diff options
| -rw-r--r-- | 01-bondi.mac | 84 |
1 files changed, 41 insertions, 43 deletions
diff --git a/01-bondi.mac b/01-bondi.mac index 9240820..e9133ab 100644 --- a/01-bondi.mac +++ b/01-bondi.mac @@ -14,11 +14,11 @@ for i : 1 thru 4 do for j : 1 thru 4 do ( /* Einstein equations tensor */ for i : 1 thru 4 do for j : 1 thru 4 do ( - E_ab[i,j] : expand(G_ab[i,j] - 8*%pi*T_ab[i,j]))$ + E_ab[i,j] : expand(G_ab[i,j] - 8*%pi*T_ab[i,j])); /* Einstein equations tensor with mixed indices */ for i : 1 thru 4 do for j : 1 thru 4 do ( - uE_ab[i,j] : expand(contr(ug[i,k]*E_ab[k,j], [k])))$ + uE_ab[i,j] : expand(contr(ug[i,k]*E_ab[k,j], [k]))); /* solve(uE_ab[1,2]=0, diff(β,r))[1], */ @@ -31,7 +31,7 @@ uE_ab[1,3]=0, /* uE_ab[1,4]=0 is trivial, would determine A' otherwise */ /* Determine B */ uE_ab[1,1]=0 -]; +]$ /* Evolution equations */ ev_eqs : [ @@ -42,7 +42,7 @@ E_ab[4,4]=0 /* Fulfilled trivially */ /* E_ab[3,4]=0 */ /* E_ab[4,3]=0 */ -]; +]$ /* Suplementary conditions */ sc_eqs : [ @@ -50,9 +50,9 @@ uE_ab[2,1]=0, uE_ab[2,3]=0 /* Fulfilled trivially */ /* uE_ab[2,4]=0 */ -]; +]$ -hs_eqs : trigreduce(hs_eqs); +hs_eqs : trigreduce(hs_eqs)$ /* Series ansatz */ @@ -62,9 +62,6 @@ expansion_order : 6; /* Sequence elements inherit dependance on u and θ */ depends([c, H, M, N, γc, βc, Ac, Bc], [u, θ])$ -/* Simplify ahead to speed up calculation */ -/* γβ_facts : [H=0, γc[2]=0]$ */ - /* Main series ansatz */ build_main_ansatz(n) := block([], [ γ = c/r + neg_pow_series(γc, r, 2, n), @@ -85,38 +82,41 @@ funcs : [A, B, β, γ, β_expr, γ_expr]; coeffs : γβ_facts; /* Equation 1 : determine β */ -e_hs_eqs[1] : expand_equation(solve(hs_eqs[1], diff(β, r))[1], coeffs); -ce_hs_eqs[1] : coeff_equations(e_hs_eqs[1]); +e_hs_eqs[1] : expand_equation(solve(hs_eqs[1], diff(β, r))[1], coeffs)$ +ce_hs_eqs[1] : coeff_equations(e_hs_eqs[1])$ + β_coeffs : block([len, lst], len : length(ce_hs_eqs[1]), lst : makelist(solve(ce_hs_eqs[1][i][2], βc[i]), i, 1, len), - apply(append, lst)); + apply(append, lst))$ -coeffs : append(coeffs, β_coeffs); +coeffs : append(coeffs, β_coeffs)$ /* Equation 2 : determine A */ -e_hs_eqs[2] : expand_equation(expand(exp(2*β)*hs_eqs[2]), coeffs); -ce_hs_eqs[2] : coeff_equations(e_hs_eqs[2]); +e_hs_eqs[2] : expand_equation(expand(exp(2*β)*hs_eqs[2]), coeffs)$ +ce_hs_eqs[2] : coeff_equations(e_hs_eqs[2])$ + A_coeffs : block( [vals : []], for i : 1 thru expansion_order do ( if not(i=3) then ( vals : append(vals, solve(subst(vals, ce_hs_eqs[2][i][2]), Ac[i])))), - ev(vals, eval, diff, expand)); + ev(vals, eval, diff, expand))$ coeffs : append(coeffs, A_coeffs)$ /* Equation 3 : determine B */ -e_hs_eqs[3] : expand_equation(hs_eqs[3], coeffs); -ce_hs_eqs[3] : coeff_equations(e_hs_eqs[3]); +e_hs_eqs[3] : expand_equation(hs_eqs[3], coeffs)$ +ce_hs_eqs[3] : coeff_equations(e_hs_eqs[3])$ + B_coeffs : block( [vals : []], - for i : 1 thru expansion_order do ( + for i : 1 thru expansion_order-1 do ( if i=2 then ( vals : append(vals, [Bc[0]=-2*M])) else ( vals : append(vals, solve(subst(vals, ce_hs_eqs[3][i][2]), Bc[i-2])))), - ev(vals, eval, diff, expand)); + ev(vals, eval, diff, expand))$ coeffs : append(coeffs, B_coeffs)$ @@ -129,42 +129,40 @@ following form. */ depends(C, [u, θ]); extra_coeffs : [γc[3]=C-c^3/6, Ac[3]=2*N + 3*c*diff(c,θ) + 4*c^2*cot(θ)]$ -coeffs : append(coeffs, extra_coeffs); +coeffs : append(coeffs, extra_coeffs)$ -e_ev_eqs[2] : expand_equation(ev_eqs[2], coeffs); -ce_ev_eqs[2] : coeff_equations(e_ev_eqs[2]); +e_ev_eqs[2] : expand_equation(expand(exp(2*γ)*exp(2*β)*ev_eqs[2]), coeffs)$ +ce_ev_eqs[2] : coeff_equations(e_ev_eqs[2])$ -eq_quad : expand(4*solve(ce_ev_eqs[2][1][2], diff(C,u))[1]); - -/* Convoluted way to show that the results for C_u are consistent */ -e_ev_eqs[1] : expand_equation(ev_eqs[1], coeffs); -ce_ev_eqs[1] : coeff_equations(e_ev_eqs[1]); - -eq_quad_alt : trigreduce(expand(trigsimp(4*solve(ce_ev_eqs[2][2][2], diff(C,u))[1]))); -gammac4 : trigreduce(expand(trigsimp(trigreduce(expand(solve(ce_ev_eqs[1][2][2], diff(γc[4],u))))[1]))); -eq_quad_alt_res : expand(subst(gammac4, eq_quad_alt)) +eq_quad : block( + [eq : 4*solve(ce_ev_eqs[2][1][2], diff(C,u))[1]], + trigreduce(expand(trigsimp(eq)))); /* Supplementary conditions */ /* Supplementary Condition 1 */ -e_sc_eqs[1] : expand_equation(sc_eqs[1], coeffs); +e_sc_eqs[1] : expand_equation(expand(exp(2*β)*sc_eqs[1]), coeffs)$ + eq_mono : block( [eq : coeff(e_sc_eqs[1], r, -2)=0], - trigreduce(expand(solve(eq, diff(M,u))))); + eq : solve(eq, diff(M,u))[1], + trigreduce(expand(trigsimp(eq)))); /* Supplementary Condition 2 */ -e_sc_eqs[2] : expand_equation(sc_eqs[2], coeffs); +e_sc_eqs[2] : expand_equation(expand(exp(2*β)*sc_eqs[2]), coeffs)$ + eq_dipo : block( [eq : coeff(e_sc_eqs[2], r, -2)=0], - trigreduce(expand(-3*solve(eq, diff(N,u))))); - + eq : -3*solve(eq, diff(N,u))[1], + trigreduce(expand(trigsimp(eq)))); -/* BUG: should not generate terms with r^-3 and so on ... */ -/* Check if bug is related to expansion of powers */ -block([expansion_order : 2], - expand_equation(sc_eqs[1], coeffs)) +/* Convoluted way to show that the results for C_u are consistent */ -block([expansion_order : 2], - expand_equation(sc_eqs[2], coeffs)) +e_ev_eqs[1] : expand_equation(ev_eqs[1], coeffs); +ce_ev_eqs[1] : coeff_equations(e_ev_eqs[1]); + +eq_quad_alt : trigreduce(expand(trigsimp(4*solve(ce_ev_eqs[2][2][2], diff(C,u))[1]))); +gammac4 : trigreduce(expand(trigsimp(trigreduce(expand(solve(ce_ev_eqs[1][2][2], diff(γc[4],u))))[1]))); +eq_quad_alt_res : expand(subst(gammac4, eq_quad_alt)) |