Reference: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Problem 12.55.
The gradient of a scalar function is a covariant vector since it transforms as
We can therefore regard the gradient operator on its own as a covariant vector, so it should have a contravariant counterpart. In flat space, the only change in switching from covariant to contravariant is that the time component changes sign. Given that the Lorentz transformation for a contravariant four-vector is
the transformations for the covariant four-vector are obtained by lowering all indices and replacing the time components by their negatives:
where we multiplied the equation through by . The corresponding inverse transformations are obtained by replacing by :
Thus the inverse covariant transformations are the same as the forward contravariant transformations.
The contravariant gradient is so
The transformations for each value of are then
Thus transforms like a contravariant vector.