> restart:

> grtw();

> read `/usr/local/MapleVR5/Grtii(5)/Lib/pertutils.mpl`;

> qload(lpschw):

> grcalc(g(up,up));

> gralter(g(up,up),linpert,simplify,factor);

Component simplification of a GRTensorII object:

Applying routine linpert to object g(up,up)

Applying routine simplify to object g(up,up)

Applying routine factor to object g(up,up)

> grdisplay(g(up,up));

> grcalcalter(R(dn,dn),13);

Simplification will be applied during calculation.

Applying routine `Apply constraints repeatedly` to object g(dn,dn,pdn)

Applying routine `Apply constraints repeatedly` to object Chr(dn,dn,dn)

Applying routine `Apply constraints repeatedly` to object Chr(dn,dn,up)

Applying routine `Apply constraints repeatedly` to object R(dn,dn)

> grmap(R(dn,dn),subs,diff(u(theta),theta$2)=
-cos(theta)/sin(theta)*diff(u(theta),theta)-j*u(theta),`x`);

Applying routine subs to R(dn,dn)

> b1:=hcollect(expand(coeff(collect(grcomponent(R(dn,dn),[r,theta]),epsilon),epsilon,1)),
ufuncs,lperts);

> c1:=hcollect(coeff(collect(grcomponent(R(dn,dn),[t,theta]),epsilon),epsilon,1),
ufuncs,lperts);

> d1:=hcollect(expand(coeff(collect(grcomponent(R(dn,dn),[t,r]),epsilon),epsilon,1)),
ufuncs,lperts);

> e1:=hcollect(expand(coeff(collect(grcomponent(R(dn,dn),[r,r]),epsilon),epsilon,1)),
ufuncs,lperts);

> f1:=hcollect(expand(coeff(collect(grcomponent(R(dn,dn),[t,t]),epsilon),epsilon,1)),
ufuncs,lperts);

> a1:=hcollect(coeff(collect(grcomponent(R(dn,dn),[theta,theta]),epsilon),epsilon,1),
ufuncs,lperts);

> g1:=kfactor(hcollect(coeff(collect(grcomponent(R(dn,dn),[phi,phi]),epsilon),epsilon,1)
,ufuncs,lperts),(cos(theta)-1)*(cos(theta)+1));

> hcollect(a1+g1/sin(theta)^2,ufuncs,lperts);

> factor(hcollect(a1-g1/sin(theta)^2,ufuncs,lperts));

> restart:

> grtw();

> read `/usr/local/MapleVR5/Grtii(5)/Lib/pertutils.mpl`;

> qload(qschw);

> grcalc(g(up,up));

> gralter(g(up,up),quadpert,simplify,factor);

Component simplification of a GRTensorII object:

Applying routine quadpert to object g(up,up)

Applying routine simplify to object g(up,up)

Applying routine factor to object g(up,up)

> grdisplay(g(up,up));

> grcalcalter(R(dn,dn),13);

Simplification will be applied during calculation.

Applying routine `Apply constraints repeatedly` to object g(dn,dn,pdn)

Applying routine `Apply constraints repeatedly` to object Chr(dn,dn,dn)

Applying routine `Apply constraints repeatedly` to object Chr(dn,dn,up)

Applying routine `Apply constraints repeatedly` to object R(dn,dn)

> gralter(R(dn,dn),expand);

Component simplification of a GRTensorII object:

Applying routine expand to object R(dn,dn)

> grdisplay(R(dn,dn));

> p2 := 5/2*(3*cos(theta)^2-1)/2;

> newH(r,t) := (2*r^2-6*m*r-3*m^2)/(r-2*m)/(2*r+3*m)*chi(r,t)+r^2/(r-2*m)*eta(r,t);

> Kdot(r,t) := 6*(r^2+m*r+m^2)/r^2/(2*r+3*m)*chi(r,t)+eta(r,t);

> a1:=coeff(collect(grcomponent(R(dn,dn),[theta,theta]),epsilon),epsilon,1):

> g1:=coeff(collect(grcomponent(R(dn,dn),[phi,phi]),epsilon),epsilon,1):

> ag1:=int(expand(a1-g1/sin(theta)^2)*(2*cot(theta)*diff(p2,theta)+2*(2+1)*p2)*sin(theta),
theta=0..Pi);

> leqn[1]:= {h[2](r,t) = solve(ag1,h[2](r,t))};

> a2:=subs(leqn[1],a1):

> g2:=subs(leqn[1],g1):

> grmap(R(dn,dn),subs,leqn[1],'x');

Applying routine subs to R(dn,dn)

> gralter(R(dn,dn),simplify,expand);

Component simplification of a GRTensorII object:

Applying routine simplify to object R(dn,dn)

Applying routine expand to object R(dn,dn)

> b1:=coeff(collect(grcomponent(R(dn,dn),[theta,r]),epsilon),epsilon,1):

> leqn[2]:=termsimp(collect(1/3*int(b1*diff(p2,theta)*sin(theta),theta=0..Pi),lperts));

> c1:=coeff(collect(grcomponent(R(dn,dn),[theta,t]),epsilon),epsilon,1):

> leqn[3]:=termsimp(collect(1/3*int(c1*diff(p2,theta)*sin(theta),theta=0..Pi),lperts));

> d1:=coeff(collect(grcomponent(R(dn,dn),[t,r]),epsilon),epsilon,1):

> leqn[4]:=termsimp(collect(int(d1*p2*sin(theta),theta=0..Pi),lperts));

> e1:=coeff(collect(grcomponent(R(dn,dn),[r,r]),epsilon),epsilon,1):

> leqn[5]:=termsimp(collect(int(e1*p2*sin(theta),theta=0..Pi),lperts));

> f1:=coeff(collect(grcomponent(R(dn,dn),[t,t]),epsilon),epsilon,1):

> leqn[6]:=termsimp(collect(int(f1*p2*sin(theta),theta=0..Pi),lperts));

> leqn[7]:=termsimp(collect(int((a2+g2/sin(theta)^2)*p2*sin(theta),theta=0..Pi),lperts));

> lsub1:={diff(k(r,t),t,t) = termsimp(collect(solve(expand((r-2*m)^2*leqn[5]+r^2*leqn[6]
-leqn[7]/r*(r-2*m)),diff(k(r,t),t,t)),lperts))};

> aa1:=simplify(coeff(collect(grcomponent(R(dn,dn),[theta,theta]),epsilon),epsilon,2),
trig):

> gg1:=simplify(expand(1/sin(theta)^2*(coeff(collect(grcomponent(R(dn,dn),[phi,phi]),
epsilon),epsilon,2))),trig):

> zz1:=simplify(expand(gg1-aa1));

> zz2:=int(zz1*(2*cot(theta)*diff(p2,theta)+2*(2+1)*p2)*sin(theta),theta=0..Pi);

> qeqn1:= {H[2](r,t) = solve(zz2,H[2](r,t))};

> qeqn[1]:=qeqn1;

> grmap(R(dn,dn),subs,qeqn1,`x`);

Applying routine subs to R(dn,dn)

> aa1:=simplify(coeff(collect(grcomponent(R(dn,dn),[theta,theta]),epsilon),epsilon,2),
trig):

> gg1:=simplify(expand(1/sin(theta)^2*(coeff(collect(grcomponent(R(dn,dn),[phi,phi]),
epsilon),epsilon,2))),trig):

> qeqn[2]:=termsimp(collect(int((gg1+aa1)*p2*sin(theta),theta=0..Pi),qperts)):

> bb1:=simplify(coeff(collect(grcomponent(R(dn,dn),[theta,r]),epsilon),epsilon,2),trig):

> qeqn[3]:=termsimp(collect(int(bb1*diff(p2,theta)*sin(theta),theta=0..Pi),qperts)):

> cc1:=simplify(coeff(collect(grcomponent(R(dn,dn),[t,theta]),epsilon),epsilon,2),trig):

> qeqn[4]:=termsimp(collect(1/3*int(cc1*diff(p2,theta)*sin(theta),theta=0..Pi),qperts)):

> dd1:=simplify(coeff(collect(grcomponent(R(dn,dn),[r,t]),epsilon),epsilon,2),trig):

> qeqn[5]:=termsimp(collect(int(dd1*p2*sin(theta),theta=0..Pi),qperts)):

> ee1:=simplify(coeff(collect(grcomponent(R(dn,dn),[r,r]),epsilon),epsilon,2),trig):

> qeqn[6]:=termsimp(collect(expand(int(ee1*p2*sin(theta),theta=0..Pi)),qperts)):

> #ff1:=simplify(coeff(collect(grcomponent(R(dn,dn),[t,t]),epsilon),epsilon,2),trig):

> #ff2:=termsimp(collect(int(ff1*p2*sin(theta),theta=0..Pi),qperts)):

> qsub1:={diff(H[0](r,t),t)=solve(qeqn[4],diff(H[0](r,t),t))};

> qeqn[1];

> undiff(qeqn[4]);

> eta1:=qeqn[5]:

> eta2:=termsimp(collect(subs(qsub1,eta1),qperts)):

> eta3:=termsimp(collect(simplify(subs(H[1](r,t)=newH(r,t),diff(K(r,t),t)=Kdot(r,t),
diff(K(r,t),r,t)=diff(Kdot(r,t),r),eta2)),zfuncs)):

> eta4:=eta(r,t)-termsimp(collect(solve(eta3,eta(r,t)),zfuncs)):

> eta5:=termsimp(collect(subs({diff(h[0](r,t),r,r)=solve(leqn[5],diff(h[0](r,t),r,r))},eta4),
zfuncs)):

> eta6:=termsimp(collect(subs({diff(h[0](r,t),r)=solve(leqn[2],diff(h[0](r,t),r)),
diff(k(r,t),r,t)=solve(leqn[4],diff(k(r,t),r,t))},eta5),zfuncs)):

> ug:={diff(k(r,t),r,r)=termsimp(collect(solve(subs(diff(h[0](r,t),r)=
solve(leqn[2],diff(h[0](r,t),r)),subs(lsub1,solve(leqn[6],diff(h[0](r,t),r,r))-
solve(leqn[5],diff(h[0](r,t),r,r)))),diff(k(r,t),r,r)),lperts))}:

> eta7:=termsimp(collect(subs(ug,eta6),zfuncs)):

> etaeqn:=eta7:

> etasource:=kfactor(mcollect(mcollect(mcollect(mcollect(mcollect(mcollect(mcollect(
mcollect(mcollect(mcollect(expand(eval(subs(eta(r,t)=0,chi(r,t)=0,etaeqn))),
{h[1](r,t),h[0](r,t)}),{h[1](r,t),diff(h[1](r,t),t)}),{h[1](r,t),diff(k(r,t),r)}),
{h[0](r,t),diff(k(r,t),t)}),{diff(h[1](r,t),t),diff(k(r,t),t)}),{k(r,t),h[1](r,t)}),
{k(r,t),diff(k(r,t),t)}),{diff(k(r,t),r),diff(h[0](r,t),t)}),{k(r,t),diff(h[0](r,t),t)}),
{h[0](r,t),diff(h[0](r,t),t)}),(r-2*m)/7/(2*r+3*m)):

> etafinal:={eta(r,t) = termsimp(collect(solve(eval(subs(h[0](r,t)=0,h[1](r,t)=0,k(r,t)=0,
etaeqn)),eta(r,t)),zfuncs))-etasource};

> atmp1:=diff(qeqn[2],t):

> atmp2:=termsimp(collect(expand(subs(qsub1,diff(qsub1,r),atmp1)),{diff(K(r,t),t,t,t)})):

> atmp3:=termsimp(collect(expand(eval(subs(H[1](r,t)=newH(r,t),
diff(K(r,t),t)=Kdot(r,t),diff(K(r,t),t,r)=diff(Kdot(r,t),r),diff(K(r,t),t,r,r)=diff(Kdot(r,t),r,r),atmp2))),zfuncs)):

> atmp4:=termsimp(collect(expand(eval(subs(etafinal,atmp3))),zfuncs)):

> coefa:=simplify(coeff(atmp4,diff(chi(r,t),t,r,t),1));

> btmp1:=termsimp(collect(expand(subs(qsub1,diff(qsub1,r),diff(qeqn[3],t))),
{diff(H[1](r,t),t,t),diff(K(r,t),t,t,t)})):

> btmp2:=termsimp(collect(expand(eval(subs(H[1](r,t)=newH(r,t),diff(K(r,t),t)=Kdot(r,t),
diff(K(r,t),r,t)=diff(Kdot(r,t),r),btmp1))),zfuncs)):

> btmp3:=termsimp(collect(expand(eval(subs(etafinal,btmp2))),zfuncs)):

> coefb:=simplify(coeff(btmp3,diff(chi(r,t),r,t,t),1));

> ftmp1:=termsimp(collect(expand(atmp4/coefa-btmp3/coefb),{chi(r,t),diff(chi(r,t),r),
diff(chi(r,t),t),diff(chi(r,t),t,t),diff(chi(r,t),r,r),diff(chi(r,t),r,t),diff(chi(r,t),r,t,t),
diff(chi(r,t),r,r,r)})):

> ftmp2:=termsimp(collect(expand(eval(subs(k(r,t)=0,h[1](r,t)=0,h[0](r,t)=0,ftmp1))),
{chi(r,t),diff(chi(r,t),r),diff(chi(r,t),r,r),diff(chi(r,t),t,t),diff(chi(r,t),r,r,r)})):

> chi1:=termsimp(collect(diff(chi(r,t),t,t)-solve(ftmp1,diff(chi(r,t),t,t)),zfuncs)):

> chieqn:=ftmp1:

> nops(chieqn);

> termsimp(collect(expand(eval(subs(k(r,t)=0,h[1](r,t)=0,h[0](r,t)=0,
chieqn/coeff(chieqn,diff(chi(r,t),t,t),1)))),{chi(r,t),diff(chi(r,t),r),
diff(chi(r,t),r,r),diff(chi(r,t),t,t),diff(chi(r,t),r,r,r)}));

> zetasub:= {chi(r,t) = zeta(r,t)+2/7*(r^2/(2*r+3*m)*k(r,t)*diff(k(r,t),t)+k(r,t)^2)};

> zeta1:=eval(subs(zetasub,chi1)):

> nops(expand(zeta1));

> new2:=termsimp(collect(expand(leqn[4]-leqn[3]/2/m*3),lperts));

> leqn8:={diff(h[1](r,t),r) = termsimp(collect(solve(new2,diff(h[1](r,t),r)),lperts))};

> new3:=termsimp(collect(expand(solve(leqn[3],diff(h[1](r,t),r))-solve(diff(epxand(
leqn[4]/coeff(leqn[4],h[1](r,t),1)),r),diff(h[1](r,t),r))),lperts));

> leqn9:={h[1](r,t) = termsimp(collect(solve(new3,h[1](r,t)),lperts))};

> leqn10:={diff(h[1](r,t),t)=termsimp(collect(expand(solve(leqn[2],diff(h[1](r,t),t))),lperts))};

> aux1:=termsimp(collect(expand(solve(leqn[2],diff(h[1](r,t),t))-solve(diff(leqn[4],t),
diff(h[1](r,t),t))),lperts));

> aux2:=termsimp(collect(expand(diff((r-2*m)*r*leqn[4]/3,r))-leqn[3],lperts));

> aux3:=termsimp(collect(expand(solve(leqn[2],diff(h[1](r,t),t))-solve(leqn[7],
diff(h[1](r,t),t))),lperts));

> tmpaux:=termsimp(collect(expand(solve(leqn[5],diff(h[1](r,t),r,t))-solve(leqn[6],
diff(h[1](r,t),r,t))),lperts));

> tmpaux2:=termsimp(collect(expand(solve(tmpaux,diff(h[1](r,t),t))-solve(leqn[2],
diff(h[1](r,t),t))),lperts));

> tmpaux3:=termsimp(collect(expand(solve(tmpaux2,diff(k(r,t),t,t))-solve(leqn[7],
diff(k(r,t),t,t))),lperts));

> aux4:=termsimp(collect(expand(solve(tmpaux3,diff(h[1](r,t),t))-solve(leqn[2],
diff(h[1](r,t),t))),lperts));

> aux5:=termsimp(collect(solve(solve(diff(leqn[3],t),diff(h[1](r,t),r,t))-solve(leqn[5],
diff(h[1](r,t),r,t)),diff(h[1](r,t),t))-solve(leqn[2],diff(h[1](r,t),t)),lperts));

> aux6:=termsimp(collect(expand(solve(aux5,diff(k(r,t),t,t))-solve(aux3,diff(k(r,t),t,t))),
lperts));

> psisub:={h[0](r,t) = (psi(r,t) - r/3*k(r,t))*3*(2*r+3*m)/r/(r-2*m) +r*diff(k(r,t),r)};

> aux7:=termsimp(collect(expand(subs(psisub,aux4)),lperts union {psi(r,t),
diff(psi(r,t),r)}));

> zsource1:=expand(simplify(subs(leqn8,zeta1))):

> nops(zsource1);

> zsource2:=expand(simplify(eval(subs(leqn10,zsource1)))):

> nops(zsource2);

> zsource3:=expand(simplify(subs(leqn9,zsource2))):

> nops(zsource3);

> zsource3a:=expand(simplify(eval(subs(diff(k(r,t),t,t,r)=solve(aux1,diff(k(r,t),t,t,r)),
zsource3)))):

> zsource4:=expand(simplify(eval(subs(diff(k(r,t),t,t)=solve(aux3,diff(k(r,t),t,t)),
zsource3a)))):

> zsource5:=expand(simplify(eval(subs(diff(k(r,t),r,r,t)=solve(aux2,diff(k(r,t),r,r,t)),
zsource4)))):

> zsource6:=expand(simplify(eval(subs(diff(h[0](r,t),t,t)=solve(aux6,diff(h[0](r,t),t,t)),
zsource5)))):

> nops(zsource6);

> zsource7:=expand(simplify(eval(subs(psisub,zsource6)))):

> nops(zsource7);

> zsource8:=expand(simplify(eval(subs(k(r,t)=solve(aux7,k(r,t)),zsource7)))):

> nops(zsource8);

> zetaeqn:=zsource8:

> s_ren := make_s():

> sprime:=make_s_mu():

> psisub;

> nops(zetaeqn);

> final1:=mcollect(expand(zetaeqn),{psi(r,t),diff(psi(r,t),r,r)}):

> final2:=mcollect(final1,{psi(r,t),diff(psi(r,t),t)}):

> final3:=mcollect(final2,{diff(psi(r,t),t),diff(psi(r,t),r,r,r)}):

> final4:=mcollect(final3,{diff(psi(r,t),r),diff(psi(r,t),t)}):

> final5:=mcollect(final4,{diff(psi(r,t),t),diff(psi(r,t),t,r)}):

> final6:=mcollect(final5,{psi(r,t),diff(psi(r,t),r)}):

> final7:=mcollect(final6,{diff(psi(r,t),r),diff(psi(r,t),t,r,r)}):

> final8:=mcollect(final7,{diff(psi(r,t),t),diff(psi(r,t),r,r)}):

> final9:=mcollect(final8,{psi(r,t),diff(psi(r,t),r,t)}):

> final10:=mcollect(final9,{diff(psi(r,t),r),diff(psi(r,t),r,t)}):

> final11:=mcollect(final10,{psi(r,t),diff(psi(r,t),t,r,r)}):

> final12:=mcollect(final11,{diff(psi(r,t),r),diff(psi(r,t),r,r)}):

> final13:=mcollect(final12,{psi(r,t)}):

> final14:=mcollect(final13,{diff(psi(r,t),t)}):

> final15:=mcollect(final14,{diff(psi(r,t),r)}):

> final16:=mcollect(final15,{diff(psi(r,t),r,r)}):

> final17:=mcollect(final16,{diff(psi(r,t),r,t)}):

> nops(final17);

> zetasource:=subs((r-2*m)=mu,(2*m-r)=-mu,(2*r+3*m)=lambda,
kfactor(eval(subs(zeta(r,t)=0,-final17)),12/7*(r-2*m)^3/(2*r+3*m)));

> final18:=subs((r-2*m)=mu,(-r+2*m)=-mu,(2*r+3*m)=lambda,final17):

> final19:=kfactor(collect(final18,{zeta(r,t),diff(zeta(r,t),r),diff(zeta(r,t),r,r),
diff(zeta(r,t),t,t)}),12/7*mu^3/lambda):

> simplify(expand(final19-final18));

> termsimp(simplify(expand(subs(zeta(r,t)=0,lambda=(2*r+3*m),mu=(r-2*m),
final19)+s_ren)));

> termsimp(simplify(expand(subs(zeta(r,t)=0,lambda=(2*r+3*m),mu=(r-2*m),final19)
+sprime)));

> kernelopts(cputime);