Four-vector

A four-vector is a four element vector that is a rank 1 class tensor in that its element notation expression carries 1 index and it transforms according to a tensor transformation. For example a four vector expressed as a contravariant rank 1 tensor $T^\mu$ transforms according to

$T'^{\mu}=\frac{\partial x' ^{\mu}}{\partial x^{\nu}}T^{\nu}$

and the four vector expressed as a covariant rank 1 tensor $T_{\mu}$ transforms according to

$T'_{\mu}=\frac{\partial x^{\nu}}{\partial x' ^{\mu}}T_{\nu}$

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