diff options
| -rw-r--r-- | bondi.mac | 159 | ||||
| -rw-r--r-- | sphere.mac | 57 | ||||
| -rw-r--r-- | tmp.mac | 29 |
3 files changed, 229 insertions, 16 deletions
@@ -1,37 +1,164 @@ -debugmode: true; - +/* 1920x1080 - HD screen */ +linel: 230; +/* 3840x1600 - Ultrawide screen */ +linel: 470; +/* 1366x768 - Poor man's thinkpad */ linel: 150; +/* 3840x1600 - Ultrawide screen with work on side */ +linel: 350; + +derivabbrev: true; + load("ctensor"); dim: 4; cframe_flag: false; -ct_coords: [u, r, theta, phi]; +ct_coords: [u, r, θ, ϕ]; -depends([U, V, _beta, _gamma], [u, r, theta]); +depends([U, V, β, γ], [u, r, θ]); lg : zeromatrix(4, 4); -lg[1, 1] : V * r^-1 * exp(2 * _beta) - U^2 * r^2 * exp(2 * _gamma); +/* du du */ +lg[1, 1] : V * r^-1 * exp(2 * β) - U^2 * r^2 * exp(2 * γ); -lg[1, 2] : exp(2 * _beta); -lg[2, 1] : exp(2 * _beta); +/* du dr */ +lg[1, 2] : exp(2 * β); +lg[2, 1] : exp(2 * β); -lg[1, 3] : U * r^2 * exp(2 * _gamma); -lg[3, 1] : U * r^2 * exp(2 * _gamma); +/* dr dθ */ +lg[1, 3] : U * r^2 * exp(2 * γ); +lg[3, 1] : U * r^2 * exp(2 * γ); -lg[3, 3] : - r^2 * exp(2 * _gamma); -lg[4, 4] : - r^2 * exp(- 2 * _gamma) * sin(theta)^2; +/* dθ dθ */ +lg[3, 3] : - r^2 * exp(2 * γ); + +/* dϕ dϕ */ +lg[4, 4] : - r^2 * exp(- 2 * γ) * sin(θ)^2; /* Load metric and compute inverse */ cmetric(); -/* For some reason some components must be simplified by hand ... */ +/* Force simplification of some expressions */ ug[2,2] : expand(ug[2,2]),simp; +/* mcs[i, j, k]: Christoffel symbols of the second kind (one upper index - k) */ +christof(false); +/* ric[i, j]: R_ab Ricci tensor*/ +ricci(false); +/* uric[i, j]: R^a_b Ricci tensor */ +uricci(false); + +/* Bondi - Eq. 17 */ +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); + +/** Main equations **/ + +/* Main equations, Bondi - Eqs. 22 - 25 */ +eq1 : ric[2, 2] = 0, expand; +eq2 : - 2 * r^2 * ric[2, 3] = 0, expand; +eq3 : ric[3, 3] * exp(2 * (β - γ)) - r^2 * uric[4, 4] * exp(2 * β) = 0, expand; +eq4 : - uric[4, 4] * exp(2 * β) * r^2 = 0, expand; + +depends([H], [u, θ]); +depends([γ1, γ2, γ3, γ4], [u, θ]); +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]; + + +/* Derive β from eq1; Replace in eq1 -> β2 = - γ1^2 / 4, β1 = 0*/ +tmp: combine(ev(subst(ansatz, eq1),derivlist(r),expand)); +/* Solve for β_1 */ +solve(coeff(tmp, 1/r, 3), β1); +/* Solve for β_2 */ +solve(coeff(tmp, 1/r, 4), β2); + +/* 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))$ +/* 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)))$ +/* Solve for V_{-1} */ +solve(trigsimp(ev(subst([U1=2*diff(H,θ)*exp(2*H)], coeff(tmp2, r, 0)),derivlist(θ))), VV1); + + +/* subst(ansatz, eq4),derivlist(r,θ),expand; */ + /* - Calculate Christoffel symbols - mcs[i, j, k]: Christoffel symbols of the second kind (one upper index - k) +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)) */ -christof(false); -ricci(false); + +/** 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; +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]; + +/* 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; diff --git a/sphere.mac b/sphere.mac new file mode 100644 index 0000000..b58cd57 --- /dev/null +++ b/sphere.mac @@ -0,0 +1,57 @@ +/* 1920x1080 - HD screen */ +linel: 230; +/* 3840x1600 - Ultrawide screen */ +linel: 470; +/* 1366x768 - Poor man's thinkpad */ +linel: 150; + +derivabbrev: true; + +load("ctensor"); + +dim: 2; +cframe_flag: false; +ct_coords: [θ, ϕ]; + +lg : zeromatrix(2, 2); +/* dθ dθ */ +lg[1, 1] : 1; + +/* dϕ dϕ */ +lg[2, 2] : sin(θ)^2; + +/* Load metric and compute inverse */ +cmetric(); + +/* mcs[i, j, k]: Christoffel symbols of the second kind (one upper index - k) */ +christof(false); + + +depends(c, [u, θ]); + +Ndown: zeromatrix(2, 2); +Ndown[1, 1] : 2*diff(c, u); +Ndown[2, 2] : -2*diff(c, u)*sin(θ)^2; + +f(m, n) := sum(ug[m, a] * sum(ug[n, b] * Ndown[a, b], b, 1, 2), a, 1, 2); + + +Nup: zeromatrix(2, 2); +for i: 1 thru 2 do for j: 1 thru 2 do Nup[i, j]: f(i, j); + + + + + +X: sum(sum(Nup[A,B]*Ndown[A,B], A, 1, 2), B, 1, 2); + +Q: zeromatrix(2, 1); +for i: 1 thru 2 do Q[i]: sum(diff(Nup[i,B], θ) + + sum(mcs[B, D, i] * Nup[D, B], D, 1, 2) + + sum(mcs[B, D, B] * Nup[i, D], D, 1, 2), B, 1, 2); + +Y: diff(Q[1], θ) + sum(sum(mcs[A, B, A], A, 1, 2) * Q[B], B, 1, 2),expand; + + +/* M_u */ +- (1/8)*X + (1/4)*Y,expand; @@ -0,0 +1,29 @@ +part('(Ndown[a, b]), ); +raise(tensor, ) + + +args('(Ndown[a, b])); + + +raise(tensor, index) := sum(ug[''index, m] * ''tensor, m, 1, 2); + +raise('(Ndown[a, b]), 'a); +--> sum(ug[]) + +:lisp (format t "~A" #$'(Ndown[a,b])$) + +:lisp (mfuncall '$mnctimes $ug $ug) + +:lisp (displa (mfuncall 'mnctimes $ug $ug)); + +:lisp #$mcs[1,1,1]$; + +mcs[1,1,1] + +ug . ug; + +:lisp (format t "~A" #$c . d$) + +'Ndown[a, b] + +('Ndown[a,b]) |