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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw
():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.70~(R5)G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,31~May~1998G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayl
l~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-1998~by~th
e~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\oLatest~version~avai
lable~from:~http://astro.queensu.ca/|irgrtensor/G" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 54 "read `/usr/local/MapleVR5/Grtii(5)/Lib/pertu
tils.mpl`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "qload(qschw):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
2Default~spacetimeG%&qschwG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~t
he~qschw~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7&%\"rG%&thetaG%$phiG%\"tG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",,*&*(,&\"\"\"F-*&,&*&%(epsilonGF-
-&%\"hG6#F'6$%\"rG%\"tGF-F-*&)F1F'F(-&%\"HGF5F6F-F-F-,&*$)-%$cosG6#%&t
hetaGF'F(\"\"$!\"\"F-F-#F-F'F-%#~dGF-)F7%#2~GF-F(,&F-F-*&%\"mGF(F7!\"
\"!\"#FNF-*.,&*&F1F(-&F46#F-F6F-F-*&F:F(-&F=FUF6F-F-F-F>F(FHF()F7%\"~G
F-%#d~GF-)F8FZF-F-**)F7F'F(,&F-F-*&,&*&F1F(-%\"kGF6F-F-*&F:F(-%\"KGF6F
-F-F-F>F(FGF-FHF()FDFJF-F-*,FhnF()-%$sinGFCF'F(FinF(FHF()%$phiGFJF-F-*
*FKF-,&F-F-*&,&*&F1F(-&F46#\"\"!F6F-F-*&F:F(-&F=F`pF6F-F-F-F>F(#FFF'F-
FHF()F8FJF-FF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,ConstraintsG7*/*$)
%(epsilonG\"\"$\"\"\"\"\"!/*$)F)\"\"%F+F,/*$)F)\"\"&F+F,/*$)F)\"\"'F+F
,/*$)F)\"\"(F+F,/*$)F)\"\")F+F,/*$)F)\"\"*F+F,/*$)F)\"#5F+F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 295 "As can be seen from the form of t
he metric, the lower case functions are the first order perturbations,
 and the upper case functions are the second order perturbations. To b
egin our analysis we again calculate the exact contravariant metric te
nsor, which is then quadratically perturbed using ``" }{TEXT 258 10 "q
uadpert()" }{TEXT -1 5 "''.\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "
grcalc(g(up,up)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"
$H%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "gralter(g(up,up)
,quadpert,simplify,factor);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component \+
simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 "
 " }}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine quadpert to object \+
g(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine simplify to \+
object g(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine facto
r to object g(up,up)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$
\"$0$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "grdisplay(g(up
,up)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~qschw~spacetime:G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<Contravariant~metric~tensorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#upGF&" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/*&)%#g~G%\"rG\"\"\")%!GF'F(,$*&*&,&F'!\"\"%\"mG\"\"#F(
,8!\"%F(*(-&%\"hG6#F16$F'%\"tGF(%(epsilonGF()-%$cosG6#%&thetaGF1\"\"\"
\"\"'*&F;FAF5FA!\"#*()F;F1FA)F5F1FA)F=\"\"%FA!\"**&FFFAFGFAF/*(FFFAFGF
AF<FAFB*&FFFA-&%\"HGF8F9F(FD*&FFFA)-&F76#F(F9F1FAF(*(FFFAFNFAF<FAFB*(F
FFAFRFAF<FA!\"'*(FFFAFRFAFHFA\"\"*F(FAF'!\"\"#F(FI" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#/*&)%#g~G%\"rG\"\"\")%!G%\"tGF(,$*(%(epsilonG\"\"\",&
*$)-%$cosG6#%&thetaG\"\"#F/\"\"$!\"\"F(F(,.**F.F(-&%\"hG6#F(6$F'F+F(-&
F>6#\"\"!F@F(F2F/!\"$**F.F/F<F/-&F>6#F7F@F(F2F/F8*&F.F/-&%\"HGF?F@F(!
\"#*(F.F/F<F/FAF/F(*(F.F/F<F/FGF/F9F<FNF(#F9\"\"%" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/*&)%#g~G%&thetaG\"\"\")%!GF'F(,$*&,2\"\"%F(*(%(epsilon
GF(-%\"kG6$%\"rG%\"tGF()-%$cosG6#F'\"\"#\"\"\"!\"'*&F0F;F1F;F:*()F0F:F
;)F7F.F;)F1F:F;\"\"**(F?F;FAF;F6F;F<*(F?F;-%\"KGF3F(F6F;F<*&F?F;FAF;F(
*&F?F;FEF;F:F;*$)F4\"\"#F;!\"\"#F(F." }}{PARA 12 "" 1 "" {XPPMATH 20 "
6#/*&)%#g~G%$phiG\"\"\")%!GF'F(,$*&,2\"\"%F(*(%(epsilonGF(-%\"kG6$%\"r
G%\"tGF()-%$cosG6#%&thetaG\"\"#\"\"\"!\"'*&F0F<F1F<F;*()F0F;F<)F7F.F<)
F1F;F<\"\"**(F@F<FBF<F6F<F=*(F@F<-%\"KGF3F(F6F<F=*&F@F<FBF<F(*&F@F<FFF
<F;F<*()F4\"\"#F<,&!\"\"F(F7F(\"\"\",&F7F(F(F(\"\"\"!\"\"#FNF." }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&)%#g~G%\"tG\"\"\")%!GF'F(,$*&*&%\"r
GF(,8!\"%F(*(%(epsilonGF(-&%\"hG6#\"\"!6$F.F'F()-%$cosG6#%&thetaG\"\"#
\"\"\"!\"'*&F2F?F3F?F>*()F2F>F?)F3F>F?F9F?\"\"'*(FCF?)F:\"\"%F?FDF?!\"
**&FCF?FDF?!\"\"*&FCF?-&%\"HGF6F8F(F>*&FCF?)-&F56#F(F8F>F?F(*(FCF?FMF?
F9F?F@*(FCF?FQF?F9F?F@*(FCF?FQF?FGF?\"\"*F(F?,&F.FK%\"mGF>!\"\"#FKFH" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "\nWith these quadratically per
turbed contra/co-variant components of the metric we can now calculate
 the perturbed Ricci tensor to second order.\n\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "grcalcalter(R(dn,dn),13); " }}{PARA 6 "" 1 "" {TEXT 
-1 50 "Simplification will be applied during calculation." }}{PARA 6 "
" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 70 "Applying routine \+
`Apply constraints repeatedly` to object g(dn,dn,pdn)" }}{PARA 6 "" 1 
"" {TEXT -1 71 "Applying routine `Apply constraints repeatedly` to obj
ect Chr(dn,dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 71 "Applying routine `Ap
ply constraints repeatedly` to object Chr(dn,dn,up)" }}{PARA 6 "" 1 "
" {TEXT -1 66 "Applying routine `Apply constraints repeatedly` to obje
ct R(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"%kh!\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gralter(R(dn,dn),expand
);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTen
sorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" 
{TEXT -1 42 "Applying routine expand to object R(dn,dn)" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$;#!\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "grdisplay(R(dn,dn)); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9For~the~qschw~spacetime:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%0Covariant~RicciG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%\"RG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%\"rG\"\"
\"&%!GF'F)%K~86717~words.~Exceeds~grOptionDisplayLimitG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%\"rG\"\"\"&%!G6#%&thetaGF)%K~28579~w
ords.~Exceeds~grOptionDisplayLimitG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&&%#R~G6#%\"rG\"\"\"&%!G6#%\"tGF)%K~53004~words.~Exceeds~grOptionDi
splayLimitG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%&thetaG\"\"
\"&%!GF'F)%K~87057~words.~Exceeds~grOptionDisplayLimitG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%&thetaG\"\"\"&%!G6#%\"tGF)%K~20438~w
ords.~Exceeds~grOptionDisplayLimitG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/*&&%#R~G6#%$phiG\"\"\"&%!GF'F)%K~72053~words.~Exceeds~grOptionDispla
yLimitG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%\"tG\"\"\"&%!GF
'F)%K~82672~words.~Exceeds~grOptionDisplayLimitG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 61 "\nFor completeness, the definitions of the Legendr
e function, " }{TEXT 259 15 "P_2(cos(theta))" }{TEXT -1 139 ", that we
 will need to calculate the projections listed in Table 1, and the sta
ndard Zerilli substitutions, that we will need to eliminate " }{TEXT 
257 3 "H_1" }{TEXT -1 7 ", and, " }{TEXT 260 5 "Kdot " }{TEXT -1 17 ",
 appear below.\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "p2 := 5/2*(3*
cos(theta)^2-1)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&*$)-%$co
sG6#%&thetaG\"\"#\"\"\"#\"#:\"\"%#!\"&F0\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 82 "newH(r,t) := (2*r^2-6*m*r-3*m^2)/(r-2*m)/(2*r+
3*m)*chi(r,t)+r^2/(r-2*m)*eta(r,t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>-%%newHG6$%\"rG%\"tG,&*&*&,(*$)F'\"\"#\"\"\"F/*&%\"mG\"\"\"F'F3!\"'
*$)F2F/F0!\"$F3-%$chiGF&F3F0*&,&F'F3F2!\"#\"\"\",&F'F/F2\"\"$\"\"\"!\"
\"F3*&*&F.F0-%$etaGF&F3F0F;FAF3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 61 "Kdot(r,t) := 6*(r^2+m*r+m^2)/r^2/(2*r+3*m)*chi(r,t)+eta(r,t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%KdotG6$%\"rG%\"tG,&*&*&,(*$)F'
\"\"#\"\"\"\"\"\"*&%\"mGF1F'F1F1*$)F3F/F0F1F1-%$chiGF&F1F0*&)F'\"\"#F0
,&F'F/F3\"\"$\"\"\"!\"\"\"\"'-%$etaGF&F1" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 135 "\nFollowing the prescription set out in Table 1, we can \+
now extract the linear contributions to the seven non-trivial Ricci co
mponents. " }}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "a1:=coeff(collect(grcomponent(R(dn,dn),[theta,theta])
,epsilon),epsilon,1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "g1
:=coeff(collect(grcomponent(R(dn,dn),[phi,phi]),epsilon),epsilon,1):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "ag1:=int(expand(a1-g1/si
n(theta)^2)*(2*cot(theta)*diff(p2,theta)+2*(2+1)*p2)*sin(theta),\nthet
a=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ag1G,&-&%\"hG6#\"\"#6$
%\"rG%\"tG\"#7-&F(6#\"\"!F+!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "leqn[1]:= \{h[2](r,t) = solve(ag1,h[2](r,t))\};" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%%leqnG6#\"\"\"<#/-&%\"hG6#\"\"#6$%\"rG%\"tG-&
F,6#\"\"!F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a2:=subs(leq
n[1],a1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "g2:=subs(leqn[
1],g1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "grmap(R(dn,dn),s
ubs,leqn[1],'x');" }}{PARA 6 "" 1 "" {TEXT -1 33 "Applying routine sub
s to R(dn,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gralter(R(
dn,dn),simplify,expand);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component sim
plification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " 
}}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine simplify to object R(d
n,dn)" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine expand to objec
t R(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"&iY\"!\"
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "b1:=coeff(collect(grco
mponent(R(dn,dn),[theta,r]),epsilon),epsilon,1):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 85 "leqn[2]:=termsimp(collect(1/3*int(b1*diff(p2,
theta)*sin(theta),theta=0..Pi),lperts));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%%leqnG6#\"\"#,**&*&%\"mG\"\"\"-&%\"hG6#\"\"!6$%\"rG%\"tGF,\"
\"\"*&,&F3!\"\"F+F'\"\"\"F3\"\"\"!\"\"!\"#-%%diffG6$F-F3F,-F>6$-%\"kGF
2F3F8*&*&F3F,-F>6$-&F/6#F,F2F4F,F5F7F;F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "c1:=coeff(collect(grcomponent(R(dn,dn),[theta,t]),eps
ilon),epsilon,1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "leqn[3
]:=termsimp(collect(1/3*int(c1*diff(p2,theta)*sin(theta),theta=0..Pi),
lperts));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%leqnG6#\"\"$,**&*&-&%
\"hG6#\"\"\"6$%\"rG%\"tGF/%\"mGF/\"\"\"*$)F1\"\"#F4!\"\"\"\"#-%%diffG6
$-&F-6#\"\"!F0F2!\"\"-F;6$-%\"kGF0F2FA*&*&,&F1FAF3F9F/-F;6$F+F1F/F4F1F
8FA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "d1:=coeff(collect(gr
component(R(dn,dn),[t,r]),epsilon),epsilon,1):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 69 "leqn[4]:=termsimp(collect(int(d1*p2*sin(theta)
,theta=0..Pi),lperts));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%leqnG6#
\"\"%,*-%%diffG6%-%\"kG6$%\"rG%\"tGF/F0!\"\"*&-&%\"hG6#\"\"\"F.\"\"\"*
$)F/\"\"#F8!\"\"\"\"$*&-F*6$-&F56#\"\"!F.F0F8F/F<F7*&*&,&%\"mGF=F/F1F7
-F*6$F,F0F7F8*&F/\"\"\",&F/F1FH\"\"#\"\"\"F<F1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 66 "e1:=coeff(collect(grcomponent(R(dn,dn),[r,r]),
epsilon),epsilon,1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "leq
n[5]:=termsimp(collect(int(e1*p2*sin(theta),theta=0..Pi),lperts));" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%%leqnG6#\"\"&,2*&-&%\"hG6#\"\"!6$%
\"rG%\"tG\"\"\"*&,&F0!\"\"%\"mG\"\"#\"\"\"F0\"\"\"!\"\"!\"$*&-%%diffG6
$F*F0F2F4F:F5*&*&F0\"\"\"-F>6%-&F,6#FBF/F0F1FBF2F4F:FB*&*&-F>6$-%\"kGF
/F0FB,&F6\"\"$F0!\"#FBF2*&F4\"\"\"F0\"\"\"F:F5*&*&F6FB-F>6$FEF1FBF2*$)
F4\"\"#F2F:F5-F>6$F*-%\"$G6$F0F7#FBF7-F>6$FLFgnF5*&*&)F0F7F2-F>6$F*-Fh
n6$F1F7FBF2*$)F4\"\"#F2F:Fjn" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 66 "f1:=coeff(collect(grcomponent(R(dn,dn),[t,t]),epsilon),epsilon,1
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "leqn[6]:=termsimp(col
lect(int(f1*p2*sin(theta),theta=0..Pi),lperts));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>&%%leqnG6#\"\"',2*&*&,&%\"rG!\"\"%\"mG\"\"#\"\"\"-&%\"
hG6#\"\"!6$F,%\"tGF0\"\"\"*$)F,\"\"$F8!\"\"!\"$*&*&F+F8-%%diffG6$F1F,F
0F8*$)F,\"\"#F8F<F0-FA6$-%\"kGF6-%\"$G6$F7F/F-*&*&F+F8-FA6%-&F36#F0F6F
,F7F0F8F,F<F-*&*(F.F0F+F8-FA6$FHF,F0F8*$)F,\"\"$F8F<F-*&*&,&F.\"\"$F,!
\"#F0-FA6$FQF7F0F8*$)F,\"\"#F8F<F-*&*&)F+F/F8-FA6$F1-FK6$F,F/F0F8*$)F,
\"\"#F8F<#F-F/-FA6$F1FJFio" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
87 "leqn[7]:=termsimp(collect(int((a2+g2/sin(theta)^2)*p2*sin(theta),t
heta=0..Pi),lperts));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%%leqnG6#\"
\"(,0-&%\"hG6#\"\"!6$%\"rG%\"tG\"\"#*&-%%diffG6$F)F/\"\"\",&F/!\"\"%\"
mGF1F6!\"#-%\"kGF.\"\"%*&*&)F/\"\"$\"\"\"-F46$F;-%\"$G6$F0F1F6FBF7!\"
\"F8*&-F46$F;F/F6,&F9FAF/F:F6F1*&-F46$-&F+6#F6F.F0F6F/F6F:*(-F46$F;-FF
6$F/F1F6F/FBF7FBF6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 223 "\nWhile th
is completes our formal analysis of the linear problem, these seven eq
uations, must now be combined in some unobvious ways in order to facil
itate the reduction of the source terms in the second order calculatio
ns.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "lsub1:=\{diff(k(r,t),t,t)
 = termsimp(collect(solve(expand((r-2*m)^2*leqn[5]+r^2*leqn[6]\n-leqn[
7]/r*(r-2*m)),diff(k(r,t),t,t)),lperts))\};" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%&lsub1G<#/-%%diffG6$-%\"kG6$%\"rG%\"tG-%\"$G6$F.\"\"#
,,*&*&,&F-!\"\"%\"mGF2\"\"\"-&%\"hG6#\"\"!F,F9\"\"\"*$)F-\"\"$F?!\"\"!
\"#*&*&)F6F2F?-F(6$F:F-F9F?*$)F-\"\"$F?FCF7*&*&F6F?F*F9F?*$)F-\"\"$F?F
CF2*&*(F6F?,&F8F9F-F7F9-F(6$F*F-F9F?*$)F-\"\"$F?FCF9*&*&F6F?-F(6$-&F<6
#F9F,F.F9F?*$)F-\"\"#F?FCFD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "\n
In order to determine the second order PDEs (" }{TEXT 261 11 "qeqn_\{1
..7\}" }{TEXT -1 67 ") we simply repeat the linear analysis, but now s
electing only the " }{TEXT 262 9 "epsilon^2" }{TEXT -1 209 " component
s of the Ricci tensor. The second order perturbation functions will, n
ecessarily, appear in these PDEs in exactly the same manner as the lin
ear perturbation functions appear in the leqn_i equations. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "aa1:=s
implify(coeff(collect(grcomponent(R(dn,dn),[theta,theta]),epsilon),eps
ilon,2),\ntrig):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "gg1:=s
implify(expand(1/sin(theta)^2*(coeff(collect(grcomponent(R(dn,dn),[phi
,phi]),\nepsilon),epsilon,2))),trig):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "zz1:=simplify(expand(gg1-aa1));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$zz1G,6-&%\"HG6#\"\"#6$%\"rG%\"tG#\"\"$F**&-&F(6#\"\"
!F+\"\"\")-%$cosG6#%&thetaGF*\"\"\"F.*$)-&%\"hG6#F5F+F*F;#!\"$F**$)-&F
@F3F+F*F;F.F1FB*&)F7\"\"%F;FEF;\"\"**&F=F;FIF;!\"**&F&F5F6F;FB*&F6F;FE
F;#!#@F**&F=F;F6F;#\"#@F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "zz2:=int(zz1*(2*cot(theta)*diff(p2,theta)+2*(2+1)*p2)*sin(theta),t
heta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$zz2G,*-&%\"HG6#\"\"
#6$%\"rG%\"tG!#7*$)-&%\"hG6#\"\"!F+F*\"\"\"#F.\"\"(-&F(F4F+\"#7*$)-&F3
6#\"\"\"F+F*F6#F;F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "qeqn
1:= \{H[2](r,t) = solve(zz2,H[2](r,t))\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&qeqn1G<#/-&%\"HG6#\"\"#6$%\"rG%\"tG,(*$)-&%\"hG6#\"
\"!F,F+\"\"\"#!\"\"\"\"(-&F)F5F,\"\"\"*$)-&F46#F=F,F+F7#F=F:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "qeqn[1]:=qeqn1;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%%qeqnG6#\"\"\"<#/-&%\"HG6#\"\"#6$%\"rG%\"tG,
(*$)-&%\"hG6#\"\"!F/F.\"\"\"#!\"\"\"\"(-&F,F8F/F'*$)-&F7F&F/F.F:#F'F=
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "grmap(R(dn,dn),subs,qeq
n1,`x`); " }}{PARA 6 "" 1 "" {TEXT -1 33 "Applying routine subs to R(d
n,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "aa1:=simplify(coef
f(collect(grcomponent(R(dn,dn),[theta,theta]),epsilon),epsilon,2),\ntr
ig): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "gg1:=simplify(exp
and(1/sin(theta)^2*(coeff(collect(grcomponent(R(dn,dn),[phi,phi]),\nep
silon),epsilon,2))),trig):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
77 "qeqn[2]:=termsimp(collect(int((gg1+aa1)*p2*sin(theta),theta=0..Pi)
,qperts)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "bb1:=simplif
y(coeff(collect(grcomponent(R(dn,dn),[theta,r]),epsilon),epsilon,2),tr
ig):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "qeqn[3]:=termsimp(c
ollect(int(bb1*diff(p2,theta)*sin(theta),theta=0..Pi),qperts)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "cc1:=simplify(coeff(collect(
grcomponent(R(dn,dn),[t,theta]),epsilon),epsilon,2),trig):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "qeqn[4]:=termsimp(collect(1/3*int(c
c1*diff(p2,theta)*sin(theta),theta=0..Pi),qperts)):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 82 "dd1:=simplify(coeff(collect(grcomponent(R
(dn,dn),[r,t]),epsilon),epsilon,2),trig):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 70 "qeqn[5]:=termsimp(collect(int(dd1*p2*sin(theta),the
ta=0..Pi),qperts)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "ee1:
=simplify(coeff(collect(grcomponent(R(dn,dn),[r,r]),epsilon),epsilon,2
),trig):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "qeqn[6]:=termsi
mp(collect(expand(int(ee1*p2*sin(theta),theta=0..Pi)),qperts)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "qsub1:=\{diff(H[0](r,t),t)=s
olve(qeqn[4],diff(H[0](r,t),t))\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#
>%&qsub1G<#/-%%diffG6$-&%\"HG6#\"\"!6$%\"rG%\"tGF1,$*&,0*&-F(6$-&F,6#
\"\"\"F/F0F;)F0\"\"#\"\"\"\"\"(*(F6F>%\"mGF;F0F;!#9*&-F(6$-%\"KGF/F1F;
F<F>!\"(*&F8F;FAF>\"#9*(-&%\"hGF-F/F;-F(6$FLF1F;F<F>\"\"$*(-%\"kGF/F;-
F(6$FSF1F;F<F>F=*(-&FNF:F/F;-F(6$FXF1F;F<F>!\"$F>*$)F0\"\"#F>!\"\"#F;F
?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "qeqn[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#/-&%\"HG
6#\"\"#6$%\"rG%\"tG,(*$)-&%\"hG6#\"\"!F*F)\"\"\"#!\"\"\"\"(-&F'F3F*\"
\"\"*$)-&F26#F;F*F)F5#F;F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "undiff(qeqn[4]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'undiffG6#,0
*&*&,&%\"rG!\"\"%\"mG\"\"#\"\"\"-%%diffG6$-&%\"HG6#F.6$F*%\"tGF*F.\"\"
\"F*!\"\"F+-F06$-&F46#\"\"!F6F7F+-F06$-%\"KGF6F7F+*&*&F2F.F,F.F8*$)F*
\"\"#F8F9F-*&-&%\"hGF>F6F.-F06$FJF7F.#\"\"$\"\"(*&-%\"kGF6F.-F06$FSF7F
.#F-FQ*&-&FLF5F6F.-F06$FYF7F.#!\"$FQ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 78 "\n\nThe derivation of the first Zerilli equation begins with th
e elimination of " }{TEXT 263 6 "Hdot_0" }{TEXT -1 7 " using " }{TEXT 
264 6 "qeqn_4" }{TEXT -1 9 ", in the " }{TEXT 265 8 "R^2_\{rt\}" }
{TEXT -1 136 " component of the Ricci tensor. We then make the previou
sly defined Zerilli substitutions, and make various substitutions invo
lving the " }{TEXT 266 6 "leqn_i" }{TEXT -1 300 " equations. (The actu
al substitutions are somewhat arbitrary. We choose to eliminate certai
n perturbation functions so as to reproduce the results presented in G
L). The resulting expression is called etaeqn in our Maple worksheet. \+
To simplify this rather awkward expression, we employ a new routine, \+
" }{TEXT 267 14 "``mcollect()''" }{TEXT -1 366 ", that collects specif
ied terms that are multiplied together. This routine could, for exampl
e, collect together terms that have a factor of, ab, but not, say, a^2
, or b^2. A more complete description of this routine appears in Appen
dix A. (Readers who are familiar with the computer algebra environment
 will, no doubt, already see the utility of such a procedure.)\n\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eta1:=qeqn[5]:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 49 "eta2:=termsimp(collect(subs(qsub1,eta1),qper
ts)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "eta3:=termsimp(co
llect(simplify(subs(H[1](r,t)=newH(r,t),diff(K(r,t),t)=Kdot(r,t),\ndif
f(K(r,t),r,t)=diff(Kdot(r,t),r),eta2)),zfuncs)):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 62 "eta4:=eta(r,t)-termsimp(collect(solve(eta3,et
a(r,t)),zfuncs)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "eta5:
=termsimp(collect(subs(\{diff(h[0](r,t),r,r)=solve(leqn[5],diff(h[0](r
,t),r,r))\},eta4),\nzfuncs)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 146 "eta6:=termsimp(collect(subs(\{diff(h[0](r,t),r)=solve(leqn[2],d
iff(h[0](r,t),r)),\ndiff(k(r,t),r,t)=solve(leqn[4],diff(k(r,t),r,t))\}
,eta5),zfuncs)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "ug:=\{
diff(k(r,t),r,r)=termsimp(collect(solve(subs(diff(h[0](r,t),r)=\nsolve
(leqn[2],diff(h[0](r,t),r)),subs(lsub1,solve(leqn[6],diff(h[0](r,t),r,
r))-\nsolve(leqn[5],diff(h[0](r,t),r,r)))),diff(k(r,t),r,r)),lperts))
\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eta7:=termsimp(colle
ct(subs(ug,eta6),zfuncs)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "etaeqn:=eta7:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "\nIn order \+
to produce a simplified final expression, we extract the source terms \+
from etaeqn (by setting the second order perturbation functions to zer
o) and then simplify the result by repeated application of  " }{TEXT 
268 14 "``mcollect()''" }{TEXT -1 4 ". \n\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 104 "esource1:=mcollect(expand(eval(subs(eta(r,t)=0,ch
i(r,t)=0,etaeqn))),             \{h[1](r,t),h[0](r,t)\}):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "esource2:=mcollect(esource1,\{h[1](
r,t),diff(h[1](r,t),t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "esource3:=mcollect(esource2,\{h[1](r,t),diff(k(r,t),r)\}):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "esource4:=mcollect(esource3,
\{h[0](r,t),diff(k(r,t),t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 64 "esource5:=mcollect(esource4,\{diff(h[1](r,t),t),diff(k(r,t),t)
\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "esource6:=mcollect(
esource5,\{k(r,t),h[1](r,t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 53 "esource7:=mcollect(esource6,\{k(r,t),diff(k(r,t),t)\}):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "esource8:=mcollect(esource7,
\{diff(k(r,t),r),diff(h[0](r,t),t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "esource9:=mcollect(esource8,\{k(r,t),diff(h[0](r,t),t
)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "esource10:=mcollec
t(esource9,\{h[0](r,t),diff(h[0](r,t),t)\}):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "etasource:=kfactor(esource10,(r-2*m)/7/(2*r+3*m)
):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "\nIt is now a simple matte
r to put these results together to arrive at the first Zerilli equatio
n for the second order calculation:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 128 "etafinal:=\{eta(r,t) = termsimp(collect(solve(eval(subs(h[0](r,
t)=0,h[1](r,t)=0,k(r,t)=0,\netaeqn)),eta(r,t)),zfuncs))-etasource\};" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)etafinalG<#/-%$etaG6$%\"rG%\"tG,&
*&*&-%%diffG6$-%$chiGF)F*\"\"\",&F*!\"\"%\"mG\"\"#F4\"\"\"F*!\"\"F6*&*
&,&F*F4F7!\"#F4,6*&-&%\"hG6#\"\"!F)F4-F06$FAF+F4F6*&-%\"kGF)F4FFF9F8*(
F*F4-F06$FIF*F4FFF9F4*&FIF9-F06$FIF+F4F8*&*&-&FC6#F4F)F4FIF9F9F*F:\"\"
%*&*(F7F4FSF9FLF9F9F*F:F>*&FSF9-F06$FSF+F4F4*&*(F5F9FAF9FSF9F9*$)F*\"
\"#F9F:F>*&*(FOF9F7F9FAF9F9F5F:F>*&*()F*F8F9FOF9FZF9F9F5F:F4F4F9,&F*F8
F7\"\"$F:#F6\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "\nThis agree
s with the result found in [1]. " }}{PARA 0 "" 0 "" {TEXT -1 107 "\nTh
e development of  the second order Zerilli ``wave equation'' follows a
 similar procedure to that of the " }{TEXT 256 9 "etafinal " }{TEXT 
-1 45 "result. We first take the time derivative of " }{TEXT 269 6 "qe
qn_2" }{TEXT -1 12 ", eliminate " }{TEXT 270 6 "Hdot_0" }{TEXT -1 5 " \+
and " }{TEXT 272 8 "Hdot^'_0" }{TEXT -1 7 " using " }{TEXT 271 6 "qeqn
_4" }{TEXT -1 53 ", make the Zerilli substitutions, and substitute for
 " }{TEXT 273 3 "eta" }{TEXT -1 64 " from our first Zerilli result. We
 then repeat this process for " }{TEXT 274 6 "qenq_3" }{TEXT -1 47 " a
nd add the resulting equations such that the " }{TEXT 276 9 "chi^\{..'
\}" }{TEXT -1 64 " term vanishes. The result of these operations, whic
h is called " }{TEXT 275 7 "chieqn " }{TEXT -1 120 "in our worksheet, \+
is essentially the raw form of the result we are seeking. This express
ion is, however, rather large.\n\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "atmp1:=diff(qeqn[2],t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
87 "atmp2:=termsimp(collect(expand(subs(qsub1,diff(qsub1,r),atmp1)),\{
diff(K(r,t),t,t,t)\})):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 
"atmp3:=termsimp(collect(expand(eval(subs(H[1](r,t)=newH(r,t),\ndiff(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),\natmp2))),zfuncs)):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 68 "atmp4:=termsimp(collect(expand(eval(subs(etafinal,
atmp3))),zfuncs)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "coefa
:=simplify(coeff(atmp4,diff(chi(r,t),t,r,t),1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&coefaG,$*$)%\"rG\"\"#\"\"\"!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 118 "btmp1:=termsimp(collect(expand(subs(qsub1,
diff(qsub1,r),diff(qeqn[3],t))),\n\{diff(H[1](r,t),t,t),diff(K(r,t),t,
t,t)\})):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "btmp2:=termsi
mp(collect(expand(eval(subs(H[1](r,t)=newH(r,t),diff(K(r,t),t)=Kdot(r,
t),\ndiff(K(r,t),r,t)=diff(Kdot(r,t),r),btmp1))),zfuncs)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "btmp3:=termsimp(collect(expand(eval
(subs(etafinal,btmp2))),zfuncs)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "coefb:=simplify(coeff(btmp3,diff(chi(r,t),r,t,t),1));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&coefbG,$*&*$)%\"rG\"\"#\"\"\"F+
,&F)!\"\"%\"mGF*!\"\"\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
204 "ftmp1:=termsimp(collect(expand(atmp4/coefa-btmp3/coefb),\{chi(r,t
),diff(chi(r,t),r),\ndiff(chi(r,t),t),diff(chi(r,t),t,t),diff(chi(r,t)
,r,r),diff(chi(r,t),r,t),diff(chi(r,t),r,t,t),\ndiff(chi(r,t),r,r,r)\}
)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "ftmp2:=termsimp(col
lect(expand(eval(subs(k(r,t)=0,h[1](r,t)=0,h[0](r,t)=0,ftmp1))),\n\{ch
i(r,t),diff(chi(r,t),r),diff(chi(r,t),r,r),diff(chi(r,t),t,t),diff(chi
(r,t),r,r,r)\})):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "chi1:=
termsimp(collect(diff(chi(r,t),t,t)-solve(ftmp1,diff(chi(r,t),t,t)),zf
uncs)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "chieqn:=ftmp1:" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "nops(chieqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$@%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "\nAlthough, at this point we can \+
at least verify that it reproduces the correct linear Zerilli equation
 when the source terms are set to zero.\n\n\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 203 "termsimp(collect(expand(eval(subs(k(r,t)=0,h[1](r,t)
=0,h[0](r,t)=0,\nchieqn/coeff(chieqn,diff(chi(r,t),t,t),1)))),\{chi(r,
t),diff(chi(r,t),r),\ndiff(chi(r,t),r,r),diff(chi(r,t),t,t),diff(chi(r
,t),r,r,r)\}));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**&*(,&%\"rG!\"\"%
\"mG\"\"#\"\"\",**$)F)\"\"$\"\"\"F/*&F'F+)F)F*F0\"\"'*&)F'F*F0F)F+\"\"
%*$)F'F/F0F6F+-%$chiG6$F'%\"tGF+F0*&)F'\"\"%F0),&F'F*F)F/\"\"#F0!\"\"!
\"'*&*(-%%diffG6$F9F'F+F&F0F)F0F0*$)F'\"\"$F0FCF**&*&-FH6$F9-%\"$G6$F'
F*F+)F&F*F0F0*$)F'\"\"#F0FCF(-FH6$F9-FR6$F<F*F+" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 237 "\nThis is just the linear Zerilli wave equation wit
h the quadrapole potential, as required.\n\nIn order to make contact w
ith the results presented in GL, we now perform a ``gauge'' transforma
tion and work with the new perturbation function " }{TEXT 277 4 "zeta
" }{TEXT -1 3 ".\n\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 84 "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)\};" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(zetasubG<#/-%$chiG6$%\"rG%\"tG,(-%%zetaGF)\"\"\"*&*()F*\"\"#
\"\"\"-%\"kGF)F/-%%diffG6$F5F+F/F4,&F*F3%\"mG\"\"$!\"\"#F3\"\"(*$)F5F3
F4F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "zeta1:=eval(subs(ze
tasub,chi1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "nops(expan
d(zeta1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$h\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 " This substitut
ion has considerably reduced the number of terms in the expression. " 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "n
ew2:=termsimp(collect(expand(leqn[4]-leqn[3]/2/m*3),lperts));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%new2G,*-%%diffG6%-%\"kG6$%\"rG%\"tG
F,F-!\"\"*&*&,&%\"mG\"\"#F,\"\"$\"\"\"-F'6$-&%\"hG6#\"\"!F+F-F5\"\"\"*
&F2\"\"\"F,\"\"\"!\"\"#F5F3*&*&,(*$)F2F3F=\"\"'*&F2F5F,F5!\")*$)F,F3F=
F4F5-F'6$F)F-F5F=*(F,\"\"\",&F,F.F2F3\"\"\"F2\"\"\"FA#F.F3*&*&FQF5-F'6
$-&F:6#F5F+F,F5F=*&F2\"\"\"F,\"\"\"FA#F4F3" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 85 "leqn8:=\{diff(h[1](r,t),r) = termsimp(collect(solve
(new2,diff(h[1](r,t),r)),lperts))\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%&leqn8G<#/-%%diffG6$-&%\"hG6#\"\"\"6$%\"rG%\"tGF0,(*&*(-F(6%-%\"kG
F/F0F1F.F0F.%\"mGF.\"\"\",&F0!\"\"F9\"\"#!\"\"#F=\"\"$*&*&,&F9F=F0F@F.
-F(6$-&F,6#\"\"!F/F1F.F:F;F>#F<F@*&*&,(*$)F9F=F:\"\"'*&F9F:F0F:!\")*$)
F0F=F:F@F.-F(6$F7F1F.F:*$)F;\"\"#F:F>#F.F@" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 150 "new3:=termsimp(collect(expand(solve(leqn[3],diff(h
[1](r,t),r))-solve(diff(epxand(\nleqn[4]/coeff(leqn[4],h[1](r,t),1)),r
),diff(h[1](r,t),r))),lperts));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%
new3G,2*&*(,&%\"mG\"\"(%\"rG!\"$\"\"\"F+F--%%diffG6%-%\"kG6$F+%\"tGF+F
4F-\"\"\",&F+!\"\"F)\"\"#!\"\"#F7\"\"$*&*&-&%\"hG6#F-F3F-F)F-F5*&F+\"
\"\"F6\"\"\"F9F8*&F+F5-F/6%-&F@6#\"\"!F3F+F4F-#F-F;*&*&,&F)F-F+!\"#F--
F/6$FHF4F-F5F6F9#F8F;*&*&,(*$)F)F8F5F;*&F)F5F+F5F-*$)F+F8F5F7F--F/6$F1
F4F-F5*$)F6\"\"#F5F9#FPF;*&*&-F/6%F1-%\"$G6$F+F8F4F-)F+\"\"%F5F5*$)F6
\"\"#F5F9F:*&*(F^oF5FenF5FXF5F5*$)F6\"\"#F5F9#!\"%F;*&*(F^oF5)F+F;F5F)
F5F5*$)F6\"\"#F5F9#FdoF;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 
"leqn9:=\{h[1](r,t) = termsimp(collect(solve(new3,h[1](r,t)),lperts))
\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&leqn9G<#/-&%\"hG6#\"\"\"6$%
\"rG%\"tG,,*&*(-%%diffG6%-%\"kGF,F-F.F+,&%\"mG\"\"(F-!\"$F+)F-\"\"#\"
\"\"F=F8!\"\"#F+\"\"'*&*(-F36%-&F)6#\"\"!F,F-F.F+F;F=,&F-!\"\"F8F<F+F=
F8F>#FJF@*&*(-F36$FEF.F+,&F8F+F-!\"#F+F-F+F=F8F>#FJ\"\"$*&*(,(*$)F8F<F
=FS*&F8F+F-F=F+*$F;F=FJF+-F36$F5F.F+F-F=F=*&F8\"\"\"FI\"\"\"F>#F+FS*&*
(FIF=-F36%F5-%\"$G6$F-F<F.F+)F-FSF=F=F8F>F?" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 95 "leqn10:=\{diff(h[1](r,t),t)=termsimp(collect(expan
d(solve(leqn[2],diff(h[1](r,t),t))),lperts))\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'leqn10G<#/-%%diffG6$-&%\"hG6#\"\"\"6$%\"rG%\"tGF1,(*
&*&%\"mGF.-&F,6#\"\"!F/F.\"\"\"*$)F0\"\"#F:!\"\"\"\"#*&*&,&F0!\"\"F5F?
F.-F(6$F6F0F.F:F0F>FC*&*&FBF:-F(6$-%\"kGF/F0F.F:F0F>F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "aux1:=termsimp(collect(expand(solv
e(leqn[2],diff(h[1](r,t),t))-solve(diff(leqn[4],t),\ndiff(h[1](r,t),t)
)),lperts));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%aux1G,0*&*&%\"mG\"
\"\"-&%\"hG6#\"\"!6$%\"rG%\"tGF)\"\"\"*$)F0\"\"#F2!\"\"\"\"#*&*&,&F0!
\"\"F(F7F)-%%diffG6$F*F0F)F2F0F6F;*&*(F0F),&F(\"\"$F0F;F)-F=6$-%\"kGF/
-%\"$G6$F1F7F)F2F:F6#F;FB*&*&F:F2-F=6$FEF0F)F2F0F6F)*&-F=6$F*FGF)F0F2#
F)FB*&*&)F0FBF2-F=6%FEF0FGF)F2F:F6FR*&*()F0F7F2FVF2F(F2F2F:F6#!\"#FB" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "aux2:=termsimp(collect(ex
pand(diff((r-2*m)*r*leqn[4]/3,r))-leqn[3],lperts));" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%%aux2G,.*&,&%\"mG\"\"&%\"rG!\"$\"\"\"-%%diffG6%-%\"
kG6$F*%\"tGF*F3F,#F,\"\"$*&,&F*!\"\"F(\"\"#F,-F.6%-&%\"hG6#\"\"!F2F*F3
F,#F8F5-F.6$F<F3#\"\"%F5-F.6$F0F3#F9F5*(F*F,F(F,-F.6%F0-%\"$G6$F*F9F3F
,FH*&)F*F9\"\"\"FJFQFA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "
aux3:=termsimp(collect(expand(solve(leqn[2],diff(h[1](r,t),t))-solve(l
eqn[7],\ndiff(h[1](r,t),t))),lperts));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%aux3G,,*&*&,&%\"rG!\"\"%\"mG\"\"#\"\"\"-&%\"hG6#\"\"!6$F)%\"t
GF-\"\"\"*$)F)\"\"#F5!\"\"F-*&-%\"kGF3F5F)F9!\"#*&*&)F)F,F5-%%diffG6$F
;-%\"$G6$F4F,F-F5F(F9#F-F,*&*&,&F+F-F)F*F--FB6$F;F)F-F5F)F9F**&F(F5-FB
6$F;-FE6$F)F,F-#F*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "tm
paux:=termsimp(collect(expand(solve(leqn[5],diff(h[1](r,t),r,t))-solve
(leqn[6],\ndiff(h[1](r,t),r,t))),lperts));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'tmpauxG,,*&-&%\"hG6#\"\"!6$%\"rG%\"tG\"\"\"*$)F-\"\"
#F/!\"\"\"\"'*&*&F-\"\"\"-%%diffG6$-%\"kGF,-%\"$G6$F.\"\"#F7F/,&F-!\"
\"%\"mGF@F3F7*&*&FAF7-F96$F;F-F7F/*$)F-\"\"#F/F3F@*&-F96$-&F)6#F7F,F.F
/F-F3F@*&*&FAF/-F96$F;-F>6$F-F@F7F/F-F3F7" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 109 "tmpaux2:=termsimp(collect(expand(solve(tmpaux,diff
(h[1](r,t),t))-solve(leqn[2],\ndiff(h[1](r,t),t))),lperts));" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%(tmpaux2G,,*&*&,&%\"mG\"\"#%\"rG\"\"$\"\"
\"-&%\"hG6#\"\"!6$F+%\"tGF-\"\"\"*$)F+\"\"#F5!\"\"!\"\"*&*&,&F+F:F)F*F
--%%diffG6$F.F+F-F5F+F9F-*&*&)F+F*F5-F?6$-%\"kGF3-%\"$G6$F4F*F-F5F=F9#
F:F**&*&F=F5-F?6$FFF+F-F5F+F9!\"#*&F=F5-F?6$FF-FI6$F+F*F-FK" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "tmpaux3:=termsimp(collect(e
xpand(solve(tmpaux2,diff(k(r,t),t,t))-solve(leqn[7],\ndiff(k(r,t),t,t)
)),lperts));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(tmpaux3G,.*&*(,&%\"
mG\"\"\"%\"rG\"\"#F*,&F+!\"\"F)F,F*-&%\"hG6#\"\"!6$F+%\"tGF*\"\"\"*$)F
+\"\"%F6!\"\"!\"%*&*&)F-F,F6-%%diffG6$F/F+F*F6*$)F+\"\"$F6F:\"\"%*&*&F
-F6-%\"kGF4F*F6*$)F+\"\"$F6F:F;*&*(F-F6,&F)\"\"(F+F;F*-F@6$FHF+F*F6*$)
F+\"\"$F6F:!\"#*&*&F-F6-F@6$-&F16#F*F4F5F*F6*$)F+\"\"#F6F:F,*&*&F>F6-F
@6$FH-%\"$G6$F+F,F*F6*$)F+\"\"#F6F:FV" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 107 "aux4:=termsimp(collect(expand(solve(tmpaux3,diff(h[1
](r,t),t))-solve(leqn[2],\ndiff(h[1](r,t),t))),lperts));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%%aux4G,,*&-&%\"hG6#\"\"!6$%\"rG%\"tG\"\"\"F-!
\"\"\"\"%*&*&,&F-!\"\"%\"mG\"\"#\"\"\"-%%diffG6$F'F-F8F/F-F0F5*&-%\"kG
F,F/F-F0F7*&*&,&F6\"\"&F-!\"$F8-F:6$F=F-F8F/F-F0F8*&F4F/-F:6$F=-%\"$G6
$F-F7F8F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "aux5:=termsim
p(collect(solve(solve(diff(leqn[3],t),diff(h[1](r,t),r,t))-solve(leqn[
5],\ndiff(h[1](r,t),r,t)),diff(h[1](r,t),t))-solve(leqn[2],diff(h[1](r
,t),t)),lperts));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%aux5G,0*&*&,(*
$)%\"rG\"\"#\"\"\"\"\"$*&%\"mG\"\"\"F+F1!\"'*$)F0F,F-F,F1-&%\"hG6#\"\"
!6$F+%\"tGF1F-*&)F+\"\"#F-F0\"\"\"!\"\"!\"\"*&*(,&F+FAF0F,F1,&F+F1F0F1
F1-%%diffG6$F5F+F1F-*&F+\"\"\"F0\"\"\"F@F1*&*&F*F--FG6$-%\"kGF:-%\"$G6
$F;F,F1F-F0F@F1*&*(FDF-,&F0F1F+FAF1-FG6$FPF+F1F-*&F0\"\"\"F+\"\"\"F@F,
*&*&)FDF,F--FG6$F5-FS6$F+F,F1F-F0F@#FAF,*&*&FinF--FG6$FPF\\oF1F-F0F@F1
*&*&-FG6$F5FRF1F*F-F-F0F@#F1F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 99 "aux6:=termsimp(collect(expand(solve(aux5,diff(k(r,t),t,t))-sol
ve(aux3,diff(k(r,t),t,t))),\nlperts));" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%%aux6G,0*&*&,(*$)%\"rG\"\"#\"\"\"\"\"&*&%\"mG\"\"\"F+F1!#9*$)F
0F,F-\"#5F1-&%\"hG6#\"\"!6$F+%\"tGF1F-*$)F+\"\"%F-!\"\"F1*&*(,&F+!\"\"
F0F,F1,&F+F1F0F1F1-%%diffG6$F6F+F1F-*$)F+\"\"$F-F@FD*&*&FCF--%\"kGF;F1
F-*$)F+\"\"$F-F@!\"%*&*(FCF-,&F0F1F+FDF1-FG6$FNF+F1F-*$)F+\"\"$F-F@FS*
&*&)FCF,F--FG6$F6-%\"$G6$F+F,F1F-*$)F+\"\"#F-F@#F1F,*&*&FhnF--FG6$FNF[
oF1F-*$)F+\"\"#F-F@!\"#-FG6$F6-F\\o6$F<F,#FDF," }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Following GL, we now
 elimanate h_0 in favour of psi using:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "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)\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'psisubG<#/-&%\"hG6#\"\"!6$%\"rG%\"t
G,&*&*&,&-%$psiGF,\"\"\"*&F-F5-%\"kGF,F5#!\"\"\"\"$F5,&F-\"\"#%\"mGF;F
5\"\"\"*&F-\"\"\",&F-F5F>!\"#\"\"\"!\"\"F;*&F-F?-%%diffG6$F7F-F5F5" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "This is the relation between the \+
Zerilli function and the metric perturbation functions.\n\n\n" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "aux7:=termsimp(collect(expand(subs(
psisub,aux4)),lperts union \{psi(r,t),\ndiff(psi(r,t),r)\}));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%aux7G,(*&*&,(*$)%\"rG\"\"#\"\"\"\"
\"\"*&%\"mGF.F+F.F.*$)F0F,F-F.F.-%$psiG6$F+%\"tGF.F-*&)F+\"\"$F-,&F+!
\"\"F0F,\"\"\"!\"\"!#=*&*&,&F+F,F0\"\"$F.-%%diffG6$F3F+F.F-*$)F+\"\"#F
-F=FB*&*&FAF--%\"kGF5F.F-*&F+\"\"\"F:\"\"\"F=FB" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "zsource1:=expand(simplify(subs(leqn8,zeta1))):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nops(zsource1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$@#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "zsource2:=expand(simplify(eval(subs(leqn10,zsource1))
)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nops(zsource2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$)>" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "zsource3:=expand(simplify(subs(leqn9,zsource2))):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nops(zsource3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"$(G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 101 "zsource3a:=expand(simplify(eval(subs(diff(k(r,t),t,t,r)=solve
(aux1,diff(k(r,t),t,t,r)),\nzsource3)))):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 97 "zsource4:=expand(simplify(eval(subs(diff(k(r,t),t,t
)=solve(aux3,diff(k(r,t),t,t)),\nzsource3a)))):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 100 "zsource5:=expand(simplify(eval(subs(diff(k(r,
t),r,r,t)=solve(aux2,diff(k(r,t),r,r,t)),\nzsource4)))):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "zsource6:=expand(simplify(eval(sub
s(diff(h[0](r,t),t,t)=solve(aux6,diff(h[0](r,t),t,t)),\nzsource5)))):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nops(zsource6);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#()" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "zsource7:=expand(simplify(eval(subs(psisub,zsource6))
)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "nops(zsource7);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#&*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "zsource8:=expand(simplify(eval(subs(k(r,t)=solve(aux7
,k(r,t)),zsource7)))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "z
etaeqn:=zsource8:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "s_ren \+
:= make_s():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sprime:=mak
e_s_mu():" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "After making this s
ubstitution, and a fairly extensive simplification of the source terms
 using the linear perturbation equations, we arrive at the unsimplifie
d answer zetaeqn.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "nops(zetaeqn
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$a\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 31 "\nHere again we make use of the " }{TEXT 278 14 "``mcol
lect()''" }{TEXT -1 34 " routine to simplify our result.\n\n" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 64 "final1:=mcollect(expand(zetaeqn),\{psi(r,
t),diff(psi(r,t),r,r)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "final2:=mcollect(final1,\{psi(r,t),diff(psi(r,t),t)\}):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "final3:=mcollect(final2,\{di
ff(psi(r,t),t),diff(psi(r,t),r,r,r)\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "final4:=mcollect(final3,\{diff(psi(r,t),r),diff(psi(r
,t),t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "final5:=mcoll
ect(final4,\{diff(psi(r,t),t),diff(psi(r,t),t,r)\}):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 53 "final6:=mcollect(final5,\{psi(r,t),diff(p
si(r,t),r)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "final7:=m
collect(final6,\{diff(psi(r,t),r),diff(psi(r,t),t,r,r)\}):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "final8:=mcollect(final7,\{diff(psi(
r,t),t),diff(psi(r,t),r,r)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 55 "final9:=mcollect(final8,\{psi(r,t),diff(psi(r,t),r,t)\}):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "final10:=mcollect(final9,\{d
iff(psi(r,t),r),diff(psi(r,t),r,t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "final11:=mcollect(final10,\{psi(r,t),diff(psi(r,t),t,
r,r)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "final12:=mcolle
ct(final11,\{diff(psi(r,t),r),diff(psi(r,t),r,r)\}):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "final13:=mcollect(final12,\{psi(r,t)\}):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "final14:=mcollect(final
13,\{diff(psi(r,t),t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "final15:=mcollect(final14,\{diff(psi(r,t),r)\}):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "final16:=mcollect(final15,\{diff(psi(r,t)
,r,r)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "final17:=mcoll
ect(final16,\{diff(psi(r,t),r,t)\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "nops(final17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#
U" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We now make one final set of
 substitutions, " }{TEXT 279 9 "mu=(r-2m)" }{TEXT -1 7 ", and, " }
{TEXT 280 14 "lambda=(2r+3m)" }{TEXT -1 78 ", and extract the source t
erms from the second order Zerilli wave equation:\n\n\n" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 126 "zetasource:=subs((r-2*m)=mu,(2*m-r)=-mu,(2*r+
3*m)=lambda,\nkfactor(eval(subs(zeta(r,t)=0,-final17)),12/7*(r-2*m)^3/
(2*r+3*m)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+zetasourceG,$*&*&)%
#muG\"\"$\"\"\",D*&*(,(*$)%\"rG\"\"#F+F**&%\"mG\"\"\"F2F6\"\"&*$)F5F3F
+\"\"'F6-%%diffG6$-%$psiG6$F2%\"tGF2F6-F<6$F>-%\"$G6$F2F3F6F+*$)F2\"\"
&F+!\"\"#\"\"%F**&*(F5F+F>F6-F<6%F>FDFAF6F+*&)F2\"\"$F+%'lambdaG\"\"\"
FJ!\"\"*&*(,&F8F6F0!\"#F6-F<6$F>FAF6FBF+F+*(F)\"\"\")F2\"\"$F+FT\"\"\"
FJF6*&*&)-F<6%F>F2FAF3F+FTF6F+*&F)\"\"\")F2\"\"#F+FJ#FVF**&*&FenF+-F<6
$F>-FE6$F2F*F6F+*$)F2\"\"#F+FJFeo*&*(,.*$)F2F7F+\"#K*&)F2FLF+F5F+\"#))
*&)F2F*F+F9F+\"$'H*&F1F+)F5F*F+\"$5&*&F2F+)F5FLF+\"$h&*$)F5F7F+\"$q#F6
F>F+F;F+F+*(F)\"\"\")F2\"\"(F+)FT\"\"#F+FJFZ*&*(,&F5\"\"(F2!\"$F6F;F+F
_oF6F+*&F)\"\"\")F2\"\"$F+FJFeo*&*(,**$FipF+F3*&F1F+F5F+FL*&F2F+F9F+\"
\"**$F\\qF+F:F6F>F+FBF+F+*&)F2\"\"'F+FT\"\"\"FJ!\"%*&*(,(F0\"\")F4\"#7
F8F]rF6F>F+F_oF+F+*(F)\"\"\")F2\"\"%F+FT\"\"\"FJF6*&*(,.Fbp\"$7\"Fep\"
$![Fhp\"$#pF[q\"$i(F^q\"$T%Faq\"$W\"F6F>F+FenF+F+*()F)\"\"#F+)FT\"\"$F
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$)F2\"\"%F+FJF_u*&*&,.FbpFZFep!\"*Fhp!\"'F[qF3F^q\"#:FaqF[vF6)F>F3F+F+
*()F)\"\"#F+)F2\"\")F+FT\"\"\"FJ!#7*&*&),(F0F6F4F6F8F6F3F+)FenF3F+F+*(
)F2\"\"%F+)F)\"\"$F+FT\"\"\"FJFcv*&*(FgvF6FenF+F_oF+F+*&)F)\"\"#F+)F2
\"\"$F+FJF_s*&*(,*Ffr!#=FgrFLFhr\"#LFjr\"#[F6FenF+F;F+F+*()F)\"\"#F+)F
2\"\"%F+FT\"\"\"FJFeo*&*&,*FfrFdsFgr\"#OFhr\"#fFjr\"#!*F6)F;F3F+F+*&F)
\"\"\")F2\"\"'F+FJF_uF6F+FTFJ#FdsF]r" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 26 "\nThis is our final result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "final18:=subs((r-2*m)=mu,(-r+2*m)=-
mu,(2*r+3*m)=lambda,final17):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 123 "final19:=kfactor(collect(final18,\{zeta(r,t),diff(zeta(r,t),r),
diff(zeta(r,t),r,r),\ndiff(zeta(r,t),t,t)\}),12/7*mu^3/lambda):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "simplify(expand(final19-fina
l18));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 89 "termsimp(simplify(expand(subs(zeta(r,t)=0,lamb
da=(2*r+3*m),mu=(r-2*m),\nfinal19)+s_ren)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&*(,(\"\"\"F'%\"mG\"\"#%\"rG!\"\"F'),(*$)F*F)\"\"\"F
'*&F(F'F*F'F'*$)F(F)F0F'F)F0)-%%diffG6$-%$psiG6$F*%\"tGF;F)F0F0*&)F*\"
\"%F0),&F*F)F(\"\"$\"\"#F0!\"\"#\"$W\"\"\"(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 90 "termsimp(simplify(expand(subs(zeta(r,t)=0,lambda=(
2*r+3*m),mu=(r-2*m),final19)\n+sprime)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
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