Diffeomorphic Invariance implies Poincare Invariance?

In summary, Diffeomorphic Invariance is the invariance of a theory under general coordinate transformations. The Poincare group is a subset of these transformations, but not an active diffeomorphism as it does not change the underlying geometry. In GR, the distinction between active and passive diffeomorphisms is still present, with active transformations changing the actual geometry while passive transformations only changing the coordinates used to describe it. However, mathematically there is no difference between the two. The Poincare group is the group of isometries for a specific spacetime, but not all spacetimes have isometries.
  • #106
WannabeNewton said:
This is true only if the map is an isometry by definition. I gave you an easy counter example in the above post. A diffeomorphism between riemannian manifolds is not in general an isometry!

Granted, this is what this construction does.

micromass said:
Are you claiming that all diffeomorphisms are isomtries?? I think there are many counterexamples for this statement. Wbn gave one already.

TrickyDicky said:
I believe Nanaki( and not only him, this thread seems to be going in circles because some distinctions are being overlooked) is falling into two of the mistakes I warned against in a previous post, conflating local diffeomorphisms with local isometries and also local diffeomorphisms with diffeomorphisms.
When you have a diffeomorphism that preserves the metric, you have an isometry, this has been sufficiently stressed by WN and micromass, the problem is that in GR, as commented already by haushofer, atyy and me, the diffeomorphisms alluded by the term "diffeomorphism invariance", have to do with "no prior geometry" and are related to gauge invariance so can't be promoted to isometries.
So if we want to talk about geometry we must restrict ourselves to the local geometry, that is local isometries, these are not bijective but are injective and preserve curvature which is important for a physics theory that identifies gravity with curvature.
Now local isometries are just local diffeomorphisms that pullback the metric tensor. Maybe some of the confusion of Nanaki comes from the fact that local diffeomorphisms induce by the inverse function theorem a linear isomorphism(thus this one is bijective) at each point of the manifold.

I really don't know what your definition of a diffeomorphism is then. Mine makes no mention of geometry at all. It's a pure mapping between manifolds. See e.g. Spivak. Isn't this the type of diffeomorphism GR is invariant under? If you go to Riemannian manifolds and you include your type of transformations, where you deviate from using the pullback metric, then obviously you're going to violate things.
 
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  • #107
NanakiXIII said:
I really don't know what your definition of a diffeomorphism is then. Mine makes no mention of geometry at all. It's a pure mapping between manifolds. See e.g. Spivak. Isn't this the type of diffeomorphism GR is invariant under?
Sure, great then. I inferred by your posts and the replies that they were getting that you thought all diffeomorphisms were isometries.

NanakiXIII said:
If you go to Riemannian manifolds and you include your type of transformations, where you deviate from using the pullback metric, then obviously you're going to violate things.
I don't understand what you mean here. What things am I going to violate and why?

The other problem that I saw here is that not all(most but not all) that is true about Riemannian manifolds carries over exactly the same to spacetimes(pseudoriemannian manifolds), that is where the Einstein's hole argument came from basically.
 
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  • #108
NanakiXIII said:
Granted, this is what this construction does.
I'm not sure what you mean by this. Are you still trying to assert that all diffeomorphisms between Riemannian manifolds are isometries?
 
  • #109
TrickyDicky said:
Sure, great then. I inferred by your posts and the replies that they were getting that you thought all diffeomorphisms were isometries.


I don't understand what you mean here. What things am I going to violate and why?

You're violating the condition that seems to me to be crucial to doing something useful within the context of GR: since the metric is a dynamical field, you cannot simply change the metric without changing the rest of your content (field, particles) as well. If you do that, of course you end up with a different physical situation. And it's not a diffeomorphism in the sense that, while you may be acting on your manifold with a diffeomorphism, you're acting on your metric separately with another function. If someone opposes this last part of my statement (which I tried to elaborately explain in a previous post) then please point out what is wrong about it. Since a diffeomorphism has nothing to do with geometry, I really don't see how a mapping defined on your geometry can be a diffeomorphism.
 
  • #110
NanakiXIII said:
You're violating the condition that seems to me to be crucial to doing something useful within the context of GR: since the metric is a dynamical field, you cannot simply change the metric without changing the rest of your content (field, particles) as well. If you do that, of course you end up with a different physical situation. And it's not a diffeomorphism in the sense that, while you may be acting on your manifold with a diffeomorphism, you're acting on your metric separately with another function. If someone opposes this last part of my statement (which I tried to elaborately explain in a previous post) then please point out what is wrong about it.
I'm not changing the metric if I stick to a local patch of the manifold and only care about a neighbourhood of the point that interests me, with curvature, geodesic length and proper time and all physically meaningful observables in GR preserved, that is the key to my distinction between global and local isometries that you seem to be missing. Even though this distinction is explained in every book about differential topology/geometry usually in the first pages, I'm yet to see a physicist or a GR book that makes this distinction.
NanakiXIII said:
Since a diffeomorphism has nothing to do with geometry, I really don't see how a mapping defined on your geometry can be a diffeomorphism.
A diffeomorphism has nothing to do with geometry but all (global) isometries happen to be diffeomorphisms, is this what confuses you?
 
  • #111
TrickyDicky said:
I'm not changing the metric if I stick to a local patch of the manifold and only care about a neighbourhood of the point that interests me, with curvature, geodesic length and proper time and all physically meaningful observables in GR preserved, that is the key to my distinction between global and local isometries that you seem to be missing. Even though this distinction is explained in every book about differential topology/geometry usually in the first pages, I'm yet to see a physicist or a GR book that makes this distinction.
Perhaps this distinction between local and global diffeomorphisms does elude me; I'm not sure I see its significance.

I also still don't understand what part of what I said you are objecting to.

TrickyDicky said:
A diffeomorphism has nothing to do with geometry but all (global) isometries happen to be diffeomorphisms, is this what confuses you?

No. What is confusing me is how for example WannabeNewton's conformal mapping or Peter's example of changing the mass in the Schwarzschild metric can be considered diffeomorphisms. Their transformations specifically act on the metric. A diffeomorphism only acts on the manifold. Therefore, and apparently I am wrong here, but I don't see how, therefore their transformations cannot be considered diffeomorphisms (unless there is some diffeomorphism that induces these mappings, in which case I would like to see those diffeomorphisms explicitly.)
 
  • #112
The conformal isometry (the diffeomorphism defined above) itself acts on the manifold. It's pullback acts on the metric tensor. This is the same thing with an isometry too obviously: the isometry acts on the manifold but its pullback acts on the metric tensor. By your claim an isometry wouldn't even be a diffeomorphism because it acts on the manifold and not the metric tensor. No smooth map between manifolds acts on the tensor fields themselves, their pull backs and pushforwards (when definable) are what act on the tensor fields.
 
  • #113
NanakiXIII said:
Perhaps this distinction between local and global diffeomorphisms does elude me; I'm not sure I see its significance.

I must admit I was exaggerating a bit when i claimed that GR books don't make the distinction, many do, (although maybe not stressed enough given how much confusion around thse issues seems to exist), for instance when distingusihing between isometries and infinitesimal isometries.
 
  • #114
WannabeNewton said:
The conformal isometry (the diffeomorphism defined above) itself acts on the manifold. It's pullback acts on the metric tensor. This is the same thing with an isometry too obviously: the isometry acts on the manifold but its pullback acts on the metric tensor. By your claim an isometry wouldn't even be a diffeomorphism because it acts on the manifold and not the metric tensor. No smooth map between manifolds acts on the tensor fields themselves, their pull backs and pushforwards (when definable) are what act on the tensor fields.

Then it would seem I have completely misunderstood what the pullback and pushforward do; I was under the impression that indeed, the pullback on the metric defined an isometry. Having a look at the Wiki page, I see this. That seems to define an isometry. Is it incorrect or am I interpreting it wrong?
 
  • #115
NanakiXIII said:
Then it would seem I have completely misunderstood what the pullback and pushforward do; I was under the impression that indeed, the pullback on the metric defined an isometry. Having a look at the Wiki page, I see this. That seems to define an isometry. Is it incorrect or am I interpreting it wrong?

If ##\varphi:M\rightarrow M## is a diffeomorphism and if ##g## is a metric. Then ##\varphi## defines an isometry between ##(M,g)## and ##(M,\varphi^*g)##.
What we are saying is that ##\varphi## is not an isometry between ##(M,g)## and ##(M,g)##.
 
  • #116
micromass said:
If ##\varphi:M\rightarrow M## is a diffeomorphism and if ##g## is a metric. Then ##\varphi## defines an isometry between ##(M,g)## and ##(M,\varphi^*g)##.
What we are saying is that ##\varphi## is not an isometry between ##(M,g)## and ##(M,g)##.

Right. That's what I figured, but I fail to see what kind of physical significance you are trying to attach to this. Are you trying to say that there are non-isometries that still leave GR invariant?

I also think that, if you define [itex]\phi : M \to M[/itex], you cannot just promote this to a map [itex]\phi : (M,g) \to (M,g')[/itex] without specifying what the action on [itex]g[/itex] is, in general. As I said earlier, you should define an implied map on the geometry or specify that you're not doing anything to it. I was using the implied map defined by the pullbacks and pushforwards on your tangent bundle, because that seems to be general practice. I don't see any sense in allowing random mappings on [itex]g[/itex] and then finding that it doesn't leave your system invariant. If that is your thought experiment, that's fine, but my answer remains the same: I don't think that's what diffeomorphism invariance in GR is about.

As this indeed seems to be just a matter of semantics, we can drop it.
 
  • #117
FedEx said:
Cause the certainly larger set of RS(M)=RM(M)/Diff(M) corresponds to the same physical situation.

micromass said:
If ##\varphi:M\rightarrow M## is a diffeomorphism and if ##g## is a metric. Then ##\varphi## defines an isometry between ##(M,g)## and ##(M,\varphi^*g)##.
What we are saying is that ##\varphi## is not an isometry between ##(M,g)## and ##(M,g)##.

NanakiXIII said:
I also think that, if you define [itex]\phi : M \to M[/itex], you cannot just promote this to a map [itex]\phi : (M,g) \to (M,g')[/itex] without specifying what the action on [itex]g[/itex] is, in general. As I said earlier, you should define an implied map on the geometry or specify that you're not doing anything to it. I was using the implied map defined by the pullbacks and pushforwards on your tangent bundle, because that seems to be general practice. I don't see any sense in allowing random mappings on [itex]g[/itex] and then finding that it doesn't leave your system invariant. If that is your thought experiment, that's fine, but my answer remains the same: I don't think that's what diffeomorphism invariance in GR is about.

Is the following be correct?

We say diffeomorphisms are the gauge group of GR, since every diffeomorphism corresponds to an isometry, provided we move manifold and metric.

When we use the term isometry in GR, we usually refer to diffeomorphisms which move the manifold without moving the metric, so not every diffeomorphism is an isometry, eg. finding isometries of the Schwarzschild solution is finding symmetries via Killing vectors.
 
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  • #118
atyy said:
Is the following be correct?

We say diffeomorphisms are the gauge group of GR, since every diffeomorphism corresponds to an isometry, provided we move manifold and metric.
I don't think this is correct. When Diff(M) is viewed as a gauge , the manifold is seen as a bare differentiable manifold, so isometries are left out.
atyy said:
When we use the term isometry in GR, we usually refer to diffeomorphisms which move the manifold without moving the metric, so not every diffeomorphism is an isometry, eg. finding isometries of the Schwarzschild solution is finding symmetries via Killing vectors.
When the term isometry is used in GR (i.e. those infinitesimally generated by KV) is actually infinitesimal isometries that are meant. Otherwise Diff(M) invariance couldn't be thought of as a gauge invariance. Remember gauge symmetries and spacetime symmetries ( those determined by global isometries) are not the same thing. The EFE only fixes the local geometry (thus we only need infinitesimal isometries) not the global spacetime topology/geometry.
 
<H2>1. What is diffeomorphic invariance?</H2><p>Diffeomorphic invariance is a concept in mathematics and physics that describes the property of a system or equation to remain unchanged under a smooth, continuous transformation. In simpler terms, it means that the system will produce the same results regardless of how it is stretched, compressed, or deformed.</p><H2>2. How does diffeomorphic invariance relate to Poincare invariance?</H2><p>Diffeomorphic invariance is a more general concept, while Poincare invariance is a specific case of it. Poincare invariance describes the property of a physical system to remain unchanged under the four-dimensional transformations of space and time, including rotations, translations, and boosts. Since these transformations are smooth and continuous, they also fall under the category of diffeomorphic transformations.</p><H2>3. Why is diffeomorphic invariance important in physics?</H2><p>Diffeomorphic invariance is important in physics because it allows us to describe physical phenomena in a way that is independent of the specific coordinates or frames of reference used. This means that the laws of physics will hold true regardless of how we choose to observe or measure them, making our understanding of the universe more consistent and universal.</p><H2>4. Can diffeomorphic invariance be violated?</H2><p>Yes, diffeomorphic invariance can be violated in certain situations. This can happen when the transformation is not smooth or continuous, or when there are external forces acting on the system that can cause deformations. In these cases, the results of the system may change depending on the specific transformation used, and diffeomorphic invariance is not preserved.</p><H2>5. How is diffeomorphic invariance used in practical applications?</H2><p>Diffeomorphic invariance is used in various fields of physics, including general relativity, quantum mechanics, and fluid dynamics. It is also utilized in image processing and computer graphics to create smooth and realistic transformations of images. In addition, the concept of diffeomorphic invariance has been applied in machine learning algorithms to improve the accuracy and robustness of models.</p>

1. What is diffeomorphic invariance?

Diffeomorphic invariance is a concept in mathematics and physics that describes the property of a system or equation to remain unchanged under a smooth, continuous transformation. In simpler terms, it means that the system will produce the same results regardless of how it is stretched, compressed, or deformed.

2. How does diffeomorphic invariance relate to Poincare invariance?

Diffeomorphic invariance is a more general concept, while Poincare invariance is a specific case of it. Poincare invariance describes the property of a physical system to remain unchanged under the four-dimensional transformations of space and time, including rotations, translations, and boosts. Since these transformations are smooth and continuous, they also fall under the category of diffeomorphic transformations.

3. Why is diffeomorphic invariance important in physics?

Diffeomorphic invariance is important in physics because it allows us to describe physical phenomena in a way that is independent of the specific coordinates or frames of reference used. This means that the laws of physics will hold true regardless of how we choose to observe or measure them, making our understanding of the universe more consistent and universal.

4. Can diffeomorphic invariance be violated?

Yes, diffeomorphic invariance can be violated in certain situations. This can happen when the transformation is not smooth or continuous, or when there are external forces acting on the system that can cause deformations. In these cases, the results of the system may change depending on the specific transformation used, and diffeomorphic invariance is not preserved.

5. How is diffeomorphic invariance used in practical applications?

Diffeomorphic invariance is used in various fields of physics, including general relativity, quantum mechanics, and fluid dynamics. It is also utilized in image processing and computer graphics to create smooth and realistic transformations of images. In addition, the concept of diffeomorphic invariance has been applied in machine learning algorithms to improve the accuracy and robustness of models.

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