d1={dt,dr,dθ,dф};
d2={dT,dr,dθ,dφ};
n=4;

σ=(a^2)+(r^2);
v=-(Sqrt[2 r-(℧^2)])/(Sqrt[σ]);
dt=dT+((v dr)/(1-(v^2))) ;
dф=dφ+((a v dr)/(σ(1-(v^2))));

g11=gtt=-((-(a^2)+2 r-(r^2)-(℧^2)+(a^2) ((Sin[θ])^2))/((r^2)+(a^2) ((Cos[θ])^2)));
g22=grr=-(((r^2)+(a^2) ((Cos[θ])^2))/((a^2)-2 r+(r^2)+(℧^2)));
g33=gθθ=-(r^2)-(a^2) ((Cos[θ])^2);
g44=gφφ=-((((((a^2)+(r^2)))^2) ((Sin[θ])^2)-(a^2) ((a^2)-2 r+(r^2)+(℧^2)) ((Sin[θ])^4))/((r^2)+(a^2) ((Cos[θ])^2)));
g14=gtφ=-((a (-2 r+(℧^2)) ((Sin[θ])^2))/((r^2)+(a^2) ((Cos[θ])^2)));
g12=g13=g23=g24=g34=0;

m1={
{g11,g12,g13,g14},
{g12,g22,g23,g24},
{g13,g23,g33,g34},
{g14,g24,g34,g44}};

M1=MatrixForm[m1];
m2=FullSimplify[Table[Sum[D[d1[[k]],d2[[i]]] D[d1[[s]],d2[[j]]] m1[[k,s]],
{k,1,n},{s,1,n}],{i,1,n},{j,1,n}]];
M2=MatrixForm[m2];
M1->M2

A1={((r ℧)/((r^2)+(a^2) ((Cos[θ])^2))),0,0,-((a r ℧ ((Sin[θ])^2))/((r^2)+(a^2) ((Cos[θ])^2)))};
A2=FullSimplify[Table[Sum[D[d1[[k]],d2[[i]]] A1[[k]],
{k,1,n}],{i,1,n}]];
A1->A2