Constancy of the speed of light

[Killtech]

I am trying to get a better understanding about the constancy of the speed of light which is a well-established axiom of current day physics. for the start i want to understand how it is experimentally established and how these results are interpreted. My difficulty here is that this seems to be far less trivial then for other quantities given the definitions of the units/metrics the speed is measured in are themselves strongly dependent on that very quantity which makes for a weird interdependence that creates circles in my mind.

My understanding is that one must at least hypothetically allow the possibility of $c$ not being constant in order to verify or falsify it with an experiment. But doing so implies that the current definition of the SI meter cannot be assumed to be an ‘absolute’ measure of length since it directly depends on the speed of light. Any change to it directly changes the length of the meter. Of course this definition was chosen after the constancy was well established thus in any case one needs to use a different one instead. But here comes my problem: every other definition has a strong interdependence with the speed of light albeit the type of these dependencies vary and is not clear to me. The bigger problem is that I cannot find the right wording to google an answer for myself.

The reason why I don’t think this is a triviality is that take for example the old metric definition via the pre-SI prototype meter bar. It is a solid state and thus in rough terms a finite grid of a constant number of atoms which total size defined the meter. Properties of the grid like distances between vertices are itself mainly determined by the size of atoms it is composed of which in turn depends on the EM interaction between the electron shell and the nucleus. And since $c$ is the propagation speed of that very force it is natural to assume all atomic properties and states will be gravely affected in one way or another and with it most definitions of the two fundamental SI units (since the definitions of the second are also based on atomic states). Apart from the microscopic effects anything changing Bohr’s radius should also shrink or stretch the entire atomic grid proportionally and therefore the meter bar as a whole (or actually anything made out of atoms).

My first naïve approach to get an understanding of the impact on atoms by an altered $c$ was that the quantum mechanical solution for Hydrogen is easily reapplied for c-modified Maxwell equations and results in the Bohr radius scaling inversely to the speed of light – i.e. a lower $c$ weakens the electric field/energy of photos and thus atom sizes grow. However this approach brings a lot of other constants into play that complicate things. For example given the relations of Planck’s quantum to the photon energy I can’t find a reason that would guarantee it remaining constant in this hypothetical circumstance. One could argue that a photons energy should reduce with $c$ under the constraint of constant frequency and if for example it scaled proportionally then atoms would shrink just in a way that both definitions of the meter (current SI and prototype bar) would remain equivalent. In a similar way definitions via wavelengths of specific atomic levels might be compromised in the same way leading to the possibility of all metric definitions (know to me) remaining equivalent in all circumstances and scale with $c$.

In that case no direct measurement of the speed of light would ever be able to find a different value regardless. Thus if this scenario cannot be ruled out it would require a very different approach to proof or disproof the constancy of $c$. Or putting it the other way around: if $c$ would actually vary locally how would we experimentally detect it assuming that matter would be affected the same way as described in the scenario above?

 

[Mister T]

My understanding is that one must at least hypothetically allow the possibility of $c$ not being constant in order to verify or falsify it with an experiment.

Okay.

But doing so implies that the current definition of the SI meter cannot be assumed to be an ‘absolute’ measure of length since it directly depends on the speed of light.

The definition implies that meter sticks can be built in different places and at different times. They will be identical to each other, within the limits of precision of the construction techniques, only if the speed of light is the same every time and every place that anyone uses light to build one of those meter sticks.

If one were ever to find a difference between the lengths of these meter sticks, a difference outside the bounds of the construction errors that is, it would constitute a way to falsify the principle that the speed of light is constant.

 

[PeterDonis]

I am trying to get a better understanding about the constancy of the speed of light which is a well-established axiom of current day physics.

The first thing you need to understand is that "constancy of the speed of light" is the wrong way to look at it. The speed of light is not a dimensionless number. The relevant dimensionless number for electromagnetism is the fine structure constant, so when you ask about the possibility of the speed of light changing, what you should really be asking about is the possibility of the fine structure constant changing.

Since the fine structure constant affects all electromagnetic phenomena, it would affect not only the propagation of light in free space, but the interactions that determine things like the sizes of atoms. So working out the consequences of changes in the fine structure constant is not a trivial exercise. Particularly if you consider experiments such as you describe, to measure the speed of light by, for example, using a standardized meter stick (or something like that) as the measure of distance (and, for that matter, a clock whose operation probably depends on the properties of electromagnetism as the measure of time).

Before even trying to work out for yourself how we might test such things experimentally, you should take the time to learn what has already been done, theoretically and experimentally.

 

[Dale]

for the start i want to understand how it is experimentally established and how these results are interpreted.

You should start here:

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

 

[Killtech]

You should start here:

http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

thanks for the link. in there i just found the part i am trying to understand but in a more general context because all metrological definitions could exhibit a similar problem given that their definition indirectly depends $c$:

In 1983 the international standard for the meter was redefined in terms of the definition of the second and a defined value for the speed of light. The defined value was chosen to be as consistent as possible with the earlier metrological definitions of the meter and the second. Since then it is not possible to measure the speed of light using the current metrological standards, but one can still measure any anisotropy in its speed, or use an earlier definition of the meter if necessary.

http://www.edu-observatory.org/phys...ments.html#Measurements_of_the_Speed_of_Light

the letter sentence is what i was trying with my original post.

but more generally take for example the Michelson-Morley experiment for the two way isotropy of light-speed. the concept of the ether wind effectively elongating the way of light orthogonal to the wind thus creating a time difference in the arrival of orthogonal light pulses in the interferometer is a convincing method. but it is build on the premise that the interferometer itself is not affected by the ether. because for example the atoms the interferometer is composed of could take an elliptical shape in the wind elongating its one axis while squeezing the other. this change in size of the interferometer could effectively undo any time difference of the light pulses. therefore in this hypothetical scenario the experiment would always conclude a negative result regardless if an ether wind is present or not.

basically such kind of test would require to be done with means and method that are ensured to not exhibit an identical non-isotropy.

on the other hand this also shows a mathematical degree of freedom: using the aforementioned pre-SI meter bar as a definition of length for the MM-experiment would make the interferometer size constant by definition and therefore imply the isotropy of light mathematically - at least in a physical model based on such metrology - regardless of whether light is assumed to be isotropic or not in this though experiment. and i am not sure if such a model could even be falsified in any way as it could be mathematically equivalent and produce identical physical predictions - if its metrology satisfies the mathematical requirements of a metric i think.

If one were ever to find a difference between the lengths of these meter sticks, a difference outside the bounds of the construction errors that is, it would constitute a way to falsify the principle that the speed of light is constant.

in order to measure a difference you need a metrology. and what do you do if that metrology might be such that it exactly undoes that difference (in a similar way as in the MM example above)? and what if all your alternate metrologies are equivalent to your original one?

a metrology that is neither directly nor implicitly bound to the speed of light is hard to come by.

 

[Mister T]

in order to measure a difference you need a metrology.

But you do not need a metrology to find a difference. All you have to do is compare.

 

[Killtech]

But you do not need a metrology to find a difference. All you have to do is compare.

and how do you compare? let's assume you have bars that change their length depending on the part of space they are in. to check if they are of same length you need to measure them. or you could of course move them directly next to each other and compare them that way but this undoes the difference you are trying to detect. alternatively you could compare their length to something else. but that requires that something else does not change depending on its location - or at least it must not do so consistently to your bars.

 

[Dale]

this change in size of the interferometer could effectively undo any time difference of the light pulses.

Yes, length contraction is indeed the explanation.

 

[Dale]

and how do you compare? let's assume you have bars that change their length depending on the part of space they are in. to check if they are of same length you need to measure them. or you could of course move them directly next to each other and compare them that way but this undoes the difference you are trying to detect. alternatively you could compare their length to something else. but that requires that something else does not change depending on its location - or at least it must not do so consistently to your bars.

It seem to me that what you are realizing is that dimensionful values are all a matter of convention. I define the length of a meter such that by definition c is constant, and by definition a meter here is the same as a meter there. I am free to do that because these are dimensionful quantities that can be arbitrarily set by a human or a committee of humans. There is, as you pointed out, no way to test it independently of our conventions. The convention defines the outcome.

So the resolution of the problem is to look at dimensionless physical constants instead. In this case the fine structure constant. If it varies from time to time or from place to place then that could be physically detected independently of your choice of units.

 

[Mister T]

and how do you compare? let's assume you have bars that change their length depending on the part of space they are in. to check if they are of same length you need to measure them.

That would be an interesting thing to consider. But in your original post you asked us instead to consider whether or not the speed of light is constant. If the speed of light is different in one place than in another, then they won't be the same length.

 

[Sorcerer]

Since the second is defined by cesium, not c or the meter, maybe OP should move the goal post there? Of course all units are arbitray, which is why the real solution is the fine structure constant.
Also, why would length contraction “hide true results” with respect to the constancy of the speed of light? How can you have one without the other? Correct me if I’m wrong, but it appears that length contraction (and time dilation) is exactly WHY c is constant, as much as c being consant is why length contraction and time dialtion exist.

It seems to be an if and only if relationship.

So why would length contraction “conspire” to give the wrong answer about c being constant? You can’t have a finite and constant universal speed limit without length and time being relative, and you can’t have length and time being relative without a finite universal speed limit (which every inertial frame agrees on). Or so it seems to me. Of course I could be wrong but how else can it be so in our universe where you can’t move two directions in time and where location and direction don’t change physical laws? You can work out the math for general transformations, and you’re left with the one form of transformation equations, with either an infinite universal speed limit (Galileo) or a finite one (Lorentz).

As the universal speed limit appears to be finite, Lorentz is the correct transform, and obviously imbedded in it is length contraction and time dilation. Am I off base here?

 

[Killtech]

The first thing you need to understand is that "constancy of the speed of light" is the wrong way to look at it. The speed of light is not a dimensionless number. The relevant dimensionless number for electromagnetism is the fine structure constant, so when you ask about the possibility of the speed of light changing, what you should really be asking about is the possibility of the fine structure constant changing.

Of course all units are arbitray, which is why the real solution is the fine structure constant.

well, when looking at it alone then yes, the fine structure constant would be what i should be looking at. but that wasn't exactly my question. i am more wondering how much each of the metrological definitions already determines the result. for the current SI definition it obvious but for the older ones its not and other constants come into play if one wants to understand their interrelation with the speed of light. when looking at those it made most sense to me to consider the light-speed changing coupled with other constants strongly associated with the EM-field - everything else just seemed too odd and artificial. but looking at the fine structure written in terms of those constants the changes would then would just cancel each other out. so this, as pointed out in the discussion above, about convention and definition and to which degree the constancy of light-speed is determined on those.

Since the second is defined by cesium, not c or the meter, maybe OP should move the goal post there?

the second is lastly defined in terms of atomic states which are defined by electromagnetic interaction for which again $c$ is a fundamental quantity, along Planck's and the two electromagnetic constants. so the situation is the same as with the pre SI-meter bar definition which i mentioned in my original post.

Also, why would length contraction “hide true results” with respect to the constancy of the speed of light? How can you have one without the other? Correct me if I’m wrong, but it appears that length contraction (and time dilation) is exactly WHY c is constant, as much as c being consant is why length contraction and time dialtion exist.

It seems to be an if and only if relationship.

So why would length contraction “conspire” to give the wrong answer about c being constant? You can’t have a finite and constant universal speed limit without length and time being relative, and you can’t have length and time being relative without a finite universal speed limit (which every inertial frame agrees on). Or so it seems to me.

hmm, i think you misunderstood me here a bit. there are no "true" results to hide or conspire against. what bugs me here is that these basic metrological definitions are just set up in a way that due to their explicit or implicit dependency on $c$ they don't allow it to change one way or the another. my line of thought was that even if i consider the speed of light changing (in a specific way coupled with other constants) these definitions would just adapt accordingly and undo that change; the convention determines the result - this was however more of a technical consideration to understand how their interdependence works rather then an assumption about a physical "truth". but as metrology goes and as it was stated in this thread it is just a human chosen convention and in a sense 'arbitrary' without implications about physical "truths". they change terminology but not predictions. therefore they can only be judged in terms of practicality rather then being right or wrong.

Of course I could be wrong but how else can it be so in our universe where you can’t move two directions in time and where location and direction don’t change physical laws? You can work out the math for general transformations, and you’re left with the one form of transformation equations, with either an infinite universal speed limit (Galileo) or a finite one (Lorentz).

As the universal speed limit appears to be finite, Lorentz is the correct transform, and obviously imbedded in it is length contraction and time dilation.

an interesting question would be though whether any of the statements holds if we were to use an entirely different metrology.

 

[Dale]

an interesting question would be though whether any of the statements holds if we were to use an entirely different metrology.

They would all hold.

 

[Mister T]

what bugs me here is that these basic metrological definitions are just set up in a way that due to their explicit or implicit dependency on $c$ they don't allow it to change one way or the another.

Your concern is misplaced. I already explained to you that if $c$ varied with location, for example, a meter stick manufactured over here would have a different length than one manufactured over there. Such a difference could be observed by bringing the two sticks together. Your response was to point out that it's possible that the length of the sticks might change when you move them, and do so in such a way that we are always fooled into thinking that $c$ is the same everywhere. The thing is, if such a thing were happening is there any experiment you can think of doing that might demonstrate that this is happening?

If not, you don't have different physics. You have instead a different interpretation of the same physics.

an interesting question would be though whether any of the statements holds if we were to use an entirely different metrology.

I kind of lost track of the statements, but changing the metrology doesn't change the physics.

 

[PeterDonis]

the second is lastly defined in terms of atomic states which are defined by electromagnetic interaction for which again $c$ is a fundamental quantity, along Planck's and the two electromagnetic constants.

You're over-counting; the two electromagnetic constants and $c$ are not independent of each other, they are really only one constant; and as has already been noted, the actual electromagnetic constant is the fine structure constant, since it is dimensionless; $c$ and $\mu_0$ and $\epsilon_0$ are just reflections of different ways of choosing units.

 

[Sorcerer]

*portion of Killtech's quote and my quote removed because I'm only addressing this single question

[...]

[...]You can work out the math for general transformations, and you’re left with the one form of transformation equations, with either an infinite universal speed limit (Galileo) or a finite one (Lorentz).

As the universal speed limit appears to be finite, Lorentz is the correct transform, and obviously imbedded in it is length contraction and time dilation.

an interesting question would be though whether any of the statements holds if we were to use an entirely different metrology.

I believe they do, as no particular metrology is used to get them.

I'm going to put in a lot of work here to show this, so I hope you actually follow (this isn't new, but I've never seen this many steps not skipped before, and believe me this was painstaking). I'm keeping this extremely general (about as general as I am capable of doing), and at no point are any specific units used. Please follow and see if you agree that at no point was any unit convention used until the very end.First, assume that Newton's law of inertial holds good when an object is instantaneously at rest with respect to a coordinate system. Or rather, such a system exists for any object. Not an unreasonable assumption (basically we're assuming the existence of inertial rest frames. We okay so far?). So let the inertial frame S exist, and let x, t be the coordinates of one object, and let S' be the inertial frame of another object moving with speed v in the positive x direction relative to S, and let x',t' be its coordinates. Still no choices of units or anything else.

Next arrange for the two systems to coincide at the origin (as this is just choosing reference frames, it is generally valid).

These two fames must be related linearly in general by the form dx' = Adx + Bdt and dt' = Cdx and Ddt, with A,B,C and D as constants for a given value of v. All that's assumed here is isotropy and the law of inertia holding. Still no metrology.

For simplicity, we're assuming that the second object is at rest in S' at the origin of the x',t' coordinates, so x' = 0. Therefore 0 = Adx + Bdt, and hence dx/dt = -B/A = v, which means B = -Av. Next do some algebraic manipulation:

Ax = x' - Bt
Ax = x' - B(t' - Cx)/D
Ax = x' - Bt'/D + BCx/D
Ax - BCx/D = x' - Bt'/D
ADx - BCx = Dx' - Bt'
(AD - BC)x = Dx' - Bt'

and doing a similar thing with t:

Dt = t' - Cx
Dt = t' - C(x' - Bt)/A
Dt = t' - Cx'/A + BCt/A
Dt - BCt/A = t' - Cx'/A
ADt - BCt = At' - Cx'
(AD - BC)t = At' - Cx'Now consider the first object moving from S'. It's moving in the other direction, so
x/t = [(Dx' - Bt')/(AD-BC)]/[(At' - Cx')/(AD - BC)]
x/t = (Dx' - Bt')/(At' - Cx')
x/t = (Dv' - B')/(A' - Cv')

x/t = (Dv' - B')/(A' - Cv')

but since the first object is at rest in S, we can let x/t = 0, giving

0 = (Dv' - B')/(A' - Cv')
Dv' + B = 0
v' = dx'/dt' = B/D

Hence, B = -Dv, and then drawing from v = -B/A, it must be that,

B = -D(-B/A)
B = D(B/A)
1 = D/A
A = D

So go back and plug B = -Av and A = D back in:

(AD - BC)x = Dx' - Bt'
(A² + vAC)x = Ax' + Avt'
(A² + vAC)x = A(x' + vt')

Now the only way for this to work both ways in a physically meaningful way (where the transformation looks the same except swapping the sign on the v) is for (AD - BC) = (A² + vAC) to equal 1.

So, solve for C:
A² + vAC = 1
vAC = 1 - A²
C = (1 - A²)/(vA)Then, a very general linear transformation between inertial coordinates that has the reciprocal nature we would expect from a real transformation can be found:

(go back to the very first thing, but I won't write it in differential form):

x' = Ax + Bt
x' = Ax - Avt
x' = A(x - vt)

and

t' = Cx + At
t' = At + Cx (allowed because of the commutative property... and done because this is usually how it is written)
t' = At + x(1 - A²)/(vA)
t' = At - x(A²-1)/(vA) (factored out a -1 from the parentheses)
t' = At -vx(A²-1)/(v²A) (factored out a v from the parentheses, same reason as above)
t' = A(t - vx(A²-1)/(v²A²) (factored out an A, because again this will be how it is usually written)
t' = A(t - vx(A²-1)/(v²A²)

So, the two basic, very general transformation equations are:

x' = A(x - vt)
and

t' = A(t - vx(A²-1)/(v²A²))

Now to make things easier to read, we can let (A² - 1)/(v²A²) = k.
For the x coordinate, solve for A, because it's A that is the coefficient multiplying what's in the parentheses, and we want this to look neater:

(A² - 1)/(v²A²) = k
A²kv² = A² - 1
A² - A²kv² = 1
A² (1 - kv²) = 1
A² = 1/(1 - kv²)
A = 1/√(1 - kv²)Then quickly looking to simplify t' using k and the solving of A:

t' = A(t - vx(A²-1)/(v²A²)
t' = A(t -vxk)
t' = A(t - kvx) (again used commutative property because that is how this will end up looking as it is usually written)
t' = (t - kvx)√(1 - kv²)

So, in easier to see form, the two transformation equations are:

x' = (x - vt)/√(1 - kv²)

and

t' = (t - kvx)√(1 - kv²)

So, this is an extremely GENERAL set of transformation equations. It doesn't tell a whole lot about the universe, because k can be anything.

To gain some insight into what k means, we'll have to use a third coordinate system; the first two have speed v between them, so let the third have speed u between the second set. This will make the transformation from the first to the third a combination of the other two, this time with two k's, k_v and k_u. The new coordinate system is S'', and it's coordinates are given by x'' and t''. So, using substitution (like a composite function) - note this is so messy I'm going to use LaTex:

$\frac{x''}{t''} = \frac{x - \frac{u + v}{1 + k_v uv}t}{\frac{1+k_u uv}{1+k_vuv}t - \frac{k_uu + k_vv}{1 + k_vuv}x}$

Now, because this has to follow the same form as the prior transformation, which had t' = (t - kvx)/√(1 - kv²), the coefficient on t in the denominator has to be 1. Otherwise the transformation would not hold generally for all inertial systems, which would make it inconsistent with this entire derivation.

Thus,

$\frac{1+k_u uv}{1+k_vuv} = 1$

which MUST mean that k_u = k_v = k.

For similar reasons, the coefficient on the t in the numerator must be the speed between first and third coordinate systems. Let this speed be denoted w. Thus,

$w=\frac{u + v}{1 + kuv}$

Therefore the transformation between the first system and the third is:

$\frac{x''}{t''} = \frac{x -wt}{t - \frac{ku + kv}{1 + kuv}x}$

$\frac{x''}{t''} = \frac{x -wt}{t - k\frac{u + v}{1 + kuv}x}$

$\frac{x''}{t''} = \frac{x -wt}{t - kwx}$

and so you can see this holds generally, which means there is only one k to worry about.

So, whatever k is, ultimately you can make it anything you want with a suitable choice of coordinates/units. I'm not making any particular choice other than looking at the three important values everyone considers for every math problem like this: -1, 0 and 1. Anything else is really irrelevant as it would just be a translation of some form.

So if you let k = 0,

x' = (x - vt)√(1 - kv²)
x' = (x - vt)√(1 - 0⋅v²)
x' = (x - vt)

t' = (t - kvx)√(1 - kv²)
t' = (t - 0⋅vx)√(1 - 0⋅v²)
t'=t

out pops the Galilean transformation.
If you let k = -1 you get this:

x' = (x - vt)√(1 - kv²)
x' = (x - vt)√(1 - (-1)v²)
x' = (x - vt)√(1 + v²)

t' = (t - kvx)√(1 - kv²)
t' = (t - (-1)vx)√(1 - (-1)v²)
t' = (t + vx)√(1 + v²)

Interesting, but I'll leave this alone for now.If you let k = 1, you get this:

x' = (x - vt)√(1 - kv²)
x' = (x - vt)√(1 - v²)

t' = (t - kvx)√(1 - kv²)
t' = (t - vx)√(1 - v²)

No assumptions about specific measuring devices were made here, and whatever the case, obviously v cannot be g̶r̶e̶a̶t̶e̶r̶ ̶t̶h̶a̶n̶ greater than or equal to 1. In terms of mathematical structure, this is, of course, the Lorentz transformation.
As far as I can tell, this is pretty close to as general as you can get, and it leaves us with three options: the Galileo transformation, some weird thing that I've never seen before that likely has no correlation with reality (for k = -1), and the Lorentz transformation. For k = -1, again, I've never personally seen anything at all that follows that form, and I'm assuming no physical measurement has ever given us anything like that, so I am discounting it. If you have a good, physical reason for keeping it, please share.

If not, that means there are ONLY two possibilities: either the Galileo transformation or the Lorentz transformation. Which depends upon your choice for k. We've yet to speak of any specific method of measurement.Now, on to that part. In both the Galileo and Lorentz case there is an absolute speed limit. It's easy to see in the Lorentz case: the absolute speed limit is v = 1 (note, again, no specific units decided). In the Galileo case, there appears to be no limit imposed by the equation itself, which means the limit is infinity.

So again, we have two choices: a FINITE universal speed limit or an INFINITE universal speed limit. If the speed limit is finite, the Lorentz transformation is the right one, and it is irrelevant which choice of units you use, as again, this entire derivation was done without making any unit assumptions or measuring conventions. Only assumptions about isotropy and Newton's laws holding instantaneously in a reference frame were made.
So... experimentally speaking, is there a finite fastest speed? If the answer is yes, then the Lorentz transformation is the correct one, irrespective of whatever units or measuring convention you use.
*edited when I realized the obvious that v cannot EQUAL 1, let alone be greater than it, for the case where k = 1. No division by zero and no imaginary numbers allowed here.

 

[Sorcerer]

Quick note about my last post and the "discounted" case of k = -1. There is no real value that would cause you to divide by zero or get an imaginary number for v, but if the universal speed limit is infinite then you end up with indeterminate forms (∞/∞), so I'll take that as another reason to discount it, besides the fact that no measurement has ever actually shown that transformation to be correct (as far as I am aware), while countless have confirmed Galileo (in the low speed limit) and Lorentz (in the high speed limit).

 

[m4r35n357]

I'm going to put in a lot of work here to show this, so I hope you actually follow (this isn't new, but I've never seen this many steps not skipped before, and believe me this was painstaking).

Nice! I did a similar thing for my own purposes based on Reflections on Relativity. One thing that I added that was not explicitly mentioned in that article is a simple demonstration of the invariance of the spacetime interval, like:

$$
t'^2 - x'^2 = \frac{(t^2 - x^2) - v^2(t^2 - x^2)}{1 - v^2} = t^2 - x^2
$$

Not as many steps as you might have put, but it might be worth considering adding something along these lines to your text for regular re-use ;)

 

[Sorcerer]

Nice! I did a similar thing for my own purposes based on Reflections on Relativity. One thing that I added that was not explicitly mentioned in that article is a simple demonstration of the invariance of the spacetime interval, like:

$$
t'^2 - x'^2 = \frac{(t^2 - x^2) - v^2(t^2 - x^2)}{1 - v^2} = t^2 - x^2
$$

Not as many steps as you might have put, but it might be worth considering adding something along these lines to your text for regular re-use ;)

I’m pretty sure I remember someone posting that last week, and it was actually the germ of my last post. Looking at it now, they did skip a LOT of steps, and it looks like there is a tiny typo* (but then I might be the one who is wrong), but that link is part of why I considered doing all this mind numbing work.

But the point I’m hoping is made (and one of our guys with a grad degree can say for sure) is that it doesn’t matter which way you go about doing it: you’re still left with only two transformation equation options (really one, with one particular choice about the universal speed limit distinguishing them), and this occurs completely irrespective of unit or measurement conventions, as far as I can tell.

*It looks like they left off a t’ in this:

(AD - BC)t = -Cx + At’

having it as (AD - BC)t = -Cx + A

But I might have followed incorrectly. If they didn’t make a typo, then I messed something up in my post, because I had (AB - BC)t = At’ - Cx. But if so I would have to have made a “double mess up” to correct for it, because I ended with the Lorentz transformation.EDIT- actually I think the typo is they meant for the x’ to be v’. They (presumably) were at this step:

x/t = (Dx' - Bt')/(At' - Cx')

and divided by t’ in the numerator and denominator, but forgot to change the x’ to v’. That’s what I did anyway, without skipping steps, so I’m assuming that’s right. Doing that and setting x/t = 0 gives v’ = B/D. Of course it doesn’t matter because with x/t = zero when you multiply by the denominator it goes away anyway. Anyway that is a great source all around.

 

[Sorcerer]

Hey, I know this should have been obvious, but I just realized that k = 1/c². I don’t know why that surprised me but it’s pretty cool. If c → ∞ then k = 0 and we have the Galileo transform, and the square keeps the units right. More insight every day. Love it. At some point, however, I’m going to have to stop this stuff and actually learn to use these transformation laws to solve physics problems though, lol.

 

[Killtech]

[...]
First, assume that Newton's law of inertial holds good when an object is instantaneously at rest with respect to a coordinate system. Or rather, such a system exists for any object. Not an unreasonable assumption (basically we're assuming the existence of inertial rest frames. We okay so far?). So let the inertial frame S exist, and let x, t be the coordinates of one object, and let S' be the inertial frame of another object moving with speed v in the positive x direction relative to S, and let x',t' be its coordinates. Still no choices of units or anything else.
[...]

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.

 

[Sorcerer]

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.

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 an old thread here about it:

https://www.physicsforums.com/threads/measurable-spaces-vs-topological-spaces.558644/

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.

 

[Mister T]

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.

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.

 

[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.

 

[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.

 

[PeterDonis]

the real world is made of elementary particle that apparently can't be split further

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.

 

[Sorcerer]

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.

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.

 

[PeterDonis]

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

"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 continuous

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.

which IMHO invalidates the claim that models of spacetime depend upon the real world in any way

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.

or at least certainly not upon metrology

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.

 

[Killtech]

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.

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.

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.

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.

 

[Sorcerer]

"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.

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

 

[Sorcerer]

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.

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.

 

[Killtech]

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.

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".

 

[PeterDonis]

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.

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

Doing that however can change the geometry of the space in a massive way

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.

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

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.

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.

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.

 

[Sorcerer]

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".

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 ∑a_kx^k 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.

 

[Killtech]

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

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.

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.

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.
[…]
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.

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.