Attempting to understand diffeomorphisms

["Don't panic!"]

I am relatively new to the concept of differential geometry and my approach is from a physics background (hoping to understand general relativity at a deeper level). I have read up on the notion of diffeomorphisms and I'm a little unsure on some of the concepts.

Suppose that one has a diffeomorphism $\phi : M\rightarrow M$ from a manifold onto itself, then a point $p\in M$ is mapped to a different point $p'\in M$ by $\phi$ as follows $$p\mapsto\phi (p)=p'$$
I get that in general these will then be two distinct points on the manifold, but I've heard diffeomorphisms are analogous to active coordinate transformations?! Is this because one can choose a coordinate chart $(U, \psi)$ in the neighbourhood of $p$, i.e. $p\in U\subset M$ , such that the point $p$ has coordinate values $$\psi (p)=\lbrace x^{\mu}(p)\rbrace$$ and then map this point to a distinct point $p'$ via the diffeomorphism $\phi$ . Having done so, can one then use a pullback mapping, $\phi^{\ast}$ to pullback the coordinates in the neighbourhood of $p'$ , $\lbrace \tilde{x}^{\mu}(p')\rbrace$ to the coordinates [itex]\lbrace x^{\mu}(p)\rbrace[itex] in the following manner $$ x^{\mu}(p)= (\phi^{\ast}\tilde{x})^{\mu}(p')$$ In this sense, is it that although $p$ and $p'$ are distinct points on the manifold, due to the diffeomorphism [$\phi$ they can be assigned the same coordinate values?

 

[WWGD]

The idea is that a diffeomorphism preserves both the topological and differentiable structure of a manifold, unlike a plain homeomorphism, which preserves only the topological structure. So, if I
understood you correctly, if M,N are diffeomorphic, you can say there is a global, i.e., single-chart change of coordinates between the two.

 

["Don't panic!"]

Sorry I hadn't realized the last part hadn't rendered properly. So is the point of a diffeomorphism that it maps between two distinct points that share the same coordinate labels, i.e. they map to the same coordinates in $\mathbb{R}^{n}$. I'm trying to understand it in the context of general relativity really; is the point that if two points on (in general) different manifolds $M$ and $N$ are related by a diffeomorphism then they can be represented by the same point in $\mathbb{R}^{n}$ and so any tensorial quantity should describe the same physical quantity when evaluated at either point on the two manifolds? Is this what is meant when it's said that GR is diffeomorphism invariant?

 

[Fredrik]

I don't know exactly what "diffeomorphism invariance" is supposed to mean. A reasonable guess is that the mathematical structure that represents space and time in the theory (a smooth manifold) is of a type such that the isomorphisms between such structures are diffeomorphisms.

My posts in this thread may help you with the active/passive stuff.

Regarding the LaTeX errors in post #1. You should always use the preview feature. If a mistake still slips by, you have 11 hours and 40 minutes to edit it.

 

["Don't panic!"]

Regarding the LaTeX errors in post #1. You should always use the preview feature. If a mistake still slips by, you have 11 hours and 40 minutes to edit it.

Thanks for the tips.

As regards to diffeomorphism, is the point that a diffeomorphism $\phi : M\rightarrow M$, from a manifold to itself, maps a point $p\in M$ to another point $p'= \phi (p)\in M$ on the manifold. Would it be correct to say that it deforms the manifold in a sense, such that it changes its geometry? Also, if one has a tensorial quantity $T$ that has a value $T(p)$ at $p\in M$ then $\phi$ defines a pushforward map $\phi_{\ast}$ that "drags" $T$ across the manifold to the point $p'$ where it has value $(\phi_{\ast}T) (\phi (p))$?!

 

[WWGD]

I don't know exactly what "diffeomorphism invariance" is supposed to mean. A reasonable guess is that the mathematical structure that represents space and time in the theory (a smooth manifold) is of a type such that the isomorphisms between such structures are diffeomorphisms. <snip>

.

I would think that diffeomorphism invariance, as applied to properties means that the property is preserved under diffeomorphisms, similar to the meaning of topological invariance, as in , e.g., Hausdorff is a topological property or topological invariant since if X is Hausdorff and Y is homeomorphic to X, then Y must also be Hausdorff.

 

[lavinia]

Would it be correct to say that it deforms the manifold in a sense, such that it changes its geometry?

No. A smooth manifold need not have any geometry. A diffeomorphism does not def

Thanks for the tips.

As regards to diffeomorphism, is the point that a diffeomorphism $\phi : M\rightarrow M$, from a manifold to itself, maps a point $p\in M$ to another point $p'= \phi (p)\in M$ on the manifold. Would it be correct to say that it deforms the manifold in a sense, such that it changes its geometry?

Your question seems to come from a context that you are not stating. Perhaps you could say where your question is coming from.

The idea of a smooth manifold is different than the idea of a smooth manifold that has a geometry. The smooth manifold is determined by its coordinate charts, not by any metric. The same smooth manifold can have many different metrics and thus many different geometries.

Sometimes this can be confusing because one is often given a smooth manifold that also has a geometry. For instance, the unit sphere in 3 space is both a smooth manifold and a Riemannian manifold (manifold with a geometry that comes from a Riemannian metric). It has a definite shape. But as a smooth manifold the sphere is determined by its coordinate charts, e.g. two polar caps that overlap in a ribbon around the equator. It is diffeomorphic to an ellipsoid which also has two charts that intersect in a ribbon.

Two smooth manifolds are diffeomorphic if there is a smooth map between them that has a smooth inverse. This means that their coordinate charts have exactly the same structure. These two manifolds may or may not have geometries and if they do these geometries may be different.

There may be many diffeomorphisms between two smooth manifolds or from a manifold to itself.

 

["Don't panic!"]

Thanks for your input. I'm really trying to understand it in the context of general relativity, in particular, I've been reading these notes http://www.roma1.infn.it/teongrav/VALERIA/TEACHING/ONDE_GRAV_STELLE_BUCHINERI/AA2006_2007/DISPENSE/variational_principle.pdf , chapter 15, section 15.3, where they give a brief introduction to diffeomorphisms.

A lot of texts seem to discuss passive and active diffeomorphisms (and their relation to coordinate transformations). Is the idea that a passive diffeomorphism describes the same point on the manifold, but in two different coordinate charts, whereas an active diffeomorphism maps the point to another point (on the same manifold, or another manifold) whose coordinate values are determined with respect to the same coordinate chart as the original point?

Also, when discussing active tranformations, is it that if one has a tensorial quantity, $T$ and a diffeomorphism, $\phi$, then $\phi$ determines a pull-back mapping, $\phi^{\ast}$ that "drags" $T$ across the manifold to the region that $\phi$ maps to?

I'm slightly confused by it all and trying to understand intuitively what a diffeomorphism actually does and also what this means in the context of GR and its diffeomorphism invariance (which as far as I understand, is the statement that if two tensorial quantities are related by a diffeomorphism then they describe the same physical property. As such, the theory is independent of any particular background geometry)?!

 

[lavinia]

I think that what is being talked about is that one can look at diffeomorphisms in two ways.

One way, what Wald calls the "active diffeomorphism," is that is it a mapping of the manifold into itself that sends each point to another.

The other way, the "passive diffeomorphism" ,is to think of the mapping as defining a new set of coordinates around each point. This means that if the manifold is covered by local coordinates ,then composing these coordinates with a diffeomorphism defines a new set of coordinates. In this case one does not think of moving the points around but rather thinks of a new set of local coordinates.

In the case of two manifolds the same thing happens. The active diffeomorphism is just a mapping of points in M to points in N. The passive is using coordinates on N to define coordinates on M

 

[Fredrik]

Also, when discussing active tranformations, is it that if one has a tensorial quantity, $T$ and a diffeomorphism, $\phi$, then $\phi$ determines a pull-back mapping, $\phi^{\ast}$ that "drags" $T$ across the manifold to the region that $\phi$ maps to?

A smooth map pulls back functions and cotangent vectors, but pushes forward tangent vectors. A diffeomorphism is a smooth map with a smooth inverse, so we can use the inverse map to define e.g. the pullback of a tangent vector. We define it as the push-forward transformation associated with the inverse of the given diffeomorphism.

I wrote a follow-up (post #18) to my post (#14) about the active/passive terminology in the thread I linked to in post #4 above. It may be of some use to you.

I'm slightly confused by it all and trying to understand intuitively what a diffeomorphism actually does and also what this means in the context of GR and its diffeomorphism invariance (which as far as I understand, is the statement that if two tensorial quantities are related by a diffeomorphism then they describe the same physical property. As such, the theory is independent of any particular background geometry)?!

It would be easier to discuss this if you can find a statement like "GR is diffeomorphism invariant" (or something similar) in a book. If you reference a specific claim here, we can discuss what it means.

 

["Don't panic!"]

In the active case, it talks about the points being moved on the manifold, but we represent them with the same coordinate frame. I'm a little confused by this as I thought that in general a particular coordinate chart is only defined on a particular subset of the manifold?!

Also, with passive diffeomorphisms, does the change of coordinates correspond to swapping between two different coordinate charts overlapping the same point (i.e. they can both be used to describe the same point)?

 

["Don't panic!"]

It would be easier to discuss this if you can find a statement like "GR is diffeomorphism invariant" (or something similar) in a book. If you reference a specific claim here, we can discuss what it means.

Sean Carroll discusses it in his notes (chapter 5, page 138). As far as I can tell, it is the statement that the theory is independent of any particular geometry, as such one can use a diffeomorphism to transform, for example, the metric tensor defined on one manifold, to a different metric tensor defined on another manifold and these will lead to the same field equations (the mathematical form of the field equations is invariant under a diffeomorphism).

 

[Fredrik]

In the active case, it talks about the points being moved on the manifold, but we represent them with the same coordinate frame. I'm a little confused by this as I thought that in general a particular coordinate chart is only defined on a particular subset of the manifold?!

I wrote a comment about that in the new post (post #18) in the other thread. I think that if we're going to discuss how the components of a vector change from $p$ to $\phi(p)$, then we have to be talking about a diffeomorphism $\phi$ such that $p$ and $\phi(p)$ are both in the domain of the same coordinate system. This isn't weird when we're talking about a 1-parameter group of diffeomorphisms $\phi_t$, and we intend to take the limit $t\to 0$. (Since $\phi_0=\phi$, the coordinate distance between $p$ and $\phi_t(p)$ will be very small when $t$ is very small).

Also, with passive diffeomorphisms, does the change of coordinates correspond to swapping between two different coordinate charts overlapping the same point (i.e. they can both be used to describe the same point)?

I would assume that the idea is to use a coordinate system $x:U\to\mathbb R^n$ and a diffeomorphism $\phi:x(U)\to\phi(x(U))$ to define a new coordinate system $x'=\phi\circ x$.

 

["Don't panic!"]

Thanks for all your help, sorry I seem to be repeating previous discussions a bit.

It has confused me a lot as there seems to be differing opinions. I thought I understood it a bit better after reading these notes http://www.roma1.infn.it/teongrav/VALERIA/TEACHING/ONDE_GRAV_STELLE_BUCHINERI/AA2006_2007/DISPENSE/variational_principle.pdf (from page 210 onwards), but they still fail to properly explain the coordinate issue in active diffeomorphisms (their description seems very vague).

 

[Fredrik]

Sean Carroll discusses it in his notes (chapter 5, page 138). As far as I can tell, it is the statement that the theory is independent of any particular geometry, as such one can use a diffeomorphism to transform, for example, the metric tensor defined on one manifold, to a different metric tensor defined on another manifold and these will lead to the same field equations (the mathematical form of the field equations is invariant under a diffeomorphism).

This is a quote from that page:

You will often hear it proclaimed that GR is a “diffeomorphism invariant” theory. What this means is that, if the universe is represented by a manifold $M$ with metric $g_{\mu\nu}$ and matter fields $\psi$, and $\phi:M\to M$ is a diffeomorphism, then the sets $(M,g_{\mu\nu},\psi)$ and $(M,\phi_*g_{\mu\nu},\phi_*\psi)$ represent the same physical situation.​

So he's not just using the phrase. He's also explaining what it means. He elaborates a bit after this quote. In particular he says that what people have in mind when they say that GR is diffeomorphism invariant is either that there's no preferred coordinate system, or that there's no preferred metric other than the one found by solving Einstein's equation.

 

[spacejunkie]

This is a quote from that page:

You will often hear it proclaimed that GR is a “diffeomorphism invariant” theory. What this means is that, if the universe is represented by a manifold $M$ with metric $g_{\mu\nu}$ and matter fields $\psi$, and $\phi:M\to M$ is a diffeomorphism, then the sets $(M,g_{\mu\nu},\psi)$ and $(M,\phi_*g_{\mu\nu},\phi_*\psi)$ represent the same physical situation.​

So he's not just using the phrase. He's also explaining what it means. He elaborates a bit after this quote. In particular he says that what people have in mind when they say that GR is diffeomorphism invariant is either that there's no preferred coordinate system, or that there's no preferred metric other than the one found by solving Einstein's equation.

Smolin mentions in his Intro to QG course the confusion over diffeomorphism invariance in the earlier GR texts and how it was largely to blame for the failure of early attempts to quantise GR. It was sorted out by Dirac in the 1950's. Per Smolin (per Dirac):-
A spacetime in GR is an equivalence class (also called a geometry) $(g_{\mu\nu},M)$ under all diffeomorphisms on the manifold. The diffeos form an infinite dimensional group, Diff(M).

The reason for this is the relationship between active and passive diffeomorphisms. Quoting Rovelli's Quantum Gravity" p45
"The group of active diffeomorphisms acts on the space of metrics, d. The group of passive diffeomorphisms acts on functions $g_{\mu\nu}(x)$. The orbits of the first group are in natural one to one correspondence with the orbits of the second. However, the relation between the individual metrics, d, and the individual function, $g_{\mu\nu}(x)$, depends on the choice of coordinate. "

So for any two fields T and T' related by an active diffeomorphism there exists a coordinate transform that will have leave the form of the function of coordinates of T unchanged and vice versa. Since GR has the property of general covariance, ie the form of the field equations don't change under coordinate transformations, then it necessarily has diffeomorphism invariance.

Hope that helps.

 

["Don't panic!"]

Thanks for the input, I think I'll have to ponder it a bit more and consult Smolin's book.