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

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

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

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