Appendix: Explicit expressions in the coordinate transformations

From Bondi gauge to Fefferman-Graham gauge

(* We give the explicit expressions of the functions involved in the diffeomorphism between Tortoise coordinates and Fefferman-Graham coordinates, up to order 2 in ρ *)

R1[t,ϕ]= (L^2);
R2[t,ϕ] = -(E^(-2 β0[t,ϕ])) (L^4) (Derivative[0,1][U0U][t,ϕ]+U0U[t,ϕ] Derivative[0,1][φ][t,ϕ]+Derivative[1,0][φ][t,ϕ]);
R3[t,ϕ]=(1/12) (L^4) (-4-3 (E^(-2 β0[t,ϕ])) M[t,ϕ]+3 (E^(-4 β0[t,ϕ])) (L^2) (3 ((Derivative[0,1][U0U][t,ϕ])^2)+((U0U[t,ϕ])^2) (-4 Derivative[0,1][β0][t,ϕ] Derivative[0,1][φ][t,ϕ]+3 ((Derivative[0,1][φ][t,ϕ])^2)+2 Derivative[0,2][φ][t,ϕ])+2 Derivative[0,1][φ][t,ϕ] Derivative[1,0][U0U][t,ϕ]+Derivative[1,0][φ][t,ϕ] (-4 Derivative[1,0][β0][t,ϕ]+3 Derivative[1,0][φ][t,ϕ])+Derivative[0,1][U0U][t,ϕ] (-4 U0U[t,ϕ] (Derivative[0,1][β0][t,ϕ]-2 Derivative[0,1][φ][t,ϕ])-4 Derivative[1,0][β0][t,ϕ]+6 Derivative[1,0][φ][t,ϕ])+2 U0U[t,ϕ] (Derivative[0,2][U0U][t,ϕ]-2 Derivative[0,1][β0][t,ϕ] Derivative[1,0][φ][t,ϕ]+Derivative[0,1][φ][t,ϕ] (-2 Derivative[1,0][β0][t,ϕ]+3 Derivative[1,0][φ][t,ϕ])+2 Derivative[1,1][φ][t,ϕ])+2 (Derivative[1,1][U0U][t,ϕ]+Derivative[2,0][φ][t,ϕ])));

T1[t,ϕ]=(1-(E^(-2 β0[t,ϕ]))) (L^2);
T2[t,ϕ]= -(E^(-4 β0[t,ϕ])) (L^4) ((E^(2 β0[t,ϕ])) Derivative[0,1][U0U][t,ϕ]+U0U[t,ϕ] (Derivative[0,1][β0][t,ϕ]+(E^(2 β0[t,ϕ])) Derivative[0,1][φ][t,ϕ])+Derivative[1,0][β0][t,ϕ]+(E^(2 β0[t,ϕ])) Derivative[1,0][φ][t,ϕ]);
T3[t,ϕ]=(1/12) (E^(-2 (3 β0[t,ϕ]+φ[t,ϕ]))) (L^4) (-4 (E^(2 (β0[t,ϕ]+φ[t,ϕ]))) M[t,ϕ]+8 (E^(2 φ[t,ϕ])) (L^2) ((Derivative[0,1][U0U][t,ϕ])^2)+8 ((E^(4 β0[t,ϕ]))-2 (E^(2 φ[t,ϕ])) (L^2) ((U0U[t,ϕ])^2)) ((Derivative[0,1][β0][t,ϕ])^2)-4 (E^(2 φ[t,ϕ])) (L^2) Derivative[0,1][U0U][t,ϕ] (U0U[t,ϕ] (Derivative[0,1][β0][t,ϕ]-5 Derivative[0,1][φ][t,ϕ])+2 (Derivative[1,0][β0][t,ϕ]-2 Derivative[1,0][φ][t,ϕ]))-4 (E^(2 φ[t,ϕ])) (L^2) Derivative[0,1][β0][t,ϕ] (2 ((U0U[t,ϕ])^2) Derivative[0,1][φ][t,ϕ]-Derivative[1,0][U0U][t,ϕ]+2 U0U[t,ϕ] (4 Derivative[1,0][β0][t,ϕ]+Derivative[1,0][φ][t,ϕ]))+4 (E^(2 φ[t,ϕ])) (-(E^(4 β0[t,ϕ]))+(L^2) ((U0U[t,ϕ])^2) (2 ((Derivative[0,1][φ][t,ϕ])^2)+Derivative[0,2][β0][t,ϕ]+Derivative[0,2][φ][t,ϕ])+(L^2) Derivative[0,1][φ][t,ϕ] Derivative[1,0][U0U][t,ϕ]-4 (L^2) ((Derivative[1,0][β0][t,ϕ])^2)-2 (L^2) Derivative[1,0][β0][t,ϕ] Derivative[1,0][φ][t,ϕ]+2 (L^2) ((Derivative[1,0][φ][t,ϕ])^2)+(L^2) Derivative[1,1][U0U][t,ϕ]+(L^2) U0U[t,ϕ] (Derivative[0,2][U0U][t,ϕ]+2 (-Derivative[0,1][φ][t,ϕ] (Derivative[1,0][β0][t,ϕ]-2 Derivative[1,0][φ][t,ϕ])+Derivative[1,1][β0][t,ϕ]+Derivative[1,1][φ][t,ϕ]))+(L^2) Derivative[2,0][β0][t,ϕ]+(L^2) Derivative[2,0][φ][t,ϕ])+(E^(2 φ[t,ϕ])) (-1+(E^(2 β0[t,ϕ]))) (-4 (E^(4 β0[t,ϕ]))-3 (E^(2 β0[t,ϕ])) M[t,ϕ]+9 (L^2) ((Derivative[0,1][U0U][t,ϕ])^2)-12 (L^2) ((U0U[t,ϕ])^2) Derivative[0,1][β0][t,ϕ] Derivative[0,1][φ][t,ϕ]+9 (L^2) ((U0U[t,ϕ])^2) ((Derivative[0,1][φ][t,ϕ])^2)+6 (L^2) U0U[t,ϕ] Derivative[0,2][U0U][t,ϕ]+6 (L^2) ((U0U[t,ϕ])^2) Derivative[0,2][φ][t,ϕ]+6 (L^2) Derivative[0,1][φ][t,ϕ] Derivative[1,0][U0U][t,ϕ]-12 (L^2) U0U[t,ϕ] Derivative[0,1][φ][t,ϕ] Derivative[1,0][β0][t,ϕ]-6 (L^2) Derivative[0,1][U0U][t,ϕ] (2 U0U[t,ϕ] (Derivative[0,1][β0][t,ϕ]-2 Derivative[0,1][φ][t,ϕ])+2 Derivative[1,0][β0][t,ϕ]-3 Derivative[1,0][φ][t,ϕ])-12 (L^2) U0U[t,ϕ] Derivative[0,1][β0][t,ϕ] Derivative[1,0][φ][t,ϕ]+18 (L^2) U0U[t,ϕ] Derivative[0,1][φ][t,ϕ] Derivative[1,0][φ][t,ϕ]-12 (L^2) Derivative[1,0][β0][t,ϕ] Derivative[1,0][φ][t,ϕ]+9 (L^2) ((Derivative[1,0][φ][t,ϕ])^2)+6 (L^2) Derivative[1,1][U0U][t,ϕ]+12 (L^2) U0U[t,ϕ] Derivative[1,1][φ][t,ϕ]+6 (L^2) Derivative[2,0][φ][t,ϕ]));

Z1[t,ϕ]=-(E^(-2 β0[t,ϕ])) (L^2) U0U[t,ϕ];
Z2[t,ϕ]=(1/2) (L^2) (2 (E^(-2 φ[t,ϕ])) Derivative[0,1][β0][t,ϕ]-2 (E^(-4 β0[t,ϕ])) (L^2) ((U0U[t,ϕ])^2) Derivative[0,1][β0][t,ϕ]+(E^(-4 β0[t,ϕ])) (L^2) Derivative[1,0][U0U][t,ϕ]+(E^(-4 β0[t,ϕ])) (L^2) U0U[t,ϕ] (Derivative[0,1][U0U][t,ϕ]-2 Derivative[1,0][β0][t,ϕ]));
Z3[t,ϕ]= (1/12) (E^(-2 (3 β0[t,ϕ]+φ[t,ϕ]))) (4 (E^(6 β0[t,ϕ])) (L^2) NU[t,ϕ]-(L^4) ((E^(2 (β0[t,ϕ]+φ[t,ϕ]))) M[t,ϕ] U0U[t,ϕ]+(E^(2 φ[t,ϕ])) (L^2) ((U0U[t,ϕ])^3) (16 ((Derivative[0,1][β0][t,ϕ])^2)-4 Derivative[0,1][β0][t,ϕ] Derivative[0,1][φ][t,ϕ]+((Derivative[0,1][φ][t,ϕ])^2)-4 Derivative[0,2][β0][t,ϕ]+2 Derivative[0,2][φ][t,ϕ])-2 (E^(2 φ[t,ϕ])) (L^2) ((U0U[t,ϕ])^2) (2 Derivative[0,1][U0U][t,ϕ] (5 Derivative[0,1][β0][t,ϕ]-Derivative[0,1][φ][t,ϕ])-2 Derivative[0,2][U0U][t,ϕ]-16 Derivative[0,1][β0][t,ϕ] Derivative[1,0][β0][t,ϕ]+2 Derivative[0,1][φ][t,ϕ] Derivative[1,0][β0][t,ϕ]+2 Derivative[0,1][β0][t,ϕ] Derivative[1,0][φ][t,ϕ]-Derivative[0,1][φ][t,ϕ] Derivative[1,0][φ][t,ϕ]+4 Derivative[1,1][β0][t,ϕ]-2 Derivative[1,1][φ][t,ϕ])+2 (-2 (E^(4 β0[t,ϕ])) Derivative[0,2][U0U][t,ϕ]+Derivative[0,1][U0U][t,ϕ] (10 (E^(4 β0[t,ϕ])) Derivative[0,1][β0][t,ϕ]-2 (E^(4 β0[t,ϕ])) Derivative[0,1][φ][t,ϕ]+(E^(2 φ[t,ϕ])) (L^2) Derivative[1,0][U0U][t,ϕ])-6 (E^(2 φ[t,ϕ])) (L^2) Derivative[1,0][U0U][t,ϕ] Derivative[1,0][β0][t,ϕ]+4 (E^(4 β0[t,ϕ])) Derivative[1,1][β0][t,ϕ]-2 (E^(4 β0[t,ϕ])) Derivative[1,1][φ][t,ϕ]+(E^(2 φ[t,ϕ])) (L^2) Derivative[2,0][U0U][t,ϕ])+U0U[t,ϕ] (3 (E^(2 φ[t,ϕ])) (L^2) ((Derivative[0,1][U0U][t,ϕ])^2)-8 (E^(4 β0[t,ϕ])) ((Derivative[0,1][β0][t,ϕ])^2)+8 (E^(4 β0[t,ϕ])) Derivative[0,2][β0][t,ϕ]-4 (E^(4 β0[t,ϕ])) Derivative[0,2][φ][t,ϕ]-16 (E^(2 φ[t,ϕ])) (L^2) Derivative[0,1][β0][t,ϕ] Derivative[1,0][U0U][t,ϕ]+2 (E^(2 φ[t,ϕ])) (L^2) Derivative[0,1][φ][t,ϕ] Derivative[1,0][U0U][t,ϕ]+16 (E^(2 φ[t,ϕ])) (L^2) ((Derivative[1,0][β0][t,ϕ])^2)-2 (E^(2 φ[t,ϕ])) (L^2) Derivative[0,1][U0U][t,ϕ] (8 Derivative[1,0][β0][t,ϕ]-Derivative[1,0][φ][t,ϕ])-4 (E^(2 φ[t,ϕ])) (L^2) Derivative[1,0][β0][t,ϕ] Derivative[1,0][φ][t,ϕ]+(E^(2 φ[t,ϕ])) (L^2) ((Derivative[1,0][φ][t,ϕ])^2)+6 (E^(2 φ[t,ϕ])) (L^2) Derivative[1,1][U0U][t,ϕ]-4 (E^(2 φ[t,ϕ])) (L^2) Derivative[2,0][β0][t,ϕ]+2 (E^(2 φ[t,ϕ])) (L^2) Derivative[2,0][φ][t,ϕ])));

From Bondi gauge to derivative expansion

(* We give the explicit expressions of the functions involved in the diffeomorphism between Bondi coordinates and derivative expansion coordinates, up to order 1 in ρ *)

R2[t,ψ]= -(1/(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]))(E^(-2 (β0[t,ψ]+φ[t,ψ]))) (L^2) (((L^2) ((uψ[t,ψ])^2)-(E^(2 φ[t,ψ])) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]))) Derivative[0,1][U0U][t,ψ]+(E^(2 β0[t,ψ])) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][uψ][t,ψ]+2 (E^(2 β0[t,ψ])) uψ[t,ψ] (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][β0][t,ψ]-(E^(2 β0[t,ψ])) uψ[t,ψ] (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ]+U0U[t,ψ] ((L^2) uψ[t,ψ] Derivative[0,1][uψ][t,ψ]-(E^(2 φ[t,ψ])) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])) Derivative[0,1][φ][t,ψ])+(L^2) uψ[t,ψ] Derivative[1,0][uψ][t,ψ]+(E^(2 φ[t,ψ])) Derivative[1,0][φ][t,ψ]-(E^(2 φ[t,ψ])) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,0][φ][t,ψ]);

T1[t,ψ] = (((L^2) ((E^(2 β0[t,ψ]))+(L^2) ut[t,ψ]-(E^(2 φ[t,ψ])) U0U[t,ψ] Z1[t,ψ]))/((E^(4 β0[t,ψ]))-(E^(2 φ[t,ψ])) (L^2) ((U0U[t,ψ])^2)));
T2[t,ψ] = -(1/2) (E^(-2 (2 β0[t,ψ]+φ[t,ψ]))) (L^2) (2 (E^(2 (β0[t,ψ]+φ[t,ψ]))) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])) W1[t,ψ]+(L^2) (((L^2) ((uψ[t,ψ])^2)-2 (E^(2 φ[t,ψ])) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]))) Derivative[0,1][U0U][t,ψ]-4 (E^(2 β0[t,ψ])) uψ[t,ψ] Derivative[0,1][β0][t,ψ]+4 (E^(2 β0[t,ψ])) uψ[t,ψ] (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][β0][t,ψ]-U0U[t,ψ] (-(L^2) ((uψ[t,ψ])^2)+2 (E^(2 φ[t,ψ])) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]))) (2 Derivative[0,1][β0][t,ψ]+Derivative[0,1][φ][t,ψ])+4 (E^(2 φ[t,ψ])) Derivative[1,0][β0][t,ψ]+2 (L^2) ((uψ[t,ψ])^2) Derivative[1,0][β0][t,ψ]-4 (E^(2 φ[t,ψ])) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,0][β0][t,ψ]+2 (E^(2 φ[t,ψ])) Derivative[1,0][φ][t,ψ]+(L^2) ((uψ[t,ψ])^2) Derivative[1,0][φ][t,ψ]-2 (E^(2 φ[t,ψ])) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,0][φ][t,ψ]));

Z1[t,ψ]=(E^(-4 β0[t,ψ]-2 φ[t,ψ])) (L^2) ((E^(2 β0[t,ψ]+2 φ[t,ψ])) U0U[t,ψ]+(E^(2 φ[t,ψ])) (L^2) U0U[t,ψ] ut[t,ψ]-(E^(4 β0[t,ψ])) uψ[t,ψ]+(E^(2 φ[t,ψ])) (L^2) ((U0U[t,ψ])^2) uψ[t,ψ]);
Z2[t,ψ]= -(1/2) (E^(-4 (β0[t,ψ]+φ[t,ψ]))) (L^2) (-(E^(2 φ[t,ψ])) (L^2) ((U0U[t,ψ])^2) (-(L^2) ((uψ[t,ψ])^2)+2 (E^(2 φ[t,ψ])) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]))) (2 Derivative[0,1][β0][t,ψ]+Derivative[0,1][φ][t,ψ])+2 (E^(4 φ[t,ψ])) (L^2) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])) Derivative[1,0][U0U][t,ψ]+(L^2) ((uψ[t,ψ])^2) (2 (E^(4 β0[t,ψ])) Derivative[0,1][β0][t,ψ]+(E^(4 β0[t,ψ])) Derivative[0,1][φ][t,ψ]-(E^(2 φ[t,ψ])) (L^2) Derivative[1,0][U0U][t,ψ])+2 (E^(2 (β0[t,ψ]+φ[t,ψ]))) uψ[t,ψ] ((E^(2 β0[t,ψ])) W1[t,ψ]+(L^2) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])) Derivative[1,0][φ][t,ψ])+(E^(2 φ[t,ψ])) U0U[t,ψ] (2 (E^(2 (β0[t,ψ]+φ[t,ψ]))) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])) W1[t,ψ]+(L^2) (2 (E^(2 β0[t,ψ])) uψ[t,ψ] (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])) (2 Derivative[0,1][β0][t,ψ]+Derivative[0,1][φ][t,ψ])+(L^2) ((uψ[t,ψ])^2) (2 Derivative[1,0][β0][t,ψ]+Derivative[1,0][φ][t,ψ])-2 (E^(2 φ[t,ψ])) (-1+(Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])) (2 Derivative[1,0][β0][t,ψ]+Derivative[1,0][φ][t,ψ]))));

Expression of χ and ϵ in terms of Bondi functions

(* We give the explicit expressions of the functions χ and ϵ involved in the solution space of the derivative expansion in terms of the functions of the Bondi solution space under the boundary gauge fixing U0 = 0 = β0 *)

χ[t,ψ]=((E^(-5 φ[t,ψ])) ((E^(2 φ[t,ψ])) (L^2) M[t,ψ] uψ[t,ψ] ((((E^(2 φ[t,ψ]))+(L^2) ((uψ[t,ψ])^2)))^2)+(E^(2 φ[t,ψ])) NU[t,ψ] (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) ((E^(4 φ[t,ψ]))+3 (E^(2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)+2 (L^4) ((uψ[t,ψ])^4))+(L^2) (-(E^(4 φ[t,ψ])) Derivative[0,2][uψ][t,ψ]+Derivative[0,1][uψ][t,ψ] (3 ((((E^(2 φ[t,ψ]))+(L^2) ((uψ[t,ψ])^2)))^2) Derivative[0,1][φ][t,ψ]+(E^(2 φ[t,ψ])) (L^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) (-(E^(2 φ[t,ψ])) Derivative[1,0][uψ][t,ψ]+uψ[t,ψ] (3 (E^(2 φ[t,ψ]))+2 (L^2) ((uψ[t,ψ])^2)) Derivative[1,0][φ][t,ψ]))-(E^(4 φ[t,ψ])) uψ[t,ψ] (2 ((Derivative[0,1][φ][t,ψ])^2)-Derivative[0,2][φ][t,ψ]-3 (L^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][uψ][t,ψ]+(L^4) ((Derivative[1,0][uψ][t,ψ])^2)+2 (L^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][uψ][t,ψ])+(L^4) ((uψ[t,ψ])^4) (-Derivative[0,2][uψ][t,ψ]+(E^(2 φ[t,ψ])) (-2 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][φ][t,ψ]+(L^2) Derivative[1,0][uψ][t,ψ] Derivative[1,0][φ][t,ψ]+3 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][φ][t,ψ]-(L^2) Derivative[2,0][uψ][t,ψ]))+(E^(2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2) (-2 Derivative[0,2][uψ][t,ψ]+(E^(2 φ[t,ψ])) (-3 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][φ][t,ψ]+3 (L^2) Derivative[1,0][uψ][t,ψ] Derivative[1,0][φ][t,ψ]+3 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][φ][t,ψ]-(L^2) Derivative[2,0][uψ][t,ψ]))+(L^4) ((uψ[t,ψ])^5) (-2 ((Derivative[0,1][φ][t,ψ])^2)+Derivative[0,2][φ][t,ψ]+(E^(2 φ[t,ψ])) (L^2) Derivative[2,0][φ][t,ψ])-(E^(2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^3) (4 ((Derivative[0,1][φ][t,ψ])^2)-2 Derivative[0,2][φ][t,ψ]-2 (L^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][uψ][t,ψ]+(L^2) ((E^(2 φ[t,ψ])) ((Derivative[1,0][φ][t,ψ])^2)+2 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][uψ][t,ψ]-(E^(2 φ[t,ψ])) Derivative[2,0][φ][t,ψ])))))/(4 G L π ((E^(2 φ[t,ψ]))+(L^2) ((uψ[t,ψ])^2)) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]));

ϵ[t,ψ] = (1/(8 G π))(E^(-4 φ[t,ψ])) ((E^(2 φ[t,ψ])) M[t,ψ] ((E^(2 φ[t,ψ]))+2 (L^2) ((uψ[t,ψ])^2))+4 (E^(2 φ[t,ψ])) NU[t,ψ] uψ[t,ψ] (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)])+(1/((((E^(2 φ[t,ψ]))+(L^2) ((uψ[t,ψ])^2)))^2))(L^2) (-(E^(2 φ[t,ψ])) ((E^(2 φ[t,ψ]))+(L^2) ((uψ[t,ψ])^2)) ((Derivative[0,1][uψ][t,ψ])^2)+2 Derivative[0,1][uψ][t,ψ] (7 (E^(2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^3) Derivative[0,1][φ][t,ψ]+3 (L^4) ((uψ[t,ψ])^5) Derivative[0,1][φ][t,ψ]+(E^(4 φ[t,ψ])) uψ[t,ψ] (4 Derivative[0,1][φ][t,ψ]-(L^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,0][uψ][t,ψ])+(E^(6 φ[t,ψ])) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,0][φ][t,ψ]+4 (E^(4 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,0][φ][t,ψ]+2 (E^(2 φ[t,ψ])) (L^4) ((uψ[t,ψ])^4) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,0][φ][t,ψ])+(E^(6 φ[t,ψ])) (2 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][uψ][t,ψ]-(L^2) ((Derivative[1,0][uψ][t,ψ])^2)-2 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][uψ][t,ψ])+2 (E^(2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^3) (-2 Derivative[0,2][uψ][t,ψ]+(E^(2 φ[t,ψ])) (-4 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][φ][t,ψ]+3 (L^2) Derivative[1,0][uψ][t,ψ] Derivative[1,0][φ][t,ψ]+5 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][φ][t,ψ]-2 (L^2) Derivative[2,0][uψ][t,ψ]))+2 (L^4) ((uψ[t,ψ])^5) (-Derivative[0,2][uψ][t,ψ]+(E^(2 φ[t,ψ])) (-2 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][φ][t,ψ]+(L^2) Derivative[1,0][uψ][t,ψ] Derivative[1,0][φ][t,ψ]+3 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][φ][t,ψ]-(L^2) Derivative[2,0][uψ][t,ψ]))-2 (E^(4 φ[t,ψ])) uψ[t,ψ] (Derivative[0,2][uψ][t,ψ]+(E^(2 φ[t,ψ])) ((Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][φ][t,ψ]-2 (L^2) Derivative[1,0][uψ][t,ψ] Derivative[1,0][φ][t,ψ]-2 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][φ][t,ψ]+(L^2) Derivative[2,0][uψ][t,ψ]))+2 (L^4) ((uψ[t,ψ])^6) (-2 ((Derivative[0,1][φ][t,ψ])^2)+Derivative[0,2][φ][t,ψ]+(E^(2 φ[t,ψ])) (L^2) Derivative[2,0][φ][t,ψ])-(E^(2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^4) (9 ((Derivative[0,1][φ][t,ψ])^2)-4 Derivative[0,2][φ][t,ψ]-4 (L^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][uψ][t,ψ]+(L^2) ((E^(2 φ[t,ψ])) ((Derivative[1,0][φ][t,ψ])^2)+4 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][uψ][t,ψ]-4 (E^(2 φ[t,ψ])) Derivative[2,0][φ][t,ψ]))-(E^(4 φ[t,ψ])) ((uψ[t,ψ])^2) (5 ((Derivative[0,1][φ][t,ψ])^2)-2 Derivative[0,2][φ][t,ψ]-8 (L^2) (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[0,1][φ][t,ψ] Derivative[1,0][uψ][t,ψ]+(L^2) ((L^2) ((Derivative[1,0][uψ][t,ψ])^2)+(E^(2 φ[t,ψ])) ((Derivative[1,0][φ][t,ψ])^2)+6 (Sqrt[1+(E^(-2 φ[t,ψ])) (L^2) ((uψ[t,ψ])^2)]) Derivative[1,1][uψ][t,ψ]-2 (E^(2 φ[t,ψ])) Derivative[2,0][φ][t,ψ]))));