General Relativity/The Tensor Product

<General Relativity

If $\mathbf {T}$ and $\mathbf {S}$ are tensors of rank $n$ and $m$, then there exists a tensor $\mathbf {T} \otimes \mathbf {S}$ of rank $n+m$. The components of the new tensor (pronounced "T tensor S") are obtained by multiplying the components of the old tensors. In other words, if $\mathbf {T} =T_{\ \beta }^{\alpha }$ and $\mathbf {S} =S_{\mu \nu },$then $\mathbf {T} \otimes \mathbf {S} =T_{\ \beta }^{\alpha }S_{\mu \nu }$.

For example, if T and S are two contravariant, one-rank tensors, then their tensor product is a two-rank, contravariant tensor.

More to come...