Vector fields, flows and tensor fields

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Discussion Overview

The discussion revolves around the generalization of vector fields and their flows to tensor fields, specifically rank-2 tensor fields, within the context of physics. Participants explore the implications of this generalization, particularly in relation to isometries, the stress-energy tensor, and the concept of flows in different dimensions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that vector fields generate flows, which are one-parameter groups of diffeomorphisms, and question how this concept extends to rank-2 tensor fields.
  • Others argue that tensor fields do not generalize to flows in the same way, as they lack the directional arrows associated with vector fields.
  • A metric tensor is discussed as not generating isometries, but rather being preserved by diffeomorphisms, leading to confusion over terminology.
  • Some participants suggest that the stress-energy tensor may relate to flow in terms of curvature and time evolution, but express hesitation in using the term "flow" due to complexities in general relativity.
  • There is a proposal to consider fields of planes instead of vectors, with references to the Frobenius theorem regarding the realizability of these planes as tangent to surfaces.
  • Participants discuss the implications of foliability in relation to tensor fields and the existence of n-parameter groups of flows along n-surfaces.

Areas of Agreement / Disagreement

Participants generally do not agree on the generalization of flows from vector fields to tensor fields, with multiple competing views on the nature of isometries and the interpretation of the stress-energy tensor. The discussion remains unresolved regarding the appropriate terminology and conceptual framework.

Contextual Notes

There are limitations in the discussion regarding the definitions of flows, isometries, and the assumptions underlying the interpretation of tensor fields. The relationship between the dimensionality of tensor fields and their isometry groups is also not fully resolved.

  • #31
TrickyDicky said:
Right, I knew this proof, that is why I said that I agreed a general order two tensor is not necessarily the tensor product of two vectors, a general second order tensor can be constructed with the sum of ##n^2## tensor products.

But I think you might have misunderstood what this proof implies from what Schutz write in the Appendix. You seemed to imply that the tensor product of two vectors has at most 2n independent components, and therefore by counting the independent components of a tensor one might deduce if it is the result of a tensor product or not. This is not correct. Just do the calculation, multiply two vectors using the outer product, you get a second order tensor with ##n^2## independent components. Granted you have only used 2n components to build it and that is why a general tensor is not necessarily the outer product of two vectors, but still after the product the ##n^2## components formed from the 2n are considered to be independent.

This doesn't make sense to me...If I can use the counting to prove that a general rank 2 tensor cannot be expressed as a dyad, why can't I use this counting to prove that a specific rank 2 tensor cannot be expressed as a dyad?

Can you come up with an example where my reasoning fails? A rank two tensor with more than 8 independent components that can be expressed as the direct product of 2 vectors alone?
 
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  • #32
Matterwave said:
This doesn't make sense to me...If I can use the counting to prove that a general rank 2 tensor cannot be expressed as a dyad, why can't I use this counting to prove that a specific rank 2 tensor cannot be expressed as a dyad?
You can. What you can't is take an arbitrary rank 2 tensor and decide whether it is a dyad or a sum of dyads just based on the number of independent components of the tensor, this is using the proof backwards.
Can you come up with an example where my reasoning fails? A rank two tensor with more than 8 independent components that can be expressed as the direct product of 2 vectors alone?
The best way is to compute it as I said, and realize that the fact that you use only 8 initial components to construct 16 components doesn't mean that just 8 of the final 16 are independent, if this was so you should be able to say which 8 are the independent ones, but you can't.
 
  • #33
I guess it is true the whole thing doesn't make much sense with tensor fields, certainly not at the level of smooth manifolds where the usual flows for vector fields are defined. Perhaps after introducing a metric and curvature it could make more sense for certain geometries' isometry groups.
 
  • #34
I don't see any way to make it work...
 

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