# Time measurement problem in regions of space with very strong spacetime curvature

• I
• Killtech

#### 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?

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.

vanhees71
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.

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vanhees71
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.

dextercioby, Killtech and vanhees71
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.

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.

PeterDonis, vanhees71 and Dale
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.

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dextercioby and 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.

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.

Ibix, phinds and vanhees71
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.

dextercioby, Dale and 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.

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

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##.

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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.

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.

Motore
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.

Motore
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.

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?

Motore
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##?

Dale
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##.

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Motore and PeterDonis
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##.

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 theory. As you say, this concept of yours stems from the field of mathematical logic. But science is not mathematical logic.

A theory 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##.