Symmetric Part of a Mixed (1,1) Tensor

In summary, a mixed tensor is basically a standard linear transform from basic linear algebra, but changing the order of the indexes can effect the transformation. For example, if you have a mixed tensor with an upper index first and a lower index second, you can "lower" or "raise" the indices with the help of the metric to get a (0,2) tensor. However, without a metric you cannot properly "symmetrise" a mixed tensor.
  • #1
wgempel
2
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I have read in a couple of places that mixed tensors cannot be decomposed into a sum of symmetric and antisymmetric parts. This doesn't make any sense to me because I thought a mixed (1,1) tensor was basically equivalent to a standard linear transform from basic linear algebra. I am also having trouble with finding anything that shows how changing the order of the indexes effects a mixed tensor and how basic coordinate transforms affect mixed tensors with permuted indexes. The background is basic relativity so I usually use lorentz transforms and minkowski metric for examples. Here is my idea of the symmetric part of a mixed tensor M.

[itex] \frac{1}{2}\left[ {M^\alpha}_\beta + {N^\alpha}_\beta \right] [/itex]

where (i think)

[itex] {N^\alpha}_\beta = g_{\nu\beta}{M^\nu}_\mu g^{\mu\alpha} = {M_\beta}^\alpha[/itex]

I am having trouble seeing how to transform it to a new coordinate system because every book I have (and I have about 10) shows the coordinate transform matrices and mixed tensors with upper and lower indices directly above/below each other, so I am not sure how they apply.
 
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  • #2
When you have a metric like the [itex]g_{\mu\nu}[/itex] here you can identify the upper and lower indices with one another by using the metric. Essentially what you always do when performing calculations in this index formalism is to "lower" or "raise" indices with the help of the metric. So in this case it make perfect sense to talk about a symmetric (1,1) tensor because you can identify it with a (0,2) tensor by using the metric and this (0,2) tensor you can use symmetrisation or anti-symmetrisation upon. So if one has a metric one usually writes indices of a symmetric (1,1) tensor directly above one another because the ordering makes no difference. Of course you cannot do this for non-symmetric tensors, e.g. assume [itex]T[/itex] non-symmetric:
[tex] T_{\nu}^{\ \alpha}=g^{\alpha\mu}T_{\nu\mu}\neq g^{\alpha\mu}T_{\mu\nu}=T^{\alpha}_{\ \nu} [/tex]
It is true that in the case where you do not have a metric (e.g. affine differential geometry) the "symmetrisation" of a mixed (1,1) tensor can not be properly defined. Because in this case the tensor can only be viewed as a linear map from the tangent to the cotangent space (or vice versa, depending on the ordering of indices) and not as a bilinear map from the tangent space to the reals. Without a metric you cannot canonically identify the tangent and the cotangent space.
 
  • #3
OK, that makes sense to me, thanks.
 

1. What is the symmetric part of a mixed (1,1) tensor?

The symmetric part of a mixed (1,1) tensor is the part of the tensor that remains unchanged when its components are swapped. This means that if we interchange the two indices of a mixed (1,1) tensor, the symmetric part will remain the same.

2. How is the symmetric part of a mixed (1,1) tensor calculated?

The symmetric part of a mixed (1,1) tensor is calculated by taking the average of the tensor and its transpose. This is done by adding the tensor and its transpose together and dividing by 2.

3. What is the physical significance of the symmetric part of a mixed (1,1) tensor?

The symmetric part of a mixed (1,1) tensor is used to represent physical quantities that are independent of the coordinate system. This means that they have the same value regardless of the orientation of the axes used to measure them.

4. How is the symmetric part of a mixed (1,1) tensor used in physics?

The symmetric part of a mixed (1,1) tensor is used in many areas of physics, including mechanics, electromagnetism, and general relativity. It allows us to describe physical quantities, such as force and stress, in a way that is independent of the coordinate system used.

5. Can the symmetric part of a mixed (1,1) tensor be zero?

Yes, the symmetric part of a mixed (1,1) tensor can be zero. This occurs when the tensor is antisymmetric, meaning that it changes sign when its indices are swapped. In this case, the symmetric part will be zero, and the tensor will only have an antisymmetric part.

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