r[τ_]:=Sqrt[(((c T[τ]))^2)-((x[τ])^2)] ArcTanh[x[τ]/T[τ]/c];
t[τ_]:=Sqrt[((T[τ])^2)-(((x[τ]/c))^2)];

"dr"==FullSimplify[(r'[τ])/.x[τ]->x/.T[τ]->T/.x'[τ]->dx/.T'[τ]->dT]
"dt"==FullSimplify[(t'[τ])/.x[τ]->x/.T[τ]->T/.x'[τ]->dx/.T'[τ]->dT]

Quit[]

x[τ_]:=c t[τ] Sinh[r[τ]/t[τ]/c];
T[τ_]:=t[τ] Cosh[r[τ]/t[τ]/c];

"dx"==FullSimplify[(x'[τ])/.r[τ]->r/.t[τ]->t/.r'[τ]->dr/.t'[τ]->dt]
"dT"==FullSimplify[(T'[τ])/.r[τ]->r/.t[τ]->t/.r'[τ]->dr/.t'[τ]->dt]

Quit[]

d1={dT,dx};
d2={dt,dr};
n=2; H0=1;

g11=gtt=(c^2);
g12=gtr=0;
g22=grr=-1;

m1={{g11,g12},{g12,g22}};

"Milne, Minkowski Koordinaten"
M1=MatrixForm[m1];
Subscript["g",μσ]->M1

dT=dt Cosh[(r/(c t))]+(((-dt r+dr t) Sinh[(r/(c t))])/(c t));
dx=(dr-((dt r)/t)) Cosh[(r/(c t))]+c dt Sinh[(r/(c t))];

m2=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}];

"Milne, FLRW Koordinaten"
M2=MatrixForm[FullSimplify[((m2)/.T->t Cosh[r/t/c]/.x->c t Sinh[r/t/c]/.t->1/H), c>0&&H>0&&r>0]];
Subscript["g",μσ]->M2

Quit[]

d1={dt,dr};
d2={dT,dx};
n=2; H0=1; 

g11=gtt=(c^2)-(H^2)(r^2);
g12=gtr=H r;
g22=grr=-1;

m1={{g11,g12},{g12,g22}};

"Milne, FLRW Koordinaten"
M1=MatrixForm[m1];
Subscript["g",μσ]->M1

dt=(((c^2) dT T-dx x)/((c^2) (Sqrt[(T^2)-((x^2)/(c^2))])));
dr=((c (dx T-dT x)+((c^2) dT T-dx x) ArcCoth[((c T)/x)])/(Sqrt[(c T-x) (c T+x)]));

m2=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}];

"Milne, Minkowski Koordinaten"
M2=MatrixForm[FullSimplify[((m2)/.H->1/t/.t->(Sqrt[(T^2)-((x^2)/(c^2))])/.r->(Sqrt[(c^2) (T^2)-(x^2)]) ArcTanh[(x/(c T))]), c>0&&T>0&&x>0]];
Subscript["g",μσ]->M2

Quit[]