Exploring the Trace of a Fourth Rank Tensor in Index Notation

[binbagsss]

What is the general expression for the trace of a fourth rank tensor? Do you sum over possibilities of contractions with some factor?

So, for instance, for the Riemann tensor, it is given by:

$\eta_{ab}\eta_{cd}R^{acbd}$

due to these being independent contractions due to the symmetry properties the Riemann tensor obeys.

But what would it be for a general fourth rank tensor?

Thanks

 

[weirdoguy]

Is it even defined for tensors of rank higher than 2?

 

[fresh_42]

What is the general expression for the trace of a fourth rank tensor? Do you sum over possibilities of contractions with some factor?

So, for instance, for the Riemann tensor, it is given by:

$\eta_{ab}\eta_{cd}R^{acbd}$

due to these being independent contractions due to the symmetry properties the Riemann tensor obeys.

But what would it be for a general fourth rank tensor?

Thanks

Why do you want to know this? Or better: what is the trace to you?

The trace is not only a formula. We can define it for matrices by the characteristic polynomial, for field extensions, or what we get if we differentiate the determinant at $1$. It is an invariant quantity (versus change of basis).

You can artificially define a trace. $$V\otimes V\otimes V\otimes V \cong \underbrace{(V\otimes V)}_{=:W}\otimes \underbrace{(V\otimes V)^*}_{=:W^*}
$$
is a matrix, i.e. an endomorphism of $W.$ As such, it has a trace.

​

See https://en.wikipedia.org/wiki/Tensor_contraction for the "official" generalization. You will find a better explanation on the German version https://de.wikipedia.org/wiki/Tensorverjüngung of it. If you use Chrome, then right-click on the page for a translation. It will give you at least the important sentence:

Applications can be found e.g. B. in the theory of relativity[3] (see also length contraction), mechanics[4] etc.[5]

with corresponding links that is not part of the English version (or not in that wording).

 

[binbagsss]

I am not too familiar with tensor product notation etc, is is possible to answer using tensor index notation?

why? looking at tensor decompositions of fourth rank tensors.

 

[fresh_42]

I am not too familiar with tensor product notation etc, is is possible to answer using tensor index notation?

why? looking at tensor decompositions of fourth rank tensors.

See the two Wikipedia pages for index notation and the links I quoted.