Can tensors always commute with each other or are there exceptions?

[emma83]

Hello,

I am still having a hard time with tensors...
The answer is probably obvious, but is it always the case (for an arbitrary metric tensor g_{\mu \nu} that g_{ab}g_{cd}=g_{cd}g_{ab} ?

I was trying to find a formal proof for that, and was wondering if we could use the relations:
(1) g_{ab}=\frac{\partial x^{c}}{\partial x^{a}} \frac{\partial x^{d}}{\partial x^{b}} g_{cd}
(2) g_{cd}=\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{b}}{\partial x^{d}} g_{ab}

And then multiply the left-hand side of (1) and (2) together and use the fact that the fractions of partial derivatives commute with the metric tensor and cancel each other to get:
g_{ab}g_{cd}<br /> =(\frac{\partial x^{c}}{\partial x^{a}} \frac{\partial x^{d}}{\partial x^{b}} g_{cd}) (\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{b}}{\partial x^{d}} g_{ab})<br /> = \frac{\partial x^{c}}{\partial x^{a}}\frac{\partial x^{a}}{\partial x^{c}} \frac{\partial x^{d}}{\partial x^{b}} \frac{\partial x^{b}}{\partial x^{d}} g_{cd} g_{ab}<br /> = g_{cd}g_{ab}

Does that make sense ?!

Thanks for your help...

 

[malawi_glenn]

in

<br /> g_{ab}g_{cd}=g_{cd}g_{ab}<br />

g_{ab} is just a number, and so is g_{cd}, and numbers commute.

for instance, g_{02} = 0, g_{00} = 1 (or g_{00} = -1) depending on which your original definition of metric is)

 

[emma83]

Argh! Thanks a lot, again I mixed up "tensors" with "tensor components"...
So the commutation relation I wrote holds for the tensor components of any tensor (not only the metric), doesn't it ?

On the other hand, the tensor (objects) do not necessarily commute with each other, right ?

 

[malawi_glenn]

no the tensor objects can be non-commuting things like matricies, e.g.

\sigma ^{\mu \nu} = \frac{i}{2}[\gamma ^\mu, \gamma^\nu]
where the gamma's are dirac 4x4 matricies