diff options
| -rw-r--r-- | README.md | 10 | ||||
| -rw-r--r-- | bondi.mac | 92 | ||||
| -rw-r--r-- | sphere.mac | 65 | ||||
| -rw-r--r-- | tmp.mac | 29 |
4 files changed, 91 insertions, 105 deletions
diff --git a/README.md b/README.md new file mode 100644 index 0000000..6daae89 --- /dev/null +++ b/README.md @@ -0,0 +1,10 @@ +The present source code was written to reproduce the original results from + +"Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system" +1962 - Bondi, Hermann and Van der Burg, M. G. J. and Metzner, A. W. K. + +and is intended to be used with maxima. The scripts require a common lisp +implementation supporting UTF-8 due to the use of greek characters. + +Maxima can be downloaded from: +https://maxima.sourceforge.io/ @@ -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)); @@ -7,8 +7,8 @@ linel: 150; derivabbrev: true; +/** **/ load("ctensor"); - dim: 2; cframe_flag: false; ct_coords: [θ, ϕ]; @@ -24,34 +24,71 @@ lg[2, 2] : sin(θ)^2; cmetric(); /* mcs[i, j, k]: Christoffel symbols of the second kind (one upper index - k) */ -christof(false); +christof(true); +ricci(true); + +/** **/ +load("itensor"); + +imetric(g); +idim(2); +metricconvert: true; + +/** **/ +depends([c, M, N, P], [u, θ]); -depends(c, [u, θ]); +Cdown: zeromatrix(2, 2); +Cdown[1, 1] : 2*c; +Cdown[2, 2] : -2*c*sin(θ)^2; 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); - +f(m, n, T) := sum(ug[m, a] * sum(ug[n, b] * T[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); +for i: 1 thru 2 do for j: 1 thru 2 do Nup[i, j]: f(i, j, Ndown); + +Cup: zeromatrix(2, 2); +for i: 1 thru 2 do for j: 1 thru 2 do Cup[i, j]: f(i, j, Cdown); + +/** Mass-loss formula **/ +mass_aspect_eqn: Mu = (1/4)*covdiff(Nu([],[A,B]), B, A) +-(1/8)*Nd([A,B], [])*Nu([], [A,B]); +Mu: block([Cu,Cd,Nu,Nd,expr], + expr: ic_convert(mass_aspect_eqn), + Cu: Cup, Cd: Cdown, Nu: Nup, Nd: Ndown, + expand(ev(expr))); -X: sum(sum(Nup[A,B]*Ndown[A,B], A, 1, 2), B, 1, 2); +/** Momentum-loss formula **/ +momentum_aspect_eqn: Pu([A], []) = covdiff(M([],[]), A) ++ (1/8)*covdiff(Cu([],[B,C])*Nd([C,B],[]), A) ++ (1/4)*covdiff(covdiff(Cu([], [B, C]), B, A) + - g([],[C,E])*g([],[B,D])*covdiff(Cd([A,B],[]), D, E), C) ++ (1/4)*covdiff(Nu([], [B,C])*Cd([A,C],[]) - Cu([],[B,C])*Nd([A,C],[]), B) +- (1/4)*Nu([], [B,C])*covdiff(Cd([B,C],[]), A); -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; +block([Cu,Cd,Nu,Nd,expr], + expr: ic_convert(eqn), + Cu: Cup, Cd: Cdown, Nu: Nup, Nd: Ndown, + ev(expr), + Pu[1]),expand; +sachs_coeff: -(2/3)*g([],[A,B])*(P([B], []) + Cd([B,C],[])*Ubu([],[C]) + βb([],[],B)); +bondi_coeff: 2*N + 3*c*diff(c,θ) + 4*c^2*cot(θ); -/* M_u */ -- (1/8)*X + (1/4)*Y,expand; +Nu: block([expr, X, Cu, Cd, Nu, Nd, Ubu, βb, N], + expr: ic_convert(sachs_coeff=X([], [A])), + expr: subst([Cu=Cup, Cd=Cdown, Nu=Nup, Nd=Ndown], expr), + Ubu[1]: -(diff(c, θ) + 2*c*cot(θ)), + βb: -(1/4)*c^2, + expand(ev(expr)), + N: rhs(first(solve(X[1]=bondi_coeff, N))), + subst(diff(P[1],u)=Pu[1], diff(-3*N, u))); diff --git a/tmp.mac b/tmp.mac deleted file mode 100644 index 4bd6efe..0000000 --- a/tmp.mac +++ /dev/null @@ -1,29 +0,0 @@ -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]) |