General Relativity/The Tensor Product

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.

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