From the use of the Minkowski metric, converting between contravariant and covariant
Where we have already defined the covariant term,
and the contravariant term
From quantum mechanics we know that partial differential is a linear operator. Following the rules of matrix multiplication this implies that the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector, and the derivative with respect to a covariant coordinate transforms as a contravariant vector.
Since it is an operator, the dot product of two partial differentials yields an operator known as the D'Alembert operator.