I Is a Manifold with a Boundary Considered a True Manifold?

[jbergman]

So in this special case (i.e. constant scalar curvature manifold) the Lie Derivative of the metric tensor $g$ along every tangent vector at every point is always null (strictly speaking we need a vector field $X$ along which calculate the Lie derivative: any smooth vector field will do the job).

I don't think this is correct. Given any tangent vector you can construct a KVF that agrees with the tangent vector at that point, but the converse is not true that any vector field with matching tangent vector at that point has 0 Lie Derivative, see my example of the dilation of the plane.

 

[PeterDonis]

So which is actually the distinguishing feature of a KVF and its integral curves?

Killing's equation. This is the sort of question you should be looking up for yourself.

 

[PeterDonis]

So, in Euclidean space if we pick a generic curve (that is not the integral curve of any KVF) the Lie Derivative of the metric tensor $g$ along the curve's tangent vector field $X$ does not vanish, namely $\mathcal{L}_{X} g \neq 0$. Yet the metric tensor $g$ does not change along that curve.

Correct. I strongly suggest that you do the math instead of relying on your intuition.

 

[PeterDonis]

Given any tangent vector you can construct a KVF that agrees with the tangent vector at that point, but the converse is not true that any vector field with matching tangent vector at that point has 0 Lie Derivative

Yes, exactly. A tangent vector at a point is not sufficient by itself to define a global curve (and hence a global vector field that has that curve as an integral curve). There are an infinite number of curves passing through a given point with a given tangent vector at that point.

 

[PeterDonis]

the Lie Derivative of the metric tensor along every tangent vector

Does not exist. You don't take Lie derivatives along vectors, you take them along curves.

Again, I strongly suggest that you do the math instead of relying on your intuition.

 

[cianfa72]

Does not exist. You don't take Lie derivatives along vectors, you take them along curves.

Sorry, to take Lie derivatives we need not just a curve but a family of them (i.e. a congruence).

In other words the vector field has to be defined in a neighborhood of each point along the curve (i.e. off the curve).

 

[PeterDonis]

to take Lie derivatives we need not just a curve but a family of them

In other words, you don't take Lie derivatives along vectors, you take them along curves, as I said. To do it, the curves you take them along need to be part of a congruence. Yes, that's true. And your earlier statement about taking Lie derivatives along vectors is still wrong.

If you are that familiar with Lie derivatives, you should be able to do the math yourself regarding all the things you are asking about in this discussion, instead of using your intuition to make wrong claims that we then have to correct.

 

[cianfa72]

Maybe I misinterpreted post #118 from @PAllen that in case of constant scalar curvature manifold every tangent direction at every point is a killing direction.

 

[PeterDonis]

Maybe I misinterpreted post #118 from @PAllen that in case of constant scalar curvature manifold every tangent direction at every point is a killing direction.

Perhaps you did. What he meant is that there an integral curve of some KVF passing through every point in every direction. But, as has been pointed out, the converse is not true: not every curve in every direction passing through a given point is an integral curve of some KVF. A direction (i.e., vector in the tangent space) at a point is not sufficient by itself to define a curve; for every tangent vector at a given point there are an infinite number of curves through that point that have that tangent vector, and only some of those curves will be integral curves of a KVF.

Once more: you need to take the time to actually do the math for this instead of using your intuition.

 

[cianfa72]

What he meant is that there an integral curve of some KVF passing through every point in every direction.

Sorry, but the above statement is always true or is true only in case of constant scalar curvature ?

 

[Ibix]

Sorry, but the above statement is always true

Clearly not: consider that there is no KVF pointing radially in Schwarzschild spacetime.

 

[PeterDonis]

the above statement is always true or is true only in case of constant scalar curvature ?

The latter, since that was the context of the original statement by @PAllen that started this subthread.

 

[ergospherical]

Well, this thread got quite badly side-tracked somehow.

 

[PAllen]

A simple example of a congruence in Minkowski spacetime that does not define a (timelike) kvf is the Milne congruence. This is despite the fact that each curve of the congruence is a timelike geodesic. What is true is that each curve of this congruence is a component integral curve of a different timelike kvf.

 

[jbergman]

Sorry, but the above statement is always true or is true only in case of constant scalar curvature ?

Personally, I think it's better to think about Isometries and flows than to think about the Killing vector fields or at least at first.

As explained on wikipedia, an isometry is a smooth mapping of a manifold of to itself that preserves the notion of distance.

A flow is a family of diffeomorphisms parameterized by some parameter like $t$ that very continuously with that parameter and whose derivatives with respect to $t$ at each point are tangent vectors and very smoothly and hence are associated with a vector field. In the case that family of diffeomorphisms associated with a flow are Isometries then the generating vector field is a Killing vector field.

The isometries are easy to understand intuitively for Euclidean Space or a sphere. For Euclidean Space you just have rotations, translations and reflections. Rotations and Translations can be parameterized to define flows. The generators of these are the KVFs. Now there are many other diffeomorphisms of $\mathbb{R}^n$ that aren't isometries. The vector fields associated with these are not KVFs.

 

[PeterDonis]

As explained on wikipedia, an isometry is a smooth mapping of a manifold of to itself that preserves the notion of distance.

It might be worth expanding on this a bit.

An isometry is a mapping $M$ of a manifold to a manifold (which might be the same one or might not) such that, for any pair of points $p$, $q$, taken by the mapping to points $M(p)$, $M(q)$, the distance from $p$ to $q$ is the same as the distance from $M(p)$ to $M(q)$. So it's not that the metric itself stays the same: the distances between mapped points have to stay the same.

So, for example, in Minkowski spacetime, translations, rotations, and boosts are all isometries of the manifold into itself. But the Milne transformation, the mapping corresponding to the Milne congruence described by @PAllen in post #134, is not; distances between corresponding points are changed by the transformation. And that's true even though the Minkowski metric itself stays the same along each curve in the congruence.

 

[cianfa72]

So it's not that the metric itself stays the same: the distances between mapped points have to stay the same.

ok, now it makes sense to me.

distances between corresponding points are changed by the transformation. And that's true even though the Minkowski metric itself stays the same along each curve in the congruence.

ok got it.

Just to give an example that helps intuition, I drew the following picture for the Euclidean plane:

Take the triangle ABC, the blue and red arrows represent two of the three sides. Consider its flow along the dotted red curves (i.e. along the congruence filling the Euclidean plane). Each curve in the congruence is parametrized by the its euclidean length $s$.

We can see that even if the metric tensor $g$ does not change along the congruence's curves, the shape of the triangle along the flow for the same parameter $s$ actually changes. Hence the curves in the congruence are not integral curves of any KVF.

 

[PAllen]

I definitely agree it is more intuitive to approach kvf's starting from a congruence, which defines a vector field, and then require that any two points connected by a geodesic, when 'flowed' along the congruence by a fixed interval along congruence lines, results in a geodesic of the same length as before. Then showing that this property being true for any pair of points and any flow amount, is equivalent to more abstract definitions applied to the vector field itself. Note, of course, that none of this language involves coordinates (though building coordinates adapted to a killing flow ensures that certain features are present in the expression of the metric in those coordinates). Also note that none of this says anything about diffeomorphisms or coordinate changes.

However, I am quite confused by what any of this has to do with isometries. My understanding has been that for any diffeomorphism, when a pullback is applied to the metric, the result is an isometry. In particular, if you consider the coordinate transform from standard Minkowski coordinates to Milne coordinates, with the associated Milne metric, then for any pair of points connected by a geodesic in standard Minkowski coordinates, the result after transform, is still a geodesic of the same length when using the Milne metric.

 

[cianfa72]

However, I am quite confused by what any of this has to do with isometries. My understanding has been that for any diffeomorphism, when a pullback is applied to the metric, the result is an isometry.

Yes, the point is that an isometry is a special diffeomorphism such that the pullback of the metric tensor along it results in the same metric tensor defined at the point where the pullback is done (i.e. the Lie derivative of the metric tensor along the flow defined by the isometry is null).

 

[PAllen]

Yes, the point is that an isometry is a special diffeomorphism such that the pullback of the metric tensor along it results in the same metric tensor defined at the point where the pullback is done.

Yes, but any diffeomorphism has an associated pullback, and in GR, this is the only sense in which diffeomorphisms are applied. Also, this doesn't answer in any way, what this has to do with kvfs. That is claim I don't understand.

 

[jbergman]

I definitely agree it is more intuitive to approach kvf's starting from a congruence, which defines a vector field, and then require that any two points connected by a geodesic, when 'flowed' along the congruence by a fixed interval along congruence lines, results in a geodesic of the same length as before. Then showing that this property being true for any pair of points and any flow amount, is equivalent to more abstract definitions applied to the vector field itself. Note, of course, that none of this language involves coordinates (though building coordinates adapted to a killing flow ensures that certain features are present in the expression of the metric in those coordinates). Also note that none of this says anything about diffeomorphisms or coordinate changes.

However, I am quite confused by what any of this has to do with isometries. My understanding has been that for any diffeomorphism, when a pullback is applied to the metric, the result is an isometry. In particular, if you consider the coordinate transform from standard Minkowski coordinates to Milne coordinates, with the associated Milne metric, then for any pair of points connected by a geodesic in standard Minkowski coordinates, the result after transform, is still a geodesic of the same length when using the Milne metric.

These questions are somewhat subtle so I want to try and give my answer. First, I don't think that GR is a diffeomorphism invariant theory. Let $R=(M,g)$ and $R'=(M', g')$ then a diffeomorphism is just a smooth invertible map $f: M \rightarrow M'$. It says nothing about the relationship between metrics. When you impose the additional condition on the pullback of the metric $g=f^{*}g'$, you are describing an isometry. I think it's just an example of loose terminology in physics.

See this discussion on https://physics.stackexchange.com/q...folds-physically-equivalent?noredirect=1&lq=1.

Now, in the case of space time there is really only one Universe (at least in this context), so conceptually we may want to restrict to maps to the same manifold $f: M \rightarrow M$ which leads to the auto-isometries which are isometries of a manifold to itself. In the case of the Milne coordinates we could define Minkowski space to be our pseudo-riemannian manifold, and just view the Milne coordinates as just one coordinate chart for that manifold. With the intrinsic definition of a manifold you have the manifold and then chart maps. This makes more sense for things like a sphere embedded in 3 space, but you can also have a 4-dimensional manifold, Minkowski space with alternative coordinate charts.

 

[jbergman]

These questions are somewhat subtle so I want to try and give my answer. First, I don't think that GR is a diffeomorphism invariant theory. Let $R=(M,g)$ and $R'=(M', g')$ then a diffeomorphism is just a smooth invertible map $f: M \rightarrow M'$. It says nothing about the relationship between metrics. When you impose the additional condition on the pullback of the metric $g=f^{*}g'$, you are describing an isometry. I think it's just an example of loose terminology in physics.

See this discussion on stackexchange.com.

Now, in the case of space time there is really only one Universe (at least in this context), so conceptually we may want to restrict to maps to the same manifold $f: M \rightarrow M$ which leads to the auto-isometries which are isometries of a manifold to itself. In the case of the Milne coordinates we could define Minkowski space to be our pseudo-riemannian manifold, and just view the Milne coordinates as just one coordinate chart for that manifold. With the intrinsic definition of a manifold you have the manifold and then chart maps. This makes more sense for things like a sphere embedded in 3 space, but you can also have a 4-dimensional manifold, Minkowski space with alternative coordinate charts.

And also vector fields generate flows which give us auto-diffeomorphisms and in the case of KVFs auto-isometries.

 

[cianfa72]

And also vector fields generate flows which give us auto-diffeomorphisms and in the case of KVFs auto-isometries.

Yes definitely, that was my point too.

 

[PeterDonis]

See this discussion on stackexchange.com.

Your link here just goes to the stackexchange home page.

 

[PeterDonis]

I am quite confused by what any of this has to do with isometries.

Wald, Appendix C, basically defines an isometry to be a transformation generated by a vector field that satisfies Killing's equation, i.e., the Lie derivative of the metric along the vector field is zero. This might be an example of a difference between physics terminology and mathematics terminology.

 

[PAllen]

These questions are somewhat subtle so I want to try and give my answer. First, I don't think that GR is a diffeomorphism invariant theory. Let $R=(M,g)$ and $R'=(M', g')$ then a diffeomorphism is just a smooth invertible map $f: M \rightarrow M'$. It says nothing about the relationship between metrics. When you impose the additional condition on the pullback of the metric $g=f^{*}g'$, you are describing an isometry. I think it's just an example of loose terminology in physics.

Well, if you want, I can provide dozens of quotes calling GR a diffeomorphism invariant theory. Further, the use of pulllbacks or pushforwards does not in any way restrict the mapping between manifolds, per se. It simply adds a rule that metric on each is determined by the mapping combined with the metric on the other. I have a whole shelf of GR textbooks, and not one ever considers a diffeomorphism without also using a pullback or pushforward.

See this discussion on stackexchange.com.

This link doesn't go to any discussion, just the home page.

What am confused about is not addressed in this post. That is, what is the connection between diffeomorphims and kvfs?? The latter are equivalently, particular congruences generating a vector field, or particular vector fields (with associate integral curves), that meet certain properties. None of these properties specify a diffeomorphism at all.

There is a passing comment in the wikipedia entry on isometries to kvf's as generators. Unfortunately, there is no further explanation or reference associated, and I don't understand it. I would guess there is something to it, but I remain search of any explanation. A related confusion, is that a pseudo-Riemannian manifold may have no kvfs at all (e.g. I think this is true of some of Misner's mixmaster universes). Yet one may readily define isometric mappings between manifolds, one of which is mixmaster (which would, of course, make the other one mixmaster as well). If I am right that a pseudo-Riemannian manifold may have no kvfs, I genuinely don't understand the comment about kvfs as generators of isometries.

 

[ergospherical]

That is, what is the connection between diffeomorphims and kvfs??

- A vector field $v$ induces a flow $\phi_t$ which shifts points a parameter distance $t$ along the integral curves of $v$.
- If $v$ is smooth then $\phi_t$ is a diffeomorphism (because then the maps $\phi^{-1}_t = \phi_{-t}$ and $\phi_{t_1} \cdot \phi_{t_2} = \phi_{t_1 + t_2}$ are both smooth).
- If $v$ is also a Killing vector field, that is if $L_v g = 0$, then the map $\phi_t$ is an isometry.

 

[PAllen]

Wald, Appendix C, basically defines an isometry to be a transformation generated by a vector field that satisfies Killing's equation, i.e., the Lie derivative of the metric along the vector field is zero. This might be an example of a difference between physics terminology and mathematics terminology.

That's not what I see in section C.1. It states that a diffeomorphism along with using a pullpack for the metric is an isometry.

[added] Reading further to section C.3 of the appendix, what I see that one can arrive at the notion of a kvf from a one parameter group of isometries from a manifold to itself. Every part of this statement is importang - one parameter, and group. This is not a different definition of isometry, but a specialized usage of the concept.

 

[PeterDonis]

That's not what I see in section C.1. It states that a diffeomorphism along with using a pullpack for the metric is an isometry.

See section C.3.

 

[PAllen]

See section C.3.

See what I added above, before seeing your response.