# I Constancy of the speed of light

1. Sep 8, 2018

### Killtech

Sorry, I had little time to respond lately but I have read your posts with interest and I appreciate you taking the time to post this. However my question of metrology seems to be mostly about the fundamental assumptions you base your calculation on. So let me explain what I had in mind, when I made my remark.

All physical axioms are an abstracted and generalized form of observations/measurements made. Newton’s inertia law you started with is no different. From a pure mathematical point of view it isn’t really a simple axiom but rather comes as one big package including basic assumptions for the vector space (among other) that is needed to even formulate the law. Now the axioms of vector space itself don’t come from nowhere but are too associated with real observations: i assume that they are the properties of the metrology used and each can be experimentally verified to hold true. And even the most basic assumption, that we can describe the world in terms of numeric quantities requires a metrological system which measurements conforms to axioms of mathematical fields and the real numbers. So these most fundamental tools are bound to the metrology.

So let’s change to an ancient first day metrology: using a foot of a living person as a measure of length. It abides by axioms of a metric and combined with other geometrical observations will yield a vector space. But It comes with a nasty disadvantage: a field measured to be 100 feet long may become only 98 feet a year later due to the foots owner aging. A pure mathematician would therefore conclude the field has shrunken over the year. While it sounds like a joke from a pure technical point of view it is a valid conclusion since it does not create any formal contradictions. We just wouldn’t ever stick to describing the world in this way because of it immense impracticality. But if it is possible, then for the sake of argument let’s use it for now.

Equipped with a metrology we are now able to measure inertia and translate them into mathematical formalism which enables us to check Newtown’s law. And here comes the trouble: all objects will very slowly lose inertia over the years since they move few fractions of a feet less in the same time. The pure mathematician would therefore say that Newton implies the presence of a universal gradient force that shrinks the entire universe thus a rest frame does not really exist (no object can be at rest… apart from the foot itself and other living beings which… ‘have a force of their own to counteract the universal one’).

Even if this example metrology is good for nothing else then to make the world look funny I simply don’t see any logical or formal reason which makes it incorrect. Of course it is useless for any practical purpose but an argument of convenience does not invalidate it from a technical point of view. So if you see any formal or logical fault in this line of thought, please point it out. Because if it is not technically wrong it would show that the metrological definitions and choices have much greater and non-trivial implications that just changing units – or at least I don’t think common sense of what units are supposed to be, allows them to make use of all degrees of freedom given by the metrological choice. And this is what I am exploring here.

2. Sep 8, 2018

### Sorcerer

I have to completely disagree with your basic premise, and in particular, your initial one. Mathematical abstractions are not in the least bit bound to metrology. I'd say they are merely historically tied to them, and share no actual physical dependence on them other than they seem to require our physical brains to exist as thoughts. Otherwise, one would be stretching the definition of metrology to include essentially everything. Even Newton’s inertial reference frames - they don’t actually exist, anywhere!

As far as I can tell, the only thing true about that intial assumption you have made is that the reasoning that led to abstract mathematical ideas like vectors was originally inspired by real objects. But those types of mathematical abstractions as they are developed now have nothing inherently to do with the real world, other than we can use the idealized abstractions to make accurate guesses about we’ll measure.

Mathematically speaking, vectors are just lists of numbers, and we can’t even make a true one-to-one correspondence between, say, the real numbers and actual objects. We can’t do that because the universe is (apparently) finite and the real numbers are infinite*, furthermore, it’s possible that the real universe isn’t even continuous, which obviously also precludes a true one-to-one correspondence with anything real. Not to mention, vectors have no units in and of themselves.

*even if the universe is infinite, it would still be an infinity with a smaller carnality than the real numbers, so once again, there could not be a one-to-one correspondence (because the real numbers are not quantized while energy levels and the like are).

There is no bijection between the real numbers and the real universe, I'd argue. We say π is the ratio between the diameter of a circle and its circumference, but does there even exist a real circle in the universe? I don't think there does, based on the fact that matter appears to be constructed from elementary particles. It seems to me that the integers or even ℚ may have a one-to-one correspondence with nature, but not ℝ, nor ℤ nor any higher number system.

Bottom line: vectors, reference frames, cartesian coordinate systems, etc, are not real things outside of the human mind. They are made up tools that allow us to make good guesses about what our measuring devices will find. Regarding the correlation between the math of special relativity and the physical universe, you can derive the general coordinate transformation which leads to the Lorentz transformation using nothing but topology/group theory (separating by choosing positive values of a particular constant- the k I derived earlier, but key point: any positive value will give the Lorentz transformation, just changing the units); but you can also derive all manor of transformations that have nothing to do with reality the same way.

Math might have arisen from human beings contemplating the physical world, but it really has nothing to do with physical world inherently as it stands today.

Case in point: Right now a few physicists are looking at octonions as a possible scaffold to find a complete unified field theory, but there is nothing about these eight dimensional abstractions that arose from contemplations on the physical world. They are neither commutative nor associative (two things that appear to be true in the real world); they were not deduced from observation or any physical intuition. They are the creation of mathematicians messing around with abstraction in their curiosity to see how far they could push the bounds.

Mathematicians in dark rooms created these things out of thin air, and then physicists found out about them and started seeing if they could be useful for physical theory. In other words, your central argument that math arises from physical theory only is wrong, as the process sometimes goes in the exact reverse.

And before we go further, keep in mind that the rabbit hole goes a lot deeper than octonions. For starters, there are the sedenions, which are 16 dimensional abstractions that seem to have no use at all at this point except as toys for mathematicians to play with (such as having a zero divisor, as in Moreno’s work for two non-zero x and y multiply togethet to get zero):

https://arxiv.org/pdf/q-alg/9710013.pdf

In what reality can two non-zero things multiply together to get zero? Certainly not THIS reality. In light of things like that (and there are so many other examples), in my opinion, to claim that octonions, sedonions and all the other abstract creations of mathematicians are “metrology” is to more or less claim that all human reasoning is, which lies FAR beyond the actual definition of metrology.

Heck, even topology strays far from any dependence on metrology. It’s entirely qualitative! Things like distance are irrelevant in topology.

A discussion on the difference between measure theory and topology:

https://lamington.wordpress.com/2009/06/16/measure-theory-topology-and-the-role-of-examples/

And just to clarify further, this is the definition of metrology: the science of measurement.

You can take aspects of topology and apply them to metrology, but in and of itself things like length and other measurements, or standards, are not relevant in topology. But how far are you willing to stretch the definition of metrology?

So in short, just in basic principles I believe your premise is wrong. I also believe it is wrong with respect to my own derivation, that your initial claim that abstract notions such as coordinates arose from contemplation about the physical universe means nothing more about them than a footnote in their history. Points do not exist in the real world. Lines do not exist. Rays do not exist. Zero divisors certainly don’t exist. Nor is it necessary to think about units or measurements in all branches of mathematics. Topology is concerned with things like open sets or closed sets, for example, not angles and distances. All these are just platonic ideals that for limited situations approximate something physical. They are not in any way inherently tied to the physical, or to units, etc.

Last edited: Sep 8, 2018
3. Sep 8, 2018

### Mister T

I don't see why. He could just as easily conclude that the foot grew. There's nothing in the math says the length of the foot is constant.

When you measure the length of an object what you are really doing is measuring the ratio of the length of that object to a unit length. The value of that ratio can be changed by changing either the numerator or the denominator.

You seem to be saying that the SI unit length, called the meter, might be changing in such a way that we can't detect it. In other words, Nature behaves "as if" it changes in such a way that the changes can't be detected. Nature behaves "as if" the laws of physics are valid, but in reality those laws are not valid. The argument even has a name. It's called the as-if argument.

4. Sep 8, 2018

### Sorcerer

Also regarding the field and the pure mathematician; a pure mathematician would not be making a measurement. She would picture an idealized scenario of a field in a Euclidean plane and define some arbitrary base unit with which to assign and equally arbitrary number labeled as “length” to.

If the ratio between the field and the unit measurement changed, the mathematician would be the one doing it, by introducing a different kind of metric where the ratio between the field and the unit device was dependent upon some parameter. There are no rules for mathematicians other than to be logically consistent.

5. Sep 9, 2018

### Sorcerer

Sheesh I said carnality two posts ago instead of cardinality. Thanks, smartphones. The real numbers appears to be a larger cardinality than the real world, as the real world is made of elementary particle that apparently can't be split further, while the real numbers have no limits on how small an interval between two of them you want.

6. Sep 9, 2018

### Staff: Mentor

But it is also made of spacetime, which as far as we can tell is a continuum, meaning it has the same cardinality as the real numbers.

There are speculations in quantum gravity that spacetime might be discrete at a small enough scale (something like the Planck scale), but those are just speculations. Our best current theories all model spacetime as a continuum.

7. Sep 9, 2018

### Sorcerer

Yeah but would I be wrong to say that objects do not as far as we can tell?

In any event, we don't know for certain that spacetime is continuous, and certainly have not measured it as such, which IMHO invalidates the claim that models of spacetime depend upon the real world in any way (except where we choose to make assumptions about them consistent with what we observe), or at least certainly not upon metrology.

8. Sep 9, 2018

### Staff: Mentor

"Objects" are ultimately made of quantum fields, and quantum field theory is built on a spacetime continuum. So if we're talking about fundamentals, I would not make the claim that "objects" are discrete.

We don't know for certain that spacetime is discrete either. But all of our best current theories, which have been confirmed by many experiments, are built using a continuous spacetime. Nobody has a theory that makes accurate predictions and is built on a discrete spacetime. So the best description of our current state of knowledge is that spacetime is continuous as far as we can tell.

I have no idea what you're talking about here. Our "models of spacetime" most certainly do "depend upon the real world", since we test them by doing experiments in the real world, and the models make accurate predictions about the results of those experiments.

Metrology is, as far as I can tell, a completely separate issue from the issue of whether spacetime is continuous or not. Which probably means that the latter issue should be discussed in a separate thread if you want to continue to discuss it.

9. Sep 9, 2018

### Killtech

Yes, measuring always takes two things into relation. The one thing you measure and the other thing that you measure it with – be it a ruler, light waves of a laser distance meter or a different measuring apparatus. And each measurement device needs to be calibrated against, well the metrological definition. There is no absolute way to measure something. Its always relative to a real reference which provides you with that unit length.

Here you are right. There are no experiments verifying all the axioms of the real numbers and there is little reason to check those more exotic ones for as long as we don’t make use of these in physics. To that end they are irrelevant and we could thus use a more suited set, true. However if you have an experiment determining a distance indirectly by measuring and adding partial sections of it, the correctness of this method relies on the validity of (some) filed axioms. These have therefore a direct link to out theory and need to be tested as well.

In any case, in order to make predictions about the real world we need to model experimental setups properly in terms of the theory and that requires a translations or mapping of real parts of the setup to the variables representing them in the model. That creates a link between mathematical abstractions and reality. For the most part this mapping is obvious but that makes it so much harder to understand where it actually is rooted in. My premise – or better my assumption – was that the metrology provides us with an important part of this translation. My impression from your responses to my example however is that there seems to be only one “true” way of translation in terms of physics rather than this being a just a modelling degree of freedom. If so I would like to understand that.

10. Sep 9, 2018

### Sorcerer

This thread is about someone who believes that the results of special relativity are dependent upon metrology.

11. Sep 9, 2018

### Sorcerer

What I am saying is that Minkowski space does not depend on measurements for its formal derivation, only on a choice for whether k=1/c2 is negative, zero (Galileo) or positive (Lorentz).

To verify whether or not it is locally true in the real world requires a measurement, but really the only one is that is needed is to measure the existence of a finite speed limit- regardless of whatever unit convention is used.

12. Sep 10, 2018

### Killtech

The metric of the Minkowski space of general relativity is still not arbitrary. It is has to be consistent with measurement results. On the other hand given a metric space mathematically one can exchange its metric for another and work with that instead. There is no unique choice for the metric. Doing that however can change the geometry of the space in a massive way but that is still just a transformation - much more general then a coordinate transformation considering that even equations formulated on such a metric manifold in a coordinate free way will also transform. Even so it does not affect predictions of anything modeled in this way.

Now going back to physics we use a very specific metric to describe the world. I want to understand where this metric originates from and how it is exactly linked to real measurement results - which is needed to make any predictions that are verified experimentally. I thought that the metrological definitions are actually its origin - not in the way it was derived historically but in a practical sense how the link between the model predictions and actual measurements works. This is why i was looking at the implications this setup might have.

Furthermore there is the question why we use only this metric and never another. The reaction to my example of basing a metric for example on the length of a living persons foot makes me realize that there seems to be only one "right" choice. So if that metric is uniquely determined i want to know how it comes to be because i would think it is up to how one prefers to setup the model rather then something dictated by nature. That is the difference between saying "we model physics in a Minkowski space (with the specific metric) because it is the most convenient way to compare the predictions with experiments" or "the Minkowski space is the way nature really is".

Last edited: Sep 10, 2018
13. Sep 10, 2018

### Staff: Mentor

Only in the sense that the mathematical form of the line element will change if you change coordinates. See below.

The geometry of the space is physically measurable; you can't change it. Any coordinate transformation you make, while it can change the mathematical form of the line element, cannot change any measurable values, so it can't change the geometry of the space.

You thought wrong. Experimental predictions are predictions of invariants--things that don't depend on your choice of coordinates. And your choice of coordinates includes your choice of units. No observable quantity depends on your choice of units.

If it seems like an observable quantity depends on your choice of units--for example, the weight you observe when you step on a scale standing on the Earth's surface appears to depend on whether your scale is calibrated in pounds or Newtons--that is because you have failed to realize that, as was said earlier in this thread, the number you read off the measuring device--in this case, the scale--is actually a ratio; in the case of the scale, it's the ratio of your weight to a standard weight. Changing units changes the standard, but that just means that your scale is now measuring a different ratio--the ratio of your weight to the standard one-Newton weight instead of the standard one-pound weight--which means it is measuring a different invariant, from the standpoint of physics. So changing units does not change the values of any invariants; it just changes which invariants--which observable quantities calculated by the theory--correspond to the reading on your measuring device.

There is only one right choice for the actual geometry of spacetime; we make physical measurements of that geometry to find out what that right choice is.

There is no unique "right" choice for units of measurement; you can measure distances in feet, meters, furlongs, light-years, parsecs, or whatever you want, and similarly for other quantities with units. None of this changes the geometry of spacetime at all.

14. Sep 10, 2018

### Sorcerer

Well, the entire point of my derivation is that the general flat spacetime set of coordinate transformation equations mentioned already has Minkowski space imbedded in it. All you have to do is choose a particular class of numbers for a constant (any positive number). Kind of like how the general equation for polynomials ∑akxk has x2 embedded in it - you just choose zero for every constant except for the one that must be unity.

Now why you “have to” choose positive numbers for that constant is indeed dependent on measurement (even Minkowski said something like that, if I recall). But not on units, and that can be seen by the fact that any value of constant k in the aforementioned derivation that is positive will give you the Lorentz transformation.

So why? Because, without respect to any units, the speed of light is finite and is (apparently) the maximal speed. If you want to believe that everything conspires to make it “as if” Minkowski space is right, but there is a “true” rest frame that is undetectable, all you’ve done is added something superfluous to Minkowski space that is impossible to detect. You might as well say unicorns are the reason. All we know is that no matter what units or measurement technique used, there is a finite maximal speed, which means k > 0.

15. Sep 10, 2018

### Killtech

Okay, to clarify my statement before let’s go into math a bit. Take $\mathbb{R}^2$ or a smooth two dimensional subset of $\mathbb{R}^n$. We can use it with default Euclidean metric or chose to equip it with one based on the maximum norm i.e. $||x||_\infty = max(x_i)$. In any case we can derive the matching metric tensor for each choice and yield two Riemannian manifolds. However as it is the same set everything formulated on one can be translated to an equivalent on the other via the identity bijection (well, some smoothness issues aside given the choice of 2nd metric). Practically this means I can swap them if it is more convenient for certain calculations and can swap them back afterwards to have the results formulated for both.

Now a change of metric has some impact: using the Euclidean metric the distance between $d_2(\begin{pmatrix}0 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 2\end{pmatrix})$ is twice as long as $d_2(\begin{pmatrix}0 \\ 0\end{pmatrix}, \begin{pmatrix}\sqrt 0.5 \\ \sqrt 0.5\end{pmatrix})$ and this ratio has no unit anymore. No change of coordinates nor units can affect that obviously - It remains a statement invariant under both. But using the $d_\infty$ metric the same ratio changes to $2 \sqrt 2$. In fact apart from the topology very little will remain invariant under an exchange of (equivalent) metrics.

Therefore an exchange of the metric is not the same as a change of units or coordinates. But it begs the question how the metric practically relates to measurements.

And how is the geometry of space measured? This is what I want to understand and why there is only one metric that it can be describe with.

The measurement part here is important and the reason why I though metrology was the thing to look into. SI definitions consist of two parts. Apart from defining a unit it also describes a basic real thing (like time between periods of Cesium hyperfine level radiation; practically an experimental setup) any measurement of that dimension is based on and related to. Every measurement method then can be validated against this basic setup and must yield consistent results. The properties of this underlying real thing however induce a metric for that quantity/dimension (e.g. what the dimensionless formulation of "twice as much" means for the quantity) and since all measurement were validated against the same metrology it means any measurement of this dimension will reproduce the same metric. That at least is my understanding of the relation of the metric to measurements so correct me if i am wrong.

In consequence any change of metrological basing from one “real thing” to another that changes over time or location in relation to the original one would therefore yield a different metric for that dimension which is far more than a change of units.

My naïve assumption was that I can swap the metric of my model as I am used to do in math but I also have to swap my system of measurement to be based on a “real thing” that behaves according to the new metric. Otherwise I lose the ability to directly compare experimental results with model predictions and will require an additional transformation step instead.

16. Sep 10, 2018

### Staff: Mentor

This is the "identity" for the topological space $\mathbb{R}^2$. But it is not the identity for the manifold "Euclidean plane", because this transformation changes the geometry from the Euclidean plane to something else. So you need to be very clear about exactly what a given transformation does and does not keep the same. At least part of your confusion appears to come from failing to do this.

By the distances between points. In the case of the manifold $\mathbb{R}^2$, specifying the distance according to your chosen metric between all pairs of points fixes the geometry. And your transformation above does not preserve this: if we take two points A and B, the distance between them according to the Euclidean metric is not the same as the distance between them according to your alternate metric. So again, your transformation changes the geometry.

If you are really concerned about keeping things independent of your choice of units, you can rephrase all of the above in terms of ratios of distances between different pairs of points: for example, take two pairs of points, (A, B) and (C, D), and look at the ratio of the distances between them according to the two different metrics. Everything I said still applies: if any of these ratios change, you have changed the geometry.

Yes.

Yes. And this is an example of what I said above, that your transformation changes the geometry.

I'm not sure I can make sense of your specific comments about metrology; I think you have confused yourself by failing to pay proper attention to the points I've made above. But I'll try to illustrate what I've said with an example, since you mention SI units: the definition of the SI second in terms of the radiation emitted by a specific hyperfine transition in Cesium is the equivalent of picking a particular pair of points (A, B) in spacetime and calling the spacetime distance between them (which in this case is a time, since the two points are timelike separated) the "standard" distance, and expressing all other spacetime distances as ratios with that standard distance. The SI definition of the meter then just extends this to spacelike distances as well as timelike distances, by fixing the speed of light--which is really the chosen ratio of "space distance" units to "time distance" units--to a particular value. But we could choose different definitions of the second and the meter (and we previously did), without changing any ratios of spacetime distances, i.e., without changing the spacetime geometry. All we would change is which particular spacetime distances we called the "standard" ones.

17. Sep 11, 2018

### Killtech

Okay, it seems I am missing something here. Perhaps I am not seeing the elephant in the room, but why would a change of geometry be a problem? I see it affecting calculations but not predictions.

I mean sure, a bijection between the two manifold is indeed not entirely trivial as I might have indicated but it exists nevertheless. The identity of the topological space is enough to translate every point $x$ from one manifold to another. Therefore and if the topology is the same it can also translates any scalar function $f(x)$ trivially and all coordinate charts $\phi$ remain conveniently the same, too. Furthermore any vector in the tangent space at $x$ can be decomposed into a linear combination of the chart gradients $d\phi_x$ which yields a bijection for the tangent space since the term is defined on both manifolds. I think in more general context this is called a pushforward in diff geo. Translating equations is a bit trickier since their coordinate free formulation is metric specific – however upon picking coordinates it becomes metric independent which yields a clear translation. The coordinate free formulation can be then derived from there and does not depend on the coordinates chosen for translation (you can do it with the pushforward instead). That leaves us with nothing which we can’t find a bijection for thus I don’t see where any information could get lost or changed irrecoverably in the process.

Therefore purely mathematically speaking I see no problem in changing the geometry. Well, at least as long the metrics are mathematically equivalent (i.e. same topology). So I don’t see any possible way such transformation could have any impact on the physical predictions as there is always an equivalent formulation of everything for both manifolds. It works like a distorting mirror so to speak.

And I would have believed that any kind of transformation that leaves all possible predictions of a model unchanged should be physically acceptable. A change of metric and therefore geometry should have this property or at least I am not able to think of anything that would be predicted wrongly. I mean because anything that can be expressed in coordinates will remain identical, equation must yield identical solutions. And even the distance ratios I used earlier can be predicted correctly in another metric – but since distances/measurement results have a transformation behavior under a metric swap one needs to apply it first: while a distance ratios change, a quotient $\frac {f(\mathbf x, \mathbf y)} { f(\mathbf x, \mathbf z) }$, where $f$ is directly derived from the pushforward and the base metric, does not. Again I don’t see anything I would miscalculate in a different metric.

The above may also explain my prior focus on metrology. The used measurement system always induces a metric those results are valid for – and they need to be transformed first before they can be used in a different geometry. Alternatively one can setup a measurement system that is compliant with the metric chosen. If there is nothing naturally behaving accordingly there is the possibility create one artificially. And even for something as stupid as the maximum norm one can do that in reality if one is stubborn enough: gyroscopes can be used to find the three axis outlined by the metric such that one can construct a device that measures distances optically at first, then decomposes it into the directions provided by the gyros and finally displays only the maximum component. Notably there is no unit transformation between SI meter and such a $d_\infty$ meter since they don’t transform by value (only though the pushforward/pullback instead as a function of the point-pairs).

Hmm, maybe I should try to formulate Newtown’s kinematic equations in the $d_\infty$ metric and see if I get any wrong solutions. Sometimes when just talking about stuff one can be blind for the most obvious error but when one does the calculus it gets hard to miss.

18. Sep 11, 2018

### Staff: Mentor

Then you see it wrong. If you change the geometry, you change the predictions of observable quantities. You can't change the geometry while holding the predictions of all observables constant; the geometry is one of the observables.

Both geometries are valid solutions of the mathematical equations, yes. (At least, I'm assuming they are; I haven't checked the particular example of the $d_\infty$ metric you give.) But these two solutions are different physically; they make different predictions about physical observables.

At this point I'm not sure how else to help you. I've said the above before, and so have other people, repeatedly in this thread, but you still don't seem to grasp these fundamental points.

19. Sep 12, 2018

### Boing3000

And we do that all the time

Equivalent ? The metric is the same, or not. It cannot change the geometry without changing the topology. Antarctica is not the biggest continent on earth...

I think you are mistaking change of coordinate with change of metric. Maybe this link will help you out

Measurements don't induces metric. Metrology is not geometry. It is grounded in physical apparatus... and measurement.

20. Sep 16, 2018

### Killtech

Hmm, that was a great idea to try to express the metric swap in terms of how it affects geometric observations! Basically this means to look at how the geodesic equation of a test particle in the original geometry transforms to the new one. Since each trajectory solution must remain identical (because this transformation is defined via a bijection) while the Christoffel symbols change (even for the identical coordinates) additional new terms appear. These terms exactly compensate the change in geometry. But as these equation has a direct analogy to Newton’s law the (predominant) new term would be identified as a global force vector field on the new manifold. Therefore any real object which movement is described by a geodesic solution in one geometry would still behave identically in the other. However, its description as being in a rest frame falls apart (i.e. not a geodesic solution in the new metric) while all actual observations and predictions that can be made about its time evolution remain untouched.

Besides, I have found that in newer publications there are equivalent formulations of GR in different geometries like in this paper.

As for my remarks made about metrology, the quote I took from the link Boing3000 provided seems to comply with my understanding between measurement and metric. Is it wrong or am I misunderstanding it somehow?

Sorry, the world equivalent was misplaced here. Since the metrics i used in my examples before were based on norms i actually referred to norm equivalence (see line 10). Equivalent norms induce the same topology. Same applies to metrics though the terminology is a bit different. Still, two metrics can yield the same topology and yet a very different geometry. The topology merely governs what a continuous (or smooth) function is but has little impact on the shape of the space apart from global properties like its homotopy group of a manifold (e.g. torus or sphere). The metric transformations i considered would definitively preserve the topology but they would e.g. not preserve the ratios of distances.

I am sure I do not mistake it with a change of coordinates. But thanks for the link you posted. The best answer to the question asked in your link contains an interesting formulation which very much summarizes exactly what I though the connection between metrology and metric was (it somewhat contradicts your second statement):

I tried to generalize this thought by exchanging the arbitrary ruler with what uniquely determines the validity of any distance measuring method including the ruler – which I assumed was in the end the metrology.

On the other hand if rulers measure the metric, then this begs the question if it possible to come up with some device that measures another one. And this seems to be within our technical possibilities to do practically, at least for some simple example cases: See my example (see paragraph 5) for the $d_\infty$ metric. The lack of practical use aside, would it not measure a different metric?