The 4D Gradient Operator \bgroup\color{black}$ {\partial\over \partial x_\mu}$\egroup

This also transforms like a vector for us. For example the derivative of a scalar function \bgroup\color{black}$ f$\egroup, \bgroup\color{black}$ {\partial f\over\partial x_\mu}$\egroup is a 4-vector. We do need to be careful about the sign on the time derivative. In GR, we keep track of the signs using the metric tensor \bgroup\color{black}$ g_{\mu\nu}$\egroup but we do not wish to introduce this complication here.

We choose to use the \bgroup\color{black}$ ict$\egroup crutch to keep the calculations simple for now. If \bgroup\color{black}$ x_0=ict$\egroup, one can see that \bgroup\color{black}$ {\partial\ \over\partial x_0}$\egroup introduces a sign change due to the \bgroup\color{black}$ i$\egroup in the denominator. (In GR... vectors can have upper or lower indices and we use the metric tensor to raise and lower them.)

Jim Branson 2012-10-21