# General Relativity/The Tensor Product

Jump to navigation Jump to search

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

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...