Coordinate Systems in Modern Interpretation of Relativity

1. Oct 23, 2008

STFiction

Hello,
I've really been enjoying reading these forums the last couple of weeks, and finally decided to register to ask a question.

This is an earnest question about what the modern interpretation is, and how I and another student of relativity can learn more about the modern understanding.

I ran into another poster from this forum elsewhere and while we both understand the basic concepts of relativity, there are some stark disagreements on things like coordinate systems and modern interpretations of relativity.

Since there appear to be many physicists here, I was wondering if you could help us. I want us both to understand relativity better and if either of us, I or him, are misunderstanding I'd like to correct that so we don't accidentally spread incorrect information.

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Here is my understanding:

My understanding is that the modern interpretations of relativity consider only coordinate system independent geometric objects or quantities as "real"/"physical". Therefore I would say that in the twin's paradox, most of the confusion comes from the time dilation equation comparing a proper time of a clock (a coordinate system independent quantity) to the coordinate time of a coordinate system (which is explicitly coordinate system dependent). So I would say the modern interpretation is that "time dilation" is just a coordinate system dependent effect and not physical. Wheeler's relativity books appear to me to be stating this.

When asked "but doesn't one twin 'really' age less", my answer would be, yes their proper times (length of world lines) are indeed different (coordinate system independent). The magnitude of their four-velocity through spacetime however was always c, so the clocks always ran at the same rate ... but they didn't travel the same distance through spacetime. In Euclidean geometry, if two people leave a point at a constant speed and meet up later at another point at the same time, then they must have travelled the same distance. This is not so in the Minkowski geometry of spacetime. Just because the coordinate time at the end of the two paths was the same does not mean the spacetime path length was the same despite the magnitude of their 'speed' through spacetime was the same.

Similarly, considering the length contraction of an inertial rod, different coordinate systems will give a different coordinate length. But the proper length is always the same. So this too would be a coordinate system dependent effect and not physical.

(As I found discussed on this forum previously, a 'rigidly' accelerating rod (and thus appearing to contract/expand according to an inertial coordinate system) must have the proper acceleration different on the front and end of the rod (as can be seen explicitly in Rindler coordinates)... so in this sense (not the one usually discussed and/or meant by 'length contraction') the contraction is real/physical.)

Basically, because we are free to choose any coordinate system, only coordinate system independent quantities should be considered 'real'/'physical' in modern interpretations of relativity.

So that is my understanding of the modern interpretations of relativity and its relation to coordinate systems.

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Here is his understanding:

Before I get to the opposing view, let me state a couple blatant things:
- I don't really understand his view, so I will use quotes to help avoid misrepresenting them. Since I do not understand, this may still not be the best (I've asked him to join in on the conversation).
- I realize that there can be multiple valid interpretations as long as they are self consistent. I do NOT want to get into a discussion of philosophy or ontology. This is an earnest question about what the modern interpretation is, and how I and another student of relativity can learn more about the modern understanding.

His comments regarding coordinate systems (the context is I often give an example coordinate system to show something he considers 'physical' is indeed coordinate system dependent)

So there seems to be a misunderstanding of what a coordinate system even is, and what its role in physics is. I have failed at explaining this to him, so maybe I'm misunderstanding his point. If someone else could offer advice here it would be helpful.

He argues that coordinate dependent things are still "real"/"physical". So he can claim both:
"that the other's clock is running slower"
and
"the clock is not physically changed"
at the same time.

This seems like falling for the twin's paradox, so when asked for clarification, the response is:
"It is simply nonsense to say 'both clocks are running slower than the other'. "
So he appears to both reject and accept coordinate dependent quantities as 'real'/'physical' depending on the scenario.

When asked why, if one accepts coordinate depedent quantities as 'real', you can't say both clocks are running slow, his answer is:
"Actually you would have to be in two places at the same time to do this."
and states his solution to the twin's paradox is that
"Proper relativistic comments can only be made from the viewpoint of an inertially moving observer considered to be stationary."

When asked what coordinate system he uses to refer to whether someone is "stationary", he feels no coordinate system is referred to implicitly or otherwise with this statement.

-----------

The quotes above are focussing on the disagreements. And therefore probably cast a negative light. This person is not a crackpot though. They understand the basics and can do some calculations with Lorentz transformations. I just feel that they missed the essence of relativity, and worse have self-conflicting ways of resolving the famous 'paradoxes'.

I fully understand that I may be incorrect as well.
Either way, please please help guide us to better understanding. If you could refer to specific textbooks (both of us own MTW, and there are probably other textbooks we own in common as well), that would be very very helpful. I thought MTW would be good as I have heard some students jokingly refer to it as the relativity 'bible', so I figured we could consider that a good representation of the modern understanding of relativity (no?).

Quoting pertinent sections or "assigning us problems" from the book to help us learn would also be great ... thanks!

2. Oct 23, 2008

Fredrik

Staff Emeritus
Wow, that's a long post, with a few too many questions at once. It would take too long to answer all of it.

First of all, you should always keep in mind that a coordinate system is just a function $x:M\rightarrow\mathbb R^4$, where M is the spacetime manifold. It has to satisfy a few more technical requirements to be considered a coordinate system, but those aren't really important here.

The time dilation equation describes the relationship between the time between two events in one inertial frame and the time between the same two events in another inertial frame.

It is, but you could say that it's forced upon us by the physics. (To be more precise, by the fact that space and time can be represented by Minkowski space). The same thing goes for Lorentz contraction too.

The four-velocity is defined with magnitude c, so that can't really be used to explain anything.

SR is a theory of physics which consists of a mathematical model (Minkowski space), and a bunch of postulates (that for some reason are never mentioned explicitly) that identifies things in the real world with things in the model. The most obvious of those postulates is that what a clock measures is the proper time of the path that represents its motion. So the correct final ages of the twins is an immediate consequence of a postulate.

I disagree. Let the rod stop accelerating after a while. Now we're back to the situation you described as just a coordinate effect, but it was a physical effect that caused it?! Something is wrong here.

Born rigidity is equivalent to having the distance between two (infinitesimally) nearby points remain constant in every local inertial frame associated with the world line of one of those points. This is what I would describe as no physical contraction.

3. Oct 23, 2008

atyy

Last edited: Oct 23, 2008
4. Oct 23, 2008

Ich

I agree. Kip Thorne lectures classical physics in coordinate-independent terms, introducing coordinates when useful. Physics is expressed as the relationship of geometric objects.
I still agree, but there is a caveat:
Standard coordinates in special relativity always have an operational definition. If you always state (even implicitly) which coordinate system you refer to, the coordinate-dependent quantity is unambiguously defined as a geometrical object.
Those quantities still have enough potential for confusion, because people tend to regard physical quantities as properties of an object, not as a statement about the relationship of certain objects, even if they are. Just take energy as an example, one is tempted to ask what the true energy of an object is, and to make unsubstantiated conclusions, like "If you go too fast, do you become a black hole?". Not because energy were ill-defined or unphysical, but because it is seen as a property, not a relation.

Let me add that I think that "time dilation" is a concept for advaced learners only, as a convenient tool to shorten calculations. It is definitely not helpful to understand relativity when it is taken as a basic point. Still physical, but extremely clouding the geometric nature of physics.

So, I think you are basically right, but your friend's comments are not formally wrong, if he knows the implications. There seems to be a misunderstanding about this property/relation issue. But he will definitely benefit when he learns to abandon coordinates and draw on white instead of squared paper. The squares are useful, but not necessary.

EDIT:
To elaborate, when your friend has no problem with the other clock "ticking slower", that's (hopefully) because he somehow does not see the "ticking rate" as a property of the respective clock, but as a relation between both. But there is a problem with the wording, because such a rate colloquially means a property of the clock. This is misleading, and fortunately there is a covariant property of the clocks that hits the nail on the head: four-velocity is, as you correctly stated, of invariant magnitude, but points in different directions. So, in loose wording, both clock's time does evolve at equal rates, but in different directions. Both directions are a priori on an equal footing, it is just that when you specify where the clocks will meet again (i.e. at what velocity) you single out one direction as the straight one and the other as a detour, whith shorter elapsed time. That's a far better viewpoint than time dilation, at least in my eyes.

Last edited: Oct 23, 2008
5. Oct 23, 2008

STFiction

Wow!
Thank you everyone so much. So many comments so quickly after posting: This forum is great!

Let me rephrase to make sure I understand.
A coordinate system is just a mapping from the set of spacetime points to an n-tuple of Real numbers (where n is the dimension).

Is that correct?
Given that, I'm guessing the 'technical requirements' have something to do with continuity or analyticity or something? (I don't want to pull the thread of topic, so if someone could recommend a source I'll just go read up on it.)

But it doesn't relate the time between any two events, or any two inertial frames. It is a very specific condition.

t' = t Sqrt[1-v^2/c^2]

Is relating the proper time t' of a clock, to the coordinate time t of an inertial coordinate system in which the clock is moving at constant speed v.

Maybe we're saying the same thing and agreeing. If not, I think I misunderstood the comment.

Hmm... That is an interesting way of looking at it.
I think this is related to Ich's comments about how inertial frames can be operationally defined and therefore the coordinates have a relation to a geometric quantity.

Would this be a fair summary:
We can take as postulates that magnitude of four-velocity is c, and proper time is length of worldline.
Or, we can define time and length operationally (like with a standard clock or ruler as Einstein did) and derive thet magnitude of four-velocity is c and proper time is length of wordline.

Maybe I'm misunderstanding Ich.
It seems both are self-consistent.

In 'rigid acceleration' the proper length is constant. But to maintain this during motion, there is a necessary physical effect that the proper acceleration is different for the different ends. If not supplied by a carefully arranged external force, then it will necessarily be supplied by an internal stress (Such as would happen with an object stationary in Rindler coordinates).

The example that I read here was of the Bell's spaceship thought experiment. For the distance between the spaceships to change, there must be a tension in the string.

This does not mean the 'usual' length contraction, which just relates measurements of an inertial rod between frames, is physical. That is a different scenario.

6. Oct 23, 2008

atyy

I think of coordinates as very fundamental, since spacetime itself is defined as something on which you can put many different sets of smooth coordinates.

Also, isn't the dimension of spacetime specified as the number of coordinates we need to specify the location of an event?

7. Oct 23, 2008

Ich

You are right, time dilation compares time between three events:
a) start
b) stop at the other clock's location
c) the event that is simultaneous to b) but at the same place as the origin in the first frame
You compare a-b with a-c. That's important.

8. Oct 23, 2008

STFiction

The coordinate system is merely how you label the events in spacetime. So I wouldn't agree with that. Using the way Fredrik defined a coordinate system above, it is clear the manifold is not defined by coordinate systems.

That is my understanding as well.

I have reread what you wrote several times above.
I think you've outlined a quite plausible explanation of the main-errors here:
- I didn't realize if you specify a coordinate system operationally that the coordinates are therefore related to geometric quantities.
- (and maybe) He doesn't realize that it is only this relation to geometric quantities, and not the values of the coordinates themselves, that have a physical meaning.

I do hope he decides to join the conversation. He prefers not to think of anything in terms of geometric quantities. So I'm not sure I can convey that well to him. Actually, to be honest, I'm worried he will read that and just think: 'see, coordinate systems are physical things' and try to leave it at that ... which as I understand it, is a seriously incorrect over-simplification of what you said (right?).

Is there a particular chapter/section of MTW that he and I can study that discusses stuff like this?

I'll also take some time to read through those notes you linked to.
Thanks again.

It seems to me that event C is superfluous. All that is needed is the proper time between events a,b, and the difference in the time component of the coordinate representation for events a and b in the inertial coordinate system.

Or are you just trying to point out a possible way for relating the coordinate times to something geometric? In that case I would agree we would at least need one more event (on the other clocks worldline). I think we would also need some paths in spacetime to connect b,c to help specify this coordinate dependent 'simulteanity' through some geometric contrivances.

Just to be clear though, even if we view it that way, it still isn't a relation between just those two clocks, right? Using what you told me above, we can probably relate the coordinate time to the proper time of another clock and some other geometric things to help pin down 'simultaneity' in that specific coordinate system.

So it is relating the proper time of one clock, to whatever series of manipulations of geometric objects that are needed to geometrically define a coordinate time which necessarily will have to include more than just the proper time another clock (since the events in question are not on its worldline).

Right? Or am I misunderstanding / reading into what you said too much?

In the end, it seems we've merely traded arbitrary choice of coordinate system for arbitrary choice of how to geometrically relate the two clocks. Yes, the individual geometric relations are real, but none involve the clocks solely unless they have two points in common on their worldlines.

Last edited: Oct 23, 2008
9. Oct 23, 2008

Naty1

Great questions...not easy..I'll try to tackle one:

I'm going to STRONGLY disagree to make a point. The above is, I believe, overly simplistic yet not "wrong". It depends on your frame of reference. To illustrate: Here are a few quotes frpm THE RIDDLE OF GRAVITATION, 1992, by Peter Bergmann, a former student of Einsteins:

(my emphasis)

There is no single Lorentz frame to define the "true" distance between astronomical bodies for use in a gravitational force computation. (paraphrase from Bergmann)

Note: Mass, energy,time, distance seem coordinate dependent; electric charge and lightspeed are not!! Is not something "physical" going on here??

So I will argue that trying to clarify what's "physical" is (a) coordinate dependent without any final and absolute right or wrong answer and (b) trying to do so is like trying to argue whether a photon or electron is a wave OR a particle....the "real" world has some fundamental ambiguities.

Ambiguity and uncertainty are with us!!!

Last edited: Oct 23, 2008
10. Oct 23, 2008

Fredrik

Staff Emeritus
Yes.

The conditions on $x:U\rightarrow\mathbb R^4$ is that

* U is an open subset of M
* x is continous
* x-1 exists and is continous

(A function from an open subset of a topological space to an open subset of a topological space that satisfies those three conditions is called a homeomorphism, probably just to intimidate the n00bs).

The set A of all coordinate systems is called an atlas. It must also satisfy a few conditions:

* The domains of definition of all the coordinate systems must cover M. If we write $A=\{x_\alpha:U_\alpha\rightarrow\mathbb R^n|\alpha\in I\}$, where I is just some set that can be mapped bijectively onto A, this condition is just

$$\bigcup_{\alpha \in I}U_\alpha=M[/itex] * If $x:U\rightarrow\mathbb R^4$ and $y:V\rightarrow\mathbb R^4$ are both in A, then [tex]x\circ y^{-1}:y(U\cap V)\rightarrow x(U\cap V)$$

is $C^\infty$. (That means that all of its nth order partial derivatives exist for arbitrary n).

In SR, we can always take the domain of definition to be Minkowski space instead of some proper subset.

I would say that t' is the coordinate time between two events in one frame and that t is the coordinate time between the same two events in another frame. That's probably the best way to think of it (because it doesn't look as asymmetric as your description), but there's of course nothing that prevents us from taking the two events to be points on the world line of a clock, where the clock is displaying times t0 and t0+t'.

I would call those "definitions", not postulates. If it's just math, it's a definition. If it relates the math to something in the real world, it's a postulate, or a consequence of other postulates.

We have to do that too. Those "definitions" are the true postulates of the theory. They tell us how measurements in the real world correspond to mathematical things in the model.

I think this is a simplified definition of Born rigid motion that only holds when the acceleration is constant. When it isn't, you have to change "proper length" to "proper distance between infinitesimally nearby points on the rod, in the co-moving local inertial frames".

This is true, but those internal forces will just keep the distance between an atom and its closest neighbors the same in all of the the co-moving inertial frames. And that's exactly what they will do if the object isn't being accelerated. The only new physical effect here is an external force that's pushing or pulling the rod, and the propagation of those forces through the rod. Those additional forces are only responsible for the acceleration, not for the contraction. That's why I think we can consider the contraction a coordinate effect in this case too.

This is how I think of that problem: The rockets are assumed to be identical in every way, so their world lines must be identical. (Otherwise there would be something fundamentally different about their starting positions, but that contradicts SR, since translations are isometries of the Minkowski metric). This means that the distance between them remains the same in the original rest frame. But after a while, the speed in the original rest frame is v>0, so we're looking at a Lorentz contracted string which is still the same length as when it wasn't Lorentz contracted. That means that it must have been forcefully stretched. We assumed at the start that the string will break if any force is applied, so the conclusion is that the string breaks.

Last edited: Oct 23, 2008
11. Oct 23, 2008

Fredrik

Staff Emeritus
Yes, it is.

A smooth manifold is defined as a Hausdorff topological space with a maximal $C^\infty$ atlas (as defined in my previous post). So the coordinate systems are a very important part of the definition.

"Maximal" means there are no more functions that you can add to the atlas without having it violate one of the conditions in the definition.

Last edited: Oct 23, 2008
12. Oct 23, 2008

STFiction

Fredrik,
Thank you for taking the time to type that up and to help explain things.

Hmm... I'm confused now. That seems like a circular definition. Manifold depends on coordinate system to define, and coordinate system is a function mapping points on the manifold to the set Reals^n.

I'll need to ruminate on this some, as I feel I'm missing something basic.

The last two make sense, for it requires it to be continuous and one-to-one.
The first I'm having trouble understanding the significance of. Why does it need to be an open subset?

Strangely enough a math friend of mine was just explaining homeotopy and homotopy to me the other day. I can kind of see it intuitively now, but am still having trouble with the details. I still have a lot of math to learn.

Okay, that wording does make more sense. Duly noted.

It IS different from when the object isn't being accelerated though. For the proper accelerations are different on each end of the rod. The proper forces are different as well. If the external force is uniform, then the internal forces cannot be. This is a physical effect.

So yes, the proper distance between atoms stays the same, but the internal forces necessary for this are different (similar to if we had an infinitely strong elastic string in Bell's spaceship problem, "contracting" the distance between the two spaceships ... if those forces are not there, the proper acceleration is the same and the length will not follow the "length contraction" formula according to any inertial frame).

13. Oct 23, 2008

atyy

In mathematics, a thing is defined as what operations you can perform on it and have everything remain consistent. The thing is a good model of our world if we have physical operations that consistently correspond to the mathematical operations. So for example, Newtonian physics and special relativity both claim that we can set up cartesian coordinate systems in which the laws look nice and simple. We would clearly have difficulty doing this if rigid bodies did not exist. We could use light rays to get straight lines, but how would we make a protractor or theodolite to measure angles? So it is a very physical thing to claim that our world is modelled by a mathematical manifold which is defined as a thing on which you can put smooth coordinates.

14. Oct 23, 2008

Fredrik

Staff Emeritus
What I called M in #10 is really just a topological space. It we want to nitpick, it's the pair (M,A) that should be called a manifold. But that terminology is only useful in the definition. When we're done with it, we can be a bit sloppy and call M a manifold. (I didn't say what M was in #10, but I used the sloppy terminology in #2).

This is similar to how a group is defined as a pair (G,*) where the * is a map from GxG into G with certain properties. We usually refer to G as the group. E.g the set of 2x2 matrices isn't really a group, but we'll call it a group because it's painful to say that the pair (X,*) is a group where X is the set of 2x2 matrices and * is matrix multiplication.

It's just a convenience that makes the whole mathematical concept easier to deal with. It certainly doesn't have anything to do with the physics. I can't immediately think of something that would be weird without the requirement that U is open, but I probably could if I gave it some time. It definitely has something to do with this definition of a continuous function: $f:X\rightarrow Y$ is said to be continuous if $f^{-1}(E)$ is open for each open $E\subset Y$.

Only on a large scale. The difference is small when you consider nearby points on the rod and goes to zero as you let the distance between them go to zero. So everything is the same locally.

Yes, that's one of the large-scale effects. It's a consequence of the fact that the coordinate systems in which "everything is the same" is a different coordinate system for each atom along the length of the rod.

I'm not sure what you mean by "uniform" here. Do you mean that the coordinate acceleration in the original rest frame is the same at every part of the rod? That would forcefully stretch the rod so that its length is constant in the orignal rest frame even though it gets Lorentz contracted. Do you mean that the external force is constant and applied to a point on the rod (e.g. it's rear endpoint)? This is the scenario I've been describing.

They are different in the original rest frame, but not in the co-moving local inertial frames.

15. Oct 23, 2008

Fredrik

Staff Emeritus
Agreed. The reason is of course that the result of all our measurements are numbers.

There actually are no rigid bodies in SR. Turn a bicycle upside down and give one of the wheels a spin. The wheel is now getting Lorentz contracted along its circumference, but it stays the same length in the original rest frame (we can be sure of that because the symmetry of the situation guarantees that all the points along the circumference have identical world lines), so it must be getting forcefully stretched. This means that anything that rotates isn't even approximately rigid (or even approximately Born rigid).

16. Oct 23, 2008

atyy

So how do we measure angles? I always imagined it's ok as long as it "bounces back" :rofl: to its original shape after acceleration. Same with length, so we can accelerate a "rigid" rod to some constant velocity, transform to its rest frame and deduce length contraction.

17. Oct 23, 2008

Fredrik

Staff Emeritus
atyy, I'll try to think about your question tomorrow, if no one else has answered it by then.

Right now I'd just like to say something about my endorsement of the view that time dilation and Lorentz contraction are "just coordinate effects". It isn't correct to say that they are just coordinate effects. Imagine a rod at rest in some inertial frame. The world lines of its endpoints are two straight lines in all frames. In the (spacetime diagram of the) rest frame, they are vertical. When I measure the length of the rod using a ruler, the result is the proper distance from an event A on the world line of one of the endpoints to an event B on the world line of the other endpoint, along a line of constant time coordinate. Another observer would disagree about which lines have a constant time coordinate, so the result of his measurement would be the proper distance along a line from A to B', where B' is some other event on the world line of the second endpoint.

These statements about which events are involved in the measurements are coordinate independent statements. They refer to events, not to coordinates of events. So they are definitely statements about something physical.

So it definitely isn't right to say that time dilation and Lorentz contraction are just coordinate effects.

Last edited: Oct 24, 2008
18. Oct 24, 2008

Ich

Yes, seriously incorrect.
b) - a) is a vector, call it k. Now, comparing the time component of k in different coordinate systems gives time dilation - if and only if those are standard IS of special relativity.
Most generally, comparing coordinates in different systems tells you nothing at all about the phenomena. You could imagine a system where 1 "coordinate second" equals 2 "real seconds", and by comparing coordinate times you get a result that has nothing to do with any reality.
Only in SR, coordinate time has this defined geometrical meaning: You project the events orthogonally onto your time axis, and call the position of this new, projected event its coordinate time. That's my event c). Now, c) - a) is a different vector (the projected one), and its length is different from |k|.
Geometrically, the length of k in the system of an observer with four-velocity u is expressed most easily as the dot product $$u_ak^a$$. Time dilation between two observers with four-velocities u and v is simply $$u_av^a$$. This shows that it is symmetric, and that it expresses a relation between both vectors, namely the angle they span (called rapidity).
If your friend decides to ingnore these points of view, he has no chance tounderstand relativity, especially the general theory.

No, you only have to project event a), too, and work with four events instead. There's no problem with the angle between vectors at different points in flat spacetime.
It is true that the physical realisation of that projection involves special procedures, but two clock are enough to measure the effect.

19. Oct 24, 2008

atyy

One way to think of this question is to ask: What equipment and what procedures with the equipment would I need to provide experimental evidence that Euclidean geometry is a good model for pictures we draw on a piece of paper? The most famous pieces of equipment are the straight edge and compass. Experimentally, one should ask, if my paper appears to violate Euclidean geometry, is it the paper's fault, or is my ruler somehow not really straight? It's interesting to compare various axiomatic formulations of Euclidean geometry and ask: at which point do you need a ruler with length, or a protractor that measures angles? Also, if we wish to introduce coordinates so that "x2+y2=constant" describes a circle, how do we do this? If our circle appears to violate Euclidean geometry, is it the paper's fault, or because our compass wasn't rigid enough?

Edit: Right there you can see the meaning of coordinates. They are clearly physical in the sense that we do worry whether we've drawn them correctly, but they are not physical in the sense that two different sets of coordinates can describe the same physical circle.

Edit: I just realised some might ask the question in reverse: how do you provide experimental evidence that the paper is a good (approximate) model for Euclidean geometry?

Edit: In the spirit of the above, I guess actually it doesn't mean anything to say that a piece of paper is modelled by Euclidean geometry. We can only ask if the paper and specified physical straight edges, compasses, rulers and protractors and operations are modelled by Euclidean geometry. It's conceivable that it may be Euclidean or not depending on which physical objects and operations you define to be straight edges. Here's a way to "demonstrate" that your paper is not Euclidean: http://comp.uark.edu/~strauss/papers/hypcomp.pdf.

Last edited: Oct 24, 2008
20. Oct 25, 2008

STFiction

Okay, the more I think about this, the more I get confused. Yes, even a coordinate dependent measurement can be related back to geometric quantities. And it turns out any operationally defined coordinate systems can be related to geometric quantities as well.

But I think there may still be a problem here in my understanding, for I still disagree here. Somehow my mind is making a disctinction between some geometric quantities as physical and some as 'arbitrary choices'. For instance consider the two parallel lines corresponding to the world lines of the ends of an inertial rod as Fredrik suggested. Choose any pair of points from each line and the proper length is the same. To me: This seems, 'real'. That length seems 'physical'.

Now consider the coordinate length as measured by an inertial reference frame. The choice of coordinate system is completely arbitrary, this doesn't seem 'real'/'physical'. Same with the coordinate time example Ich gave above. Yes, the geometric invariant $u_a k^a$ gives the relation of the proper time along one clocks path to the coordinate time of some inertial reference frame with spatial origin moving at velocity $u$. But the choice of that coordinate system u is arbitrary.

Yes, you can calculate that geometric object in any coordinate system and get the same value, but it has absolutely no experimental consequence. It doesn't have any physical meaning anymore than the choice of coordinate system does.

To help illustrate the problem I am having conceptually, let me choose two geometric objects that lie on opposite spectrums of this (and let's stick to classical physics for now, I don't want to have to worry about commuting observables, etc. right now):
Geometric invarient A: The number of electron paths intersecting a specified spacetime path(ie. the number of electrons hitting an object between two events).
Geometric invarient B: A scalar field I define to everywhere be 1.

Those are both geometric invariants. Regardless of coordinate system, the value of the field B at a point, or the value of A will be agreed upon.

Yet I could have chosen the geometric object for B, where the field is everywhere 2. While again a geometric invariant, there are an infinite number of such geometric objects, and none have any effect on an experiment ... so I would call B not physical (despite being a geometric invariant).

Similarly, what we decide to call the coordinate time between events is as arbitrary as our choice of coordinate system (obviously), and in no way effects any experiments. Yes, $u_a k^a$ is a geometric object, but that is true for any choice of u and as such doesn't seem to have any physical meaning.

So I don't think we can consider any geometric quantity as physical.
As is probably clear from above, I'm starting to get confused on what is and isn't physical or an observable in SR now.

But I'm currently falling on the side of still calling time dilation just a coordinate system dependent effect... yes, the usual time dilation equation is for inertial reference frames, which come about due to their choice of spacetime labels preserving as many symmetries of Minkowski space as possible, so in this sense the time dilation equation is intimately related to the geometry of spacetime. But ultimately, these are just a subset of possible reference frames, and an infinite set at that, and nature cannot possibly be affected by how we choose to label it... there is nothing physical that actually makes one clock run slower than the other... for time dilation is not a direct comparison between two clocks, but between a clock and some unmeasurable quantity we associate with another clock.

If this is all incorrect, please help me understand what is considered physical in the modern interpretation of SR. For the current discussion seems to really really be blurring the lines between geometric and coordinate dependent representations.