I Is GR about fixed curved background or dynamical?

[mieral]

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?

 

[PeterDonis]

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.

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.

 

[RockyMarciano]

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.

 

[Haelfix]

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.

 

[RockyMarciano]

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.

 

[Haelfix]

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.