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Vector fields, flows and tensor fields

  1. May 6, 2014 #1
    Vector fields generate flows, i.e. one-parameter groups of diffeomorphisms, which are profusely used in physics from the streamlines of velocity flows in fluid dynamics to currents as flows of charge in electromagnetism, and when the flows preserve the metric we talk about Killing vector fields and isometries are defined and used i.e. in GR.

    How exactly is this generalized to the case of tensor fields?, let's concentrate on rank-2 tensor fields that are most often found in physics. Intuitively one would think they would generate two-parameters groups of diffeomorphisms?. Say for instance if the tensor field is a metric tensor(thinking about Klein's Erlangen program), they would generate isometries in the form of transformations like translations, rotations, reflections involving two directions? For other two-rank tensors that have physical significance , say the stress-energy tensor, what kind of (local)"flow" would it generate?
     
    Last edited: May 6, 2014
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  3. May 7, 2014 #2
    I don't think it does generalize, or, if it does, you wouldn't call it a flow. Flow means you follow the arrows. But in general, tensor fields don't have arrows to follow.

    A metric tensor doesn't generate isometries. Things isometries with respect to them. Also, the fact that it's a rank-2 tensor has absolutely nothing to do with having 2-parameters in its isometry group. For example, the isometries of n-dimensional space will have MUCH more than 2-parameters, obviously.

    The stress energy tensor is doing something a little like generating a flow, if you interpret it from that viewpoint. It's telling you something about the curvature of space-time. It's a little like time-evolution for PDE's like Maxwell's equations or the wave equation or something. However, calling it a "flow" seems to rely on a well-defined notion of time, and time is sort of tricky to get your hands on in general relativity, so it's a bit different, in general, although I think, in special cases, you can get away with interpreting it that way--telling you how the metric on some space-like surfaces changes over time. The Cauchy problem: I give you the geometry of a space-like surface, you tell me the rest of the geometry. Similar to an initial value problem in ODE (or in PDE, like with the wave equation if I tell you the configuration of a rubber sheet at t = 0, you tell me what happens in the rest of time). If you want to call it a flow, maybe it's sort of a flow in the configuration space of the rubber sheet. And maybe in GR, you might think of it as a flow through the space of geometries for your space-like 3-surface, I guess (don't quote me on it), but I am hesitant to call it a flow.

    A more appropriate generalization would be to ask what would happen if you had a field of planes, rather than a field of vectors. Can the planes be realized as the tangent planes to a surface, just vector fields can be realized as the tangent vectors to a curve? Not so easily. This question is addressed by the Frobenius theorem.

    Not a particularly good article, but here it is:

    http://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology [Broken])
     
    Last edited by a moderator: May 6, 2017
  4. May 7, 2014 #3
    Hi, homeomorphic, thanks for taking a stab at answering this.

    Sure, I wouldn't call it flow either, flows as we understand it are one-dimensional, and as you also write below, I had in mind something more like surfaces or planes.
    Well, to make it clearer, an isometry is any diffeomorphism that preserves the metric tensor, so yes it might be misleading to use the terminology for vector fields in terms of generating flows.
    Couldn't understand this sentence, can you rephrase it?

    Yes, I was more thinking about the idea of surfaces or planes, in the sense that a tensor of rank-2 may have whatever dimensionality but there is a sense in wich it is bidimensional just like n-dimensional vectors are unidimensional.

    Well, I was rather thinking that the stress-energy tensor field assures local conservation of momentum-energy, while in contrast other quantities like momentum or energy are quantities that by Noether theorem are conserved by admitting the flow(one-parameter groups of diffeomorfisms) of certain vector fields.
    Yes, this is the kind of visualization I had in mind. Frobenius theorem is quite interesting and pertinent here, it seems to suggest one has to have a space with certain requirements in terms of foliability to make more concrete what I intend with tensor fields.
     
  5. May 7, 2014 #4
    Typo. Things ARE isometries with respect to them.


    Hmm. Well, I suspect there are analogues of that sort of thing in GR, but I don't know GR well enough.

    There might be something interesting along those lines to be found here in Track 2:

    http://math.ucr.edu/home/baez/qg-winter2001/

    Not sure it's exactly what you are looking for, but it does discuss some things, like what is the analogue of momentum in configuration spaces of fields, like those of GR, so I imagine it might help.

    Well, the foliability has more to due to with which plane-fields you choose than the space they live in. There's some theorem of Thurston that a compact manifold in which there exists a distribution (plane-field) has a foliation of the same dimension.
     
  6. May 7, 2014 #5

    Ben Niehoff

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    If you do have a foliation by n-surfaces, then you always have an n-parameter group of flows along those n-surfaces. This essentially pops out of the Frobenius theorem, because in order to have a foliation by n-surfaces, you must have n vector fields $X_i$ which are closed under the Lie bracket. These vector fields then generate n flows, each of which preserves the family of n-surfaces.
     
  7. May 7, 2014 #6
    Thanks for the Baez reference, it is not what I was looking for but it is always refreshing to read.

    I'm still interested in knowing if some analogue of integral curves for vector fields is defined for tensor fields, something like (for the case of rank-2 tensors) "integral surfaces".
    Anybody?
     
  8. May 7, 2014 #7
    Hi Ben I missed this while typing.
    So basically one can define such "integral n-surfaces" generated by tensor fields of rank n when that foliation is available?
     
  9. May 7, 2014 #8

    Matterwave

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    Not in a direct way. The vector fields under which a tensor field is invariant form a Lie algebra. These vector fields define integral curves which mesh into submanifolds which foliate the manifold.

    The latter part of the statement is Frobenius' theorem. The first part of the statement can be seen from direct calculation.
     
  10. May 7, 2014 #9

    Ben Niehoff

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    No, I'm afraid that idea doesn't make sense. One can talk about a group of flows generated by a collection of vector fields ##X_i##. But each flow is still generated by a vector field.

    You might be able to make some sense of the notion of "flows" generated by antisymmetric tensors, but I suspect it will still only make sense when those tensors can be written as a wedge product of 1-tensors (i.e. vectors).

    The notion of a "flow" is just a very 1-dimensional thing.
     
  11. May 7, 2014 #10
    Yes I can see that , still is there not a way to make it more direct? I'm thinking specifically about rank-2 tensors here, and relying on their multiple role as bilinear map(higher order function) and operator-transformation-local diffeomorphism.


    I already agreed the term flow is not right here just for the reason you mention.
    Let's limit the discussion for simplicity to rank-2 tensors.Their role as transformations-mappings-diffeomorphisms as I said above seems to allow a more direct way to relate tensor fields and diffeomorpisms.
    Yes, well that is basically done in inverse-square Gauss laws relying on the properties of antisymmetric tensors, Euclidean space, Stokes and all that. wich allows to ultimately reduce it all to vector fields and their flows. I'm trying to twist it a bit further.
     
  12. May 8, 2014 #11

    Matterwave

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    A general tensor field of rank 2 can be expressed as:

    $$T=\sum_{i,j}T^{ij}e_i\otimes e_j$$

    How do you propose to turn this into a flow, or a diffeomorphism? Perhaps for tensors of the form:

    $$T'=V\otimes W$$

    One might be able to use the mesh of integral curves to attempt something, but 1. this equation does not hold in general, and 2. the integral curves will in no way be guaranteed to mesh to form a sub manifold unless ##[V,W]=aV+bW##. Even if the integral curves do mesh to form a family of 2 dimensional sub manifolds, what is the flow from one to the other? The sub manifolds, as foliations, are disjoint, and so you'd need a third vector field to define diffeomorphisms from one to the other.

    I don't see any advantages to be had from such a narrow definition.
     
  13. May 8, 2014 #12
    Hmmm, I'm afraid we are not talking about the same thing, I'm not saying the same notion of integral curve-flow must be retained in the generalization to tensor fields. It would be the analogue just in the sense of relating fields and diffeomorphisms.So it wouldn't really look like a flow, but then again it is relatively easy to visualize vector fields and their flows, while it is not so easy to visualize even second rank tensor fields.
    For instance, think of translations in the Euclidean plane, they are maps R2→R2, and they are two-parameter diffeomorphisms(that are also isometries since they preserve the metric), how would one relate them to a rank-2 tensor field, there appears to be a more direct way than the usual decomposing the group in their x and y directions and from the one-parameter get the vector fields, in the sense that the transformation itself can be seen as a tensor(universal property), but this doesn't exactly capture what I had in mind.
    But I realize it is not as easy as I had pictured at first, and I'm not sure it can be fully generalized.
     
  14. May 8, 2014 #13
    Hmmm, I'm afraid we are not talking about the same thing, I'm not saying the same notion of integral curve-flow must be retained in the generalization to tensor fields. It would be the analogue just in the sense of relating fields and diffeomorphisms.So it wouldn't really look like a flow, but then again it is relatively easy to visualize vector fields and their flows, while it is not so easy to visualize even second rank tensor fields.
    For instance, think of translations in the Euclidean plane, they are maps R2→R2, and they are two-parameter diffeomorphisms(that are also isometries since they preserve the metric), how would one relate them to a rank-2 tensor field, there appears to be a more direct way than the usual decomposing the group in their x and y directions and from the one-parameter get the vector fields, in the sense that the transformation itself can be seen as a tensor(universal property), but this doesn't exactly capture what I had in mind.
    But I realize it is not as easy as I had pictured at first(wich implied changing the staticity of the integral curves in flows by a more dynamic picture of the curves-thus producing surfaces in their sweeping motion, I thought this could appeal to string theorists as it puts the accent in one dimensional rather than 0-dimensional objects-, and I'm not sure it can be fully generalized.
     
    Last edited: May 8, 2014
  15. May 8, 2014 #14

    Matterwave

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    How would you use a rank 2 tensor field to map a point P on a manifold to another point Q, maintaining that this is a diffeomorphism? What is your conception of this operation? Because none come to mind to me.

    The important property that allows a vector field to define a diffeomorphism is that any point P lies on one and only one congruence (the congruences don't intersect each other). This allows you to define a map from P to Q which lies along the congruence. I see no generalization of this phenomenon to rank 2 tensor fields. What did you have in mind?
     
  16. May 8, 2014 #15
    No, I'm not trying to map fixed points, the idea is rather mapping curves that lie on parametrized congruences of surfaces instead of curves. This is what I had in mind but I don't know how to make it more explicit or even if it is feasible at all.
     
  17. May 8, 2014 #16

    Ben Niehoff

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    Think about this another way:

    Let's say you have a 2-parameter family of diffeomorphisms ##\varphi_{st} : M \to M##. By "2-parameter family" we mean that for each ##s, t##, we have a unique diffeomorphism. (This will turn out to be very restrictive!)

    So, if we hold ##s = s_0## fixed, then we are left with a 1-parameter family of diffeomorphisms ##\varphi_{s_0 t}##, which is just a flow along some vector field ##X##. Similarly, if we hold ##t = t_0## fixed, we get the 1-parameter family ##\varphi_{s t_0}##, which is just a flow along some vector field ##Y##.

    Finally, since we claim that each ##\varphi_{st}## corresponds to exactly one diffeomorphism, we must in fact have that ##X## and ##Y## commute!

    [tex][X,Y] = 0[/tex]
    So even if you start with the idea of a "2-parameter family of diffeomorphisms", it turns out that this idea is even more restrictive than the idea of a group of flows on the leaves of a foliation. Because in order to have a foliation, ##[X,Y]## need only be closed into ##a X + b Y##; it needn't be zero.
     
  18. May 8, 2014 #17
    Yes, I can see what you mean, Ben. Thanks for the clarifying post.
    The idea is quite restrictive rather than a generalization as I thought. Only under very specific conditions it can be achieved.
    I'll think some more about it just in case I see if there is any potential situation where this could be applicable or if I think up some other question.
     
  19. May 8, 2014 #18
    I was thinking about the two restrictive conditions you wrote above and since I already mentioned the stress-energy tensor as a possible example I was wondering if being a both symmetric and divergenceless tensor field would meet those criteria or at least get close to it(I guess the dimensionality of the space would also matter by the Frobenius theorem?).
     
  20. May 8, 2014 #19

    Matterwave

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    A diffeomorphism is always mapping points to points. If you map a line to a line, you are still mapping points on that line to another point on another line. Point P to point Q is the most basic and most general diffeomorphism map you can obtain.

    As regards to your questions about the stress-energy tensor "meeting the requirements" put forth by Ben, Ben's restrictions are on the vector fields generating each specific flow. The stress energy tensor cannot be expressed as the simple direct product of two vectors. Two 4-vectors has a total of 8 independent components, while the Stress-energy tensor has 10 independent components in general.
     
  21. May 8, 2014 #20
    If you map a vector in one direction to a vector in another direction with a tensor I guess you are in some sense also mapping points but not fixed points in the sense I think you mean by P and Q.
    The vanishing commutator involves both vector fields
    The tensor product of two 4-vectors would have 16 components, being symmetric they are reduced to 10.
     
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