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{SECT 0 {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 15 "limiteleven.mws" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 48 "The special case v(theta)=-1 , with constraints," }}
}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 36 "Notation diff(v(thtea),theta
$n)=v[n]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 35 "the following substitution is
 made." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "P(t,theta) = P[i
nfinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[in
finity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[in
finity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/
4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,
2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(th
eta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"PG6$%\"tG%&thetaG,,-&F%6#%)in
finityG6#F(\"\"\"*&-%\"vGF.F/-%#lnG6#F'F/!\"\"*&#F/\"\"%F/**)-%$expG6#
F*\"\"#F/)-%%diffG6$-&%\"QGF,F.-%\"$G6$F(F?F?F/)F'F?F/)F3F?F/F/F/*&-&%
\"VG6#F/F.F/FJF/F/*(,(*(-F=6#,$*&F?F/F*F/F/F/-&%$psiG6#FFF.F/FAF/F/*&#
F/F9F/*$F@F/F/F6*&#F/F9F/-FB6$F1FGF/F6F/FJF/F3F/F//-FFF&,(FDF/*&FXF/)F
',$*&F?F/F1F/F/F/F/*&#F/F?F/*(FAF/FaoF/F3F/F/F/" }}}{EXCHG {PARA 0 "" 
0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "gr
tw();" }}{PARA 6 "" 1 "" {TEXT -1 25 "Scalar invariant library." }}
{PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 1997." }}{PARA 6 "
" 1 "" {TEXT -1 25 "`Differential Invariants`" }}{PARA 6 "" 1 "" 
{TEXT -1 29 "`Last modified Jan. 20, 1995`" }}{PARA 6 "" 1 "" {TEXT 
-1 41 "`Basis/tetrad related object definitions`" }}{PARA 6 "" 1 "" 
{TEXT -1 31 "`Last modified 23 January 2001`" }}{PARA 6 "" 1 "" {TEXT 
-1 25 "`Last built 27 May, 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last
 built 27 May, 1999`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~V
ersion~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~De
nis~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyr
ight~1994-2001~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%in
Latest~version~available~from:~http://grtensor.phy.queensu.ca/G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(6)/MetricsG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(gowdy);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%2Default~spacetimeG%&gowdyG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9For~the~gowdy~spacetime:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"x
G6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7&
%\"tG%&thetaG%#x1G%#x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elem
entG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",,**-%$ex
pG6#,$*&#F(F'F(-%&gammaG6$%\"tG%&thetaGF(!\"\"F(F4#F6F'%#~dGF()F4%#2~G
F(F6**F+F(F4F7F8F()F5F:F(F(**F4F(-F,6#-%\"PGF3F(F8F()%#x1GF:F(F(*2F'F(
F4F(F>F(-%\"QGF3F(F8F()FC%\"~GF(%#d~GF()%#x2GFHF(F(*(,&*(F4F(F>F()FEF'
F(F(*&F4F(F>F6F(F(F8F()FKF:F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%,
ConstraintsG7&/-%%diffG6$-%&gammaG6$%\"tG%&thetaGF-,**(F-\"\"\")-%$exp
G6#-%\"PGF,\"\"#F1)-F(6$-%\"QGF,F-F8F1!\"\"*&F-F1)-F(6$F6F-F8F1F>*(F-F
1F2F1)-F(6$F<F.F8F1F>*&F-F1)-F(6$F6F.F8F1F>/-F(6$F*F.,&**F8F1F-F1FIF1F
AF1F>*,F8F1F-F1F2F1F:F1FEF1F>/-F(6$F6-%\"$G6$F-F8,(-F(6$F6-FU6$F.F8F1*
&FAF1F-F>F>*&-F46#,$*&F8F1F6F1F1F1,&*$F9F1F1*$FDF1F>F1F1/-F(6$F<FT,*-F
(6$F<FZF1*&F:F1F-F>F>*(F8F1FAF1F:F1F>*(F8F1FIF1FEF1F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(WeylSq);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%*CPU~Time~G$\"#%*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "gralter(_,13,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Co
mponent simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" 
{TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "Applying routine `Apply c
onstraints repeatedly` to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 
40 "Applying routine expand to object WeylSq" }}{PARA 6 "" 1 "" {TEXT 
-1 40 "Applying routine factor to object WeylSq" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%*CPU~Time~G$\"#Z!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 453 "grmap(_,subs,P(t,theta) = P[infinity](theta)-v(theta
)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(thet
a,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](
theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](thet
a),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,thet
a) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infini
ty](theta),`$`(theta,2))*t^(2*v(theta))*ln(t),`x`);" }}{PARA 6 "" 1 "
" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "" 
0 "" {MPLTEXT 0 21 42 "Keep all derivatives of Q[infinity](theta)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "grmap(_,subs,diff(Q[infinity
](theta),theta$5)=Q[5],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying \+
routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "g
rmap(_,subs,diff(Q[infinity](theta),theta$4)=0,`x`);" }}{PARA 6 "" 1 "
" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 53 "grmap(_,subs,diff(Q[infinity](theta),theta$3)=
0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylS
q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "grmap(_,subs,diff(Q[in
finity](theta),theta)=Q[1],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Apply
ing routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
53 "grmap(_,subs,diff(Q[infinity](theta),theta$2)=0,`x`);" }{MPLTEXT 
0 21 0 "" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to Weyl
Sq" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 32 "Keep all deriv
atives of v(theta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap
(_,subs,diff(v(theta),theta$4)=v[4],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 
31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),theta$3)=v[3],`x`);" }}
{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),t
heta$2)=v[2],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine su
bs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,sub
s,diff(v(theta),theta$1)=v[1],`x`):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Ap
plying routine subs to WeylSq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 
21 37 "Keep all derivatives of psi[Q](theta)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 59 "grmap(_,subs,diff(psi[Q](theta),`$`(theta,2))=ps
i[Q2],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to W
eylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "grmap(_,subs,diff(
psi[Q](theta),`$`(theta,3))=psi[Q3],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 
31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "grmap(_,subs,diff(psi[Q](theta),`$`(theta,4))=psi[Q4]
,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {MPLTEXT 0 21 17 "Apply constraints" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "grmap(_,subs,diff(psi[Q](theta),theta)=0,`x`):" }}
{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "grmap(_,subs,psi[Q](theta)=0
,`x`):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "grmap(_,subs,v(theta)=-1,`x`):" }}{PARA 6 "" 1 "
" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 
48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "
" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand
 to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine fac
tor to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$
\"$M#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "denom(grcompon
ent(WeylSq,[]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"&%Q;\"\"\")-
%$expG6#,$*&#F&\"\"#F&-%&gammaG6$%\"tG%&thetaGF&!\"\"F.F&F2F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "factor(subs(t=0,numer(grcomp
onent(WeylSq,[]))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"&Ob'\"\"
\",&*(-%$expG6#-&%\"PG6#%)infinityG6#%&thetaGF&-F*6#\"\"!F&&%$psiG6#%#
Q2GF&F&&%\"vG6#F&!\"\"F&,&F(F&F:F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "core:=
simplify(limit(factor(t*grcomponent(WeylSq,[])/(exp(gamma(t,theta)))),
t=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coreG,&*&\"\"%\"\"\")&%\"
vG6#F(\"\"#F(!\"\"*(F'F(-%$expG6#,$*&F-F(-&%\"PG6#%)infinityG6#%&theta
GF(F(F()&%$psiG6#%#Q2GF-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "Ww:=core*exp(gamma(t,theta))/t;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#WwG*(,&*&\"\"%\"\"\")&%\"vG6#F)\"\"#F)!\"\"*(F(F)-%$expG6#,$*&F.
F)-&%\"PG6#%)infinityG6#%&thetaGF)F)F))&%$psiG6#%#Q2GF.F)F)F)-F26#-%&g
ammaG6$%\"tGF<F)FGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "sub
s(v[1]=diff(v(theta),theta$1),psi[Q2]=diff(psi[Q](theta),`$`(theta,4))
,Ww);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*&\"\"%\"\"\")-%%diffG6$-
%\"vG6#%&thetaGF/\"\"#F'!\"\"*(F&F'-%$expG6#,$*&F0F'-&%\"PG6#%)infinit
yGF.F'F'F')-F*6$-&%$psiG6#%\"QGF.-%\"$G6$F/F&F0F'F'F'-F46#-%&gammaG6$%
\"tGF/F'FMF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" }
}{PARA 6 "" 1 "" {TEXT -1 63 "\{\\frac \{ \\left( -4\\, \\left( \{\\fr
ac \{d\}\{d\\theta\}\}v \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 
70 " \\right)  \\right) ^\{2\}+4\\,\{e^\{2\\,P_\{\{\\infty \}\} \\left
( \\theta \\right) \}" }}{PARA 6 "" 1 "" {TEXT -1 63 "\} \\left( \{\\f
rac \{d^\{4\}\}\{d\{\\theta\}^\{4\}\}\}\\psi_\{\{Q\}\} \\left( \\theta
" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\right)  \\right) ^\{2\} \\right) \+
\{e^\{\\gamma \\left( t,\\theta \\right) \}\}\}\{" }}{PARA 6 "" 1 "" 
{TEXT -1 3 "t\}\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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2 33 1 1 }