Please read the following carefully. The point of the following is to distinguish between [tex]T^{\mu}_{\mbox{ } \nu}[/tex] and [tex]T_{\mbox{ }\mu}^{\nu}[/tex] which clearly involves a metric tensor. But when you want to go from component manipulation to matrix operations you have to be careful. Components are scalars and the products are commutative, but the matrix products they represent are not.(adsbygoogle = window.adsbygoogle || []).push({});

Setup: let M be a manifold and p a point on the manifold. Let V be the tangent space at p and V* be the cotangent space at p. Let [tex]T:V^*\times V^*\to R[/tex] be a (2,0) tensor. Choosing a basis for V we can talk about the components of T: [tex]T^{\mu\nu}[/tex]. Let g be the metric tensor so [tex]g:V\times V\to R[/tex] and the inverse [tex]g^{-1}:V^*\times V^*\to R[/tex].

If we only "fill" one of the slots of T we can create maps like [tex]T:V^*\to V[/tex] and we can also do the same for g (think Riesz rep. theorem) [tex]g:V\to V^*[/tex].

So in that sense we can create some new maps: [tex]gT:V^*\to V^*[/tex] and [tex]Tg:V\to V[/tex].

My Claim:

the components of gT are [tex]g_{\mu\nu}T^{\nu\lambda}=T_{\mu}^{\mbox{ }\lambda}[/tex] while the components of Tg are [tex]T^{\nu\lambda}g_{\lambda\mu}=T^{\nu}_{\mbox{ }\mu}[/tex]. Moreover I claim that to work out the matrix products you use gT and Tg as guidelines. A professor disagrees however, he says he's never seen the metric act from the right (and usually in component form it is written on the left).

How, using matrices would you lower the first or second index on [tex]T^{\mu\nu}[/tex] and does the order matter? Looking at the above in a coordinate free manner with composition of functions it does seem to matter.

Thank you in advance.

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# Raising/Lowering indices and matrix multiplication

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