SetAttributes[ϵ,Constant];
SetAttributes[M,Constant];
SetAttributes[κ,Constant];

t:=dt[t1-t2];
r:=dr[r1-r2];
dt'[x_]:=1;
dr'[x_]:=1;
u:=U[r1,r2,η];
s:=S[t,r1,r2,η];
v:=V[r1,r2,η];

Derivative[1,0,0,0][S][t_,r1_,r2_,η_]:=-t;
Derivative[0,m_,n_,o_][S][t_,r1_,r2_,η_]:=Derivative[m,n,o][V][r1,r2,η];

G1=u/s;
G2=κ*s/v*Sinh[κ*v]/(Cosh[κ*v]-Cosh[κ*t]);

Format[dt[t1-t2]]:="t";
Format[dr[r1-r2]]:="r";
Format[U[r1,r2,η]]:="u";
Format[S[t,r1,r2,η]]:="s";
Format[V[r1,r2,η]]:="v";

g1rule={g1[s,u]:>G1,D[g1[s,u],{s,m_},{u,n_}]:>D[G1,{s,m},{u,n}]}

g2rule={g2[s,v,t]:>G2,Derivative[m_,n_,o_][g2][s,v,t]:>D[G2,{s,m},{v,n},{t,o}]}

G=g1[s,u]*g2[s,v,t];

box[A_]:=-D[A,{t1,2}]+b[r1]^2*D[A,{r1,2}]+2*b[r1]^2/r1*D[A,r1]-2*(η+1)*b[r1]/r1^2*D[A,η]-(2η+η^2)*b[r1]/r1^2*D[A,{η,2}];

box2[A_]:=-D[A,{t2,2}]+b[r2]^2*D[A,{r2,2}]+2*b[r2]^2/r2*D[A,r2]-2*(η+1)*b[r2]/r2^2*D[A,η]-(2η+η^2)*b[r2]/r2^2*D[A,{η,2}];

dAdB[A_,B_]:=-D[A,t1]*D[B,t1]+b[r1]^2*D[A,r1]*D[B,r1]-(2*η+η^2)*b[r1]/r1^2*D[A,η]*D[B,η];

dAdB2[A_,B_]:=-D[A,t2]*D[B,t2]+b[r2]^2*D[A,r2]*D[B,r2]-(2*η+η^2)*b[r2]/r2^2*D[A,η]*D[B,η];

ddAddB[A_,B_]:=-dAdB2[D[A,t1],D[B,t1]]+b[r1]^2*dAdB2[D[A,r1],D[B,r1]]-(2*η+η^2)*b[r1]/r1^2*dAdB2[D[A,η],D[B,η]];

vrule={((Derivative[1,0,0][V][r1,r2,η])^2)->(1+(((2 η+(η^2)) b[r1] ((Derivative[0,0,1][V][r1,r2,η])^2))/(r1^2)))/((b[r1])^2)};

urule={u:> 1+ϵ};

crule={κ*v:>c1,κ*t:>c2};

Temp1=Expand[box[G]-R[r1]*G/6/.g1rule/.g2rule]/.vrule/.urule/.crule//Expand

Temp2=Sum[Series[Temp1[[i]],{v,0,-2}]//Normal,{i,Dimensions[Temp1][[1]]}]//Expand

arule={Cosh[c1]-Cosh[c2]:> Sinh[c1]/A}

Temp3=Temp2+A*κ/v^2*(2*dAdB[u,v]+u*box[v]-2*u/v)/.urule/.arule//Expand