The formula gives ##T^{\mu nu}##. Isn't that just like saying ##T^{\mu \nu}## are components, rather than a rank - 2 tensor? Why can't ##T^{ij}## denote the 3d matrix running over the indices 1,2,3, rather than the case ##\mu =1,2,3,4##? what else should I use to write this then?
So you're...