z1c[x_,y_,z_,η_]:=-α x/η;
z2c[x_,y_,z_,η_]:=-α y/η;
z3c[x_,y_,z_,η_]:=-α z/η;
z4c[x_,y_,z_,η_]:= -α^2/η;

z1s[x_,y_,z_,t_]:=Exp[t/α]x;
z2s[x_,y_,z_,t_]:=Exp[t/α]y;
z3s[x_,y_,z_,t_]:=Exp[t/α]z;
z4s[x_,y_,z_,t_]:=-α*Exp[t/α];

T1[A_,B_]:=Solve[{z1c[A[[1]],A[[2]],A[[3]],A[[4]]]==z1s[B[[1]],B[[2]],B[[3]],B[[4]]],z2c[A[[1]],A[[2]],A[[3]],A[[4]]]==z2s[B[[1]],B[[2]],B[[3]],B[[4]]],z3c[A[[1]],A[[2]],A[[3]],A[[4]]]==z3s[B[[1]],B[[2]],B[[3]],B[[4]]],z4c[A[[1]],A[[2]],A[[3]],A[[4]]]==z4s[B[[1]],B[[2]],B[[3]],B[[4]]]},A][[1]]

c1={xc1,yc1,zc1,η1};
c2={xc2,yc2,zc2,η2};
s1={x1,y1,z1,t1};
s2={x2,y2,z2,t2};

Vs[A_,B_]:=A/.T1[A,B]

V1=Vs[c1,s1]//Simplify
V2=Vs[c2,s2]//Simplify

transrule={c1[[1]]-> V1[[1]],c1[[2]]-> V1[[2]],c1[[3]]-> V1[[3]],c1[[4]]-> V1[[4]],c2[[1]]-> V2[[1]],c2[[2]]-> V2[[2]],c2[[3]]-> V2[[3]],c2[[4]]-> V2[[4]]};

Js[A_,B_]:=Table[D[Vs[A,B][[i]],B[[j]]],{i,4},{j,4}]

J1=Js[c1,s1]//Simplify
J2=Js[c2,s2]//Simplify

J1//Simplify//MatrixForm

gc1={{α^2/η1^2,0,0,0},{0,α^2/η1^2,0,0},{0,0,α^2/η1^2,0},{0,0,0,-α^2/η1^2}};

gc2={{α^2/η2^2,0,0,0},{0,α^2/η2^2,0,0},{0,0,α^2/η2^2,0},{0,0,0,-α^2/η2^2}};

gs1=Table[Sum[gc1[[a1,a2]]*J1[[a1,b1]]*J1[[a2,b2]],{a1,4},{a2,4}]/.transrule,{b1,4},{b2,4}]//Simplify;
gs2=Table[Sum[gc2[[a1,a2]]*J2[[a1,b1]]*J2[[a2,b2]],{a1,4},{a2,4}]/.transrule,{b1,4},{b2,4}]//Simplify;

gs2//Simplify//MatrixForm

Ntttt=Exp[-4*t1/α]*Exp[-4*t2/α]/(4Pi^4*α^8(Exp[-t1/α]-Exp[-t2/α])^8)

Series[Ntttt/.t2->0,{t1,0,-4}]//Normal