Undergrad Mathematical Truth of Physically Observable Quantities

[Replusz]

TL;DR

dear All,
I am just looking for a bit of a background on "any physically observed quantity must be coordinate independent".

I assume this is true because using a passive coordinate transformation of the coordinate system should not effect how we measure something. I don't know if this is enough, hence if my original statement is just trivial, or if there is some deeper underlying thing lurking.

Is the statement true at all mathematically?
Thank you!

 

[Ibix]

What you measure is what you measure. A theory that says "because I'm using spherical polars instead of Cartesian coordinates your measurements change" isn't physically plausible.

 

[PeroK]

Summary:: dear All,
I am just looking for a bit of a background on "any physically observed quantity must be coordinate independent".

I assume this is true because using a passive coordinate transformation of the coordinate system should not effect how we measure something. I don't know if this is enough, hence if my original statement is just trivial, or if there is some deeper underlying thing lurking.

Is the statement true at all mathematically?
Thank you!

It's not really a question of mathematics, it's a question of physics. Modern theories focus on coordinate-independent quantities, also called invariant quantities. These are the quantities that ought to appear in your laws of physics.

There's an element of this in Newton's theories as well. Newton's second law, $F = ma$, deals with three quantities that are invariant across all inertial reference frames. The conservation of energy and momentum likewise holds in all inertial reference frames. But, the measurements of energy and momentum themselves are not invariant. The laws of physics, therefore, deal with conservation of energy and momentum.

There are lots of important things you can measure, but ultimately the laws of physics must depend only on the aspects of those measurements that are invariant across all reference frames.

 

[Replusz]

It's not really a question of mathematics, it's a question of physics. Modern theories focus on coordinate-independent quantities, also called invariant quantities. These are the quantities that ought to appear in your laws of physics.

There's an element of this in Newton's theories as well. Newton's second law, $F = ma$, deals with three quantities that are invariant across all inertial reference frames. The conservation of energy and momentum likewise holds in all inertial reference frames. But, the measurements of energy and momentum themselves are not invariant. The laws of physics, therefore, deal with conservation of energy and momentum.

There are lots of important things you can measure, but ultimately the laws of physics must depend only on the aspects of those measurements that are invariant across all reference frames.

When saying invariant, what exactly do you mean? For example mass in classical physics is coordinates independent, but position is not. Neither is velocity...

 

[jbriggs444]

When saying invariant, what exactly do you mean? For example mass in classical physics is coordinates independent, but position is not. Neither is velocity...

For instance, the distance between two [relatively motionless] points is invariant in classical physics. It does not depend on coordinate choice. Nor does it depend on the state of motion of the selected inertial reference frame.

Whether the points are relatively motionless is also an invariant fact of the matter which can be agreed upon regardless of coordinate system or reference frame.

 

[Ibix]

I think there's a moderately subtle point in the philosophy of science here. A theory is expressed as pure maths. For example, $F=ma$ is simply a linear equation linking three quantities. To make it into useful physics I have to relate those quantities to something in the real world: $F$ is the reading on my Newtonmeter, $m$ is the reading on my weighing scale (don't ask too many questions about this example!) and $a$ is the second derivative of displacement. These are all measurements, and what the mathematics of the theory does is predict the third measurement given any two.

It's really easy to invent a theory that doesn't respect the rule that measurements must be coordinate independent. I'd like to announce Ibix Relativity, a theory which is mathematically identical to Einstein's Special Relativity. However, in my theory clock measurements correspond to the concept of coordinate time, not proper time. Hence this theory predicts measurements that are not coordinate independent. If you look at my clock and any written records of my clock measurements I predict that you will see something different from what I see!

But we know the real world doesn't work like that - you don't see my clock reading something different from me just because you start walking. If I break my clock at 3pm, everyone sees it stopped showing 3pm, whatever time they think it actually is when I broke it. So, sadly, we have to consign Ibix Relativity to the scrapheap (hopefully before I get an infraction for a personal theory), not because of any mathematical problem but because the things it predicts are not consistent with the real world. Our experience is that measurements are invariant - so any plausible theory has to have the predictions of measurements it makes be invariant too.

Edit: that's my understanding of this, anyway.

 

[PeroK]

When saying invariant, what exactly do you mean? For example mass in classical physics is coordinates independent, but position is not. Neither is velocity...

Invariant means it is the same in all reference frames. Position and velocity are not invariant, but acceleration is.

I think it's easier to understand this in Newtonian physics first. Before looking at invariance in relativity.

 

[Ibix]

When saying invariant, what exactly do you mean? For example mass in classical physics is coordinates independent, but position is not. Neither is velocity...

But you can't measure position or velocity. You can only measure distance from some point (which is invariant) or velocity relative to some chosen object defined to be at rest (which is also invariant).

 

[strangerep]

Invariant means it is the same in all reference frames. Position and velocity are not invariant, but acceleration is.

(I'm guessing that's not quite what you meant to say?)

 

[PeterDonis]

(I'm guessing that's not quite what you meant to say?)

If he meant proper acceleration, his statement is correct.

If he meant coordinate acceleration, you're right, it's not.

 

[PAllen]

(I'm guessing that's not quite what you meant to say?)

He stated Newtonian physics. Implicitly assumed is Newtonian inertial coordinates. Of course, if general coordinates are allowed, then the statement is not true.

 

[PeroK]

He stated Newtonian physics. Implicitly assumed is Newtonian inertial coordinates. Of course, if general coordinates are allowed, then the statement is not true.

Yes, exactly, and that was edited out in the reply. Thanks for pointing this out.

 

[PeterDonis]

He stated Newtonian physics.

He stated that it's easier to see in Newtonian physics. But that still leaves it somewhat ambiguous exactly what is "easier to see". The concept of acceleration that is invariant in Newtonian physics is not the same as the concept of acceleration that is invariant in relativity.

 

[Nugatory]

Several posts that weren’t helping have been removed from this thread.

Please, everyone, try to keep on topic.