diff options
Diffstat (limited to 'sphere.mac')
| -rw-r--r-- | sphere.mac | 65 |
1 files changed, 51 insertions, 14 deletions
@@ -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))); |