I Time Measurement in Extremely Curved Space Regions

[Killtech]

In the beginning, there was time - at least within the SI system. Second is the first unit defined and it has no dependency on any other unit and is only defined via a physical entity, a Caesium 133 hyperfine transition frequency. Since clocks aim to meet that standard, it effectively is a definition of a clock's tick rate.

I was having some issues with the definition of time/clocks in some extreme situation though - i.e. in regions of space with extreme curvature. The Caesium atom cannot be understood as a pointlike object in general, so what happens when the geometry becomes non-trivial over the area the atom covers? At first, even if we were to assume it affects its spectrum, it will affect the frequency standard of clocks based on it in just the same way (by def) such that the defining hyperfine frequency won't numerically change - because anything measured in units of itself must be constant, obviously.

But on the other hand, if the geometry becomes non-trivial, the orientation of the Caesium may become relevant and it's not unthinkable that the defining transition might split in a Stark/Zeeman-like effect but for gravity. That would render the SI definition not well defined. But if the concept of a clock becomes unclear, then so does the concept of proper time defined by it, which is tied to the temporal part of the metric tensor. This in turn means the geometry becomes blurred, yet it is fundamental to even establish what theoretically happens to Caesium in such circumstance. A recursive problem.

In physics, time is defined by its measurement: time is what a clock reads. But this needs a clarification of what even counts as a clock. Given how a clock uniquely implies a part of the metric tensor and in return the temporal part of the tensor uniquely implies what that clock measures in all circumstance, these concepts could be seen as equivalent. But there is no one clock. Nature did not choose the Caesium clock standard - we did - and i realized that there are different choices of reference oscillators usable as a standard but they exhibit different behavior. The same problem persists in geometry: given a metrizable topological manifold, there is no unique choice of a metric (meaning geometry) either but instead there are many topologically equivalent options (that aren't isometric).

Naively my first idea was to use the time of a remote clock in a friendly region: i.e. utilizing theoretical radio clocks adjusted for signal travel time. I found the Geocentric Coordinate Time (TCG; French) somewhat aligns with that idea - except I was thinking of local (platonic) devices that are able to measure the value of TCG anywhere. Taking measurements of such a TCG clock device at face value, time passes at a different rate then for Caesium clocks at various locations, since TCG simulates a Caesium clock shielded from any local gravity influence (conveniently resolving all issues that come with it). But if i were to apply the concept of proper time to these clocks, i end up with something else, yielding a different metric tensor and consequently measuring the same reality in a different geometry.

Every event has a well defined TCG time though, hence it as a valid measure of time. Besides, the definition of second specified that Caesium must be shielded from influence like electric fields, yet must not be shielded from gravity - that seems like an practical yet arbitrary choice. Or Actually it has to be "unperturbed", so technically, if extreme gravity becomes a problem, maybe even TCG is then more in line with that spec? Lacking a theoretical specification what a clock is, what is preventing us to use TCG device as an alternative clock standard? As in not just a coordinate, but a real time unit alternative to the second?

More generally, how are we supposed determine what the "right" rate is at which time passes at a location? Given two clocks, how are we supposed to decide which gives the "right" time?

Alternatively i figured that Einsteins postulates may implicitly define a time standard/clock for which they always hold - there will be only one proper time and length (i.e. metric tensor) such that Maxwell always keeps its well known form. i would think the theoretical clock this implies is always well defined even if Caesium might not be able to follow it in all circumstance. Now the issue with a pure theoretical clock is that a theory based on it cannot be compared with experimental data using another clock. one would first have to calculate the local SI Caesium transition frequency within that theory to understand how the two clocks are related and derive a transformation along each worldline. But again, that's a transformation of physics between two geometries.

How does modern physics approach this issue?

 

[PeterDonis]

if the concept of a clock becomes unclear, then so does the concept of proper time defined by it

No, you have things backwards. The concept of proper time, in the theory, is logically prior to the concept of a clock that measures it. So even under conditions where our definition of the clock might not work, such as in a region with very strong spacetime curvature, the concept of proper time still works. We might not know how to construct a device that measures it, but the theoretical model that uses it still works just fine.

If you want to object that the theory might not be testable in such a region, since we would have no way of obtaining any observational data to compare with its predictions, that is true if we define "testable" in terms of what we can currently measure with our current technology, but it is still beside the point. We can always change our definition of the SI second, or our physical model of what device we call a "clock", as our technology and physical understanding improves. (Indeed, the current definition of the SI second is precisely the result of such improvements over time.)

In physics, time is defined by its measurement: time is what a clock reads.

No, this is not what defines time. What defines proper time in GR is geometry: it is arc length along a timelike curve. A "clock" is a device that, to some approximation, reads proper time along its worldline; but such a concept cannot even be defined without the underlying geometric concept.

a clock uniquely implies a part of the metric tensor

It does no such thing. You don't even have to have a timelike coordinate at all in order to write down the metric tensor.

given a metrizable topological manifold, there is no unique choice of a metric (meaning geometry) either but instead there are many topologically equivalent options (that aren't isometric).

This is true as a matter of abstract mathematics, but irrelevant as a matter of physics. In GR, for example, we don't just pull a metric out of thin air. We compute a metric by solving the Einstein Field Equation with a given stress-energy tensor. Sure, any such metric we compute will be just one of an infinite number of mathematical possibilities for the same underlying topological manifold; but that is irrelevant to our construction of a physical model, because we are using other data besides just the nature of the topological manifold to determine the metric in our model.

my first idea was to use the time of a remote clock in a friendly region

This is obviously wrong given what I said above about what clocks do. A clock cannot measure time elsewhere than on its own worldline.

Every event has a well defined TCG time

Wrong. Try extending your "TCG time" inside the horizon of the black hole at the center of our galaxy. It won't work.

how are we supposed determine what the "right" rate is at which time passes at a location? Given two clocks, how are we supposed to decide which gives the "right" time?

By understanding that the theoretical model is a model of much more than time. It's a model of spacetime geometry. Proper time, as already noted above, is a geometric property: arc length along a timelike worldline. Obviously you can't consider such a thing in isolation; you have to consider it in the context of the entire model.

Note, btw, that since arc length along a timelike worldline is an affine parameter, there is no single "right" proper time. Given a proper time $\tau$ along a particular worldline, $\pi = a \tau + b$, where $a$ and $b$ are real numbers, is an equally valid proper time along the same worldline. So if you are trying to compare two clocks, you need to take that into account.

Einsteins postulates may implicitly define a time standard/clock for which they always hold - there will be only one proper time and length (i.e. metric tensor) such that Maxwell always keeps its well known form

This is obviously false as well, since Maxwell's Equations can be formulated in any curved spacetime.

 

[Dale]

what happens when the geometry becomes non-trivial over the area the atom covers?

We have no empirical data from this regime and no particularly promising theories either. I don’t think this part of the question is worth worrying about, nor the proposed solutions (if it ain’t broke, don’t fix it).

However one other part is worthwhile.

Given two clocks, how are we supposed to decide which gives the "right" time?

Let’s take a step back.

Clocks exist. The reading that they give is a useful quantity. Useful things are often given names. In the case of clocks, the name we have given to the useful reading they produce is “proper time” or “time” for short.

So there is no intrinsic metaphysically “right” time. However, there are clocks which agree with each other well and clocks which do not. Clocks that agree with each other are more useful than clocks that don’t. And the whole point of giving time a name in the first place was because the reading of a clock is useful.

If we build several identical clocks of one type, and find that they agree with each other better than several identical clocks of another type, then the first type is a better clock than the other type. There is no need for any metaphysical concept of the “right” time in order to prefer one type over another. They agree more, so they are more useful, so we prefer them.

Now, within a given type of clock, how are we to choose among individual clocks? Again, we take several identical clocks and compare them to each other. Any that is an outlier is less preferred than ones that are typical. The agreement itself is useful, and hence preferred.

 

[Dale]

No, you have things backwards. The concept of proper time, in the theory, is logically prior to the concept of a clock that measures it.

I am not so sure about this. In order for a physical theory to be a theory it must have both a mathematical framework and a mapping to experiment (the minimal interpretation). I don’t think either part can be considered to have logical priority in the theory. Without both you don’t have a theory, so how could either have theoretical priority.

 

[PeterDonis]

In order for a physical theory to be a theory it must have both a mathematical framework and a mapping to experiment (the minimal interpretation).

I agree with this, but I would emphasize the "minimal" part. Saying that the minimal interpretation of a GR spacetime model requires one to have measuring devices called "clocks" that are treated as measuring proper time along their worldlines is a much weaker statement than saying that "clocks" have to be cesium atomic clocks that are calibrated in SI seconds. Ultimately, we decide whether actual clocks are doing a good enough job of measuring proper time by looking at the geometric implications of the measurements--do the measured arc lengths along various timelike curves match up with the spacetime geometry in the model.

I don’t think either part can be considered to have logical priority in the theory.

I would view this as follows: the theoretical claim being made is that it is possible to find actual measuring devices that behave, to a good enough approximation, like the idealized "clocks" in the model, that directly measure arc length along their timelike worldlines. Then we find, as a matter of empirical fact, that this theoretical claim turns out to be correct: we can in fact find actual devices that satisfy that theoretical claim.

But that theoretical claim, logically, comes after the theoretical model that treats spacetime as a geometric object and proper time as the geometric arc length along timelike curves. Logically, we construct that model first, and then we infer what kinds of measuring devices we would need to have in order to test it. Yes, to qualify as a physical theory we have to include the results of those inferences about measuring devices, but those results are not given to us a priori or independently, logically, of the theoretical model. We only know we need "clocks" as measuring devices because our theoretical spacetime model includes timelike curves.

 

[DaveC426913]

Is this anything but a technology shortcoming?

Our current best clocks get janky when they are in the presence of spacetime curvature over distances on the order of an atom's span.

A couple of centuries ago our clocks got janky when they were on a heeling ship. It didn't make physics grind to a halt, it just meant we needed to up our clock game.

 

[Ibix]

Is this anything but a technology shortcoming?

Well, it's worth noting that when we get to curvature on atomic scales we may need a quantum theory of gravity, because we don't understand how quantum sources of gravity work. But real atomic clocks will fail when there's significant curvature on the scale of the whole clock, and yes that is more of a technological issue than a philosophical one.

 

[Dale]

Saying that the minimal interpretation of a GR spacetime model requires one to have measuring devices called "clocks" that are treated as measuring proper time along their worldlines is a much weaker statement than saying that "clocks" have to be cesium atomic clocks that are calibrated in SI seconds

I completely agree with this. SI is just a unit system, and by design it is useful for ordinary scenarios. There is no reason in principle that its limitations in other scenarios should hinder the theory. And in practice the scenario itself doesn’t arise.

But that theoretical claim, logically, comes after the theoretical model that treats spacetime as a geometric object and proper time as the geometric arc length along timelike curves.

Here, I think we may disagree. In my opinion we don’t have a theory until we have the minimal interpretation as well as the mathematical model. So with just the mathematical model we cannot make a theoretical claim because we don’t have a theory.

We agree that the minimal interpretation doesn’t require specifically SI clocks based on cesium. But in making the minimal interpretation we do specify that a timelike spacetime interval is measured with clocks and not with voltmeters, for example. Since that is part of the minimal interpretation, without it we do not have a theory.

I may be misunderstanding the point you are making.

 

[vanhees71]

Is this anything but a technology shortcoming?

Our current best clocks get janky when they are in the presence of spacetime curvature over distances on the order of an atom's span.

Indeed, and there's hope that soon we'll have a more robust "nuclear clock", where instead of an atomistic a nuclear em. transition is used as a frequency standard. That's the Thorium clock, which has a unusually soft transition in the UV regime:

https://en.wikipedia.org/wiki/Nuclear_clock

A couple of centuries ago our clocks got janky when they were on a heeling ship. It didn't make physics grind to a halt, it just meant we needed to up our clock game.

 

[PeterDonis]

when we get to curvature on atomic scales we may need a quantum theory of gravity

Not necessarily. The atomic scale is still some 20 to 25 orders of magnitude larger than the Planck scale, which is where physicists currently believe a quantum theory of gravity will be necessary.

 

[PeterDonis]

In my opinion we don’t have a theory until we have the minimal interpretation as well as the mathematical model. So with just the mathematical model we cannot make a theoretical claim because we don’t have a theory.

This is a limitation of the ordinary language we are using. Let me try restating using symbols. We have a model $M$, and we have a minimal interpretation $I$ that says which particular numbers in $M$ correspond to physically observed quantities. So, for example, $M$ might contain a 4-dimensional spacetime and a timelike curve in that spacetime, and $I$ might say that arc length along that timelike curve corresponds to the time elapsed on a clock which has that curve as its worldline. Note that we cannot even formulate $I$ until we have $M$ and know that it includes timelike curves with arc lengths; so logically, $M$ is prior to $I$.

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. Call that claim $C$. Then $C$, logically, depends on $I$ already having been formulated as above; we can't formulate the claim that "clocks" should exist in the world until we know from $I$ as stated above what "clocks" are according to the theory. So logically, $I$ is prior to $C$.

The theory, by itself, cannot tell us whether claim $C$ is true. Testing that claim is part of the empirical testing of the theory, not the theory itself. So we don't know, just as a matter of theory, that the "clocks" that appear in $I$ are in fact the "clocks" of our ordinary everyday experience (or more technologically advanced forms thereof). We only know that because we have tested the theory and found that it works.

 

[vanhees71]

I think this is the same debate we had some years ago. Of course, to have a physical theory, you need both a mathematical model and real-world measurement devices that with sufficient accuracy can realize the theoretical, idealized observables.

It's naturally most difficult for what are really fundamental notions already of the theory, and the concepts and quantification of space and time are among such notions, and the difficulty is as old as modern physics (modern in the sense of the physics starting to develop around 1600). Already in Newtonian physics it was not easy to get the theoretical, idealized conceptions of space and time formulated in a straight, consistent way. E.g., to establish the notion of an inertial frame, of course you already need the Newtonian spacetime model with space being an affine Euclidean 3D manifold and time as an independent parameter (mathematically a fiber bundle). The debate started immediately, roughly between the arch enemies Newton and Leibniz. While Newton postulated an absolute space (and also an absolute time) as a priori given, Leibniz already realized that there is no way to physically realize the absolute space, i.e., as is pretty obvious from everyday experience, physically space manifests itself in the (quantitative) spatial relations between bodies (idealized as "points"). I'm not aware whether he also doubted or discussed already a possibility that also time should be somehow "relative".

In any case the great success of Newton's space time model is of course only possible, because with some satisfactory accuracy time could be measured by (pendulum) clocks and lengths measured by yard sticks and in this way also inertial frames of reference could be realized with some accuracy too. Indeed, for most everyday-world situations a reference frame with a point of the Earth consisting in principle of a clock and three non-complanar sticks fastened at this point on Earth and being at rest relative to this point, is sufficient. That it's indeed of course a rotational non-inertial frame, is of course also clear when looking in more detail and building, e.g., a Foucault pendulum.

Of course nowadays, our time and length measurements are much more accurate, and we have general relativity as the most comprehensive spacetime model, but the general scheme is the same: We have on one hand a concise mathematical description about space and time, but to make it a physical theory about the real world it must also be possible to realize the mathematical concepts, defining (idealized) observables like durations of time and distances between bodies, etc. in terms of real-world technical equipment, i.e., "clocks" and "yardsticks" suitable to test the mathematical description and apply it to real-world objects.

So far obviously we haven't come to situations, where the realization of observations testing the GR spacetime model, was impossible. E.g., with pulsars we have utmost precise "clocks", who send (electromagnetic) signals to us which we can measure with utmost accuracy with our very precise clocks, and this "pulsar timing" results in ever more precise confimation of the predictions of GR. AFAIK it's usually done by determining the post-Newtonian parameters describing the motion of, e.g., binary-star systems, and it's found that it agrees with high precision for the parameters to 5 or 6th order of the post-Newtonian expansion. Also the predictions of the gravitational-wave signals from black-hole mergers agree with high precision with GR. So it seems as if we are really far from the point, where we are able to observe such extreme conditions that GR fails to describe all these observations.

 

[Dale]

This is a limitation of the ordinary language we are using. Let me try restating using symbols.

That is indeed helpful.

Note that we cannot even formulate $I$ until we have $M$ and know that it includes timelike curves with arc lengths; so logically, $M$ is prior to $I$.

I agree, however you left out one other important antecedent, which I think gets at my point. $I$ is a mapping between the model $M$ and experiments $E$, including previous experiments and possible subsequent experiments. Since $I$ is a mapping between $M$ and $E$, both are logically prior to $I$. But $M$ is not logically prior to $E$.

$I$, 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. Call that claim $C$.

I think this is the difference. The existence of clocks is already contained in $E$. That is not a theoretical claim, IMO, because it is independent of any theory. It belongs purely to $E$. What is the corresponding theoretical claim is that the clock readings in $E$ map to the arc length of a timelike curve in $M$.

 

[PeterDonis]

Since $I$ is a mapping between $M$ and $E$, both are logically prior to $I$. But $M$ is not logically prior to $E$.

I see what you mean. However, I would also say that $E$ is not logically prior to $M$. Both are logically prior to $I$, and all of them are therefore logically prior to claims like claim $C$ that I described in my last post.

I think this is the difference. The existence of clocks is already contained in $E$.

But now we are using the word "clocks" to refer to two different things: the "clocks" in $M$ and the "clocks" in $E$. We really should use two different terms for these two different things; say "model-clocks" (for the clocks in $M$) and "experiment-clocks" (for the clocks in $E$). The mapping $I$ gives a relationship between them, but that does not mean that, for example, you can deduce that model-clocks must exist in $M$ because experiment-clocks exist in $E$. Logically, it's perfectly possible that the readings of experiment-clocks are accounted for in some other way in the model. The knowledge that model-clocks exist in $M$ and that their readings correspond to those of experiment-clocks in $E$ only comes after we have the mapping $I$ and have verified that the model works.

 

[Dale]

I see what you mean. However, I would also say that E is not logically prior to M. Both are logically prior to I, and all of them are therefore logically prior to claims like claim C that I described in my last post.

Yes, I completely agree.

But now we are using the word "clocks" to refer to two different things: the "clocks" in M and the "clocks" in E. We really should use two different terms for these two different things; say "model-clocks" (for the clocks in M) and "experiment-clocks" (for the clocks in E).

I get your point. I think I would put clocks firmly in $E$, but the issue you raise does happen with many terms, such as proper time.

If someone asks “what is proper time?” then one valid response is “the thing a clock measures”, referring to $E$. Another equally valid response is “the spacetime interval along a timelike worldline” referring to $M$.

Ideally, we should use different words. Occasionally this leads to people getting weird impressions like we believe that the mathematical model causes the physics. So I agree that it does cause occasional headaches that could be avoided with more explicit language.

 

[Killtech]

This is a limitation of the ordinary language we are using. Let me try restating using symbols. We have a model $M$, and we have a minimal interpretation $I$ that says which particular numbers in $M$ correspond to physically observed quantities. So, for example, $M$ might contain a 4-dimensional spacetime and a timelike curve in that spacetime, and $I$ might say that arc length along that timelike curve corresponds to the time elapsed on a clock which has that curve as its worldline. Note that we cannot even formulate $I$ until we have $M$ and know that it includes timelike curves with arc lengths; so logically, $M$ is prior to $I$.

Note that any model $M$ actually is made of two things: the technical axiomatic definitions, equations etc. that define the general model framework $M_F$ and the concrete model realization $M_R$. For the issue you raise, we don't need any physical assumption other then geometry itself, so let's take Riemanns theory as $M$. $M_R$ will then be be a Riemann manifold, e.g. a sphere with radius 1, or the curved space time around earth.

Now your approach neglects that there is also a different way to look at the problem: given $M_F$ and experiments $E$ we can check if there is anything in their observations that satisfies the assumptions of $M_F$ (e.g. what are a valid measurements of time / proper time) and if so establish that as a connection with reality $I$. With that set, we can now establish $M_R$ from more observations $E'$ - that is we survey the geometry of space time around earth through measurements just having a geometry model and specifications $I$ what we accept as a measure of time (and space).

So there is no intrinsic metaphysically “right” time. However, there are clocks which agree with each other well and clocks which do not. Clocks that agree with each other are more useful than clocks that don’t. And the whole point of giving time a name in the first place was because the reading of a clock is useful.

If we build several identical clocks of one type, and find that they agree with each other better than several identical clocks of another type, then the first type is a better clock than the other type. There is no need for any metaphysical concept of the “right” time in order to prefer one type over another. They agree more, so they are more useful, so we prefer them.

Now, within a given type of clock, how are we to choose among individual clocks? Again, we take several identical clocks and compare them to each other. Any that is an outlier is less preferred than ones that are typical. The agreement itself is useful, and hence preferred.

(continuing the above response)

However, we find that with a general framework $M_F$ there is more then one thing in reality that satisfies its concepts - hence multiple $I$ are possible and valid. Riemann geometry does clarify what a measuring devices for space and time must fulfil to be usable to survey the geometry of an manifold - specifically in the context of measuring at different locations and frames. Note that clocks of one type may disagree with clocks of another, even though every clock agrees with clocks of the same type very well. You will find that $M_F$ is then applicable to both types individually. For example we can make the claim that only a clock outside earths gravity well is unperturbed, hence accept only the radio signal of such clock to measure time on earth (so we have another $I'$) - i.e. we measure time in something akin to TCG-second instead of second. In context of Riemann geometry this is perfectly fine to map it to proper time but the consequent survey of space time around earth based on $I'$ will yield another Riemann manifold $M'_R$.

Now let's extend $M_F$ with Einstein's assumption about Maxwell (including in curved spacetime; @Peter you completely misunderstood me there earlier) and call it $M'_F$. The many possible interpretations are disqualified and only one is left: if we were to measure the time it takes light in vacuum to move 1 meter there and back again, when measured in seconds we will always get the same result no matter at what location the experiment is done. But measured in TCG-seconds the results will in fact vary depending how deep we stand in earths gravity field. So only this disqualifies the TCG-second as a proper time of GR because it violates Einsteins assumptions on Maxwell.

The question what type of measurements we map to the concepts of proper time and length in the model $M'$ seem to be calibration entities to make the model fit reality. In that sense, the model $M'$ in fact precedes $I$ by dictating what result for proper time we have to measure at each frame and location and we now just need to build clocks that conform with it.

Now there is nothing wrong with such an approach, but this has implications to what proper time and $M'$ then mean. In particular we must be careful what about the model is indeed testable because many tests will merely reproduce what we put into the calibration: hypothetically if we observed an experiment with the violation of time dilatation, it would be illogical to say $M'$ is wrong but rather conclude the clock used to determine time dilation was not conforming to $M'$ specifications of proper time.

 

[Dale]

Note that clocks of one type may disagree with clocks of another, even though every clock agrees with clocks of the same type very well

I don’t think this is true.

For the rest, you have more than doubled the number of symbols, throwing in new concepts at a rapid fire pace. I have no clue what your point is.

from more observations E′ -

This is unnecessary. $E’$ is already in $E$.

any model M actually is made of two things: the technical axiomatic definitions, equations etc. that define the general model framework MF and the concrete model realization MR.

A concrete realization of $M$ would be an element of $E$, not an element of $M$.

I find your whole line of reasoning here unconvincing. Why don’t you start over and try to justify each of these new concepts you are introducing one by one.

 

[Killtech]

I find your whole line of reasoning here unconvincing. Why don’t you start over and try to justify each of these new concepts you are introducing one by one.

Hmm, sorry. I'll try; no promises

Maybe let's start with a old model, like Newtons theory: made of 3 laws, build on top of the calculus of analytic geometry available at his time and one law describing the force of gravity. All these axioms define the framework of the model $M_F$, similar like the set theory is based on 9 axioms.

Yet on their own they are unable to offer any predictions at all. Like any physical theory, Newtons model requires to know the initial conditions in order to make any predictions. We need to know the masses of planets, the sun, their initial positions and velocities so we are able to predict their movement. Only given a complete set of initial conditions, the framework gives concrete predictions for all objects at all times: a complete model of the solar system $M_R$. Think of it as the solution space for Newton's differential equations applied to the specific initial conditions.

But Newton's theory all by itself is unable to provide the initial conditions. Those have to be surveyed by measurement beforehand. For that we need to know how to map real measurements to what they correspond to in the theory. This is the role of interpretation $I$. So initially the model must be 'calibrated' by observations. Maybe the term of calibration isn't usually applied to finding the initial positions, but measuring the masses of planets, which are constants in this case is more fitting. Anyhow this process requires $I$ to precede $M$, since it is the component needed to make $M$ into a complete model for a particular application.

Looking now at another theory, Riemann's geometry, it's axiomatic set on it's own leaves a lot of things open. If we want to apply it to reality, we also have to start it up with 'initial conditions'. Since it does not make predictions about any time evolutions, the Initial conditions practically requires us to survey the geometry at every spacetime location experimentally.

If we were to strictly apply Peter's declaration that $M$ precedes $I$, we can start with an arbitrary geometry of proper dimensions $M_R$ and its metric tensor. Now that model will dictate at each worldline what a clock has to measure and we just have to build devices that reproduce these results. This approach is in fact equally valid, but because Riemann's axioms aren't complete, it works in fact for any geometry that has the same topology as the reality you survey.

did this help?

 

[PeterDonis]

Note that any model $M$ actually is made of two things: the technical axiomatic definitions, equations etc. that define the general model framework $M_F$ and the concrete model realization $M_R$.

You're going to need to be much more explicit about this split, since as far as I can tell you're making it up out of whole cloth. In the particular case under discussion, where in my formulation the model $M$ consists of a 4-d spacetime geometry and a timelike curve within that geometry, what part of that is your $M_F$ and what part is your $M_R$?

 

[PeterDonis]

Peter's declaration that $M$ precedes $I$

But "initial conditions" are part of $E$, not $I$, and I did not say that $M$ logically precedes $E$.

Also, the conditions do not have to be "initial". A better term would be "model parameters". For example, say we are considering Schwarzschild spacetime and a particular timelike curve within that spacetime. "Schwarzschild spacetime" is not a single model, it's a family of models; to pick out one particular model within the family, we have to specify the value of the parameter $M$ that appears in the metric. But that is all we have to specify. We don't have to specify anything else--in particular, we don't have to specify $I$, the general mapping between numbers in the model and observable quantities. We could get the value of $M$ we use from observation (for example, we could say it's the mass of the Earth), but that's still not the same as specifying a complete mapping $I$ between the model and observations. Nor is it the same as specifying what real world devices correspond to "clocks" (or "rulers") in the model.

 

[Dale]

like Newtons theory: made of 3 laws, build on top of the calculus of analytic geometry available at his time and one law describing the force of gravity. All these axioms define the framework of the model MF, similar like the set theory is based on 9 axioms.

Yet on their own they are unable to offer any predictions at all. Like any physical theory, Newtons model requires to know the initial conditions in order to make any predictions.

I see what you are saying here, you are making a distinction between the equations and the solutions to the equations. I am lumping them all into $M$ and you would like to split them into $M_F$ and $M_R$. I have two potential problems with that.

1) I don’t think that the claim that only the solutions to the equations can make predictions is correct. Using conservation of energy I can predict that a satellite orbits fastest at periapsis and slowest at apoapsis. No initial conditions needed. I can also predict that its speed at periapsis will be the same each orbit. There are many such predictions that can be made. Conservation laws are very powerful ways to generate predictions independent of details.

2) I don’t see the value of this separation in general. What about circuit theory where KVL and KCL aren’t differential equations and don’t need initial or boundary conditions? What about situations like Newton’s laws that can be expressed in terms of forces or in terms of a Lagrangian such that they have the same $M_R$ but distinct $M_F$? It seems to me that the distinction is not so clear or useful.

Since we can make predictions with both $M_R$ and $M_F$ (eg conservation laws) and since sometimes $M_R$ and $M_F$ are not clearly distinct (laws not in the form of differential equations) and since multiple $M_F$ correspond to the same $M_R$ I just question the value of the distinction.

Peter's declaration that M precedes I,

To be clear, this isn’t Peter’s declaration. This is a logical necessity. $I$ is a map between $M$ and $E$. So logically you must have both $M$ and $E$ before $I$. There is no avoiding this necessity.

Anyhow this process requires I to precede M,

This, in itself is a cause to question the validity of the proposal. Since $I$ is a mapping between $M$ and $E$ it cannot precede either. That your reasoning leads to this shows there was a critical flaw. I think the flaw is the attempt to split $M$.

 

[Killtech]

You're going to need to be much more explicit about this split, since as far as I can tell you're making it up out of whole cloth. In the particular case under discussion, where in my formulation the model $M$ consists of a 4-d spacetime geometry and a timelike curve within that geometry, what part of that is your $M_F$ and what part is your $M_R$?

Here you left the geometry entirely unspecified apart from the dimension, hence we do not need to any additional assumptions other then framework of Riemann plus the concrete assumption on dimension. It does not actually need any assumptions on field equations. So this is just a model framework $M_F$ that needs no kind of experimental input.

Only if you were to concretize the geometry as the Schwarzschild spacetime specifically which is already a solutions to Einsteins field equations, all statements that are not deductible from the previously explained $M_F$ but require further assumptions that are specific to the Schwarzschild case, are then $M_R$.

I see what you are saying here, you are making a distinction between the equations and the solutions to the equations. I am lumping them all into $M$ and you would like to split them into $M_F$ and $M_R$.
have two potential problems with that.

This distinction stems from the discipline of mathematical logic which i tried to adapt to Peter's nomenclature. In that sense a theory is defined by a set of axioms/postulates/definitions that always hold. Of course there can be entirely different representations of those axioms and in that case mathematical logic is concerned at verifying that those two are indeed equivalent. I might have been quite vage with my words but what i meant with $M_F$ was the part of a physical model that in mathematics is defined as a format theory. But i also used it interchangeably for the entire model this creates (in the sense of model theory in math - i.e. the entirety of statements that are true given the theory).

Initial conditions and other parameters are logically also assumptions - i.e. statements considered true. So formally they are an extensions of the original theory. But since these assumptions depend on the particular situation/experiment of study (they are not always true in $M_F$, or rather are undecided by $M_F$), those are like 'optional'/'situation' assumptions and hence treated differently. This extended theory and model i was calling $M_R$.

In physical theories there is a further noticeable distinction between those that that $M_F$ is the core part of a theory that requires no input from experiments to work and instead all of it is all left with $M_R$. I would even count the values of constants amongst the latter since they never appear in axiomatic definitions of theories - and in reality we don't know most constants exactly.

This, in itself is a cause to question the validity of the proposal. Since $I$ is a mapping between $M$ and $E$ it cannot precede either. That your reasoning leads to this shows there was a critical flaw. I think the flaw is the attempt to split $M$.

sorry, my bad, forgot the index. $M_F$ and $E$ must logically precede $I$. However, $I$ may precede $M_R$:

For example instead of assuming a Schwarzschild spacetime around a massive body, we theoretically can go the other direction and could use clocks and distance measurements $E$, interpret the results in terms of $M_F$ to determine the geometry $M_R$ around it empirically instead.

So the distinction between $M_F$ and $M_R$ is also in that the interpretation $I$ only requires the prior. The relation between $I$ and $M_R$ however quite more interesting, because in all assumptions unique to $M_R$ are not independent of $I$.

 

[Dale]

In that sense a theory is defined by a set of axioms/postulates/definitions that always hold.

I would not accept this as the definition of a scientific theory. As you say, this concept of yours stems from the field of mathematical logic. But science is not mathematical logic.

Physics is part of science, not mathematics. The minimal interpretation, $I$, is not optional when using the scientific method. Until you have that you do not have a theory. And since $I$ is a mapping between $M$ and $E$, the $M$ must be considered broadly enough to allow mapping to $E$.

Your artificial split seems counterproductive.

 

[PeterDonis]

you left the geometry entirely unspecified

My intent is that the model $M$ includes some specific geometry; I simply didn't say which one. I did not mean that the model $M$ is just "there is some unspecified 4-dimensional spacetime geometry".

Only if you were to concretize the geometry as the Schwarzschild spacetime specifically

Even that is not enough: you have to specify a particular mass $m$, as I have made clear in other posts. And yes, my intent was that the model $M$ is "concretized" in this way.

 

[vanhees71]

It's important to keep in mind that science is, as any human endeavor, a historical process. I'd say the beginning of all physics, the part of science that refers to the most fundamental notions of how to describe properties of Nature, is the discovery that there are regularities in what we experience in Nature through our senses.

If you think about it, the most regular behavior are indeed the motions of the planets around the Sun and our Moon around the Earth with respect to the "fixed stars". This explains why the first part of physics discovered and also quantified was astronomy, strongly based on the discovery of (Euclidean) geometry on Earth. Note that also time was quantified early on through the regular (quasi-)periodic motions of the Earth around the Sun and the Moon around the Earth. This was more or less completely empirical knowledge about the various periods discovered in these motions of the heavenly bodies, and this was astonishingly precise. For me the most impressive hint is the mechanism of Antikythera, which has been analyzed to be a pretty precise "clock" describing these motions of the (then known) planets and the Moon.

The next big step in the development of modern physics was the Copernical Revolution and particularly Kepler's discovery of his 3 laws of planetary motion, including first ideas on a "dynamical model", which was then fully developed with Newton's mechanics, brought already in a kind of "axiomatic system".

The success of all these applications of mathematics, starting from the use of Euclidean geometry in describing "spatial relations between physical bodies" and the idea of an "absolute time" as a parameter describing the causal order of events, of course always hinges on how you can quantify the corresponding phenomena using concrete measurement devices. For geometry that are "rigid rods" with marks on it defining distances and the discovery of "coordinate systems" defined by such rigid rods in the sense of "reference frames". For time that were, e.g., pendulum clocks. Of course, to construct these devices you also already need the mathematical model underlying it, e.g., Newtonian spacetime axiomatics. It's indeed hard to say, which came first, the mathematical model or the devices enabling the mapping between the mathematical entities with the observable phenomena in Nature.

Having established such an apparently logically closed model as Newtonian mechanics, of course one has to work out all the consequences to apply it to any kind of phenomena not yet thought about before, and in this process it happens that some phenomena do not fit, and one has to extend the model. E.g., besides Newton's general law of gravitational forces (interactions at a distance), there are electric and magnetic forces, which had to be also described, and to a certain extent that was still possible within Newtonian mechanics. Particularly in the early 19th century there was also a fully consistent model about "electricity and magnetism" based on actions at a distance between electric charges, currents and magnets.

Then with Faraday a completely new idea started to be established, the idea of local interactions and (electric and magnetic) fields. What's (for me a bit surprisingly) nearly lost in our standard physics education is the fact that besides Newton's point-particle mechanics also continuum mechanics had been developed by Euler, Lagrange, Navier, Stokes, et al. delivering also the mathematical tools (be it in terms of quaternions first and somewhat later vector calculus) develop classical electrodynamics (or rather classical electromagnetics) by Maxwell.

With this theory, however, Newtonian physics was really in danger, and as is well known, the final solution for the obvious difficulties to make Maxwell electromagnetics consistent with the Newtonian spacetime model, was special relativity, developed by Lorentz and Poincare and finally completed by Einstein and Minkowski in 1905-1908.

Then, in turn, Newton's theory of gravitation was obsolete, and the attempt to describe the gravitational interaction ended after a struggle of around 8 years with Einstein's General Relativity, which was understood as "geometrodynamics" pretty right from the beginning, although from a modern point of view it's just a dynamical theory of fields based on the (heuristic) principle of "gauge invariance", i.e., it's making the global spacetime symmetry of Minkowski space (Poincare) symmetry local, and finalizing in this way the development from Newtonian "actions at a distance" to "local interactions of fields".

One should be aware that there's a continuity in this development since the old map between the abstract mathematical ingredients and the measurement devices enabling the quantitative measurement of spatial relations between real-world objects and the temporal sequence of events is still based on Euclidean geometry for local measurements of a locally inertial observer. Of course, the spacetime geometry is now no longer Euclidean or pseudo-Euclidean Minkowski spacetime but the local version of it, i.e., pseudo-Riemannian spacetime of General Relativity, and also the measurement devices have been refined and more or less substituted by "electromagnetic/optical" devices based on precise natural standards due to quantum theory (like the Cesium standard for the definition of the second of the SI for precise time measurable by atomic clocks), but the principle is still the same since Galilei and Newton.

 

[Killtech]

I would not accept this as the definition of a scientific theory.

Bad choice of words then, this was not at all what i meant. It's very unfortunate how ambiguous the meaning of the word "theory" is without a preceding word like "scientific" or "formal".

Physics is part of science, not mathematics. The minimal interpretation, $I$, is not optional when using the scientific method. Until you have that you do not have a theory. And since $I$ is a mapping between $M$ and $E$, the $M$ must be considered broadly enough to allow mapping to $E$.

Of course, i never put that into question. It's absurd to ignore the interpretation since that is like ignoring reality when discussing science.

The mathematical terminology i mentioned was introduced to discuss what mathematics can and cannot say about scientific (and mathematical) theories. Yet, as you said so yourself any scientific theory is more then math & model because the minimal interpretation is obviously mandatory - and this part is naturally beyond mathematics which tools are not suited to adequately discuss it. Anyhow, since i know this field of math a little, please excuse if i tend to think in this way whenever someone brings the the word model into play. The confusion in terminology here arouse, because i identified a model solely be the formal assumption it makes (axioms, equations, laws, etc).

And as far as i see it, there is exactly one general model for GR, that is $M_F$ and a large family of very specific models of GR $(M_F + M_{R,n})_n$ like the Schwarzschild spacetime or simply a flat Minkowski.

The split of the model into two aspects has nothing to do with the mathematical terminology. 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 $M_F$ and $E$, at least when it is supposed to be minimal and without ambiguity. i explained it in my previous post in more detail.

 

[Killtech]

The success of all these applications of mathematics, starting from the use of Euclidean geometry in describing "spatial relations between physical bodies" and the idea of an "absolute time" as a parameter describing the causal order of events, of course always hinges on how you can quantify the corresponding phenomena using concrete measurement devices. For geometry that are "rigid rods" with marks on it defining distances and the discovery of "coordinate systems" defined by such rigid rods in the sense of "reference frames". For time that were, e.g., pendulum clocks. Of course, to construct these devices you also already need the mathematical model underlying it, e.g., Newtonian spacetime axiomatics. It's indeed hard to say, which came first, the mathematical model or the devices enabling the mapping between the mathematical entities with the observable phenomena in Nature.

Wow, thanks for the detailed read :D

There was just this part that i am compelled to thinking more about. i take the "rigid rod" is a reference to the orinasal meter rod in Paris as an exemplary of the general abstract concept of a length measure.

One should be aware that there's a continuity in this development since the old map between the abstract mathematical ingredients and the measurement devices enabling the quantitative measurement of spatial relations between real-world objects and the temporal sequence of events is still based on Euclidean geometry for local measurements of a locally inertial observer. Of course, the spacetime geometry is now no longer Euclidean or pseudo-Euclidean Minkowski spacetime but the local version of it, i.e., pseudo-Riemannian spacetime of General Relativity, and also the measurement devices have been refined and more or less substituted by "electromagnetic/optical" devices based on precise natural standards due to quantum theory (like the Cesium standard for the definition of the second of the SI for precise time measurable by atomic clocks), but the principle is still the same since Galilei and Newton.

This involvement of geometry and it's interpretation (in the sense of a mapping between model $M$ and experiment $E$) is in fact what i am so curious about.

To be more precise, the word "rigid" you used to describe the rod. Because in Riemanns geometry there is no property that would require the rod to be rigid. While it cannot have a hysteresis like effect when it travels trough space to serve as a reference for experiments at different locations, it absolute "real" length is of concern for geometry. Of course if we were to measure the rod's length then the only way to do so is to use the rod as a reference of length for itself to find out that is has exactly the constant length of one rod everywhere. Thus measured like that it appears rigid.

But let's say we are given a round body, two type of rods $r_a$, $r_b$ are are tasked to determine if the surface of the body is a sphere or an ellipsoid. All rods of type $r_a$ will be always of the same length when put next to each other and same for $r_b$. But when you were to compare $r_a$ vs $r_b$ you will notice that they change lengths relative to each other along path on the body's surface. Assuming no hysteresis like effects, that difference does not depend on the path, but only on the actual location.

If we were to use $r_a$ to measure the circumferences of the body, it appears as a sphere, but if we measure it with $r_b$ we yield the shape of an ellipsoid. Which rod types is rigid and provides us with the proper length?

My understanding of Riemann geometry is that its interpretation is all but trivial.

 

[PeterDonis]

All rods of type $r_a$ will be always of the same length when put next to each other and same for $r_b$. But when you were to compare $r_a$ vs $r_b$ you will notice that they change lengths relative to each other along path on the body's surface. Assuming no hysteresis like effects, that difference does not depend on the path, but only on the actual location.

In this situation you would not accept either type of rod as a valid measure of length without doing further experimentation. What you would be looking for in further experimentation is some physical effect that caused the relative lengths of the two types of rods to vary with position. (For example, to take an effect that Einstein made use of, the rods might be affected differently by temperature and the temperature might vary over the round body being measured.) Then you would apply a correction for that effect and see if that made the corrected lengths of the rods match everywhere.

Which rod types is rigid and provides us with the proper length?

Neither one, on the information you have given. See above.

 

[vanhees71]

It's of course clear that there are no "rigid rods" in the literal sense, only approximations. And that's why the definition of the unit for lengths has changed since the French Revolution, where the metre stick defined what 1m is. One direct way is to use the wave length of some atomic transition (similar to the frequency of the Cs-133 hyperfine transition defining the second). Indeed in 1960 one redefined the metre by the Krypton-86 standard:

https://en.wikipedia.org/wiki/Metre#Wavelength_definition

However, since it's more accurate to measure times than wave lengths already in 1983 the metre has been defined by setting the speed of light in vacuo to an exact value, using the definition of the second via the Cs-133 standard, and this hasn't changed since then for the newest SI of 2019.

 

[Killtech]

In this situation you would not accept either type of rod as a valid measure of length without doing further experimentation. What you would be looking for in further experimentation is some physical effect that caused the relative lengths of the two types of rods to vary with position. (For example, to take an effect that Einstein made use of, the rods might be affected differently by temperature and the temperature might vary over the round body being measured.) Then you would apply a correction for that effect and see if that made the corrected lengths of the rods match everywhere.

What you suggest here is however logically not entirely possible: 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. What experiments can detect however are hysteresis like effects or "lag" - e.g. temperature takes time to spread over the entirety of the rod leaving a dependence on the path the rod took - by that feature we can detect it. But for the exemplary rods, this possibility is specifically excluded.

The issue is that all that experiments can observe are in fact only relatives - how two physicals lengths relate to each other but they are never able to observe absolute changes. That makes it impossible to find the one length that doesn't change absolutely.

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. We just should be aware that model will always remain relative to the chosen rod.

 

[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 Newtown 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.