What is the Concept of Zero Double Trace in Tensor Analysis?

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Discussion Overview

The discussion revolves around the concept of "zero double trace" in tensor analysis, specifically in relation to a rank four tensor that possesses symmetries similar to the Riemann tensor. Participants seek clarification on the definition and implications of this property.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses unfamiliarity with the concept of double trace and seeks assistance.
  • Another participant inquires about the specific type and properties of the tensor in question, asking for details such as the number of indices.
  • A participant describes the tensor as having the symmetries of the Riemann tensor, noting it is rank four with indices ranging from 0 to 3.
  • Discussion includes the idea of iterated contraction, with a participant suggesting that multiple double contractions may exist depending on index permutations.
  • There is speculation that the "zero double trace" property may relate to the Weyl tensor, which is described as "completely traceless" and shares symmetries with the Riemann tensor.
  • One participant references a paper by A. Kostelecky and M. Mewes, questioning if the term "double trace" is used in a specific context within that work.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition or implications of "zero double trace," and multiple interpretations and ideas are presented without resolution.

Contextual Notes

Participants acknowledge the complexity of the term "double trace" and its potential variations, indicating that assumptions about its meaning may depend on specific contexts or definitions.

kauw7
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Hello all,

I have recently encountered a tensor which is said to have the property "zero double trace". I am unfamiliar with the concept of a double trace and was hoping someone here could help.

Thanks
 
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What kind of tensor is it? Does it have a specific name?
How many indices does it have?
 
Hi thanks for the reply.

I don't think it has a proper name, but it has all the symmetries of the Riemann tensor. It is rank four and the indices run from 0:3.

Thanks
 
Iterated contraction

This can only mean, I think, an iterated contraction, e.g.
[tex] {P^{ab}}_{cd} <br /> \mapsto {Q^a}_d = {P^{am}}_{md} <br /> \mapsto R = {Q^m}_m[/tex]
But as you can see by permuting the indices, there are in general many such double contractions!

Incidently, the obvious guess is that you are reading about the Weyl tensor, aka conformal curvature tensor, which shares all the symmetries of the Riemann tensor but which is also "completely traceless". Did you see the concurrent thread on that object?
 
Is this what everyone else thinks the mysterious 'double trace' must be? I came across it in a paper on the standard model extension by A. Kostelecky and M. Mewes...

Does anyone have any other ideas about what they could be talking about?
 

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