How to calculate the strength tensor of a vector field?

Questions
1

Dear Wolfram team:

I try to calculate :

T_{ab} = g_{ab} F_{gd} F^{gd} - F_a^g F_{bg},

where

F_{ab} = \partial_a A_b - \partial_b A_a,

So I define F_{ab} by:

s[a_, b_] := D[v[b], a] -  D[v[a], b]

But how to define F^{ab} and F_a^b ?

g_{ab} is given by:

metric =  DiagonalMatrix[{-1, 1, 1, 1}];

m[a_, b_] := metric[[a, b]];