Tensor product of a 4-vector and tensor with shared indices

[Milsomonk]

TL;DR

Basic Index gymnastics

Hi all, I'm pretty rusty on my index gymnastics, I'm wondering if someone can explain to me the correct way to take the tensor product of a general 4-vector $g^{\mu}$ and a tensor $P^{/mu\ /nu}$ in Minkowski space. The part that is troubling me is the fact that both have a $\mu$ as a contravariant index. Many thanks in advance for your help.

 

[Orodruin]

The tensor product would be a rank 3 tensor $g^\sigma P^{\mu\nu}$. The names of the indices does not have any intrinsic meaning. If you want the tensor product, you simply should not call indices in the different tensors the same.

 

[Ibix]

As Orodruin says, the product of a vector and a rank two tensor is a rank three tensor, $T^{\mu\nu\sigma}=g^\mu P^{\nu\sigma}$.

If you are expecting a single index in the output, you need to contract the vector with the tensor. First you need to lower an index, either the one on your vector or the one on the tensor you want to contract over, using the metric $\eta_{\mu\nu}$. Then you can sum over the indices you intend to contract. So $g_\mu=\eta_{\mu\nu}g^\nu$, then your final result (assuming you want to contract with the first index on $P$) is $v^\nu=g_\mu P^{\mu\nu}$.

Note that the metric in flat spacetime is denoted $\eta_{\mu\nu}$ by convention, but in curved spacetime $g_{\mu\nu}$ is usually used. You can't confuse the rank two metric tensor with a vector, but nonetheless $g$ is probably a bad choice of symbol.

 

[Milsomonk]

Thanks for your replies! I am just trying to follow the working in an old thesis. I am expecting a single index result in the form of a force so I think their use of the term tensor product was in fact a little misleading.
Thanks again, this has been very helpful!