How can we test space-time for pseudo-Riemannian-ness?

[Dale]

This does not even look like genuine stupidity. I give up.

There is no need to be rude, particularly since I am right. A measuring device produces a number, and that number is always the same regardless of what coordinate system you are using, therefore the number is a scalar.

 

[fr.jurain]

A measuring device produces a number, and that number is always the same regardless of what coordinate system you are using, therefore the number is a scalar.

Hem... How do you measure the position of a particle (in a Euclidean setting)?

 

[Dale]

Hem... How do you measure the position of a particle (in a Euclidean setting)?

Usually with three individual measurements, unless the particle is otherwise constrained.

In any case, the numbers returned by the measuring apparatus, whether one or multiple, are the same in all coordinate systems. So they transform as scalars.

 

[fr.jurain]

Usually with three individual measurements, unless the particle is otherwise constrained.

In any case, the numbers returned by the measuring apparatus, whether one or multiple, are the same in all coordinate systems. So they transform as scalars.

Yes, vectors and other multi-dimensional quantities can be measured. You do it by specifying a reference frame, and then, you get simultaneous readings -scalars assuredly, present and the same for all to see, whatever their position- which are the components of what you want measured, a 3-dim vector in the above case, in that frame, and only in that frame.

Now (with emphasis added here, not when 1st posted):

Okay... Only there's a bit of the way you'll have to do by yourself. Which is to realize:
1) all of this is done on one and same manifold M, in a neighborhood of one 4-point e, so we pick one set of coordinates around e and write everything in the basis of tangent vectors.
2) not everything that is measured is a tensor, so the best we can do is to show their components in the chosen basis.

Measurement #1: interferometry; yields $g_{ab}$, the components of the fundamental metric;
Measurement #2: gravimetry; yields $\Gamma^a_{bc}$ the components of the connection in the chosen basis, possibly under the postulate that the connection is symmetric.
These are smooth fields, meaning that we imagine we could get a continuous record of these quantities with arbitrary precision all along any finite set of paths from e to some other point e', and numerically compute derivatives, to any order and with arbitrary precision as well.
[...]
In other words, h, the gravitational potential, can obtain from the $\Gamma^a_{bc}$'s alone. It is a peculiarity of Einsteinian GR that it be equal to g.

In view of the above, can't you answer your question by yourself?

Measurements are always scalars, please show from first principles what pair of measurements you think could disagree to demonstrate two metrics.

Is there any ambiguity in my post about what is measured and what is integrated from measurement results, what is a metric tensor and what is not, what I claim can disagree whereas Einsteinian GR posits they're equal?

 

[Dale]

Is there any ambiguity in my post about what is measured and what is integrated from measurement results,

Yes, it is completely ambiguous. I have no idea what measuring device you are considering, what physical principle it operates under, and what experiment you are proposing. As far as I can tell you are avoiding answering my question and are simply assuming the consequent.

If you would like to describe how you collect some experimental data and then compute tensors from that, then that is fine, but I am interested in the first step which you have been skipping, the experimental measurements. Those are scalars. You are completely missing the description of how you physically collect those.

 

[fr.jurain]

Is there any ambiguity in my post about what is measured and what is integrated from measurement results,

Yes, it is completely ambiguous.

What a pity. Well: $g$ and $\Gamma$ are obtained by measurement; $h$ is obtained by integration.

I have no idea what measuring device you are considering, what physical principle it operates under, and what experiment you are proposing. As far as I can tell you are avoiding answering my question and are simply assuming the consequent.
If you would like to describe how you collect some experimental data and then compute tensors from that, then that is fine, but I am interested in the first step which you have been skipping, the experimental measurements. Those are scalars. You are completely missing the description of how you physically collect those.

This is not the point; not necessarily an uninteresting or irrelevant one, mark. Let me just remind you that you've had the following info:

We can make physical measurements by 2 fundamentally different mechanisms (among others):
1) interferometry; make light or a microwave emitted by electrons interfere with itself, and use the resulting pattern as a ruler or a clock;
2) weighing a test mass; tune the Lorentz force to balance fictitious forces acting on the mass.

You've also had clear indications that interferometry is how we get $g$, and weighing how we get $\Gamma$.

Now, back to the point. My question was:

In view of the above, can't you answer your question by yourself?
Is there any ambiguity in my post about what is measured and what is integrated from measurement results,
what is a metric tensor and what is not, what I claim can disagree whereas Einsteinian GR posits they're equal?

The question in question being:

Measurements are always scalars, please show from first principles what pair of measurements you think could disagree to demonstrate two metrics.

Well? Is it clear to you what I think could disagree? Is it a pair of measurements? Is it clear how I define the two metrics?

 

[Dale]

What a pity. Well: $g$ and $\Gamma$ are obtained by measurement

How, exactly?

You've also had clear indications that interferometry is how we get $g$, and weighing how we get $\Gamma$.

You think it is clear, but I don't know what measurements you intend.

Is it clear to you what I think could disagree? Is it a pair of measurements? Is it clear how I define the two metrics?

No, it is not clear to me. That is why I keep asking you so many times to explain your intended measurements.

 

[fr.jurain]

OK, one last try. Remember this?

Gee... How do you understand this then?

2 different metrics mean different physical measurements, by 2 different mechanisms, at the same point yield 2 different results whereas GR *posits* they must be equal.

I understand it as an incorrect assertion for the reasons given above. GR does not posit that two different physical measurements by two different mechanisms at the same event must be equal, and finding that they are different does not imply there are two metrics.

It's somewhat unfortunate you truncated my explanations right before the moment they had a chance to avoid a misunderstanding. The full quote is:

2 different metrics mean different physical measurements, by 2 different mechanisms, at the same point yield 2 different results whereas GR *posits* they must be equal.
1st physical measurement at point e: establish the metric at e the Pavillon de Breteuil's way;
2nd physical measurement at point e: define small loops around e, not hesitating to let 1st physical measurement at e help you specify them; general covariance sees to it it makes sense. Measure Gamma's along these loops, the balancing way. Discover the Gamma's can be integrated along the loops as Christoffel prescribed, publish paper "Einstein was right! 1st direct measurement of grav potential". Read paper 1 week later "Einstein was wrong! Discrepancies between grav potential and fundamental metrics".

It should be clear by now the 1st physical measurement in question was always meant to yield the $g_{ab}$ at e, and the 2nd was always meant to yield the $\Gamma^a_{bc}$ around e, which of course cannot be compared one for one with the $g_{ab}$. And so I never intended them to be compared that way, and it takes a good dose of benevolence to accept you could misunderstand me on that count:

GR does not posit that two different physical measurements by two different mechanisms at the same event must be equal.

Dub the statement what you want, in Einsteinian GR $\Gamma^a_{bc} = g^{ad}(\partial_{b}g_{dc} + \partial_{c}g_{db} - \partial_{d}g_{bc})/2$. Whereas the bulk and the detail of the quote, and of my subsequent posts, is I claim it can be that $\Gamma^a_{bc} \ne g^{ad}(\partial_{b}g_{dc} + \partial_{c}g_{db} - \partial_{d}g_{bc})/2$, all the while maintaining $\Gamma^a_{bc} = h^{ad}(\partial_{b}h_{dc} + \partial_{c}h_{db} - \partial_{d}h_{bc})/2$ for some $h \ne g$. No more, no less than that.
Where in Einsteinian GR $g$ is by definition what gives the tangent space at e its Minkowskian structure, thereby allowing SR to hold locally; and of course it is defined the same in my posts;
and likewise in Einsteinian GR $\Gamma$ is by definition the connection thanks to which Newton's law admits a generally covariant formulation, i. e.
$F^a = mA^a = m(\frac{dU^a}{ds} + \Gamma^a_{bc}U^bU^c)$ with F the resultant of the 4-forces acting on the test particle, m its (constant) mass, U its 4-velocity, and again it is defined the same in my posts. Again: of course, what else could it be.
If it was not clear until stated the way I just did, then sorry; just quote me in full in the future, and in return I'll make an effort to launch math formulas right from the start.

So, in answer to your question:

No, it is not clear to me. That is why I keep asking you so many times to explain your intended measurements.

Well it's really too bad it's not clear; for these are exactly the same measurements that give experimental access to $g$ and $\Gamma$ in Einsteinian GR; so if you know enough of GR to assert what's posited in it and what's not, as in:

GR does not posit that two different physical measurements by two different mechanisms at the same event must be equal

then you must necessarily know what these measurements are. Mustn't you?

 

[Dale]

you must necessarily know what these measurements are. Mustn't you?

Clearly not, which is why I have been asking for several pages. And despite more than ample opportunity, I see no indication that you know either.

Also, you have avoided answering the repeated question about if you agree that all tests for pseudo-Riemannian-ness are tests for local Lorentz invariance, or if you are thinking that it is possible to have local Lorentz invariance and still not be pseudo-Riemannian?

 

[fr.jurain]

Clearly not, which is why I have been asking for several pages. And despite more than ample opportunity, I see no indication that you know either.

Not so fast. Let me remind you that a statement by you is part of the dispute, and despite more than ample indication that we might not have the same meanings in mind when using the same words, you have not clearly committed to one yet.
The disputed part is emphasized like this here (it was not when posted):

I understand it as an incorrect assertion for the reasons given above. GR does not posit that two different physical measurements by two different mechanisms at the same event must be equal, and finding that they are different does not imply there are two metrics.

You chose to assert, not to ask. What did you consider these "two different physical measurements" to be? Were they $g$ and $\Gamma$, defined as I now define them? Or the same two, defined otherwise? Or "Any two things fr.jurain might claim can obtain from measurement"? Else, what?