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Descriptions of setups and findings involving clocks can also involve distinct observers (in the following a.k.a. receivers) which (in general) perceived, and possibly then analyze, relevant signals having originated from the given clocks. While terminology referring to a clock itself has been considered and discussed already, I'm hereby looking for terminology suitable and distinctive for receivers.

1.
Given one identifiable tick indication of a clock --
How do we call the corresponding perception indication of a receiver ?

(For lack of reference with any other suggestions, I've been referring to "the tick-response" indication of a receiver, already e.g. in the title above; and also in the following.)

2.
In case that the tick-response indications of a receiver "occured regularly", i.e. with equal, constant duration of the receiver between its successive tick-response indications corresponding to (in general successive) tick indications signalled by the clock --
How do we call the corresponding rate (a.k.a. frequency) of the receiver ?

(Surely we would not call it "rate of the clock", or "frequency of the clock", since that would be misattribution, and thereby confusing. By default, in the following I'll refer to "the reception rate" of the receiver.)

3.
In case that, moreover, the tick indications of the clock "occured regularly" as well, i.e. with equal, constant duration of the clock between its successive tick indications --
How do we call the ratio between "the reception rate" of the receiver and the tick rate of the clock ?

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  • $\begingroup$ @PM 2Ring: "If you put [...] 2 clocks next to each other, then they do tick at the same rate." -- Not at all. Instead: If you put a receiver next to a clock, then the receiver's tick-responses are in unison with the clock's ticks. If, moreover, the receiver's tick-responses occured regularly, at a particular reception rate of the receiver, then the clock had stated its tick's necessarily at equal rate as the receiver's reception rate. Nevertheless you can put several clocks next to each other which all tick at different (unequal) rates. Cmp. Synge, GR, p. 106. $\endgroup$
    – user12262
    Commented Feb 3, 2023 at 20:48
  • $\begingroup$ @PM 2Ring: "[...] caused by the difference in the gravitational potential [...]" -- Explicitly: If a clock ticked at a particular constant rate, and a separate receiver had its corresponding tick-responses at a particular constant reception rate, then the one of these two whose rate (either tick rate of the clock, or reception rate of receiver) is higher is (by definition) said to be "on lower gravitational potential". (If the response rate of the receiver is higher than the tick rate of the clock, we say that the clock had been "blueshifted" wrt. the receiver, etc., I'd suggest.) $\endgroup$
    – user12262
    Commented Feb 3, 2023 at 21:08
  • $\begingroup$ @PM 2Ring: "But if one clock's on Earth & the other's on the Moon then they run at different rates" -- Not necessarily at all. Instead, as with all comparisons of (constant) (tick) rates of two ticking clocks, $\mathfrak A \equiv (\mathcal A, t_{\mathfrak A})\}$ and $\mathfrak B \equiv (\mathcal B, t_{\mathfrak B})\}$: For any two distinct indications $A_J, A_P \in \mathcal A$ and any two distinct indications $B_K, B_Q \in \mathcal B$, [contd. ...] $\endgroup$
    – user12262
    Commented Feb 3, 2023 at 21:24
  • $\begingroup$ if$$(t_{\mathfrak A}[\, A_P\,]- t_{\mathfrak A}[\, A_J\, ]) > (t_{\mathfrak B}[\, B_Q\, ]- t_{\mathfrak B}[\, B_K\, ]) \times \left( \frac{\tau A[\, \_J,\_P\, ]}{\tau B[\,\_K,\_Q\, ]} \right) $$ then clock $\mathfrak A$'s rate was higher than clock $\mathfrak B$'s, $$~$$ where $\tau A[\, \_J,\_P\, ]$ denotes the duration of "material point" $A$ (who is promoted to being a clock $\mathfrak A$ through assignment of readings, such as "tick counts", to $A$'s indications, i.e. $$t_{\mathfrak A} : \mathcal A \rightarrow \mathbb R$$ ), from $A$'s indication $A_J$ until $A$'s indication $A_P$, etc. $\endgroup$
    – user12262
    Commented Feb 3, 2023 at 21:34
  • $\begingroup$ @PM 2Ring: Considering JEB's answer, I now realize that my second comment ("Explicitly ...") is inaccurate as it stands: I neglected to mention the additional requirement of constant (though of course not necessarily mutually equal) mutual ping durations between clock/sender and receiver (which is obviously required when discussing uniformly accelerating rigid rocket cabins, cabins or towers being held rigidly on Earth's surface, etc.) $\endgroup$
    – user12262
    Commented Feb 3, 2023 at 23:17

1 Answer 1

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It's called the Doppler Shift:

$$ f'_{\rm received} = \sqrt{\frac{1+\beta}{1-\beta}} \times f_{\rm rest} $$

This differs from the thought-experiment observed clock rate:

$$ f_{\rm observed}' = f_{\rm rest}/\gamma = (1-\beta)f'_{\rm received} $$

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  • $\begingroup$ JEB: "It's called the Doppler Shift: [...]" -- Apparently it's (the answer to 3.) called the Doppler Shift if the geometric-kinematic relations between clock (sender) and receiver can be characterized by the value of (just) one number ($\beta$); and the sender is characterized as being "(at) rest"; a.k.a. as being a member of an inertial system. More generally, it might be just "shift"; incl. "gravitational shift". (So: +1.) [contd.] $\endgroup$
    – user12262
    Commented Feb 3, 2023 at 23:05
  • $\begingroup$ JEB: "[...] $f^{\prime}_{\text{received}}$" -- Why the prime at all ?? (And I'd name/denote this rate rather: $f_{\text{receiver}}[ \, \text{sender} \, ]$, for instance.) "[...] differs from $f^{\prime}_{\text{observed}}$ [...]" -- That's a very peculiar sort of rate, since it doesn't refer to analyzing perceptions of one particular participant, but instead it appears derived from perceptions of several; namely several members of the same inertial frame. I'd rather call that the result of a measurement, $f_{\text{receiver frame}}[ \, \text{sender} \, ]$. $\endgroup$
    – user12262
    Commented Feb 3, 2023 at 23:05
  • $\begingroup$ @user12262 Perhaps measurement is better. This is a source of constant confusion in thought experiments. IRL you have to wait for signals to reconstruct reality, but in thought experiments, you, the P.I., has an infinite lattice of grad students in your inertial frame with perfect clocks and rulers furiously scribbling in their composition books, from whence you can reconstruct your 3 + 1 reality at any moment, everywhere. In this view, the clock bias (relativity of simultaneity) becomes clear and all paradoxes are resolved, except for The Andromeda Paradox. $\endgroup$
    – JEB
    Commented Feb 7, 2023 at 15:18
  • $\begingroup$ @user12262 I like clarity, so the 1st reference frame mentioned is unprimed, the 2nd gets a prime in its name, coordinates and observables, and so on to double-prime, if needed. And I don't change variable names btw frames (primes notwithstanding). If you're talking about a detection event, with a well defined position and time, it's good to say "The signal emitted at $E_{Tx}$ and received at $E_{Rx}$", you can then Lorentz transfer to $E'_{Tx}$, $E'_{Rx}$ and get quantitative results. This is not medicine, so Rx and Tx are not "Prescription" and "Treatment", rather "Receive" and "Transmit". $\endgroup$
    – JEB
    Commented Feb 7, 2023 at 15:25
  • $\begingroup$ JEB: "an infinite lattice of grad students in your inertial frame with perfect clocks and rulers furiously scribbling" -- Just to help clarify "where my question came from": I (like to explore that we) insist on Einstein's prescription of GR, and define "(joint membership in an) inertial frame" and "perfect/good/ideal clock" etc. from that. Farewell to EPS! p.s. I hate primes! ... p.p.s. physics.stackexchange.com/a/748956/12262 $\endgroup$
    – user12262
    Commented Feb 7, 2023 at 22:59

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