Solving "const. K" in Gourgoulhon's SR in General Frames

[Frank Wappler]

TL;DR Summary

É. Gourgoulhon "Spec. Rel. in General Frames", sect. 2.3.2 defines "ideal clock(s)", introducing a "proportionality factor K", for short: "between duration and number of elapsed ticks"; cmp. eq. (2.11). Question: Is K the inverse of ticking rate?

In sect. 2.3.2 "Ideal Clock", p. 33, of É. Gourgoulhon's text book on "Special Relativity in General Frames"

[... a]n ideal clock is [...] defined as a clock [C] for which [C's duration] between two [not necessarily consecutive of its] ticks [...] is equal to a constant K times the number N of elapsed ticks:

$$ \tau_C [ \, \text{tick}_{j}, \text{tick}_{(j + N)} \, ] = K_C \, N. $$
(equation (2.11); notation adapted.)

The only other reference to this "constant K" is on the following page 34, as

proportionality factor K

On this "constant proportionality factor K" I have a few closely related questions:

(1) Is it correct to understand "constant K" as being specific to each ideal clock;
e.g. constant K_C of ideal clock C, constant K_Q of ideal clock Q, etc. ?

(2) Is it then correct that those specific constants, such as K_C of ideal clock C, and K_Q of ideal clock Q,
are not outright and necessarily presumed equal; but may be found equal, or not equal, by measurement ?

(3) Is it correct to identify K_C as the inverse of the ticking rate of ideal clock C ?

(4) In case that K_C is not zero, is it then correct to identify the value 1 / K_C as value of the ticking rate of ideal clock C ?

 

[Ibix]

I think K is the period of the clock in its rest frame. An ideal clock isn't affected by acceleration, so measures elapsed time along any worldline it follows.

A pendulum clock is not ideal because its period depends on the proper acceleration of the clock and its relationship to the axis of the clock. An atomic clock is very closeto ideal because it depends on atomic transition energies.

 

[Frank Wappler]

I think K is the period of the clock [...]

I think that's (already by itself) a great answer to my question part (3) -- provided that "one tick period" (or "several consecutive tick periods") is to be understood in the well-known sense of a duration of the (ticking, ideal) clock under consideration. For that, so far: Thank you!

[...] the period of the clock in its rest frame.

Any clock under consideration (and thus, in Gourgoulhon's sense, any approximate point particle which is thereby referenced) was not necessarily prescribed being a member of an inertial frame (and thus being "at rest wrt. anyone else", or in W. Rindler's words, "sitting still wrt." anyone else), nor being itself "free" (i.e. not accelerating).

So: What meaning and what purpose does the specification "in its rest frame" have ? -- Don't you plainly mean the period duration of that clock (and of the corresponding approximate point particle) itself, and not of anyone else ??

An ideal clock isn't affected by acceleration

... so it's apparently permissable that the clock (and corresponding approximate point particle) indeed was and remained accelerating ...

so measures elapsed time along any worldline it follows.

... where Gourgoulhon's eq. (2.11) presents a specific result value of such a measurement; i.e. a specific duration value (on both sides of the equation).

Now, to rephrase the remaining parts of my question accordingly, in order to gently press you for some (educated) response on those, too, please (and thanks for that ahead, too):

(1, 2): Do you agree that of distinct ideal (ticking) clocks it either may be, or may not be, found that they had (until conclusion of the measurement) unequal periods ?

(4): Is so: Do you agree that if it has been found that the period K_C of a (ticking, ideal) clock C had been larger than the period K_Q of another (ticking, ideal) clock Q, in the course of a certain experimental trial,
then we (can) say synonymously: "clock C had ticked slower than clock Q, in that trial" ?

A pendulum clock is not ideal because its period depends on [...] An atomic clock is very close to ideal because [...]

Well, as far as I understand Gourgoulhon's definition (to which I intend to adhere, for the purpose of my question):
A (ticking) clock C is ideal if and only if:

- for any two of its (not necessarily distinct and) suitably indexed tick indications ##\text{tick}_j## and ##\text{tick}_k##, in the trial under consideration, with ##j## and ##k## whole numbers, and

- for every two (not necessarily distinct) whole numbers ##N_p## and ##N_q## such that the whole numbers ##j + N_p## as well as ##k + N_q## are index values of tick indications of this clock C, in the trial under consideration, as well

holds $$ N_q \, \tau_C \! \left[ \, \text{tick}_{j}, \text{tick}_{(j + N_p)} \, \right] = N_p \, \tau_C \! \left[ \, \text{tick}_{k}, \text{tick}_{(k + N_q)} \, \right].$$

[The period durations of an] atomic clock [...] depend[] on atomic transition energies.

Well -- before delving into dynamics (variational calculus and conjugate quantities) I'd much rather be completely sure about geometry/kinematics. So:

Do you mean that the relevant atomic transition energies of a given atomic clock in the trial under consideration are to be called constant if (or as far as) this clock, in this trial, has been found ideal as specified above ?

And referring again to the SI second definition:
Do you agree that the atoms of a given atomic clock in the trial under consideration are to be called "having been (sufficiently) unperturbed" if (or as far as) this clock, in this trial, has been found (sufficiently) ideal as specified above ?

 

[Ibix]

What meaning and what purpose does the specification "in its rest frame" have ?

I could measure the period of a moving clock. That is not K, due to time dilation.

Do you agree that of distinct ideal (ticking) clocks it either may be, or may not be, found that they had (until conclusion of the measurement) unequal periods ?

Of course. Base your time standard on something other than the hyperfine Cesium transition.

Is so: Do you agree that if it has been found that the period KC of a (ticking, ideal) clock C had been larger than the period KQ of another (ticking, ideal) clock Q, in the course of a certain experimental trial,
then we (can) say synonymously: "clock C had ticked slower than clock Q, in that trial" ?

Not in general. Consider a clock that, when at rest with respect to another, has half the period. Send it off at high speed such that the time dilation factor is four and return it. It's at least arguable that it ticked slower despite having a shorter K.

Overall, this seems like a very long winded treatment of the topic. We assert that there is an analogue of distance in Minkowski spacetime that we call interval. We can measure this using an array of inertial rods and synchronised clocks. An ideal clock shows an elapsed time equal to the interval along its path since it was zeroed, whether that path is inertial or not. A non-ideal clock may not function correctly (or at all) under acceleration or in free fall, so care is necessary in designing real experiments.

 

[Frank Wappler]

I could measure the period of a moving clock. That is not K

Referring to a specific ideal clock, C, the corresponding constant K_C is by Gourgoulhon's definition to be measured explicitly as

$$ K_C := \frac{\tau_C[ \, \text{tick}_{j}, \text{tick}_{(j + N)} \, ]}{N},$$

provided at least one suitable index value ##j## and at least one suitable increment value ##N \ne 0## can be found,
and provided this clock C turns out to have been ideal overall, as shown above;
and, as far as I understand, regardless of any further circumstance whatsoever.

So ...
(a) What could you possibly mean by "measuring the period of clock C" (under whichever circumstances imaginable), if not specifically:

- selecting two distinct suitable whole number index values ##j## and ##j + N \ne j##,

- then measuring the corresponding duration value ##\tau_C[ \, \text{tick}_{j}, \text{tick}_{(j + N)} \, ]## of clock C, and

- finally evaluating ##\frac{\tau_C[ \, \text{tick}_{j}, \text{tick}_{(j + N)} \, ]}{N},## as the result value to be reported ??

Or (b):
What could you possibly want to evaluate or find out about anything else which is not clock C and then pass that off as (your idea of) the value of constant K_C, which is a.k.a. the period of clock C (under any further circumstance whatsoever) ??

(Arguably, your astonishing interpretation of measurement might warrant posting a separate question on this topic. ...)

[...] due to time dilation.

Time dilation ...
(as far as it specifically concerns clock C, i.e. the case that

$$\frac{\tau_C[ \, \text{tick}_{j}, \text{tick}_{(j + N)} \, ]}{\ell[ \, \varepsilon_{C(j)}, \varepsilon_{C(j+N)} \, ]} < 1, $$

i.e. in the denominator with the (necessarily non-zero) value of Lorentzian distance between the (necessarily timelike separated) events ##\varepsilon_{C(j)}## and ##\varepsilon_{C(j+N)}## in which clock C (or the material point associated with clock C, resp.) had taken part)
... along with geometric/kinematic relations as a whole must of course be taken into account, as necessary, in measuring ##\tau_C[ \, \text{tick}_{j}, \text{tick}_{(j + N)} \, ]## in the first place.

But:
time dilation is hardly an excuse for not even attempting to measure ##\tau_C[ \, \text{tick}_{j}, \text{tick}_{(j + N)} \, ]## in the first place -- is it ??

Consider a clock that, when at rest with respect to another, has [...]

I can consider an ideal clock, say again specificly clock C, which has had constant non-zero period K_C under any circumstances. Such a clock is appropriately said to have had "constant tick rate", too, btw.

[...] half the period. Send it off at high speed such that the time dilation factor is four and return it.

I can consider another ideal clock Q, which has had constant non-zero period K_Q = 2 K_C under any circumstances, and which had remained a member of an inertial system throughout having been deserted, and subsequently been met again, by clock C.

Notably, since ##K_Q > K_C##, clock Q's constant tick rate would be called smaller than the constant tick rate of clock C;
and clock Q would correspondingly be said to have ticked slower than clock C (under any circumstances), and clock C ticked faster than clock Q.

Now, with your setup prescription, it's not difficult to compare the separation ("trip") duration of clock C and the corresponding separation ("wait") duration of clock Q:

$$\frac{\tau_C[ \, \_ Q(leave), \_ Q(rejoin) \, ]}{\tau_Q[ \, \_ C(leave), \_ C(rejoin) \, ]} = \frac{\tau_C[ \, \_ Q(leave), \_ Q(rejoin) \, ]}{\ell[ \, \varepsilon_{CQ(leave)}, \varepsilon_{CQ(rejoin)} \, ]} = \frac{1}{4}.$$

It's at least arguable that it [clock C had] ticked slower [than clock Q] despite having a shorter K.

It is neither uncontested to say somesuch, because it is correctly said (already, or at least as well) that
"clock C ticked faster than clock Q".

Nor is it necessary, because it can be said instead perfectly correctly, that
"clock C had ticked a shorter duration (or for short: shorter) than clock Q".

So -- No: I'm certainly not following what you suggested as arguable;
and I strongly advise against that, especially for purposes of didactics.

p.s. -- I'm looking forward to discussing the construction or identification of inertial frames in threads dedicated to this very interesting topic.