing"; and that Chou, Hume, Rosenband, Wineland (2010) either write incorrectly about that same one "thing", or they mean sth. else ... "You can think of gravitational time dilation as [...]" -- I like and reference this derivation. Commented Oct 4, 2021 at 22:52

## 2 Answers

When determining the values of fractional difference in frequency between their two clocks, as presented in Fig. 3 (B), did Chou, Hume, Rosenband, and Wineland account for redshift or blueshift, corresponding to the reported height differences and the accelerations of their clocks (as they well should have) ?

No, they were measuring the redshift. It would not make sense to “account for” the very thing that they are measuring.

Essentially, general relativity predicts a specific effect. They were measuring that effect. To “account for” it would mean to assume that it occurs and then correct for it. But it would be inappropriate to assume that it occurs before they measure it.

• Dale: "No, they were measuring the redshift." -- If Chou, Hume, Rosenband, Wineland (2010) indeed had been measuring the redshift (which is a relation between the "sending rate" of a sender, and the corresponding "receiving rate" of a receiver), then why would they expressedly (Fig. 3 (A)) write of "one clock changing its ticking rate" ?? That's either an incorrect (improper) description of gravitational shift; or it does just mean what it says (to stick to the more pleasant possibilities). Commented Oct 4, 2021 at 23:12
• @user12262 You would have to ask them why they used that specific phrasing, but it seems unobjectionable to me. It is reasonable to speak of the clock that is moved as being the one that changes.
– Dale
Commented Oct 5, 2021 at 0:57
• @user12262 This is common when people write short papers announcing experimental results. A lot of theoretical statements are not 100% correct (to make them that way would take much more space), but they're correct enough to be understood. Commented Oct 5, 2021 at 1:15

When determining the values of fractional difference in frequency between their two clocks, as presented in Fig. 3 (B), did Chou, Hume, Rosenband, and Wineland account for redshift or blueshift, corresponding to the reported height difference and the accelerations of their clocks (as they well should have) ?

No: apparently Chou, Hume, Rosenband, and Wineland did not explicitly account for gravitational shift; and arguably they did not have to.

If any wording of their article (which they may have chosen to keep their artcle suitably short) which seems suggestive of

• any one or both of their clocks "changing clock rate" can be, and is meant to be, understood instead as the rates of both clocks staying (as good as) constant, and

• one of their clocks "running faster" than the other clock can be, and is meant to be, understood instead as both clocks running (as good as) equally fast, and

• "the fractional difference in frequency between the two clocks" can be, and is meant to be, understood instead as the fractional difference between the ticking frequency of one of the clocks and the receiving frequency of a receiver of the tick signals of that clock

then the measurement described by Chou, Hume, Rosenband, and Wineland can be, and is apparently meant to be, understood as measurement of gravitational shift corresponding to the height differences of the clocks (instead of being a measurement of changing clock rates corresponding to these height differences).

If so, they wouldn't have to account for the gravitational shift corresponding to the height differences of the clocks in addition to their measurement.

However, it then remains to be investigated how Chou, Hume, Rosenband, and Wineland determined the reported "change in height by $$33~{\rm cm}$$" actually being a change in height (instead, for instance, of a change in displacement in another direction, but at the same height); or at least how they determined the corresponding systematic uncertainty.