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load("setup.mac")$
load("tensors.mac")$
load("series.mac")$
/* Equations */
/* Stress-energy tensor of the theory */
T_ab : zeromatrix(4,4)$
/* Einstein tensor */
for i : 1 thru 4 do for j : 1 thru 4 do (
G_ab[i,j] : expand(ric[i,j] - ric_s*lg[i,j]/2));
/* 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]));
/* 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])));
/* solve(uE_ab[1,2]=0, diff(β,r))[1], */
/* Hypersurface equations */
hs_eqs : [
/* Determine β */
uE_ab[1,2]=0,
/* Determine A */
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 : [
/* Determine γ */
E_ab[3,3]=0,
/* Should be fulfilled automatically (?) */
E_ab[4,4]=0
/* Fulfilled trivially */
/* E_ab[3,4]=0 */
/* E_ab[4,3]=0 */
]$
/* Suplementary conditions */
sc_eqs : [
uE_ab[2,1]=0,
uE_ab[2,3]=0
/* Fulfilled trivially */
/* uE_ab[2,4]=0 */
]$
hs_eqs : trigreduce(hs_eqs)$
/* Series ansatz */
expansion_order : 6;
/* Sequence elements inherit dependance on u and θ */
depends([c, H, M, N, γc, βc, Ac, Bc], [u, θ])$
/* Main series ansatz */
build_main_ansatz(n) := block([], [
γ = c/r + neg_pow_series(γc, r, 2, n),
β = H + neg_pow_series(βc, r, 1, n),
A = neg_pow_series(Ac, r, 1, n),
B = neg_pow_series(Bc, r, -1, n)])$
main_ansatz : build_main_ansatz(expansion_order)$
funcs : [A, B, β, γ, β_expr, γ_expr];
/**/
/* Simplify ahead to speed up calculation */
γβ_facts : [H=0, γc[2]=0];
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])$
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))$
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])$
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))$
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])$
B_coeffs : block(
[vals : []],
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))$
coeffs : append(coeffs, B_coeffs)$
/* Equation 4 : determine γ (evolution equation) */
/*
We wish M,N,C to be related to monopole, dipole and quadrupole moments
respectively, also when converted from the static case. Thus, Ac[3] as to be of
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)$
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 : 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(expand(exp(2*β)*sc_eqs[1]), coeffs)$
eq_mono : block(
[eq : coeff(e_sc_eqs[1], r, -2)=0],
eq : solve(eq, diff(M,u))[1],
trigreduce(expand(trigsimp(eq))));
/* Supplementary Condition 2 */
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],
eq : -3*solve(eq, diff(N,u))[1],
trigreduce(expand(trigsimp(eq))));
/* 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))
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