Time Measurement in Extremely Curved Space Regions

[Dale]

Which i did.

And which @PeterDonis did not. I for one have 0 interest in a historical discussion about Newton’s original theory.

If you really want to challenge that please consider reading this section of the Principa and point out text passages that support your claims.

Asking for references is fine. Please clarify by a targeted direct quote which specific claim you want references for. Keep in mind that his claims are about the modern theory, so the references need not be passages from Principia.

 

[Dale]

But that is what Newtonian mechanics predicts if you claim that TDB coordinate time is Newton's "absolute time". Its prediction is wrong.

I think I would go even further. Newton’s original (Principia) concept of absolute time, by design, has no minimal interpretation. (It is part of Newton’s personal non-minimal interpretation of his theory) As a result clocks in Newtonian mechanics (modern) are understood to measure relative time. Thus usually the unnecessary concept of absolute time is dropped and the unqualified word “time” in classical mechanics is relative time.

 

[PeterDonis]

I have not brought up the concept of proper time in terms of Newtons theory. I was pretty clear on highlighting that his original work is based on a concept of time that aligns with coordinate time.

In Newton's theory, coordinate time and proper time are the same--the time elapsed along any clock's worldline is the coordinate time. And of course that coordinate time is the same in all inertial frames, because Galilean transformations do not change coordinate time at all. That's why Newton sometimes used the term "absolute time" to describe the time in his theory. Newton didn't bother to distinguish a distinct concept of "proper time"--time elapsed along a particular clock's worldline--because he didn't think there was a distinct concept. We now know that he was wrong about that.

Newtown who takes a lot of effort to make the distinction between absolute and relative time

I have already explained why the "relative time" that Newton described has nothing to do with proper time; it was due to a lack of understanding in general of the variability of the phenomena that were then used as standards of timekeeping. As far as this discussion is concerned, Newton's "relative time" is off topic (and indeed it plays no role whatever in modern applications of Newtonian mechanics).

Newton put quite a lot of effort to find a time suitable to describe the movement of celestial bodies and his reasoning is not too different from IAU conceptions of fundamental position (ICRF) and time (TDB) systems - obviously, since those are a further refinement of the very same astronomical origins Newton refers to.

Sort of. Modern definitions of, for example, the Earth-Centered Inertial frame and the solar system barycentric frame take into account relativistic corrections, which Newton of course did not and could not. These corrections do not just involve proper time, btw; there are also simultaneity differences between the two frames.

the prescribed corrections needed to arrive at the specified ideal

Yes, I understand that Newton allowed for the possibility of having to apply corrections to a particular timekeeping method because of non-idealities in that particular method. But his "ideal" was absolute time (or "coordinate time in any Newtonian inertial frame"--as above, all of those are the same), and, as I have already said, he believed that the ideal "elapsed time" along any clock's worldline was the absolute time. And, as I have pointed out multiple times now, he was wrong about that. We know that from many experimental tests, for example the Hafele Keating experiment--if Newton were correct about absolute time, the clocks in that experiment would all have shown the same elapsed time from start to finish, but they didn't.

If you really want to challenge that please consider reading this section of the Principa and point out text passages that support your claims.

I don't think you understand the point I have been making. The fact that Newtonian mechanics has an absolute time, as I described above, has been a well known feature of Newtonian mechanics ever since it was published. Do you really need me to point out specific passages in the Principia that illustrate that fact?

If you mean that you don't understand how Newtonian mechanics predicts that the elapsed time along any clock's worldline is the same as absolute time as I described above, again, do you really need me to point out specific passages in the Principia that illustrate that fact? Every single equation that has time in it does. In terms of our earlier formalism with models and interpretations, part of Newton's interpretation $I$ of his model was that the time $t$ that appears in his equations directly reflects the elapsed time on ideal clocks. (His comments about "relative time" then allow for non-ideal clocks to be corrected appropriately--but the ideal they are correcting to is as I have just said.) Do you really need me to point out specific passages in the Principia to back that up? Isn't it obvious from the entire book?

What you don't appear to grasp is the simple fact that Newton was wrong in his belief about absolute time. We know that now; it was one of the key things we learned from relativity. But that doesn't change what Newtonian theory says. It just makes the theory wrong, at least in this respect.

 

[PeterDonis]

Newton’s original (Principia) concept of absolute time, by design, has no minimal interpretation. (It is part of Newton’s personal non-minimal interpretation of his theory)

I would say that Newton's concept of absolute time is a prediction: the prediction that elapsed times on all ideal clocks will tick at the same rate. (And, as I have said, we now know that this prediction is wrong.) So, for example, Newtonian mechanics predicts that "twin paradox" scenarios are impossible. But making that prediction involves interpreting the time $t$ in Newton's equations as absolute time, not relative time.

"Relative time" was Newton's way of acknowledging that real clocks did not always tick in absolute time and their actual readings might need to be corrected. That was because he included in "real clocks", for example, astronomical observations of the orbits of satellites around other planets, which were well known to vary in a systematic way (in modern terminology we would say that the observed times needed to be corrected for light travel time from the source), as well as observations based on the Earth's rotation, which, as we now know, does not have a constant period. Our modern way of dealing with this is to find better clocks based on phenomena that are not subject to these kinds of variability; our best current standard in that regard is atomic clocks.

As a result clocks in Newtonian mechanics (modern) are understood to measure relative time. Thus usually the unnecessary concept of absolute time is dropped and the unqualified word “time” in classical mechanics is relative time.

I'm not sure I agree. The equations involving time in modern Newtonian mechanics are the same as the ones Newton used, and the interpretation of the time $t$ in those equations should be the same as well.

 

[Dale]

I'm not sure I agree. The equations involving time in modern Newtonian mechanics are the same as the ones Newton used, and the interpretation of the time t in those equations should be the same as well.

I think that we may agree that in the mathematical framework and equations of modern Newtonian mechanics there is only one concept of time, which is indicated by the variable $t$ in those equations. I hope we also agree that, in the minimal interpretation, $t$ is mapped to the reading of a clock.

So I think our disagreement is merely whether Newton originally would call that "absolute time" or "relative time". I don’t think it matters too much, but the reason for my understanding comes from this passage:

“Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external … measure of duration by the means of motion” (emphasis added)

I take “sensible and external measure” to mean that relative time is the measurable time. In particular the word sensible. And I take “without relation to anything external” to mean that absolute time cannot be measured.

I don’t think the word choice matters since modern Newtonian mechanics only uses one time.

 

[PeterDonis]

I hope we also agree that $t$, in the minimal interpretation, is mapped to the reading of a clock.

I think this needs to be qualified further. As I understand it, $t$ in the math is mapped to the reading of an ideal clock that is at rest in a (Newtonian) inertial frame. Newton considered the paradigmatic example of such a frame to be a frame that was at rest with respect to the fixed stars, as for example in this passage (Section XIV of the Scholium):

"...if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions..."

I think the "ideal" qualifier on the clock is necessary because of, for example, passages like this (Section V of the Scholium):

"Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the apparent time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality that they may measure the celestial motions by a more accurate time."

In other words, actual clocks, or other timekeeping schemes, such as counting rotations or revolutions of astronomical bodies, might not be ideal and might require a correction to match the ideal time $t$ that is in the equations. An ideal clock would not require any such correction. (Arguably there were no actual clocks in Newton's day that approached the ideal, but of course today we do have such clocks.)

 

[Dale]

I think this needs to be qualified further. As I understand it, t in the math is mapped to the reading of an ideal clock that is at rest in a (Newtonian) inertial frame

I am fine with that qualification, although I would probably use the word “good” rather than “ideal”. With “good” meaning indistinguishable from “ideal” to within experimental uncertainty. This would allow a hourglass if the experimental uncertainty were large, but not an hourglass on its side.

 

[Killtech]

In Newton's theory, coordinate time and proper time are the same--the time elapsed along any clock's worldline is the coordinate time

As far as i read Newton's theory it mainly uses coordinate time and hence makes only predictions on that. If we have an minimal interpretation how to interpret coordinates, we don't need another representation of time, because everything is already perfectly specified by coordinates. Hence, there is no need for it to make any predictions about proper time at all.

If you really want to bring the concept into play however, we have to dig into the relative time. By Newtons time, the way time was measured by astronomers via the mean sun time had already introduced a strong dependence of the measurement on the frame and location it was conducted in - circumstantially GR adds more effects to such a dependence. Because of that, frame dependent corrections are already part of relative time.

Furthermore, he keeps the definition very open accepting any sensible measure for this purpose while you have a very specific device in mind when you talk about a clock. That's a point of contention. I don't see where the Principia would treat a Caesium clock any different from a special device that merely measures a coordinate time, apart from the corrections they need to make. The latter is clearly not a clock in your sense, however both are relative times. Both are just different devices that need different corrections to will make them agree in all frames. But because it is the coordinate time which must be derived from their measurements, the theory makes it fall on the Caesium clock to make relativistic corrections to be able to do that. And once it does, it will comply with Newtons framework. Same applies to GR proper time. Any prediction of it requires a correction by an analog to the equation of time to obtain it from absolute time.

From where do you conclude, that proper time of GR is a relative time in Newtons work that needs no further corrections?

I think that we may agree that in the mathematical framework and equations of modern Newtonian mechanics there is only one concept of time, which is indicated by the variable t in those equations. I hope we also agree that, in the minimal interpretation, t is mapped to the reading of a clock.

Also, i've been reading Elia Cartans book (ugh.. French). In fact reading it makes it apparent that Newton-Cartan is no single theory but rather a family of closely related yet distinct theories, because it has one parameter by which it identifies Newtons spacetime:

Cartan accepts Newtons absolute space and time and identifies it (as i presumed) via the coordinates. This is very significant, because he assumes that in the chosen coordinates, Newtons law of gravity takes the form of the Poisson equation. This is of course not true for just any coordinates. Note that you already yield an different theory with different predictions if you were to replace the ICRF and TDB coordinates be some Lorentz transformed version. Same problem arises when you ask for the local length of the coordinate unit vectors in meters and seconds in gravity fields (if they correct for it or not).

Therefore Cartans interpretation seems to disagree with your mapping of t to a clock. t must be the right coordinate instead, implicitly selecting a frame and a bit more to make the translation.

Otherwise you would get a theory (model + interpretation) that is self inconsistent from the get go, because clocks can measure time based on any frame and if you do not account from the disagreements from relativistic effects between those frames, you will fail to uniquely establish the initial conditions of whatever system you are studying (these will depend on how you chose to measure them) - i.e. you will fail to have a basis to make any predictions at all. In that case it is the interpretation is not minimal but highly ambiguous. If the model of a theory does not comply with the reality of clocks, you have to account for it in the interpretation to make it able to work at all.

 

[Dale]

As far as i read Newton's theory it mainly uses coordinate time and hence makes only predictions on that. … Cartan accepts Newtons absolute space and time and identifies it (as i presumed) via the coordinates.

The problem is that presumption (if you stop there) makes it non-scientific. You cannot measure coordinates. In the end, if you want a scientific theory of space and time then you have to have something in your theory that maps to clocks.

It is often true that the focus is on the math during the description of a complicated model, like Cartan’s. But in order to go beyond the model and have a scientific theory that is insufficient. I think you are getting stuck there, and I don’t think that is what those authors intended.

In classical mechanics, including Newton Cartan, a clock measures $t$ (technically $\Delta t$). The model quantity $t$ maps directly to the reading on a clock via the minimal interpretation. That quantity $t$ can also be used as a coordinate, but as an experimental science its primary role in physics is to predict the measurement of a clock.

Otherwise you would get a theory (model + interpretation) that is self inconsistent from the get go

I think this is wrong. Classical mechanics (including Newton Cartan) is self consistent considering clocks as measuring $t$.

It is just inconsistent with experiment. Your preceding argument seemed not to identify a self inconsistency but rather an inconsistency with known experimental relativistic effects.

 

[PeterDonis]

As far as i read Newton's theory it mainly uses coordinate time

The parameter $t$ in Newton's equations can be viewed, theoretically, as coordinate time in a Newtonian inertial frame. But as Newton himself explains, that makes it equivalent to Newton's absolute time, or to time kept by an ideal clock that is at rest in an inertial frame. So as a matter of interpretation of the theory, $t$ is the time kept by an ideal clock.

If you really want to bring the concept into play however, we have to dig into the relative time.

Only in the sense that, as I have said, "relative time" is Newton's acknowledgment that the methods of timekeeping that were known in his day were variable, and had to have corrections applied to have their time behave the way an ideal timekeeping device (or a "good" timekeeping device in @Dale's preferred terminology) would behave. In other words, by "relative time" Newton meant the uncorrected time kept by a timekeeping method with known variability, such as the observed periods of various astronomical rotations or revolutions. Newton never, AFAIK, claims that the $t$ in his equations is the same as "relative time", i.e., as uncorrected time kept by a variable timekeeping method. The whole point of requiring the corrections to "relative time" is to make the corrected time behave the way $t$ in his equations behaves, i.e., like absolute time, or time kept by an ideal clock at rest in an inertial frame.

 

[DanMP]

In this extreme case, each atom would be its own clock, and by your hypothesis, their readings would not all agree, so you would have to read each one separately to find the proper time along each individual atom's worldline. There would no longer be a single clock composed of many atoms that you could treat as having a single worldline with a single proper time. In the very extreme case where the spacetime curvature was large enough to disrupt the structure of the individual atoms, then that type of clock would no longer work at all. Which is to be expected: as @Dale has already pointed out, all clocks have limitations.

On Earth's surface, the spacetime curvature is large enough in order to detect time dilation due to a change in height of less than 1 meter. A caesium atomic clock can be quite big, so my question:

If the clock is a device that, to some approximation, reads proper time along its worldline, how would the clock choose what worldline to "read" [...] ?

may be valid even here on Earth.

If you consider that on Earth the worldlines are the same for all the atoms in the clock, at what spacetime curvature they are no longer the same and how would the clock "know" that and "pick" only one?

Keep in mind that the clock may be in one piece and measuring/counting in much larger spacetime curvatures than we have here.

 

[Dale]

If you consider that on Earth the worldlines are the same for all the atoms in the clock, at what spacetime curvature they are no longer the same and how would the clock "know" that and "pick" only one?

This is actually not the same issue. We have clocks that are accurate enough to detect differences in the gravitational potential over a meter or so. We do not have clocks that are anywhere near accurate enough to sense spacetime curvature over those distances. By the equivalence principle you can have a "gravitational" potential in a rocket accelerating in flat spacetime.

This is a recognized issue. The recommendation by the BIPM is to pick the worldline of a designated connector and report the proper time at that connector. Unfortunately, in the Mise en pratique for the second they use the phrasing "the non-uniformity of the gravitational field over the size of the device" when it is the non-uniformity of the gravitational potential that is at issue.

 

[vanhees71]

Indeed today's most accurate clocks (realized with frequency combs) are so accurate that a more accurate definition of a common time on Earth is impossible, because we don't know the gravitational field of the Earth well enough to correct for the gravitational time-dilation.

 

[DanMP]

This is a recognized issue. The recommendation by the BIPM is to pick the worldline of a designated connector and report the proper time at that connector. Unfortunately, in the Mise en pratique for the second they use the phrasing "the non-uniformity of the gravitational field over the size of the device" when it is the non-uniformity of the gravitational potential that is at issue.

My question was: "at what spacetime curvature they are no longer the same and how would the clock 'know' that and 'pick' only one?". What we should pick is interesting to know, thank you, but my problem is with PeterDonis's claim/definition: "A 'clock' is a device that, to some approximation, reads proper time along its worldline", so I'm interested in how exactly the clock is doing it (pick a particular worldline at a certain spacetime curvature) and why his interpretation is better than:

In order to measure time, the clock must produce/host/observe some events and then count them. There is no proper time to read, only events to count. From that we can derive the proper time.

Indeed today's most accurate clocks (realized with frequency combs) are so accurate that a more accurate definition of a common time on Earth is impossible, because we don't know the gravitational field of the Earth well enough to correct for the gravitational time-dilation.

Interesting. You seem to know a lot about clocks. Do you know if we have an atomic clock on (or around) the Moon?

 

[jbriggs444]

My question was: "at what spacetime curvature they are no longer the same and how would the clock 'know' that and 'pick' only one?".

At what space time curvature does it start to make a difference? It depends on how accurate your clock is.

How does the clock know? How does a pendulum clock pick one acceleration of gravity to use when the gravitational acceleration varies over the length of the pendulum? It will depend on the details of the clock.

There is no particular problem defining the second in terms of one sort of clock and measuring time with another.

 

[Dale]

What we should pick is interesting to know, thank you, but my problem is with PeterDonis's claim/definition: "A 'clock' is a device that, to some approximation, reads proper time along its worldline", so I'm interested in how exactly the clock is doing it (pick a particular worldline at a certain spacetime curvature)

I guess I don't understand the difference in what I answered and what you are asking. I thought I already answered that. After all, clocks are devices that we design and build. So the clock does what we choose for it to do. If we choose to measure the proper time at a given connector then we design it to do that.

Can you please explain instead of repeat? What are you looking for that I did not already provide. Are you looking for detailed design approaches for a specific type of clock?

 

[PeterDonis]

My question was: "at what spacetime curvature they are no longer the same and how would the clock 'know' that and 'pick' only one?".

You're missing the point. Under the conditions you describe, there is no longer one clock. So your question makes no sense.

I'm interested in how exactly the clock is doing it

Same answer: under the conditions you describe, there is no such thing as "the" clock.

why his interpretation is better than

My "interpretation", as you call it, is not different from yours. In the case of a cesium clock, the "events" are the cycles associated with a particular atomic transition. (I am glossing over a lot of technical detail here, but what I've said is, I think, enough for this discussion.) But any real mechanism that generates and counts such events will have a finite size; it will not be a single point. So treating any such mechanism as a single object that we can use as a clock is, strictly speaking, an approximation: the mechanism does not have just one worldline, it has a bundle of worldlines that, under normal conditions, all go together in such a way that their proper times match up, and we can treat the entire clock as having just one worldline (if you like, the worldline of its center of mass) whose proper time the clock is measuring by generating and counting its events.

But under extreme conditions, such as very strong spacetime curvature, the approximation described above breaks down: the bundle of worldlines no longer matches up and we can no longer treat them as making up a single object that we can use as a clock. As has already been pointed out, that is to be expected: all real clocks have limitations. Not working under very extreme spacetime curvatures is one such limitation.

 

[PeterDonis]

It depends on how accurate your clock is.

And on how it is constructed.

 

[DanMP]

we can treat the entire clock as having just one worldline (if you like, the worldline of its center of mass) whose proper time the clock is measuring by generating and counting its events

Now it sounds much better: we can treat the entire clock as having just one worldline, so we pick the worldline. The clock is just generating and counting its events.

Only one observation: in places with stronger spacetime curvature, the EM radiation emitted by the atoms may suffer redshift/blueshift if directed up/down, so the orientation of the clock becomes important, and when we "pick" what worldline was measured we should take this in consideration.

Thank you all for your answers.

I had one more question:

Do you know if we have an atomic clock on (or around) the Moon?

Maybe I'll open a new thread for it.

 

[PeterDonis]

we can treat the entire clock as having just one worldline, so we pick the worldline.

Yes. @Dale gave an explicit example of how that is currently done with atomic clocks.

 

[PeterDonis]

Maybe I'll open a new thread for it.

Yes, that would be appropriate.

 

[Killtech]

I think this is wrong. Classical mechanics (including Newton Cartan) is self consistent considering clocks as measuring t.

Sorry, i cannot see how with the interpretation you propose makes Newtons theory at all usable.

Consider a clock $r_1$ on earths surface and another one $r_2$ in orbit, and a last one $r_3$ passing earth with relativistic speed, all of them measuring the time it takes earth spin one full turn. You will get 3 different results. The interpretation you apply will map those 3 different values to 1, the very same absolute time, so you have a contradiction right in your interpretation. If you cannot determine even the most basic initial conditions for you model, like how fast earth initially spins, how are you supposed to work with that? This problem will be the same for every single value you need for your model - and in terms of astronomy the different methods may yield significant deviances without relativistic corrections. You cannot work with a theory that has contradictions!

Even if Newtons theory is not relativistic, it doesn't mean it is not workable. But that requires to first establish a method of interpreting measurements in terms of the model, and if in reality it turns out that we have two devices that are supposed to measure the same thing but yield different results in practice, we have to work that out first. This very case has been discussed in this thread extensively: metrology suggest to figure out what is differenten about the devices/the conditions they are in and at very least establish a corrective function in between those to make them consistent - at very least empirically (e.g. fit the bias via regression). It is that corrective function, deducted from compares between measurements, that turns your time measuring devices to be able to measure a coordinate time.

During Newton's time, the different measuring techniques were all but consistent and as i read the relative time concept, it accounts for the problems this practicality entails.

 

[Dale]

i cannot see how with the interpretation you propose makes Newtons theory at all usable.

Consider a clock … with relativistic speed

Newton’s theory is not useable at relativistic speed.

You will get 3 different results.

Not according to Newtonian mechanics. Newtonian mechanics predicts that all three produce the same result.

the very same absolute time, so you have a contradiction right in your interpretation

That is a contradiction with experiment, not a contradiction in the interpretation.

If your model + interpretation does not produce this same contradiction with experiment then it is clearly not Newton’s theory. This is a well known contradiction between Newtonian theory and experiment.

 

[PeterDonis]

Consider a clock on earths surface and another one in orbit, and a last one passing earth with relativistic speed, all of them measuring the time it takes earth spin one full turn. You will get 3 different results. The interpretation you apply will map those 3 different values to 1

No. I have already pointed this out more than once, and so has @Dale. You are confusing the interpretation of the theory with the fact that it makes wrong predictions.

Newtonian theory predicts that all three of these clocks will measure the same time for the Earth spinning one full turn. This prediction is wrong. It's not that Newton's theory takes these three actual experimental results that are different and tries to adopt some "interpretation" to gerrymander them into one theoretical value of $t$. It's that these actual experimental results falsify Newtonian theory. You appear unable to grasp this essential point.

 

[PeterDonis]

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[PeterDonis]

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