Is GR about fixed curved background or dynamical?

• I
• mieral
In summary,The formalism of particles on a fixed curved background is only an approximation. We do expect the motion of all particles to modify the 4D spacetime. So Smolin is right to emphasize background independence.
julian said:
because it has the same functional form as ##g_{ab} (x)## but belongs to a different coordinate system, it imposes a different spacetime geometry!

I don't see how this follows, because you have not said how the ##x## and ##y## coordinates are actually different. Changing from ##x## to ##y## is just changing a label. But that means that you cannot assume that the two metrics ##g_{ab}(x)## and ##g_{ab}(y)## are actually different geometries. It might turn out that the ##x## and ##y## labels actually label exactly the same points in exactly the same geometry--you just didn't realize it because you started out using two different labels.

PeterDonis said:
But doing that changes the geometric invariants, and therefore changes the physics (if we are talking about applying such an operation to a spacetime geometry). The doughnut and the coffee cup are different geometries. Similarly, two spacetimes which are related by a diffeomorphism of the kind you describe (an "active diffeomorphism", as opposed to a "passive" one) are different geometries. And different geometries in GR means different physical predictions, so there is no point in asking which points in the different geometries are "the same".

You may have to revise what is physical in light of this invariance under active diffeomorphisms. Radically change your view of the physical world.

PeterDonis said:
I don't see how this follows, because you have not said how the ##x## and ##y## coordinates are actually different. Changing from ##x## to ##y## is just changing a label. But that means that you cannot assume that the two metrics ##g_{ab}(x)## and ##g_{ab}(y)## are actually different geometries. It might turn out that the ##x## and ##y## labels actually label exactly the same points in exactly the same geometry--you just didn't realize it because you started out using two different labels.

I don't put any restrictions on the two coordinate systems (aside from the usual requirements pertaining to the differentiablilty of the manifold).

julian said:
You may have to revise what is physical in light of this invariance under active diffeomorphisms.

What invariance? The metric is not invariant under active diffeomorphisms; at least, that's what you are saying.

julian said:
I don't put any restrictions on the two coordinate systems

Then I don't see how it follows that the two spacetime geometries, ##g_{ab}(x)## and ##g_{ab} (y)##, must be different. In fact, with the conditions as you give them, it seems to me that they must be the same.

First, observe that the equation ##R_{ab} = 0##, by itself, is not one differential equation (or even one per component ##ab##). It's more like a template for an infinite number of possible differential equations. Which actual differential equation among that infinite number you are talking about depends on the metric (meaning here the function ##g_{ab}(x)##), because ##R_{ab}## is an expression involving the metric and its derivatives with respect to the coordinates. So if two coordinate charts ##x## and ##y## end up giving you exactly the same differential equation, that means the two metrics must be the same.

So now we have the following: we have two coordinate charts, ##x##, and ##y##. We have two metrics, ##g_{ab} (x)## and ##g_{ab} (y)##, which have exactly the same functional form. That means, given any point ##x##, the geometric invariants for that point are identical to the geometric invariants for the point ##y## for which ##y = x##. That, to me, means the two metrics ##g_{ab} (x)## and ##g_{ab} (y)## describe the same spacetime geometry.

If you disagree with the above, I would really like to see a concrete counterexample. Is there one that you know of?

maline
Say you have two close points ##P## and ##Q##. Say ##P## is labelled by ##x^a## and ##Q## is labelled by ##x^a + dx^a## in the ##x-##coordinates. Say ##P## is labelled by ##y^a## and ##Q## is labelled by ##y^a + dy^a## in the ##x-##coordinates. In general ##dx^a## won't be equal to ##dy^a##. As such, given that ##g_{ab} (x)## and ##\tilde{g}_{ab} (y)## have the same functional form, we will have in general that

##
##

This initially alarmed Einstein, but then he came to understand that there is a resolution but it requires a radical revision of our understanding of the physical world (in fact this revision is what Einstein was referring to when he made his remark "beyond my wildest expectations"). To understand this revision I recommend you look at Rovelli's book.

Physical observations that are made all the time, these make definite unique predictions despite active diffeomorphisms because the reference systems are coupled to the gravitational field and are part of the physical system under consideration, they transform along with the gravitational field under active diffeomorphisms in such a way as to make the whole thing work.

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There is nothing stopping you from considering the differential equation ##R_{ab} = 0## in its general form, that I can write down. I can then consider plugging in any metric I like to see if it solves it. You need to open up from this restrictive view that the differential equation depends on a particular solution.

I see where you are comming from because I've had the same thoughts. It is like what came first ##g_{ab} (x)## or ##\tilde{g}_{ab} (y)##? It is like you are saying that ##g_{ab} (x)## is more privileged and from this point on the differential equation itself must abide to it. But if we are being democratic, which we should, neither is more privileged.

julian said:
In general ##dx^a## won't be equal to ##dy^a##.

Why not?

To put the question a different way: if ##dx^2 \neq dy^a##, then what is your basis for saying that ##y^a + dy^a## describes the "same" point Q as ##x^a + dx^a##? By your own hypothesis, the geometric invariants at the point ##y^a + dy^a## are different from those at the point ##x^a + dx^a##. And the coordinate values of the two are different. So what, exactly, "stays the same" that allows you to identify the points?

PeterDonis said:
Why not?

To put the question a different way: if ##dx^2 \neq dy^a##, then what is your basis for saying that ##y^a + dy^a## describes the "same" point Q as ##x^a + dx^a##? By your own hypothesis, the geometric invariants at the point ##y^a + dy^a## are different from those at the point ##x^a + dx^a##. And the coordinate values of the two are different. So what, exactly, "stays the same" that allows you to identify the points?

A differatiable manifold admits coordinates, in particular two overlapping coordinate systems, in the absence of a distance function.

julian said:
You need to open up from this restrictive view that the differential equation depends on a particular solution.

That's not the view I am taking.

julian said:
It is like you are saying that gab(x)g_{ab} (x) is more privileged and from this point on the differential equation itself must abide to it.

That's not the view I'm taking either.

See my previous post for the view I'm taking. Basically, I see all this vague, general talk about two points being "the same" in two different metrics, when as far as I can see, there is nothing the same between them. The coordinates are different and all the geometric invariants are different. So what makes them "the same"? This question is critical to the whole argument, and yet I have seen no concrete answer to it. That is why I asked for a specific, concrete example.

To illustrate the issue another way, consider two possible diffeomorphisms, first a passive one and then an active one.

We start off with a flat 2-dimensional plane described in polar coordinates, so the coordinates are ##r, \theta## and the metric is ##g_{ab} (r, \theta) = dr^2 + r^2 d\theta^2## (I'm writing it as a line element for easier typing, the meaning should be clear).

An example of a passive diffeomorphism would be transforming to Cartesian coordinates: ##x = r \cos \theta## and ##y = r \sin \theta##. This gives a metric ##g_{ab} (x, y) = dx^2 + dy^2##. Points are "the same" in the two metrics if their coordinate values are related by the transformation formulas I just gave; so, for example, the point ##x = 1##, ##y = 1## is the same as the point ##r = \sqrt{2}, \theta = \pi / 4##. Here all of the geometric invariants are unchanged--points that are "the same" in the two metrics have the same invariants. That is why we call this a "passive" diffeomorphism.

An example of an active diffeomorphism would be using the same coordinates ##r, \theta## but changing the metric to, e.g., ##\tilde{g}_{ab} (r, \theta) = f(r) dr^2 + r^2 d\theta^2##. This changes the flat plane to a curved surface which is rotationally symmetric about the origin ##r = 0##. Points are "the same" in the two metrics if they have the same coordinate values ##r, \theta##. Here points that are "the same" in the two metrics do not have the same geometric invariants; that is why we call this an "active" diffeomorphism. But we still have to have the coordinate values in order to tell which points are "the same".

What you appear to be describing is a case where neither of the above are true: we have a point Q which is said to be "the same" in both metrics, but it has different coordinate values and different geometric invariants in the two metrics. So I don't see how it fits into either of the cases I described above.

julian said:
A differatiable manifold admits coordinates, in particular two overlapping coordinate systems, in the absence of a distance function.

But a metric is a distance function, and we are assuming we have a metric, so I don't see how this is relevant.

julian said:
To understand this revision I recommend you look at Rovelli's book.

What particular part of it best explains what you are referring to?

PeterDonis said:
What invariance? The metric is not invariant under active diffeomorphisms; at least, that's what you are saying.

The ##y-## coordinates are different from the ##x-##coordinates. We are considering a metric tensor function in the ##y-##coordinates, ##\tilde{g}_{ab} (y)##, that has the same functional form as ##g_{ab} (x)##. If you take ##\tilde{g}_{ab} (y)## and do a coordinate transformation on it taking it to the ##x-##coordinate system, then it won't have the same functional form as ##g_{ab} (x)##, and as such it won't impose the same geometry.

PeterDonis said:
Then I don't see how it follows that the two spacetime geometries, ##g_{ab}(x)## and ##g_{ab} (y)##, must be different. In fact, with the conditions as you give them, it seems to me that they must be the same.

First, observe that the equation ##R_{ab} = 0##, by itself, is not one differential equation (or even one per component ##ab##). It's more like a template for an infinite number of possible differential equations. Which actual differential equation among that infinite number you are talking about depends on the metric (meaning here the function ##g_{ab}(x)##), because ##R_{ab}## is an expression involving the metric and its derivatives with respect to the coordinates. So if two coordinate charts ##x## and ##y## end up giving you exactly the same differential equation, that means the two metrics must be the same.

I think I see what you are saying. The metric tensor function is the dependent variable of the differential equation (well set of equations really to be solved for the metric tensor function). It is a function of the independent variable. A metric tensor function, the dependent variable, doesn't affect the general form of the differential equation. When I said it is the same differential equation in the ##y-##coordinates I actually meant the differential equation has the same general form in the ##y-##coordinates as it has in the ##x-##coordinate system.

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PeterDonis said:
I don't see how this follows, because you have not said how the ##x## and ##y## coordinates are actually different. Changing from ##x## to ##y## is just changing a label. But that means that you cannot assume that the two metrics ##g_{ab}(x)## and ##g_{ab}(y)## are actually different geometries. It might turn out that the ##x## and ##y## labels actually label exactly the same points in exactly the same geometry--you just didn't realize it because you started out using two different labels.

I'm not considering a coordinate transformation. I am coming along and considering writing down the same function as ##g_{ab} (x)## but replacing ##x## with ##y##, which I denote ##\tilde{g}_{ab} (y)##, which I am free to do if I want.

This metric tensor function ##\tilde{g}_{ab} (y)## will solve the field equations in the ##y-##coordinates system because the field equations have the same general form in the ##y-##coordinates.

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PeterDonis said:
But a metric is a distance function, and we are assuming we have a metric, so I don't see how this is relevant.

I'm saying that a bare manifold, a manifold without a metric, still admits coordinates. In particular two overlapping coordinate systems and points ##P## and ##Q##, which to start off with we consider abstarctly, can be labelled in the two coordinates systems and in general ##dx^a \not= dy^a##.

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julian said:
I'm saying that a bare manifold, a manifold without a metric, still admits coordinates.

Yes, I agree with that.

julian said:
In particular two overlapping coordinate systems and points ##P## and ##Q##, which to start off with we consider abstarctly, can be labelled in the two coordinates systems and in general ##dx^a \not= dy^a##.

But, once again, how do you know that the coordinate labels ##x^a + dx^a## and ##y^a + dy^a## label the same point ##Q##?

PeterDonis said:
But, once again, how do you know that the coordinate labels ##x^a + dx^a## and ##y^a + dy^a## label the same point ##Q##?
Because it is supposed to be the overlapping point between the two coordinate systems? It is a manifold differentiability requirement, no?

PeterDonis said:
Yes, I agree with that.
But, once again, how do you know that the coordinate labels ##x^a + dx^a## and ##y^a + dy^a## label the same point ##Q##?

O.K. so a bare manifold admits coordinates. Where two coordinate charts overlap, each point will be labelled by some value in the ##x-##coordinates and another value in the ##y-##coordinates.

Then it is quite simple. Say ##P## is labelled by ##x^a (P)## and ##Q## is labelled by ##x^a(Q) = x^a (P) + dx^a## in the ##x-##coordinates. Say ##P## is labelled by ##y^a (P)## and ##Q## is labelled by ##y^a(Q)##in the ##y-##coordinates. Then we just define ##dy^a## by ##dy^a := y^a(Q) - y^a (P)##.

Because ##x-##coordinates and ##y-##coordinates are different in general we will have ##dx^a \not= dy^a##.

julian said:
Where two coordinate charts overlap, each point will be labelled by some value in the x-coordinates and another value in the y-coordinates.

Yes, sure. But none of this answers the question I have been asking: how do you know that a given point ##Q## is "the same" point in the two coordinate charts?

Your answer is basically, "because I say so". But when you are constructing a model of some actual physical system, you can't just say so.

To clarify further, note that in the Rovelli paper you linked to, when he discusses the hole argument and Einstein's two switches of position (from for general covariance to against it, then back to for it), he describes two possible resolutions of the hole argument: (i) physical theories do not respect general covariance; or (ii) there is no physical meaning to a "point in the manifold". As Rovelli makes clear, Einstein first thought that (i) was the resolution (that was his first switch), but then realized that the correct resolution was (ii) (that was his second switch).

In other words, in the absence of some additional structure on the manifold, the answer to the question I have repeatedly asked you--how do you know that a point "Q" is "the same" in two different coordinate charts--is "you don't, because points in the manifold have no physical meaning". Rovelli then discusses the additional structure that is required in order to give "points" (i.e., events) physical meaning: it basically amounts to identifying events by the values of invariants--observable properties of the various fields in the theory, and their relationships.

Mathematically, of course, you can always say that a given point ##Q## is labeled by coordinate values ##x^a + dx^a## and ##y^a + dy^a##, which are unequal, just because; in mathematics you can construct any consistent model you wish. But this discussion is about physics, not mathematics.

Peterdonis. According to atyy: "There are two different things in GR both called diffeomorphism invariance. One is the ability to use arbitrary coordinates, also called "general covariance". This is not specific to GR, and is true of all theories, even special relativity and Newtonian physics. When Smolin says "This principle implies that, unlike theories prior to general relativity, one is free to choose any set of coordinates to map spacetime and express the equations.", he seems to be referring to general covariance. However, it is not true that general covariance applies only to general relativity.
The special thing about GR is that the 4D spacetime metric is modified by matter such that specifying the spacetime metric completely specifies the distribution of energy in spacetime. This is also called background independence, because there is no fixed background that is unmodified by matter."

Peterdonis. What you and Julian were discussing now. Is it the first or second case or combined? Anyway about General Covariance.. where it is about able to use any coordinate system we please. Can you please give an example of any system or scenario of what it means unable to use any coordinate system we please or Nongeneral covariance? Thank you.

mieral said:
According to atyy

Please give a reference to the actual post and the actual thread where @atyy said this. It is very bad manners (as well as technically against the PF rules) to quote someone without giving a reference, so we can see the context of the quote. You might be seriously misrepresenting what the person you are quoting was actually trying to say.

I'll respond further only after I see the reference.

PeterDonis said:
Please give a reference to the actual post and the actual thread where @atyy said this. It is very bad manners (as well as technically against the PF rules) to quote someone without giving a reference, so we can see the context of the quote. You might be seriously misrepresenting what the person you are quoting was actually trying to say.

I'll respond further only after I see the reference.

Sorry,, here's the link.. i searched at pf the wildcards "diffeomorphism invariance" and found in message 7 atyy distinctions:

The problem as I emphasized at the beginning is that the word background independance, is one of the most heavily equivocated words in modern physics, both on these forums and in the literature. Already in this thread you have 5 different senses of the word, just for the classical theory of gravitation. One for 'no prior geometry', another for the use of the background field method, one sense for the objects that are allowed to be dynamic (varied over) vs fixed in a Lagrangian formalism, another that links it to general covariance and yet another sense where it corresponds to active but not passive transformations(even though they are mathematically equivalent for Riemann metric theories)

Note, some of these senses are mutually contradictory. For instance the background field methods primary virtue is that it preserves manifest gauge invariance(diffeomorphism invariance in the case of GR).

That's why I hate the word and instead insist on focusing on real physical properties of the equations... something that can defined and measures in an experiment. Words get no where with this stuff, especially when people get into the game of (my theory is more BI than yours)..

atyy and martinbn
PeterDonis said:
Yes, sure. But none of this answers the question I have been asking: how do you know that a given point ##Q## is "the same" point in the two coordinate charts?

Mathematically, of course, you can always say that a given point ##Q## is labeled by coordinate values ##x^a + dx^a## and ##y^a + dy^a##, which are unequal, just because; in mathematics you can construct any consistent model you wish. But this discussion is about physics, not mathematics.

If you are to formulate a physical theory you first need to establish a mathematical foundation. Given that in GR there is no a priori given geometry, when formulating the theory you are starting from a bare differentiable manifold. Differentiable manifolds come equipped with coordinate charts and when charts overlap the same point ##p## inside the overlap is labelled by different coordinate values in the two different coordinate charts. Books on GR start out describing maths of differential manifolds.

O.K. say we are working in ##x-##coordinates and we find that a metric tensor function ##g_{ab} (x)## that solves Einstein's field equations expresssed as a set of differential equations where the independent variable is ##x^a##. Given two close points ##P## and ##Q## defined by certain ##x-##coordinate values, the metric tensor function gives the distance between them.

In textbooks you are told that if you do a coordinate transformation from say ##x-##coordinates to ##y-##coordinates you get a coordinate induced metric tensor function usually denoted by ##g_{ab}'(y)## which asigns the same distance between ##P## and ##Q## in the ##y-##coordinates as ##g_{ab} (x)## did in the ##x-##coordinates.

What I'm telling you about is that things aren't as simply as this and that there are implications of general covarince that Einstein "...initially panics in front of...". There is an additional solution to Einstein's equations in the ##y-##coordinates - namely the same function as ##g_{ab} (x)## but with ##x## replaced by ##y##. This follows from the requirement that the laws of nature must be the same in all coordinates systems.

What I think you don't like is that this additional solution assigns a different distance between the two points ##P## and ##Q## than does ##g_{ab} (x)## in the ##x-##coordinates. I think this is why you keep asking how can ##Q## be the same point in the ##y-##coordinates as it is in the ##x-##coordinates: `if it says the distance from ##P## to ##Q## is different then how can ##Q## be the same point in the ##y-##coordinates?'

Rovelli also defines a new field. On page 48 of

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

in the paragraph starting "Let me repeat the same argument in a different form" Rovelli talks about a "different field" where he is using the same fact as I am using - that the laws of physics have the same for in all coordinate systems to allude that a field with the same functional form is also a solution. He actually takes the coordinate transformed (tetrad) field ##e^{'I}_\nu (y)## and then writes down a field in the ##x-##coordinates that has the same functional form as this, namely ##\tilde{e}_\mu^I (x)## - see eq (2.134). This different field will asign a different distance between the points than does the original field ##e_\mu^I (x)##. What I did was similar and has the same conclusion - the distance between ##P## and ##Q## is not determined by GR.

What you have to come to terms with is that GR does not uniquely determine the distance between two points defined by coordinates values as ##P## and ##Q## are.

If you want to make predictions that GR actually determines you must define points physically, for example as the intersection point between the world lines of two particles. This is what the diagram on page 49 is referring to in

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

julian said:
If you are to formulate a physical theory you first need to establish a mathematical foundation. Given that in GR there is no a priori given geometry, when formulating the theory you are starting from a bare differentiable manifold.

The Rovelli paper you linked to goes to some trouble to describe how to formulate GR without using the bare differentiable manifold as the underlying structure, and to make the case for why this is necessary in order to properly construct a theory of quantum gravity. (The basic reason is that the differentiable manifold turns out to be purely a manifestation of gauge choice and has no actual physical meaning.)

julian said:
What you have to come to terms with is that GR does not uniquely determine the distance between two points defined by coordinates values as ##P## and ##Q## are.

I already understand the underlying point here; I'm just phrasing it differently. I'm saying that GR tells us that "the distance between two points defined by coordinate values" has no physical meaning. The only "distance" that has physical meaning is the distance between points "defined physically", as you put it--for example, the distance between two intersections of worldlines.

I need to understand something about coordinate systems.

according to haushoffer in message 3 in https://www.physicsforums.com/threads/diffeomorphism-invariance-in-gr.485023/

"The main point is that with a fixed background, you can shift fields with respect to that background. That fixed background defines points which have a physical meaning, and this can be covered by different coordinate systems. So coordinates do not have physical meaning. Without fixed background, as in GR, you don't have this. If you transform all the physical fields, you also tranform the metric. Points loose their meaning and only distances are physically meaningful."

Actually I need an example. When it's fixed background. How come it can be covered by different coordinate system, and what does it mean. Please give something with fixed background that can be covered by different coordinate system. For example. My computer table is fixed. So what different coordinate system can cover it?
And if the background is not fixed like in classical GR. Does it mean it can't be covered by different coordinate system? and why is that?

mieral said:
My computer table is fixed. So what different coordinate system can cover it?

Cartesian coordinates and polar coordinates are two different coordinate systems that both cover the surface of your table.

mieral said:
if the background is not fixed like in classical GR. Does it mean it can't be covered by different coordinate system?

No, it means something more drastic. It means that the notion of identifying "points" (events in spacetime) by their coordinates is not physically meaningful. You have to identify points by actual observable quantities, such as the intersection of two worldlines (two objects passing each other, for example). It's very hard to visualize what this means because in order to visualize as "set of points" at all we have to attach it to some concrete object, like your computer table, and as soon as we do that we have something physical--the object--to use to identify points. So the idea of a "set of points" (a manifold) that is completely abstract and not "attached" to any object, so that there is nothing by which to identify any particular point, is not something we can easily comprehend. But in a theory like GR that does not have a fixed background, that is what is left if you take away all the physical entities, because the spacetime geometry itself is a physical entity and interacts dynamically with all the other physical entities in the theory. So you can't take away the physical entities and still have a geometry left (which is what a "fixed background" would be), which means you can't take away the physical entities and still identify any points.

mieral
Haelfix said:
The problem as I emphasized at the beginning is that the word background independance, is one of the most heavily equivocated words in modern physics, both on these forums and in the literature. Already in this thread you have 5 different senses of the word, just for the classical theory of gravitation. One for 'no prior geometry', another for the use of the background field method, one sense for the objects that are allowed to be dynamic (varied over) vs fixed in a Lagrangian formalism, another that links it to general covariance and yet another sense where it corresponds to active but not passive transformations(even though they are mathematically equivalent for Riemann metric theories)

Note, some of these senses are mutually contradictory. For instance the background field methods primary virtue is that it preserves manifest gauge invariance(diffeomorphism invariance in the case of GR).

That's why I hate the word and instead insist on focusing on real physical properties of the equations... something that can defined and measures in an experiment. Words get no where with this stuff, especially when people get into the game of (my theory is more BI than yours)..
The physics are based on a mathematical model, so we should be capable of agreeing on a mathematical term perfectly defined, like diffeomorphism invariance and see the physical consequences. But for instance in this thread there is a certain view that transition functions are not valid in GR to define the same point in two coordinate charts because they are simply math not physics, and math can model anything.
Even if all physics is based on the existence of these transition functions that define differentiability in manifolds and allow diffeomorphisms, and also are necessary to build the metric structure that according to that view is "physical", even if it requires the previous mathematical definition of diffeomorphism to be true and that it belongs to the same mathematical model. How can anyone even discuss diffeomorphism invariance if diffeomorphisms are questioned as "just math, no physics"?
Now in GR there are certain things that are not uniquely measurable, like proper distances, but are used in the model, how could one possibly introduce this as real physical properties if they are not really measurable in experiments according to the theory? We are left with diffeomorphism invariance.

Diffeomorphism invariance and general covariance are also a little bit equivocated upon, but they are much less so than BI and I'm much more comfortable discussing them with just words as the context usually makes their meaning apparent. For instance sometimes when we are in the ADM formalism, we are discussing transverse diffeomorphisms and the qualifier is dropped... the context will however make it apparent and understood.

With the concept of BI though, you can read almost any message in this thread and walk away confused. For instance Peter writes above that GR is BI. Well yes, in one reading of the word that would be completely correct, in another you would need to include the words 'written in a particular way'. Like you could write GR in the spin 2 formalism and argue that it necessarily involves the existence of fictitious 'prior geometry' that is then expanded around.

atyy
Haelfix said:
Diffeomorphism invariance and general covariance are also a little bit equivocated upon, but they are much less so than BI and I'm much more comfortable discussing them with just words as the context usually makes their meaning apparent.
Diffeomorphism invariance is a perfectly defined property of differentiable manifolds, I'm not sure what possible equivocation you mean. Moreoverit is a necessary condition to build the concept of Riemannian manifold on top of the differentiable manifold level of structure. In other words without a differentible manifold invariant under diffeomorphisms there is no way to introduce metric tensors and their invariants, so I'm not how one could reivindicate "geometric invariants" derived from the metric in the absence of diffeomorphism invariance like Peterdonis does.
With the concept of BI though, you can read almost any message in this thread and walk away confused. For instance Peter writes above that GR is BI. Well yes, in one reading of the word that would be completely correct, in another you would need to include the words 'written in a particular way'. Like you could write GR in the spin 2 formalism and argue that it necessarily involves the existence of fictitious 'prior geometry' that is then expanded around.
I insist that it would be better to go back to mathematical well defined terms, for instance what are the manifold "geometric invariants" Peterdonis refers to and what exactly is the scope of that invariance. I think some confusion is introduced if this is not specified.

I completely agree that Mathematicians do a much better job than physicists in this particular case, note that the purpose of general covariance has been a hot subject in physics since at least Einstein (google the Kretschmann objection).. My point coincides with yours though, I am all for well defined unambiguous math or absent that a tangible physical observable that can be measured in principle.

Btw, entire GR textbooks have been written without using the words diffeomorphism invariance a single time.

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