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

[TrickyDicky]

If by "preserving its metric" you mean the same as I do (#25), then see above (#24) for a counter-example of what you claim. Else, what do you mean?

No, by preserving the metric is meant that the covariant derivative of the metric vanishes not the partial derivative as you did, so you can't use a "connection" from a flat space as a counter-example.

 

[fr.jurain]

In the last section of http://arxiv.org/abs/gr-qc/9806062, Doesn't this mean that the metric is not necessarily diag(-1,1,1,1) in coordinates in which the connection coefficients disappear?

At least it was my 1st read... I tend to say it's still how I read it now.

http://arxiv.org/abs/1008.0171 has some discussion about whether a non-LC connection can be observed.

Yes, it's the more promising of those listed so far.

it's not clear to me what specific tests they have in mind.

Time to open a new thread maybe?

 

[fr.jurain]

No, by preserving the metric is meant that the covariant derivative of the metric vanishes not the partial derivative as you did

Gee... How do you understand the following, then?

By "preserving the metric at e" I mean canceling the cov. derivative of the metric tensor (also at e), or equivalently since the coordinates are normal, its partial derivatives (again, at e).

(emphasis not in the original).

So, let me add that, where the coordinates are not normal, then of course by "preserving the metric" I just mean canceling its covariant derivative.

That being said, can we agree that we agree on the meaning of "preserving the metric"? If so, then (#24) for a counter-example; where NO metric whatsoever is preserved.

 

[TrickyDicky]

You miss my point, when I said that normal coordinates imply metric preservation it was in the context of a curved manifold, as I explained in my first answer that is the whole point in GR, that because we are in a non-euclidean manifold it is important to preserve the metric, That is not the case in a flat manifold where all you need is orthonormal basis, and you only need regular partial derivatives to have covariance.since the tangent spaces at each point are isomorphic to the global space.

 

[fr.jurain]

OK, back to your point then.

Here's how I understand it: the paper you linked states the existence (and uniqueness) of normal coordinates for symmetric linear connections, and the presence of normal coordinates seem to imply the preservation of the metric, then the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique symmetric connection that preserves the metric.

Now:

the paper you linked states the existence (and uniqueness) of normal coordinates for symmetric linear connections

Do you agree with it? Do you believe that whenever a linear connection C on a manifold M is symmetric, then for any point e of M, coordinates can be found which are normal at e, or do you not?
And if you do, then do you further believe there exist symmetric connections which are not Levi-Civita, that is for every metric m on M you could specify, C does not preserve m, or do you not?

and the presence of normal coordinates seem to imply the preservation of the metric,

Do you believe it does more than seem; that is, if normal coordinates are available at any prescribed point e of manifold M, which was endowed with linear connection C, does it actually imply C preserves the metric, or does it not?
And if it does, what metric is *the* metric?

then the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique symmetric connection that preserves the metric.

I believe you assert this theorem is true. Am I right?
Are you concluding that

 

[fr.jurain]

(Continued)

Are you concluding that

Sorry, wrong button; discard that bit.

Now it seems that If I'd got your point right, I'd have exhibited the set of normal coordinates at (t, r, u, z) in my counter-example, instead of taking its existence for granted as I did. Is that it?

 

[TrickyDicky]

Do you agree with it? Do you believe that whenever a linear connection C on a manifold M is symmetric, then for any point e of M, coordinates can be found which are normal at e, or do you not?

That is shown in the paper about the EP you linked.

And if you do, then do you further believe there exist symmetric connections which are not Levi-Civita, that is for every metric m on M you could specify, C does not preserve m, or do you not?

you keep confusing this, the definition of covariant derivative and of connection is independent of the metric, it doesn't use it so how could there not exist connections that are not Levi-Civita. But that is not the same as what you say after "that is". Read the WP page on Covariant derivative:"for each metric there is a unique torsion-free covariant derivative called the Levi-Civita connection"

Do you believe it does more than seem; that is, if normal coordinates are available at any prescribed point e of manifold M, which was endowed with linear connection C, does it actually imply C preserves the metric, or does it not?

Read the previous post, this is a statement only meaningful for curved manifolds.

And if it does, what metric is *the* metric?

Strange question. The one the manifold under consideration is equipped with?

I believe you assert this theorem is true. Am I right?

Are you kidding?

 

[fr.jurain]

Are you kidding?

Only 3/4. It's clear to me what's happening: we've given different definitions to some language element we both use, and which we never imagined could be define in any other way than our own. As a result we may have to question everything blindingly obvious to ferret out the discrepancy, and I'm not amused.

I'll try one direct stab, in the hope that it's located in our respective definition of "Levi-Civita connection". If it fails it might be wiser to drop the subject.

you keep confusing this, the definition of covariant derivative and of connection is independent of the metric, it doesn't use it so how could there not exist connections that are not Levi-Civita. But that is not the same as what you say after "that is".

Before you read "that is" you were free to believe we shared the definition of Levi-Civita connection; after reading it you have to be aware s/thing is astray.

I surmise this is how you'd define it:
On a (pseudo-)Riemannian manifold (M, m0) there is exactly one Levi-Civita connection; it is the only linear connection which preserves the metric m0 everywhere on M, that is, the only linear connection C such that the covariant derivative induced by C vanishes at every point when evaluated on m0. In a plain manifold M there is no Levi-Civita connection: the notion makes only sense for a (pseudo-)Riemannian manifold.
So there are trivially zillions of connections which are not Levi-Civita, even if we restrict ourselves to symmetric ones.

My definition is: on a manifold M, a linear connection C is a Levi-Civita connection if there is a metric m on M such that C preserves m.
C is the unique linear connection with the property that it preserve this particular metric; it is symmetric, and it requires at least a bit of a proof that the converse is not true: there are symmetric connections C that, whatever the metric m, don't preserve m everywhere.
My definition of a Levi-Civita connection on the (pseudo-)Riemannian manifold (M, m0) is simple: ignore m0, and work with manifold M. However, the Levi-Civita connection preserving m0 is as special as m0, and it's typically the only connection used. It is therefore almost always referred to as "the" Levi-Civita connection, or even "the connection".


Here's the full remark on the WP page you quoted in part (http://en.wikipedia.org/wiki/Covariant_derivative):

The definition of the covariant derivative does not use the metric in space. However, for each metric there is a unique torsion-free covariant derivative called the Levi-Civita connection such that the covariant derivative of the metric is zero.

And the one-liner from the WP page on the LC connection (http://en.wikipedia.org/wiki/Levi-Civita_connection):

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

 

[TrickyDicky]

... there are symmetric connections C that, whatever the metric m, don't preserve m everywhere.

Remeber the other condition, the equivalence principle, my claim was that if the EP holds then all symmetric connections preserve the metric.
A question, in your opinion what would be the gain in introducing a metric in a manifold if you are going to use a connection that is not compatible with it if the main goal of that connection was to transport vectors in a consistent way? I think in GR it'd make no sense.

 

[Dale]

My definition of a Levi-Civita connection on the (pseudo-)Riemannian manifold (M, m0) is ...

Please stick with the standard definition, purposely using different definitions of standard terms is not helpful at all and turns the whole thread from something potentially useful into nothing more than a semantic argument.

 

[atyy]

I think Tricky Dicky and fr. jurain differ in their definition of "normal coordinates", not the LC connection. I think Tricky Dicky uses the metric to define "normal coordinates", but fr. jurain uses the connection to define them.

 

[fr.jurain]

Please stick with the standard definition, purposely using different definitions of standard terms is not helpful at all and turns the whole thread from something potentially useful into nothing more than a semantic argument.

Please quote the standard definition (with authorship) or href it. When defining Levi-Civita connection, WikiPedia makes no reference to Riemannian manifolds, they might be to amateurish for a standard though.

 

[fr.jurain]

I think Tricky Dicky and fr. jurain differ in their definition of "normal coordinates", not the LC connection. I think Tricky Dicky uses the metric to define "normal coordinates", but fr. jurain uses the connection to define them.

Why I'd never have thought of that, let's give it a try.
Normal coordinates (http://en.wikipedia.org/wiki/Normal_coordinates):

In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p.

Hence, immediately brought about by merely being "symmetric affine". I don't like the 2nd sentence, though:

In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations.

Whereas I only label the coefficients of the connection "Christoffel symbols" when the connection is Levi-Civita; because only then can I make sense of "1st kind" & "2nd kind". However, they're consistent (http://en.wikipedia.org/wiki/Christoffel_symbols, 2nd sentence):

In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols.

PlanetMath http://planetmath.org/encyclopedia/RiemannNormalCoordinates.html defines the exact same thing as "Riemann normal coordinates" in case it wasn't murky enough already; they make it clear it's not an especially Riemannian-manifold thing though, and they're careful not to throw in Christoffel symbols.

TrickyDicky, what's the definition you had in mind in your posts?

 

[Dale]

Please quote the standard definition (with authorship) or href it. When defining Levi-Civita connection, WikiPedia makes no reference to Riemannian manifolds, they might be to amateurish for a standard though.

The wikipedia entry certainly does refer to a Riemannian manifold, as do all the other references I could find.

http://en.wikipedia.org/wiki/Levi-Civita_connection
http://planetmath.org/encyclopedia/LeviCivitaConnection.html
http://mathworld.wolfram.com/Levi-CivitaConnection.html
http://mathworld.wolfram.com/FundamentalTheoremofRiemannianGeometry.html
http://en.wikipedia.org/wiki/Fundamental_theorem_of_Riemannian_geometry
http://deltaepsilons.wordpress.com/...nian-geometry-and-the-levi-civita-connection/
http://www-math.mit.edu/~mrowka/Math966notesSp05.pdf

The standard definition is pretty standard, so there are more references available. Please stick with the standard definition since it is essential to the fundamental theorem of Riemannian geometry, also referenced.

 

[fr.jurain]

Remeber the other condition, the equivalence principle, my claim was that if the EP holds then all symmetric connections preserve the metric.

It's the 1st time I read your claim stated in one sentence. Phew! (sighing with relief).
How does the EP transpose into mathematical models then? My understanding of the usual doctrine is that it only requires we be able, at any prescribed point e, to change to normal coordinates; normal for the connection of choice, that is. No harm done if they're simultanously available everywhere in a small neighbourhood of e of course, although it's regarded a mere bonus.

The connection is supposed to manifest itself in the physical world as a tangled mix of inertial and gravitational forces. These forces are fictitious, meaning we can bring them to 0 by the right change of frame. Inertial contributions to these fictitious forces cannot be sorted *locally* from the gravitational ones because of the EP: it parameterizes all possible sorting criteria in a way that makes them ineffective in the real world; the only criterium that got its WEP clearance is the comparing of Mg/Mi ratios. Elevator cars in free fall stuffed with laser interferometers and physics students are regarded realizations of tests particles, so they strictly follow geodesics (not of the metric, which although unused as you observed will almost inevitably be there, see below; geodesics of the connection).

Then, the counter-example I presented dismisses the remains of the claim. Its connection being symmetric, it offers normal coordinates hence abides by the EP as per the 1st §; yet preserves no metric whatsoever, be it Euclidean, (0, 4)-Riemannian, (1, 3)-pseudo-R; none at all. So there can't exist a theorem stating "there could be normal coordinates everywhere with connection C => actually, there exists a certain metric that C preserves."

A question, in your opinion what would be the gain in introducing a metric in a manifold if you are going to use a connection that is not compatible with it if the main goal of that connection was to transport vectors in a consistent way? I think in GR it'd make no sense.

You're right, the connection is all we need to have the things we put into our differential equations behave tensor-like. So it would seem the metric is only there to inflict physics students with migraines.
On the other hand, the human race has been living in Euclidean space and linear time (and infinite c) all the way down from Sixth Day on to early 20th c., and arguably more that 90% in it even after; so it's hard to contend lengths, angles, and time lapses do not exist. To introduce a metric is to assert our ability to measure distances (and time) locally, i. e. with measuring devices that fit entirely inside the proverbial elevator car; and that these measures account for s/thing physical, so the metric behaves the tensor way. Once it's done, then we can ask "how come the Euclidean approximation is so good?"; if it's not done, the question cannot be put into mathematical form.

It's my understanding, like yours, that GR allows only Levi-Civita (from one metric) to produce derivatives. The motivation for this choice I think is heuristic: the resulting theory would likely produce a gravitational potential with a minimum number of components; 16, reduced to 10 by symmetry. It is also so simple (grav. potential = metric, so it's a theory of one object only) that after commiting to this choice, (1) there is practically no reasonable alternative to the EFE, and (2) the theory has but a very, very Spartean choice of adjustable paramaters.
The interesting consequence regarding practical tests is that it is very hard to tweak GR in response to embarrassing experimental results.

 

[fr.jurain]

The wikipedia entry certainly does refer to a Riemannian manifold, as do all the other references I could find.

http://en.wikipedia.org/wiki/Levi-Civita_connection
http://planetmath.org/encyclopedia/LeviCivitaConnection.html
http://mathworld.wolfram.com/Levi-CivitaConnection.html
http://mathworld.wolfram.com/FundamentalTheoremofRiemannianGeometry.html
http://en.wikipedia.org/wiki/Fundamental_theorem_of_Riemannian_geometry
http://deltaepsilons.wordpress.com/...nian-geometry-and-the-levi-civita-connection/
http://www-math.mit.edu/~mrowka/Math966notesSp05.pdf

The standard definition is pretty standard, so there are more references available. Please stick with the standard definition since it is essential to the fundamental theorem of Riemannian geometry, also referenced.

So the WP folks are too amateurish after all. Not to wave away your point regarding the applicability of the fundamental theorem, their Riemann-free definition so appealed to me because it solves a particular problem. If I relinquish the definition, will you help me solve the problem? Here goes:

I need to refer to a notion, which provisionally I'll dub a schmorglub; and sometimes to a schmorglub on the (pseudo)-Riemannian manifold (M, m0); and which is is so defined:
"a schmorglub on the (pseudo)-Riemannian manifold (M, m0) is a linear connection C on the manifold M having the additional property that for some metric tensor m defined at every point on M and not otherwise specified, C is the Levi-Civita connection on the (pseudo)-Riemannian manifold (M, m)."
Please provide a more becoming term than schmorglub, to be used everywhere in its stead. The priority is clarity, so replacing every occurrence by the definition is not an option; standard terminology is particularly welcome *if* available.

I'd have thought to refer instead to "*a* Levi-Civita connection (on M, if any ambiguity is possible)" would nicely dodge the issue... too bad it's not standard.
On the other hand, maybe you'll grant me my problem is not that standard either. How many Riemannian geometry papers in the literature start by committing to a metric m0, get *the* Levi-Civita connection that preserves m0, and work with them till the end? 80%? 95%?
Whereas I acknowledge there was an original metric m0, I start by picking some connection C, recognize that it is, well, a schmorglub, and putting the fundamental theorem to work the other way round than the usual (correctly though, with due regard for your concerns), I let C implicitely, yet completely, specify a metric m.

Can we agree that, in the present instance, to allow me a bit of linguistic slack is neither to indulge in faceties, nor to cheat with the fundamental theorem? Given R-manifold (M, m0), what could "*a* or *some* Levi-Civita connection on manifold M" designate? Couldn't some extraordinary stroke of good fortune make every reader understand it as "a linear connection C on the manifold M having the additional property that for some metric tensor m defined at every point on M and not otherwise specified, C is the Levi-Civita connection on the (pseudo)-Riemannian manifold (M, m)"? And if not as that, as what else?

 

[PAllen]

I think Tricky Dicky and fr. jurain differ in their definition of "normal coordinates", not the LC connection. I think Tricky Dicky uses the metric to define "normal coordinates", but fr. jurain uses the connection to define them.

I know it is trivial to define geodesics with a connection (in fact I prefer the definition for GR since the variation doesn't really guarantee any extremal properties, so why bother). However, is there definition of normality without a metric? I've only seen a definition involving dot product, which uses the metric.

 

[fr.jurain]

I let C implicitely, yet completely, specify a metric m.

Sorry, it's not "completely", only "completely, up to a constant non-zero scalar factor".

 

[TrickyDicky]

I know it is trivial to define geodesics with a connection (in fact I prefer the definition for GR since the variation doesn't really guarantee any extremal properties, so why bother). However, is there definition of normality without a metric? I've only seen a definition involving dot product, which uses the metric.

That is my point, to measure angles (for instance right angles) you need a metric. and to set up normal coordinates you need to be able to define othogonality. Edit:here I refer specifically to Riemannian normal coordinates.

 

[Dale]

"a schmorglub on the (pseudo)-Riemannian manifold (M, m0) is a linear connection C on the manifold M having the additional property that for some metric tensor m

This doesn't make any sense to me. A (pseudo)-Riemannian manifold (M,g) has one and only one metric. You may define some other tensor field on M, but it is not a "metric tensor m". Certainly it doesn't make physical sense to call two different tensors "metrics" on the same spacetime manifold.

 

[TrickyDicky]

is there definition of normality without a metric? I've only seen a definition involving dot product, which uses the metric.

Actually there is, it relies in the fact that it is enough with having an affine connection on a differentiable manifold to define a geodesic thru a point in the manifold and then using a exponential map to the tangent space at that point you get geodesic normal coordinates.

 

[fr.jurain]

However, is there definition of normality without a metric? I've only seen a definition involving dot product, which uses the metric.

Planet Math http://planetmath.org/encyclopedia/RiemannNormalCoordinates.html has one where I don't see any dot product. It is labelled "Riemann normal coordinates" however it relies solely on the existence of geodesics, and coincides with what WikiPedia labels "geodesic normal coordinates" http://en.wikipedia.org/wiki/Normal_coordinates#Geodesic_normal_coordinates.

In both cases, it's clearly stated that
1) changing to them cancels the connection coeffs, which is crucial to be able to identify the connection coeffs with the fictitious forces of old,
2) they're available as soon as the connection is symmetric;
whereas I suspect TrickyDicky intends his "Riemann normal coordinates" to be geodesic normal coordinates in the special case where the connection is Levi-Civita.

 

[fr.jurain]

This doesn't make any sense to me. A (pseudo)-Riemannian manifold (M,g) has one and only one metric. You may define some other tensor field on M, but it is not a "metric tensor m".

A metric tensor on M is just a tensor field required to satisfy certain properties, and certainly you don't need to wait for M to be Riemannian to state what these properties are; cf. the WP definition:
http://en.wikipedia.org/wiki/Metric_tensor
They take care to mark the difference with *the* metric tensor in GR,
http://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)
which can't invoked here since the situation we're discussing is inherently outside the GR sandbox.

Certainly it doesn't make physical sense to call two different tensors "metrics" on the same spacetime manifold.

Oh it most definitely does; no less than, say, to have both a gravitational potential and an EM (=electromagnetic) 4-potential defined on the same space-time manifold. Here goes.

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.
The 1st mechanism relies on EM only, and gives us experimental access to the fundamental metric.
The 2nd mechanism relies on EM, and also on something extraneous; from times immemorial to this day this 2nd mechanism has been our only experimental access to the "fictitious forces", those we model as a linear connection now that space-time is a manifold. As far as I can fathom, which is pretty little, this mechanism is no longer used when we establish a unit of time, which thus has become only EM-based (CAUTION! not sure about that bit).

The 2nd mechanism is almost the very definition of the "fictitious forces" after digestion of the EP: they're but what puts test masses in motion when we've made sure the Lorentz forces acting on them are 0.

At this point, we have 2 mathematical objects modelling the physical situation: the fundamental metric tensor and the connection. The only constraint imposed on the latter is that normal coordinates be available, so that an EP can be recovered from the theory under construction. A side, maybe snide, observation here: in view of PAllen's post (with emphasis added by me):

I know it is trivial to define geodesics with a connection (in fact I prefer the definition for GR since the variation doesn't really guarantee any extremal properties, so why bother). However, is there definition of normality without a metric? I've only seen a definition involving dot product, which uses the metric.

and TrickyDicky's:

That is my point, to measure angles (for instance right angles) you need a metric. and to set up normal coordinates you need to be able to define othogonality. Edit:here I refer specifically to Riemannian normal coordinates.

it looks like Pr Einstein or his zealots did a nice reflex-conditioning job to let the physics folks turn 1st thing to schmor... sorry, to Levi-Civita connections, when time has come to account for the EP. Why bother indeed, if Levi-Civita's is the only casino in town.

Anyway: we're under strictly 0 mathematical obligation to let the physical connection, the one we detect by weighing, be the same as the Levi-Civita connection derived from the fundamental metric; although there are some empirical signs that they're very close to one an other. It's only natural to ask: "what are the mathematical consequences if they are equal?". Well, that was snide, too; it sure is more natural today than it was in the 1900's. The answer I understand to be "you're technically forced to adopt GR".

The next most-natural question is: "could the connection be Levi-Civita, yet not be the one derived from the fundamental metric?". In theories that answer "yes", you have 2 metrics: one is the fundamental metric, a thing of EM-only descent; the other is the gravitational potential, a thing of EM+inertial descent.
These theories have the advantage that they salvage & recycle the bulk of GR's theoretical results, at the same time allowing test results some slack, where GR only conceives null-results.

 

[Dale]

Oh it most definitely does; no less than, say, to have both a gravitational potential and an EM (=electromagnetic) 4-potential defined on the same space-time manifold. Here goes.

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.

This would imply two different laws, not two different metrics. I.e. g_{\mu\nu} J^{\mu} J^{\nu} and g_{\mu\nu} K^{\mu} K^{\nu}

Two different metrics would mean that the same physical measurement by the same mechanism would yield two distinct results, which doesn't make physical sense.

 

[fr.jurain]

This would imply two different laws, not two different metrics. I.e. g_{\mu\nu} J^{\mu} J^{\nu} and g_{\mu\nu} K^{\mu} K^{\nu}

Sorry, this time it's me not making of sense of what you wrote.

Two different metrics would mean that the same physical measurement by the same mechanism would yield two distinct results, which doesn't make physical sense.

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".

 

[Dale]

Sorry, this time it's me not making of sense of what you wrote.

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.

No, two different metrics would be g_{\mu\nu}J^{\mu}J^{\nu} and h_{\mu\nu}J^{\mu}J^{\nu}.

If you have some other mechanism by a different physical law and also by a different metric then you have g_{\mu\nu}J^{\mu}J^{\nu} \ne h_{\mu\nu}K^{\mu}K^{\nu}, which you can always attribute entirely to differences between J and K with g=h. In order to definitively attribute the difference to g and h you must use the same physical law and find that g_{\mu\nu}J^{\mu}J^{\nu} \ne h_{\mu\nu}J^{\mu}J^{\nu} which doesn't make physical sense.

Also, this does not make mathematical sense. If you have a (pseudo)-Riemannian manifold (M,g) then the term "metric" refers to g (hence "the" metric since there is only one). If you have some other tensor h which is not equal to g then h is not the metric. If you have some manifold (M,g,h,...) where g, h, etc. are all "metrics" on M then (M,g,h,...) is by definition not a (pseudo)-Riemannian manifold.

 

[fr.jurain]

No, two different metrics would be g_{\mu\nu}J^{\mu}J^{\nu} and h_{\mu\nu}J^{\mu}J^{\nu}.

Yeah right, that's what I mean: same manifold, same point, same coordinates so same tangent vectors; for I assume this is what your J^{\mu}'s are; and YET: different lengths obtain.
A bit like in "Heck, my altimeter says we're at 4792m on the summit where IGN's gravimeters said 4810m". See? One point, one and same physical path to get there, 2 different altitudes obtained by different kinds of measurements.

Also, this does not make mathematical sense. If you have a (pseudo)-Riemannian manifold (M,g) then the term "metric" refers to g (hence "the" metric since there is only one). If you have some other tensor h which is not equal to g then h is not the metric. If you have some manifold (M,g,h,...) where g, h, etc. are all "metrics" on M then (M,g,h,...) is by definition not a (pseudo)-Riemannian manifold.

Are you kidding me? I'm afraid not, although arguably I'd deserve it. So:
If you have some manifold M on which g, h, etc. are defined and they're all metric tensors (w/out double quotes) by any account sensible folks might think of, then of course (M, g), (M, h),... are (pseudo-)Riemannian manifolds. What else.
Are we clear now? Can we step back to the problem?

 

[Dale]

Yeah right, that's what I mean: same manifold, same point, same coordinates so same tangent vectors; for I assume this is what your J^{\mu}'s are; and YET: different lengths obtain.

This is exactly the situation which does not make sense physically. You cannot possibly get two different results for one physical measurement.

A bit like in "Heck, my altimeter says we're at 4792m on the summit where IGN's gravimeters said 4810m". See? One point, one and same physical path to get there, 2 different altitudes obtained by different kinds of measurements.

The altimeter and the gravimeter are based on different physical principles so the two tensors would represent different physical quantities obtained by different physical laws. In my notation above E.g. the altimiter reading would be based on J and the gravimeter reading would be based on K. Any differences would be wholly attributable to the differences between J and K and not due to the metric.

If you have some manifold M on which g, h, etc. are defined and they're all metric tensors (w/out double quotes) by any account sensible folks might think of, then of course (M, g), (M, h),... are (pseudo-)Riemannian manifolds.

Yes, but if g is not equal to h then (M,g) and (M,h) are entirely different (pseudo-)Riemannian manifolds. A (pseudo-)Riemannian manifold is not just a manifold by itself, but a manifold equipped with a metric. If you equip it with different metrics then you have different mathematical objects. So g is the metric for (M,g) and h is the metric for (M,h), but g is not a metric for (M,h) nor is h a metric for (M,g).

 

[atyy]

Bimetric theories are reviewed in http://arxiv.org/abs/grqc/0502097 .

Are they related to metric-affine theories?

 

[fr.jurain]

This is exactly the situation which does not make sense physically. You cannot possibly get two different results for one physical measurement.

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.

Please quote the post where I claimed otherwise.

The altimeter and the gravimeter are based on different physical principles so the two tensors would represent different physical quantities obtained by different physical laws. In my notation above E.g. the altimiter reading would be based on J and the gravimeter reading would be based on K. Any differences would be wholly attributable to the differences between J and K and not due to the metric.

Continuing with the metaphor, " the altimiter reading would be based on J and the gravimeter reading would be based on K" makes sense if, when discussing altimeter readings we are required to use isobares, and when discussing gravimeter readings we are required to use isopotentials. Yet it ignores the fact that we can make precise, physically meaningful, statements, without using either. Natives from Chamonix will instantly locate "la Voie Royale, the regular climb from Le Nid d'Aigle to the summit", or trace it on a particular map (of the Massif du Mont-Blanc in the current instance) for the rest of us.
The path is the same up to scale, whether it is traced on a meteorological map or on a map of the grav. anomaly; same J's, different "lengths".

So g is the metric for (M,g) and h is the metric for (M,h), but g is not a metric for (M,h) nor is h a metric for (M,g).

Glad to read it. Let's go to business then.

Bimetric theories are reviewed in http://arxiv.org/abs/grqc/0502097 .

Are they related to metric-affine theories?

As far as I understand (which is not much actually), § 2.1.1 describes the precise situation on which DaleSpam is choking: g is the fundamental metric, tau is the difference between g and the gravitational potential, Einsteinian GR is a theory where tau must be exactly zero.
And I understand "metric-affine theory" as an umbrella word for the above situation and others, where there is a fundamental metric on the 1 hand, and an affine connection on the other hand; the "bi-metric" situation being the case above, where the affine connection in question derives from the metric g+tau. Again don't get we wrong: I don't know these articles by heart, so their authors might start in horror at what I write.