Time Measurement in Extremely Curved Space Regions

[Dale]

The split of the model into two aspects has nothing to do with the mathematical terminology.

Regardless, it is a bad idea. You make a split, as a result of the split you find a problem. The solution is not to make the split.

The point i was focused on is that there are different assumptions from which a model is made have a different relation to I, particularly that I is a mapping between MF and E, at least when it is supposed to be minimal and without ambiguity.

Which is why $I$ maps between $M$ and $E$, not just $M_F$ and $E$.

Your split is artificial and according to you it introduces problems. These problems do not seem to arise without the split, so the fix is easy: don’t do the split.

 

[Dale]

What you suggest here is however logically not entirely possible

I disagree. It is entirely possible. You cannot by fiat declare that there is no physical effect which causes the discrepancy. That non-physical assertion is what leads to your non-physical conclusion.

If two different sets of physical devices produce different physical measurements then there is a physical reason. If two sets of measuring devices disagree with each other then at least one of them is sensitive to something other than the measurand.

 

[Killtech]

I disagree. It is entirely possible. You cannot by fiat declare that there is no physical effect which causes the discrepancy. That non-physical assertion is what leads to your non-physical conclusion.

Yet, that does in no way hinder us to describe the cause why one rod changes lengths relative to another. We just have to pick one (doesn't matter which) and model the world from its perspective.

So that was not the part i was referring to as impossible.

Peter suggested

In this situation you would not accept either type of rod as a valid measure of length without doing further experimentation.

But:

experiments require measurements which need a measuring system like SI is. In this simple case SI is nothing else but a choice of which rod to take, so we go in circles.

If you accept neither one as a valid measure of length, what kind of further experimentation would Peter be doing then? I don't see him being able to make any quantitative observations, let alone derive any numerical corrections for one rod.

If two different sets of physical devices produce different physical measurements then there is a physical reason. If two sets of measuring devices disagree with each other then at least one of them is sensitive to something other than the measurand.

Yes, that's obvious. But to be more precise, we can only find that the ratio of those two physical measurements is sensitive to something else. From that you are non the wiser which one is affected or if both are. And since the rods will always yield consistent results among rods of the same type, there is nothing able to resolve it.

 

[PeterDonis]

If you accept neither one as a valid measure of length, what kind of further experimentation would Peter be doing then?

I already described that.

I don't see him being able to make any quantitative observations

There are lots of things we can measure besides lengths. In my post I gave just one particular example (taken from Einstein) of an independent quantity that could be measured to see if it accounts for the different behavior of the rods.

However, what you appear to be completely oblivious of is that, in our actual, real world, your hypothetical about the rods is false. We can manufacture "rods" using very different materials and very different principles of operation (for example, consider comparing the readings of a measuring tape and a laser range finder) that all measure length the same way, that all continue to compare equally with each other as we move them around spacetime. And the fact that we can do that is what justifies our treatment of the measurements those rods give us as valid measurements of length, i.e., of a geometric quantity independent of the particular properties of particular rods. (Similar remarks would, of course, apply to clocks.)

 

[pervect]

As long as we have a smooth manifold structure, we'll have affinely parameterized geodesics. This will give us a situation where we can mark regular repeating "time intervals" along a timelike geodesic worldline which can serve as a sort of clock. I suspect it's possible to leverage this idea to mark regular intervals along a non-geodesic curve, but I don't have a specific proposal. This may be a problem for my idea, but I think it's worth putting out there anyway.

It's not clear to me what sorts of other standards might exist other than some form of matter, though one thought that occurs to me is that if we can somehow define some standard packet of energy by some sort of physical structure (the mass-energy of some sort of defined particle or transition), we can set some sort of "length" scale using plancks constant. $E = h \nu$, so energy defines a frequency which defines a time interval by inverting the frequency. So some "unit" of energy gives us some "unit" of time. This is basically the approach the cesium clock takes, it uses a specific transition energy to define the "packet size" of energy to set a scale factor for time. We'll need planck's constant to still apply, in any case, for this basic idea to work.

So smooth manifolds -> repeatable time and space intervals, and planck's constant allows us to define a "unit" of time if we have a "unit" of energy. There's one further point to make. The metric structure of General Relativity gives us the ability to compare the intervals along different geodesics - without a metric structure, with only an affine geometry, we can mark out regular intervals along a curve, but there's no way to relate the regular intervals on one curve to another. We see this issue in GR with null geodesics - we can mark regular intervals along them, but they're all associated with the number 0. So we can't set up a "unit" for the interval of a null geodesic other than zero, even though we can divide any particular null geodesic into regular intervals.

[revision]
This stream of thought can be slimmed down a lot. Basically, we need a smooth manifold structure with a metric. Then, proper time is the interval given by the metric for any curve, geodesic or not, and "distance" is the interval along a space-like curve. Null curves have an interval of zero, so, they don't have a similar notion.

Back to the problem of the unit of energy. If there is utter chaos, with no repeatable structure, this approach has difficulties. It's unclear to me what sort of use time would have in universe of total chaos with no repeatable structure though.

Of course, there are theories, such as Wheeler's "quantum foam", where the existing structure of space-time breaks down altogether, as there is no longer a smooth manifold. This is an even more fundamental issue with the very notion of the existence of time - and space. The "beyond the standard model" folks might have more/better ideas as to what the alternatives are to the classical smooth manifold structure of General Relativity.

As always, much of my thinking has been influenced by Misner, Thorne, and Wheeler. Much of it in "Gravitation", though they don't deserve any blame for where I may have gone off the rails...

 

[Dale]

If you accept neither one as a valid measure of length, what kind of further experimentation would Peter be doing then?

You could heat them and see how they change, or compress them, or accelerate them, or expose them to an EM field, or to vibrations. You perform experiments to find out why they disagree. You learn how your measuring devices work.

But to be more precise, we can only find that the ratio of those two physical measurements is sensitive to something else. From that you are non the wiser which one is affected or if both are.

No, that is not correct. You can also compare rods to others of the same type. If I heat 3 rods of type A and don’t heat 3 rods of type A and see that the heated ones differ from the unheated ones then we are completely wise that rods of type A are sensitive to temperature. This is standard metrology.

And since the rods will always yield consistent results among rods of the same type, there is nothing able to resolve it.

This is simply not true.

 

[Killtech]

However, what you appear to be completely oblivious of is that, in our actual, real world, your hypothetical about the rods is false. We can manufacture "rods" using very different materials and very different principles of operation (for example, consider comparing the readings of a measuring tape and a laser range finder) that all measure length the same way, that all continue to compare equally with each other as we move them around spacetime. And the fact that we can do that is what justifies our treatment of the measurements those rods give us as valid measurements of length, i.e., of a geometric quantity independent of the particular properties of particular rods. (Similar remarks would, of course, apply to clocks.)

You could heat them and see how they change, or compress them, or accelerate them, or expose them to an EM field, or to vibrations. You perform experiments to find out why they disagree. You learn how your measuring devices work.

We are talking past each other. I don't dispute anything you wrote here, or rather fully agree with it. Of course we must ensure all length measurement is consistent and of course we can build the knowledge to make it so, even if we were to use rod types that don't agree with each other, we will be able to correct that to ensure a consistent process. That is a core of scientific research.

No, that is not correct. You can also compare rods to others of the same type. If I heat 3 rods of type A and don’t heat 3 rods of type A and see that the heated ones differ from the unheated ones then we are completely wise that rods of type A are sensitive to temperature. This is standard metrology.

In my opening post the abstract rods we are talking about now were clocks, and in that particular case $r_a$ was a clock according to SI specifications, i.e. Caesium based. $r_b$ was the very same Caesium clock but its time was meant to be adjusted by a locally dependent correction factor following TCG concept, i.e. such that its time is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the actual clock, but so far away as to be considered outside of all gravity wells.We have established the terminology of model $M$ and it's interpretation $I$ that maps the proper time of $M$ to experimentally tangible clocks $r$. Since $I$ is a mapping, we can denote $I(r)$ the model clock that measures proper time. With that we can skip the interpretation part and replace the Caesium clock altogether by $I(r)$. Doing so makes both measuring device types implementation flawless and it is clear to begin with why they measure time differently.

Let's denote GR by $M_a$, its interpretation by $I_a$ via Caesium clocks $r_a$. We are able to calculate within the theory $I_a(r_b)$ everywhere, i.e. determine what the other device will measure. Let's call the related mapping $T_{ab}$, which for clocks is $I_a(r_a) \mapsto I_a(r_b)$ and identity for proper lengths.

What i am intrigued about is that geometry suggest that if $T_{ab}$ is smooth, there exists a model $M_b = T_{ab} (M_a)$ with an interpretation $I_b$ that maps proper time of that model to $r_b$ such that $I^{-1}_a(M_a) = I^{-1}_b(M_b)$, i.e. the correctly interpreted predictions of either model will always agree. This situation is called a commutative diagram.

Let me try rephrase that in words of others to reduce the confusion of what i intend to express:

I think that @Killtech is saying that the same physics can be described by different geometries, if you also adjust the forces (and the laws for how matter and energy affect geometry).

It's sort of trivially true. Suppose we pick some coordinate system $x^\mu$, and according to the "true" laws of physics (say, General Relativity plus some force law), the path of a particular particle is given by:

$\dfrac{d U^\mu}{d\tau} = F^\mu - \Gamma^\mu_{\nu \lambda} U^\nu U^\lambda$

where

• $U^\mu = \dfrac{dx^\mu}{d\tau}$
• $d\tau = \sqrt{g_{\mu \nu} dx^\mu dx^\nu}$
• $g_{\mu \nu} =$ the metric tensor
• $\Gamma^\mu_{\nu \lambda} = \dfrac{1}{2} g^{\mu \sigma} (\partial_\nu g_{\sigma \lambda} + \partial_\lambda g_{\nu \sigma} - \partial_\sigma g_{\nu \lambda})$
• $g^{\mu \sigma} =$ the inverse of $g_{\mu \sigma}$.

Now, let's let $g'_{\mu \nu}$ be any candidate alternative metric. Then we can in terms of $g'_{\mu \nu}$ compute an alternative proper time $\tau'$, and an alternative connection $\Gamma'^\mu_{\nu \lambda}$ and an alternative 4-velocity $U'^\mu$. Finally we can compute (using the "true" laws of physics) the expression

$F'^\mu = \dfrac{d U'^\mu}{d\tau'} + \Gamma'^\mu_{\nu \lambda} U'^\nu U'^\lambda$

Then $F'^\mu$ will in general be a messy combination of the original force vector $F^\mu$ and the original connection coefficients $\Gamma^\mu_{\nu \lambda}$ and the original 4-velocity $U^\mu$. Nevertheless, presumably it can be expressed as a function of the coordinates $x^\mu$, their derivatives with respect to the "false" proper time, $\tau'$ via $U'^\mu = \dfrac{x^\mu}{d \tau'}$, and the false metric tensor $g'_{\mu \nu}$.

where in this case alternative metric $g'_{\mu \sigma}$ belongs to $M_b$. In accordance with this thought:

Both $M$ and $I$ are part of the theory, but that means we have to be careful about what $I$ actually means. $I$, if we view it as part of the theory, leads to the theoretical claim I described before: that there should exist devices that behave like the "clocks" $I$ describes.

the "clocks" $I_b$ describes matches the devices $r_b$.On their own $M_a$ and $M_b$ may be very different models. But if each combined with its own matching interpretation yields identical predictions about reality, can either model claim the monopoly on the true proper time?

 

[PeterDonis]

In my opening post the abstract rods we are talking about now were clocks, and in that particular case $r_a$ was a clock according to SI specifications, i.e. Caesium based. $r_b$ was the very same Caesium clock but its time was meant to be adjusted by a locally dependent correction factor following TCG concept, i.e. such that its time is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the actual clock, but so far away as to be considered outside of all gravity wells.

And that means that your clock $r_b$ does not measure proper time along its worldline. It measures something else. We know that because it has to be corrected, whereas clock $r_a$ does not.

On their own $M_a$ and $M_b$ may be very different models. But if each combined with its own matching interpretation yields identical predictions about reality, can either model claim the monopoly on the true proper time?

If both models yield identical predictions about reality, then both models must agree that arc length along the timelike worldline of both clocks, $r_a$ and $r_b$ (since both are following the same worldline), is given by the reading on clock $r_a$, since that is the reading that requires no correction. Not only that, but if we bring in some other clock, $r_c$, which works on some other principle, we will find that its readings match those of clock $r_a$ and not clock $r_b$. (We know this because that's how things actually work in our actual reality. Your wristwatch and your smartphone don't have cesium clocks in them, but they still keep the same time.)

The difference between your two models $M_a$ and $M_b$, in other words, is not that they make different claims about proper time. The difference is that model $M_b$ says that, for some unexplained reason, "TCG coordinate time" is "physically meaningful" even for clocks that are not at infinity and which, without correction, do not keep TCG time (i.e., their proper time is not the same as TCG time), whereas $M_a$ says it's just a coordinate with no physical meaning for clocks not at infinity that don't keep TCG time.

 

[Dale]

at rest in a coordinate frame co-moving with the actual clock, but so far away

This is already problematic. The relative velocity between distant objects in curved spacetime is not well defined. Better just to say it is an arbitrary coordinate time, and that you construct devices which display this coordinate time, like the GPS satellite clocks do.

On their own Ma and Mb may be very different models. But if each combined with its own matching interpretation yields identical predictions about reality

I guess that is a bit of a matter of personal taste. Do you consider Newtonian mechanics and Lagrangian mechanics to be different models? How about Lagrangian vs Hamiltonian mechanics? If you consider them different models then together with the appropriate interpretations would you consider them different theories?

Personally, I wouldn’t. So I also wouldn’t consider your $M_a$ and $M_b$ to be different models. I am not sure I would even consider them to be different mathematical frameworks, any more than I would consider $F=ma$ and $a=F/m$ different models or frameworks. But I believe that is a matter of personal preference.

can either model claim the monopoly on the true proper time?

Words mean what people agree that they mean. We have clocks like what you describe: the satellite clocks in the GPS. As far as I know, nobody calls their time proper time. Also, as far as I know nobody considers using those clocks to be a different model from relativity.

 

[Killtech]

If both models yield identical predictions about reality, then both models must agree that arc length along the timelike worldline of both clocks, $r_a$ and $r_b$ (since both are following the same worldline), is given by the reading on clock $r_a$, since that is the reading that requires no correction.

Yes, but you have to be more careful, because the arc length is not measured absolutely, but rather always represented by a unit. The arc length in units of $r_a$ needs no correction when measured by $r_a$, but requires one when measured by $r_b$. The situation is reversed when the arc length is represented in units of $r_b$. It becomes more complicated when the correction varies locally. The arc length given in (locally) different units won't agree by value but these differing representation have no impact on the predictions made. We just must interpret the unit locally right.

The geometry of space time is not some abstract absolute physical entity independent of everything else, but rather a representation of the devices that are used to measure it, and in particular a description of how they locally behave. If we were to choose different devices to measure and represent the same spacetime, we yield a different geometry.

The difference between your two models $M_a$ and $M_b$, in other words, is not that they make different claims about proper time. The difference is that model $M_b$ says that, for some unexplained reason, "TCG coordinate time" is "physically meaningful" even for clocks that are not at infinity and which, without correction, do not keep TCG time (i.e., their proper time is not the same as TCG time), whereas $M_a$ says it's just a coordinate with no physical meaning for clocks not at infinity that don't keep TCG time.

They don't make different claims about proper time, no, but i would say they define the term differently to align with their interpretations. Other then that, $M_b$ is indeed somewhat arbitrary.

Only considering how much of a problem geometry is for the quantization of gravity, models like $M_b$ may be of theoretical interest. That requires to know how they map to reality in general. However, for the reasons you highlight, particular how much friendlier it is for measurement, $M_a$ will without doubt always remain central.

And while your argument about $r_b$ is generally sound, consider a clock $r_w$ that is defined similar to Caesium but instead of using a pure electromagnetic transition frequency, based as much as possible on the weak force. For example the frequency obtained from a W boson during a muon decay at rest. Such a clock is physically meaningful, albeit much harder to practically construct. Our understanding is not yet good enough to conclude the weak and electromagnetic forces behave relatively the same in gravity fields of all scales. If they were to diverge, we run into the same situation discussed here.

I guess that is a bit of a matter of personal taste. Do you consider Newtonian mechanics and Lagrangian mechanics to be different models? How about Lagrangian vs Hamiltonian mechanics? If you consider them different models then together with the appropriate interpretations would you consider them different theories?

Personally, I wouldn’t. So I also wouldn’t consider your $M_a$ and $M_b$ to be different models. I am not sure I would even consider them to be different mathematical frameworks, any more than I would consider $F=ma$ and $a=F/m$ different models or frameworks. But I believe that is a matter of personal preference.

You are right. These are different representation rather then different models but with the terminology introduced in this thread we would formally need to stick to that. But in terms of theory we can consider them to belong the same theory, but different representations of it (there is nothing suggesting that the separation of a theory into $I$ and $M$ is in any way unique).

You are right, In the end, it's just up to how we define these words - and the personal preference of the person who defines them first.

This is already problematic. The relative velocity between distant objects in curved spacetime is not well defined. Better just to say it is an arbitrary coordinate time, and that you construct devices which display this coordinate time, like the GPS satellite clocks do.

A valid point. I've taken that formulation somewhere from an explanation of how TCG was defined.

And you are right, i am better off just sticking to coordinates then bother too much about geometry. In principle i can define a general coordinate specification (must work for any spacetime) and express all laws of physics within these coordinates. Then i get a geometry independent representation of the theory, and even if the geometry becomes problematic in some regions, the coordinate representation will always be clear what the laws of physics say - i.e. there are never any inter- or extrapolation issues. That may put off the heavy load onto the interpretation but makes my life a lot easier. Though admittedly, finding a general coordinate specification where all laws of physics have at least a somewhat useable uniform form, is just about as hard.

 

[Dale]

They don't make different claims about proper time, no, but i would say they define the term differently to align with their interpretations. Other then that, Mb is indeed somewhat arbitrary.

Having thought some more about this, I don’t think $M_b$ is a different model from $M_a$ at all. They are mathematically equivalent to each other, so mathematically I wouldn’t consider them to be different any more than I would consider $F=ma$ to be a different model than $a=F/m$.

Furthermore, they don’t have a different interpretation. Both will produce the same or equivalent expression for the value measured on a clock. Both will produce the same or equivalent expression for the value measured on one of your adjusted clocks. Both will agree that clocks and adjusted clocks are different devices. Etc.

About the only difference seems to be terminology. But an experimental device is a physical object, not a word. And an interpretation is a mapping between a model and experiment, not a mapping between a model and words about experiments.

So I just don’t see them as different models. The only way that I can see it is if any mathematical manipulation is considered a new model. That is not a meaning that I am willing to accept.

 

[PeterDonis]

arc length is not measured absolutely, but rather always represented by a unit

The unit is provided by the clock that follows the worldline along which arc length is being measured. But your $r_b$ goes beyond that and applies a "correction". That means you are not measuring arc length any more; you are measuring (a better term would be calculating) something else. No amount of belaboring or obfuscation on your part will change that fact, and I see no point in continuing to discuss it.

 

[Killtech]

Having thought some more about this, I don’t think $M_b$ is a different model from $M_a$ at all. They are mathematically equivalent to each other, so mathematically I wouldn’t consider them to be different any more than I would consider $F=ma$ to be a different model than $a=F/m$.

Furthermore, they don’t have a different interpretation. Both will produce the same or equivalent expression for the value measured on a clock. Both will produce the same or equivalent expression for the value measured on one of your adjusted clocks. Both will agree that clocks and adjusted clocks are different devices. Etc.

About the only difference seems to be terminology. But an experimental device is a physical object, not a word. And an interpretation is a mapping between a model and experiment, not a mapping between a model and words about experiments.

So I just don’t see them as different models. The only way that I can see it is if any mathematical manipulation is considered a new model. That is not a meaning that I am willing to accept.

Out of curiosity then, how is Newtons model of gravity actually compared against GT?

Because, here the details of interpretation are not clear to me. One could either say Newtons description is a pure coordinate view and hance should be matched with GR's model of the solar system by the right coordinates interpreted as being the same for both models. Alternatively one could assume Newtons model uses actual distances as measured in meters and hence fails to predict the curvature of spacetime.

These two possible interpretation are very much even based on the same $r_a$ and $r_b$ discussed before (extended to lengths and not just clocks). Note that in either case Newtons model framework $M_F$ remains identical with the very same laws. What is however different is the provision of initial conditions for the solar system, mainly if initial planet positions and velocities are given in coordinates or meters and seconds. Different initial conditions make for different solutions $M_R$, the realization of the solar system model. Applying either one of the interpretations to the same model produces different theories that make different predictions.

These details of interpretations were way ahead of Newtons time, hence i don't think the original model specifies this. In particular Newton did not specify how clocks behave along trajectories nor which clocks to use invalidating neither interpretation. Though i bet, if you asked him back then how an ideal clock should behave, he would naively come up with a description that is more akin to $r_b$ then $r_a$.

Maybe this is an example why $M_a$ and $M_b$ should be distinguished. On their own, namely without knowing their exact specifications on their interpretations, it is not clear how they translate/relate to each other and hence if they are physically equivalent or not.

 

[Dale]

Out of curiosity then, how is Newtons model of gravity actually compared against GT?

They are different models. Mathematically, the Newtonian model can be derived as a limiting case of the GR model, but not vice versa. They are not isomorphic to each other.

One could either say Newtons description is a pure coordinate view and hance should be matched with GR's model of the solar system by the right coordinates

Do you have a professional scientific reference for this? I think this is not just false but obviously false.

Newtons model uses actual distances as measured in meters and hence fails to predict the curvature of spacetime

Curvature of spacetime isn’t the issue. If you use Newton Cartan gravity you have Newtonian gravity with spacetime curvature. Newton Cartan gravity has spacetime curvature and is isomorphic to Newtonian gravity. It is not isomorphic to GR.

Applying either one of the interpretations to the same model produces different theories that make different predictions.

Applying different minimal interpretations to the same model indeed would produce a different theory that makes different predictions. I do not dispute that.

What I strongly dispute is your claim that there exists any possible interpretation that would make GR and Newtonian gravity equivalent. This is an extraordinary claim and thus requires an extraordinarily high quality scientific reference to support it.

 

[Killtech]

What I strongly dispute is your claim that there exists any possible interpretation that would make GR and Newtonian gravity equivalent. This is an extraordinary claim and thus requires an extraordinarily high quality scientific reference to support it.

Well, honestly sorry but i have no idea why you think i have claimed that? It's a bit insulting tbh. Newton's gravity is intentions lacking entirely a field equation so there is no way to translate that. These models are obviously structurally incompatible.

All i wanted to point out is that on a finer look Newtons model is not a single theory because it leaves the interpretation open and there are several possible candidates that lead to different theories, but obviously all of them will be false. It's just that our understanding of GR is what points that out (which is why i mentioned it). I found it noteworthy because it creates the situation that two axiomatically identical models may not be physically equivalent due to differing interpretation.

And I was inspired a bit by Peter's approach to the interpretation as part of the theory: that $I$ prescribes that there should be devices that behave like clocks, but by the time of Newton his idea of a clock would differ. If someone defined how a classical clock is supposed to behave, we probably would find that most probably it's actually possible to construct devices that meet that specification.

 

[Dale]

Well, honestly sorry but i have no idea why you think i have claimed that? It's a bit insulting tbh.

No insult was given nor intended, I did misunderstand your point. I read this:

One could either say Newtons description is a pure coordinate view and hance should be matched with GR's model of the solar system by the right coordinates

as making exactly that claim. It seemed to me that you were saying here that by an appropriate coordinate transform Newtonian gravity should match GR. I must confess that even knowing now that you are not making that claim, I cannot understand this sentence in any other way than I did.

Newtons model is not a single theory because it leaves the interpretation open and there are several possible candidates that lead to different theories

Yes, I agree. This is generally true. A theory is a mathematical model and an associated minimal interpretation, so different minimal interpretations of the same mathematical model yield different theories.

 

[PeterDonis]

Newtons model is not a single theory because it leaves the interpretation open and there are several possible candidates that lead to different theories

Do you have a reference for this? I have never heard of multiple candidate interpretations of Newtonian physics.

 

[Killtech]

as making exactly that claim. It seemed to me that you were saying here that by an appropriate coordinate transform Newtonian gravity should match GR. I must confess that even knowing now that you are not making that claim, I cannot understand this sentence in any other way than I did.

Oh, i see the word "match" is misleading here. What i meant was merely that in order to make a comparison of two models one must first identify something that is supposed to represent the same thing. Naively i would assume one would try to identify each point in spacetime in one theory to their correspondence in the other. Considering Newtons model is based on early analytical geometry, trying to find agreeing coordinates for that purpose sounded reasonable. So it would be only the coordinates that matched but certainly not the solutions.

I don't know much about Newton-Cartan, but it seems it skips the question of how to interpret Newton altogether and instead attempts to depict gravity in the same way as GR does. But is it the same theory as Newtons original? Note that in a curved spacetime, a planets orbit has a slightly different length then in a flat geometry. Is Mercuries lack of perihelion precession in the model the same in Newton-Cartan theory as in the original?

Do you have a reference for this? I have never heard of multiple candidate interpretations of Newtonian physics.

The theory is wrong either way and the impact of diverging interpretation will be quite limited. It thus offers little benefit to iron out the details for the purpose of measurement. Therefore I don't see people spending time investigating it and i don't have a reference.

Yet, you brought up an interesting point:

Both $M$ and $I$ are part of the theory, but that means we have to be careful about what $I$ actually means. $I$, if we view it as part of the theory, leads to the theoretical claim $I$ described before: that there should exist devices that behave like the "clocks" $I$ describes.

Clocks aren't mentioned in Newtons theory, but if we were to ask how Newton might have understand time, the idea of a classical clocks would be little else then measuring the time coordinate of Cartesian spacetime coordinates. Today we can however say that such a classical view won't be reflected by Caesium based clocks so technically it would lead us to conclude that Newtons $I$ implies another device.

The question is interesting beyond Newton, because most scientific theories use some concept of time.

 

[PeterDonis]

The theory is wrong either way

AFAIK there is no "either way". I'm only aware of one interpretation of Newtonian physics. Yes, the theory is wrong under that interpretation.

i don't have a reference.

Then you shouldn't have made the claim. Please don't clutter the thread with claims that you can't back up with a reference. Particularly if, when challenged, your response is that the claim didn't really matter to the discussion anyway.

Clocks aren't mentioned in Newtons theory, but if we were to ask how Newton might have understand time, the idea of a classical clocks would be little else then measuring the time coordinate of Cartesian spacetime coordinates. Today we can however say that such a classical view won't be reflected by Caesium based clocks

You are confusing two different things. The fact that Newton's theory gives wrong predictions about how clocks actually behave does not mean Newton's theory must have had a different conception of clocks. Newton's theory accepts cesium clocks as clocks; it just makes wrong predictions about how they behave (because it makes wrong predictions about how clocks behave in general).

 

[DanMP]

A "clock" is a device that, to some approximation, reads proper time along its worldline;

Really? How is that device instructed (& capable) to do such thing?

As far as I know, a clock is a device that counts repetitive events.

 

[PeterDonis]

Really? How is that device instructed (& capable) to do such thing?

It doesn't have to be.

As far as I know, a clock is a device that counts repetitive events.

And that event count corresponds to arc length along the clock's worldline. (The type of event being counted--for example, vibrations of a particular atomic transition--gives the units in which the arc length is being measured.) That is just a fact of spacetime geometry. You might as well ask how a ruler is "instructed" to measure length.

 

[jbriggs444]

Really? How is that device instructed (& capable) to do such thing?

As far as I know, a clock is a device that counts repetitive events.

A clock is a device that measures the passage of time. One can do that by counting repetitive events. Or one can do that by observing the progress of a continuous process.

For instance, an hourglass with markings for the sand level, the progress of a child into adulthood and eventual retirement, the changing of the seasons or the angle of the sun in the sky.

Even today, demolition experts use fuses (e.g. detcord) as clocks.

 

[Dale]

Really? How is that device instructed (& capable) to do such thing?

There are many different types of clocks, each type is designed differently. Which type would you like to learn about?

As far as I know, a clock is a device that counts repetitive events.

Not necessarily. You could build a clock that measures time through the radioactive decay of some sample. And historically there have been clocks that measured time by dropping fine sand through a small opening or by burning a candle.

So while your description describes many clocks, it certainly does not cover all possible clocks nor even all extant clocks.

 

[DanMP]

A clock is a device that measures the passage of time. One can do that by counting repetitive events. Or one can do that by observing the progress of a continuous process.

For instance, an hourglass with markings for the sand level, the progress of a child into adulthood and eventual retirement, the changing of the seasons or the angle of the sun in the sky.

Even today, demolition experts use fuses (e.g. detcord) as clocks.

Ok, I agree with you, but I can't see any connection/agreement with
PeterDonis's claim: A "clock" is a device that, to some approximation, reads proper time along its worldline;

How can the hourglass read proper time? And why when is tilted on a side or used in imponderability, the reading of proper time fails?

And that event count corresponds to arc length along the clock's worldline. ...

We can make/choose this interpretation, yes, but the clock is only counting, not reading proper time.

 

[Dale]

How can the hourglass read proper time?

An hourglass consists of two large reservoirs and a narrow neck. It contains enough sand to fill one reservoir and the sand is uniform and fine enough to barely pass through the narrow neck.

When the hourglass subjected to a constant proper acceleration and oriented such that a reservoir containing sand is directly above the other, then the rate of sand falling through the neck is constant. This constant rate of sand falling can then be used to measure time.

And why when is tilted on a side or used in imponderability, the reading of proper time fails?

Because the rate of sand falling is zero. This is the same reason why the reading fails when oriented correctly but all of the sand is in the lower reservoir. This type of clock is sensitive to the amount of proper acceleration as well as its orientation with respect to the proper acceleration.

We can make/choose this interpretation, yes, but the clock is only counting, not reading proper time.

As far as I can see this is a distinction without a difference. We built the clock for the purpose of reading proper time. One way we can accomplish that purpose is by counting oscillations of a frequency standard. So in what way is “counting not reading proper time”?

To me, this claim is like saying “a bridge only supports shear stresses it does not span a river”

 

[vanhees71]

Ok, I agree with you, but I can't see any connection/agreement with
PeterDonis's claim: A "clock" is a device that, to some approximation, reads proper time along its worldline;

How can the hourglass read proper time? And why when is tilted on a side or used in imponderability, the reading of proper time fails?

An hourglass reads its proper time when gauged accordingly. It won't work as a clock in a (local) inertial (free-falling) reference frame, because then it wouldn't show any time.

Instable elementary particles with their lifetime are pretty robust clocks, as has been demonstrated in accelerator experiments.

The best prospect for a really robust clock for practical purposes is a nuclear atomic clock, i.e., the Thorium clock. Just last weak there was another breakthrough in directly observing the photons:

https://www.nature.com/articles/s41586-023-05894-z

 

[DanMP]

Because the rate of sand falling is zero. This is the same reason why the reading fails when oriented correctly but all of the sand is in the lower reservoir. This type of clock is sensitive to the amount of proper acceleration as well as its orientation with respect to the proper acceleration.

Correct, but what I asked is how is the hourglass sensitive to proper time, in order to "read" it.

As far as I can see this is a distinction without a difference. We built the clock for the purpose of reading proper time. One way we can accomplish that purpose is by counting oscillations of a frequency standard. So in what way is “counting not reading proper time”?

Counting is just counting. 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.

I would say: we built the clock for the purpose of measuring proper time (not reading proper time).

To say that clocks read proper time is like saying that rulers read distances. No, we use rulers to measure distances and we use clocks to measure time intervals (or proper time if you want).

 

[PeterDonis]

How can the hourglass read proper time?

Because the falling of sand in the hourglass is a process that happens at a certain rate--in your "counting" metaphor, it is counting grains of sand that fall, and each grain of sand falling corresponds to an increment of arc length along the hourglass's worldline.

why when is tilted on a side or used in imponderability, the reading of proper time fails?

Because no grains of sand are falling, so you have taken away the thing whose counts correspond to arc length along the hourglass's worldline.

We can make/choose this interpretation, yes, but the clock is only counting, not reading proper time.

You're quibbling. The "counting" is "reading proper time"; the counts that the clock is counting correspond to increments of arc length along the clock's worldline, and arc length along the clock's worldline is proper time.

You are making this much harder than it needs to be.

 

[PeterDonis]

To say that clocks read proper time is like saying that rulers read distances. No, we use rulers to measure distances and we use clocks to measure time intervals (or proper time if you want).

Again, you're quibbling. "Read" and "measure" are the same thing in this context.

 

[DanMP]

Again, you're quibbling. "Read" and "measure" are the same thing in this context.

Ok, maybe the fact that English is not my first language is the problem. I'm sorry about that.

Thank you all for your input and patience.