What are the restrictions when swapping between geometries in physics?

[Killtech]

My question has a strong physical context but in it's core it's a mathematical one. I haven't been meddling with diff geo for quite a long time hence I might have gotten a little rusty.

Anyhow, given a Riemann manifold $(M,g)$ I would like to take the underlying set of $M$ and give it another metric $h$, hence create a new manifold $(N,h)$. Furthermore assume I have a family of equations defining function and vector fields on $M$ that I would like to carry over to $N$. Assuming I can find a smooth map $\phi: M \rightarrow N$ shouldn't I be able to do that? Those equations are supposed to represent the laws of physics i.e. differential equations defining the time evolution of fields and whatnot. So the general question is, what are the restricting factors between switching the geometry of my physics space at will?

A simple example would be starting with $(M,g)$ being the Euclidean space in $\mathbb {R}^3$ and defining $(N,h)$ via the spherical coordinates mapping and taking canonical metric from its codomain to define $h$ thus in the spherical coordinates $h$ will be represented by an identity matrix. Well, for sake of dodging problems I take out the entire polar region.

As an example for family of equations I would just pick only Newtons equation of motion, perhaps in Earth's gravity field centered at $(0,0,0)$ (which should be the geodesic equation with a term added accounting for the gravity field). Yeah, for the sake of the joke I'd like to know if I can correctly/equivalently describe physics in a "flat earth" geometry :D. The underlying thing I want to know is whether the geometry of the physical space is just a convention where the laws of physics take the simplest form or whether there is more to it.

 

[martinbn]

I am not sure what the question is.

Anyhow, given a Riemann manifold $(M,g)$ I would like to take the underlying set of $M$ and give it another metric $h$, hence create a new manifold $(N,h)$. Furthermore assume I have a family of equations defining function and vector fields on $M$ that I would like to carry over to $N$. Assuming I can find a smooth map $\phi: M \rightarrow N$ shouldn't I be able to do that? Those equations are supposed to represent the laws of physics i.e. differential equations defining the time evolution of fields and whatnot. So the general question is, what are the restricting factors between switching the geometry of my physics space at will?

If the two manifolds are the same $M=N$, what do you mean to carry over, it is the same manifold? Also what do you mean by if you can find a smooth map $\phi: M \rightarrow N$, you already have one, the identity? What am I missing?

 

[FactChecker]

A smooth map does not imply that the mapping is 1-1. It is not clear to me that you are starting with enough assumptions to get what you want. Can you guarantee that N does not give new multiple branches in the new metric? I realize that that may be what you are asking about.

 

[Killtech]

I am not sure what the question is.

If the two manifolds are the same $M=N$, what do you mean to carry over, it is the same manifold? Also what do you mean by if you can find a smooth map $\phi: M \rightarrow N$, you already have one, the identity? What am I missing?

Well, a manifolds $M$ and $N$ are topological spaces, hence even if their underlaying sets is the same, they may differ in their topology and thus are not the same. But indeed in my case I want them to use equivalent topologies but still, as Riemann manifolds they would still differ using a different metric.

Yes, obviously the identity is a map between $M$ and $N$ but it's not guaranteed to be smooth in general. So that could be a possible pitfall to lookout. My general question is really about such pitfalls. Also talking about this with physicists turned out to that changing the metric of my space was viewed like a sacrilege. But on the other hand I am not sure what would speak purely mathematically against such an approach, even if physical interpretation of that is complicated.

 

[Killtech]

A smooth map does not imply that the mapping is 1-1. It is not clear to me that you are starting with enough assumptions to get what you want. Can you guarantee that N does not give new multiple branches in the new metric? I realize that that may be what you are asking about.

I am trying to find the possible pitfalls in the general approach and hence I am starting with rather vague assumptions and intended to specify them when needed as I go. Thus finding the right assumptions for this is part of my question.

But in general I actually want to merely exchange the geometry and not the topology (that's the aspect more interesting for physics I guess) so I could add the assumption that the two metrics should be topologically equivalent. That would of course also solve the question about the identity map being always smooth. And as physics context goes there is a bunch of other assumptions I could possibly throw in as to what metrics are even worth considering. On the other hand I don't want to add assumptions that aren't really necessary.

 

[martinbn]

Well, a manifolds $M$ and $N$ are topological spaces, hence even if their underlaying sets is the same, they may differ in their topology and thus are not the same. But indeed in my case I want them to use equivalent topologies but still, as Riemann manifolds they would still differ using a different metric.

Yes, you do say that they are the same as sets, but most likely all of the sets involved will have the same cardinality, so that is not saying much. The sets $\mathbb R$ and $\mathbb R^2$ are of the same cardinality.

Yes, obviously the identity is a map between $M$ and $N$ but it's not guaranteed to be smooth in general. So that could be a possible pitfall to lookout. My general question is really about such pitfalls. Also talking about this with physicists turned out to that changing the metric of my space was viewed like a sacrilege. But on the other hand I am not sure what would speak purely mathematically against such an approach, even if physical interpretation of that is complicated.

If $M$ and $N$ are the same manifold, the identity is smooth. Different metrics have no relevance on that.

 

[FactChecker]

If $M$ and $N$ are the same manifold, the identity is smooth. Different metrics have no relevance on that.

The original question mentioned a new manifold. I do not see where the specification of "the same manifold" originated.

PS. I see that the first mention of the manifolds being the same is in @martinbn 's post #2. I am not sure why he concluded that. Is there something in the OP that implies it? It's not clear to me what the intent of the OP is. He mentions different manifolds, but the question seems to imply generally that he only wants a different metric, with nothing like different branch points or topology.

 

[Killtech]

If $M$ and $N$ are the same manifold, the identity is smooth. Different metrics have no relevance on that.

Ehmm... Take $([1,2],d_2)$ and $([1,2],\delta)$ as the manifolds based on the same interval between 1 and 2 on the real numbers. Let $d_2(x,y) = \left\|x-y \right\|_2$ be the default Euclidean metric and let $\delta(x,y) = 0$ if $x=y$ and $\delta(x,y) = 1$ otherwise be the discrete metric. Then the manifold have different topologies induced by their metrics and hence the identity won't be so smooth in both directions. So I do not understand your answer.

Also I already think of the manifolds as Riemann manifolds though I haven't specified that so far. Given that my physical equations will include differentials make it a necessity though I guess.

 

[FactChecker]

Ehmm... Take $([1,2],d_2)$ and $([1,2],\delta)$ as the manifolds based on the same interval between 1 and 2 on the real numbers. Let $d_2(x,y) = \left\|x-y \right\|_2$ be the default Euclidean metric and let $\delta(x,y) = 0$ if $x=y$ and $\delta(x,y) = 1$ otherwise be the discrete metric. Then the manifold have different topologies induced by their metrics and hence the identity won't be so smooth in both directions. So I do not understand your answer.

Also I already think of the manifolds as Riemann manifolds though I haven't specified that so far. Given that my physical equations will include differentials make it a necessity though I guess.

Yes, these would not have the same topologies. Do you want to allow that or not?

 

[Killtech]

Yes, these would not have the same topologies. Do you want to allow that or not?

That's what I said. But you are right, I wouldn't want to allow that normally. Except when looking at the example in my original post using the spherical coordinates and the metric taken from their codomain it turns out that around the poles similar problems arise which would make the topologies differ there. Sure, one can simply cut out that region and be done with it but I have not decided how I want to best treat those issues in general. This is why i let it somewhat open for maybe someone has a better idea :).

 

[martinbn]

I think I know why I was confused. The first post says that a Riemannian manifold $(M, g)$ is given, then the set $M$ is kept the same, but a new metric is chosen. So I thought that by metric it was meant a Riemannian metric i.e. the new Riemannian manifold $(N,h)$ has the same underlying manifold but a different metric.

 

[Killtech]

Well, when it comes to diff geo these forums don't seem to be particularly active. Does anyone know a better place to discuss this kind of topic?

I've been thinking and to make my initial question more precise i want to know under which circumstance all laws of physics formulated on one manifold/metric can be equivalently represented in another geometry (on the same underlying space). Fundamentals of physics are mostly time evolution equations for all sorts of different entities living on that space and they use a lot of different derivatives. So for that matter it would seem the manifolds must have equivalent topologies because otherwise some of those derivatives might break and wouldn't have an equivalent representation everywhere.

And perhaps it is sufficient to require this only almost everywhere, since if there are problems on some null sets (like in my example between the Euclidean geometry and a topology derived from the spherical coordinates in its polar region) the laws of physics are uniquely defined even if these regions are cut out i think.

 

[Killtech]

Okay, I have figured that what i am looking for is called conformal geometry and the transformation between metrics are called Weyl transformations.

Does anyone know some good books on the topic that are freely available on the internet? I am kind of also interested in the history of this discipline and in particular Weyl's work on the topic. I couldn't so far find a good enough source that described what his motivation was to develop this exactly but i found that having a historical context very often explains a lot of the specialties of a theory that is otherwise not so intuitive to grasp. Kind of to understand the original problem that parts of the theory are aimed to solve.