diff options
Diffstat (limited to 'bondi.mac')
| -rw-r--r-- | bondi.mac | 92 |
1 files changed, 30 insertions, 62 deletions
@@ -26,7 +26,7 @@ lg[1, 1] : V * r^-1 * exp(2 * β) - U^2 * r^2 * exp(2 * γ); lg[1, 2] : exp(2 * β); lg[2, 1] : exp(2 * β); -/* dr dθ */ +/* du dθ */ lg[1, 3] : U * r^2 * exp(2 * γ); lg[3, 1] : U * r^2 * exp(2 * γ); @@ -53,7 +53,7 @@ uricci(false); sum(sum(ug[α, ε] * mcs[α, ε, 1], ε, 1, 4), α, 1, 4),expand; /** General utilities **/ -exp_taylor: taylor(exp(x), x, 0, 3); +exp_taylor(x) := subst(t=x, taylor(exp(t), t, 0, 3)); /** Main equations **/ @@ -69,12 +69,10 @@ depends([β1, β2, β3, β4], [u, θ]); depends([U1, U2, U3, U4], [u, θ]); depends([VV1, V0, V1, V2, V3, V4], [u, θ]); -γ_ansatz: γ1*r^-1 + γ2*r^-2 + γ3*r^-3 + γ4*r^-4; -β_ansatz: H + β1*r^-1 + β2*r^-2 + β3*r^-3 + β4*r^-4; -U_ansatz: U1*r^-1 + U2*r^-2 + U3*r^-3 + U4*r^-4; -V_ansatz: VV1*r + V0 + V1*r^-1 + V2*r^-2 + V3*r^-3 + V4*r^-4; -ansatz: [γ=γ_ansatz, β=β_ansatz, U=U_ansatz, V=V_ansatz]; - +ansatz: [γ = γ1*r^-1 + γ2*r^-2 + γ3*r^-3 + γ4*r^-4, + β = H + β1*r^-1 + β2*r^-2 + β3*r^-3 + β4*r^-4, + U = U1*r^-1 + U2*r^-2 + U3*r^-3 + U4*r^-4, + V = VV1*r + V0 + V1*r^-1 + V2*r^-2 + V3*r^-3 + V4*r^-4]; /* Derive β from eq1; Replace in eq1 -> β2 = - γ1^2 / 4, β1 = 0*/ tmp: combine(ev(subst(ansatz, eq1),derivlist(r),expand)); @@ -83,82 +81,52 @@ solve(coeff(tmp, 1/r, 3), β1); /* Solve for β_2 */ solve(coeff(tmp, 1/r, 4), β2); +matchdeclare (nn, lambda ([e], not is(equal(listofvars(e), [H])))); +defrule (r1, %e^nn, (partition(nn, H), exp_taylor(first(%%)) * exp(second(%%)))); + /* Derive U from eq2 */ -sub: cons(exp(2*γ - 2*β)=exp(-2*H)*subst(x=2*(γ - β + H), exp_taylor), ansatz)$ -tmp: combine(ev(subst(sub,eq2),derivlist(r,θ),expand))$ +tmp: combine(ev(apply1(subst(ansatz, eq2), r1),derivlist(r,θ),expand))$ /* Solve for U_1 */ solve(coeff(tmp, r, 1), U1); - - -matchdeclare (nn, lambda ([e], not is(equal(listofvars(e), [H])))); -defrule (r1, %e^nn, (partition(nn, H), subst([x=first(%%)], exp_taylor) * exp(second(%%)))); - -combine(ev(apply1(subst(ansatz, eq2), r1),derivlist(r,θ),expand)); -solve(coeff(%, r, 1), U1); - - /* Derive V from eq2 */ -tmp: subst(ansatz, eq3),derivlist(r,θ),expand$ -ex: part(coeff(tmp, r, 0), 1, 6, 2, 2, allbut(9))$ -sub: [exp(ex + 2*H)=exp(2*H)*subst([x=ex], taylor(exp(x), x, 0, 3)), - exp(- ex - 2*H)=exp(-2*H)*subst([x=-ex], taylor(exp(x), x, 0, 3))]$ -tmp2: combine(expand(subst(sub, tmp)))$ +append(ansatz, [U1=2*diff(H,θ)*exp(2*H)]); +tmp: combine(ev(apply1(subst(%, eq3), r1),derivlist(r,θ),expand))$ /* Solve for V_{-1} */ -solve(trigsimp(ev(subst([U1=2*diff(H,θ)*exp(2*H)], coeff(tmp2, r, 0)),derivlist(θ))), VV1); +solve(coeff(tmp, r, 0), VV1),expand; /* subst(ansatz, eq4),derivlist(r,θ),expand; */ -/* -U_ansatz: 2 * diff(H, θ) * exp(2 * H) / r -V_ansatz: r * exp(2 * H) * (1 + 2 * diff(H, θ) * cot(θ) + 4 * diff(H, θ) ^ 2 + 2 * diff(H, θ, 2)) -*/ - /** Supplementary conditions **/ depends([c, C, N, M], [u, θ]); -γ_ansatz: c*r^-1 + (C - c^3/6)*r^-3; -/* Assumption given H=0, not on paper*/ -β_ansatz: -(1/4)*c^2*r^-2; -U_ansatz: -(diff(c,θ) + 2*c*cot(θ))*r^-2 -+ (2*N + 3*c*diff(c,θ) + 4*c^2*cot(θ))*r^-3 -+ (1/2)*(3*diff(C,θ) + 6*C*cot(θ) - 6*c*N - 8*c^2*diff(c,θ) - 8*c^3*cot(θ))*r^-4; -V_ansatz: r - 2*M -- (diff(N,θ) + N*cot(θ) - diff(c,θ)^2 - 4*c*diff(c,θ)*cot(θ) - c^2*(1 + 8*cot(θ)^2)/2)*r^-1 -- (1/2)*(diff(C,θ,2) + 3*diff(C,θ)*cot(θ) - 2*C + 6*N*(diff(c,θ) + 2*c*cot(θ)) -+ 8*c*(diff(c,θ)^2 + 3*c*diff(c,θ) + 2*c^2*cot(θ)^2))*r^-2; +ansatz: [γ = c*r^-1 + (C - c^3/6)*r^-3, + β = -(1/4)*c^2*r^-2, + U = -(diff(c,θ) + 2*c*cot(θ))*r^-2 + + (2*N + 3*c*diff(c,θ) + 4*c^2*cot(θ))*r^-3 + + (1/2)*(3*diff(C,θ) + 6*C*cot(θ) - 6*c*N - 8*c^2*diff(c,θ) + - 8*c^3*cot(θ))*r^-4, + V = r - 2*M + - (diff(N,θ) + N*cot(θ) - diff(c,θ)^2 - 4*c*diff(c,θ)*cot(θ) + - c^2*(1 + 8*cot(θ)^2)/2)*r^-1 + - (1/2)*(diff(C,θ,2) + 3*diff(C,θ)*cot(θ) - 2*C + + 6*N*(diff(c,θ) + 2*c*cot(θ)) + 8*c*(diff(c,θ)^2 + 3*c*diff(c,θ) + + 2*c^2*cot(θ)^2))*r^-2]; + C_u: (2*c^2 + 2*c*M + N*cot(θ) - diff(N,θ)) / 4; -ansatz: [γ=γ_ansatz, β=β_ansatz, U=U_ansatz, V=V_ansatz]; +matchdeclare (mm, lambda ([e], true)); +defrule (r2, %e^mm, exp_taylor(mm)); /* Mass loss formula, Bondi - Eq. 35 */ tmp: combine(ev(subst(ansatz, ric[1,1]),derivlist(u,r,θ),expand))$ distrib(trigsimp(first(solve(coeff(%, 1/r, 2)=0, diff(M,u))))),expand$ sup_cond1: substpart(trigreduce(part(%, 2, 1)), %, 2, 1); - /* Bondi - Eq. 36 */ tmp: combine(ev(subst(ansatz, ric[1,3]),derivlist(u,r,θ),expand))$ -sup_cond2: distrib(trigsimp(-3*first(solve(coeff(%, 1/r, 2)=0, diff(N,u))))),expand; - -taylor(rhs(sup_cond2), x, 0, 3) - -rhs(sup_cond2) - -ex: exp(- 2*c*r^-1 - c^2*r^-2/2 + c^3*r^-3/3 - 2*C*r^-3) - -ex: exp(2*c*r^-1 + c^2*r^-2/2 - c^3*r^-3/3 + 2*C*r^-3) - -8*c*diff(c,u)*cot(θ) - (3/2)*diff(c, u)*diff(c,θ) * ex + c*diff(diff(c,u),θ)*ex/2 + diff(M, θ)*ex + 5*diff(c,u)*diff(c,θ)/2 + 9*c*diff(diff(c,u),θ)/2; - -ser: subst([x=2*c*r^-1 + c^2*r^-2/2 - c^3*r^-3/3 + 2*C*r^-3], taylor(exp(x), x, 0, 3)); -tmp2: subst([ex=ser], tmp),expand; - -tmp3: combine(tmp2); - -coeff(tmp3, 1/r, 2)=0 - -distrib(trigsimp(-3*first(solve(coeff(%, 1/r, 2)=0, diff(N,u))))),expand; +combine(expand(apply1(tmp, r2)))$ +sup_cond2: -solve(coeff(%, 1/r, 2)=0, 3*diff(N,u)); |