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 $$\mathbf{T} = T^{\alpha}_{\ \beta} $$ and $$\mathbf{S}=S_{\mu \nu}, $$then $$\mathbf{T} \otimes \mathbf{S} = T^{\alpha}_{\ \beta} S_{\mu \nu}$$.

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...