{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 14 "limitseven.mws" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 143 "The special case v(theta)=1 , diff(Q[infinify](the ta),theta)=0 for consistency and diff(v(theta),theta) = 0 diff(Q[infin ity](theta),theta$2)=0" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 64 "No tation diff(v(thtea),theta$n)=v[n] diff(Q(thtea),theta$n)=Q[n]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 "The following substitutions for P(t,theta) and Q(t,theta) are made:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "P[infinity](theta)-v(the ta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(th eta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q ](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](th eta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t)+t^4*Z( t,thea),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1 /2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t)+t^4*W(t, theta);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$,.-&%\"PG6#%)infinityG6#%&t hetaG\"\"\"*&-%\"vGF)F+-%#lnG6#%\"tGF+!\"\"*&#F+\"\"%F+**)-%$expG6#F$ \"\"#F+)-%%diffG6$-&%\"QGF'F)-%\"$G6$F*FF+F+*&#F+ F6F+*$F=F+F+F3*&#F+F6F+-F?6$F-FDF+F3F+FGF+F/F+F+*&)F2F6F+-%\"ZG6$F2%%t heaGF+F+/-FC6$F2F*,*FAF+*&FUF+)F2,$*&FF+FeoF +F/F+F+F+*&F[oF+-%\"WGFboF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "V[1](theta):=(exp(P[infinity](theta))^2*diff(Q[infinity](theta) ,`$`(theta,2))^2/4)*(3/2)-exp(2*P[infinity](theta))*psi[Q](theta)*diff (Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(thet a,2))^2-1/4*diff(v(theta),`$`(theta,2))+exp(P[infinity](theta))^2*(psi [Q](theta)+diff(Q[infinity](theta),`$`(theta,2))/4)^2+(diff(P[infinity ](theta),theta$2)+diff(v(theta),theta$2))/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"VG6#\"\"\"6#%&thetaG,,*&#\"\"$\"\")F(*&)-%$expG6# -&%\"PG6#%)infinityGF)\"\"#F()-%%diffG6$-&%\"QGF8F)-%\"$G6$F*F:F:F(F(F (*(-F36#,$*&F:F(F5F(F(F(-&%$psiG6#FAF)F(F " 0 "" {MPLTEXT 1 0 9 "latex(%);" }}{PARA 6 "" 1 "" {TEXT -1 68 "3/8\\, \\left( \{e^\{P_\{\{\\infty \}\} \\left( \\theta \\right ) \}\} \\right) ^\{2\}" }}{PARA 6 "" 1 "" {TEXT -1 65 " \\left( \{\\fr ac \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}Q_\{\{\\infty \}\} \\left( \\the ta" }}{PARA 6 "" 1 "" {TEXT -1 68 " \\right) \\right) ^\{2\}-\{e^\{2 \\,P_\{\{\\infty \}\} \\left( \\theta \\right) \}\}" }}{PARA 6 "" 1 " " {TEXT -1 67 "\\psi_\{\{Q\}\} \\left( \\theta \\right) \{\\frac \{d^ \{2\}\}\{d\{\\theta\}^\{2\}\}\}Q_\{\{" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\infty \}\} \\left( \\theta \\right) -1/4\\, \\left( \{\\frac \{d^\{2 \}\}\{d\{\\theta\}" }}{PARA 6 "" 1 "" {TEXT -1 70 "^\{2\}\}\}Q_\{\{\\i nfty \}\} \\left( \\theta \\right) \\right) ^\{2\}+ \\left( \{e^\{P_ " }}{PARA 6 "" 1 "" {TEXT -1 67 "\{\{\\infty \}\} \\left( \\theta \\ri ght) \}\} \\right) ^\{2\} \\left( \\psi_\{\{Q\}\}" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\left( \\theta \\right) +1/4\\,\{\\frac \{d^\{2\}\}\{d \{\\theta\}^\{2\}\}\}Q_\{\{\\infty " }}{PARA 6 "" 1 "" {TEXT -1 70 "\} \} \\left( \\theta \\right) \\right) ^\{2\}+1/4\\,\{\\frac \{d^\{2\} \}\{d\{\\theta\}^\{" }}{PARA 6 "" 1 "" {TEXT -1 40 "2\}\}\}P_\{\{\\inf ty \}\} \\left( \\theta \\right) " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{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 "`Differe ntial 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 Januar y 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~Version~1.79~(R4)G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%06~February~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayll~LakeG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~the~authors.G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~available~from:~ http://grtensor.phy.queensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4 c:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qlo ad(gowdy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&g owdyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~gowdy~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&%\"tG%&thetaG%#x1G%#x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$) %$~dsG\"\"#\"\"\",,**-%$expG6#,$*&#F(F'F(-%&gammaG6$%\"tG%&thetaGF(!\" \"F(F4#F6F'%#~dGF()F4%#2~GF(F6**F+F(F4F7F8F()F5F:F(F(**F4F(-F,6#-%\"PG F3F(F8F()%#x1GF:F(F(*2F'F(F4F(F>F(-%\"QGF3F(F8F()FC%\"~GF(%#d~GF()%#x2 GFHF(F(*(,&*(F4F(F>F()FEF'F(F(*&F4F(F>F6F(F(F8F()FKF:F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%,ConstraintsG7&/-%%diffG6$-%&gammaG6$%\"tG%&t hetaGF-,**(F-\"\"\")-%$expG6#-%\"PGF,\"\"#F1)-F(6$-%\"QGF,F-F8F1!\"\"* &F-F1)-F(6$F6F-F8F1F>*(F-F1F2F1)-F(6$F*&F-F1)-F(6$F6F.F8F1F>/ -F(6$F*F.,&**F8F1F-F1FIF1FAF1F>*,F8F1F-F1F2F1F:F1FEF1F>/-F(6$F6-%\"$G6 $F-F8,(-F(6$F6-FU6$F.F8F1*&FAF1F-F>F>*&-F46#,$*&F8F1F6F1F1F1,&*$F9F1F1 *$FDF1F>F1F1/-F(6$FF>*(F8F1FAF1F:F1F>*(F8F1F IF1FEF1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "psi[Q](theta) := (Int((1/2*diff(P[infinity](theta),`$`(theta,2))+1/2*diff(P[infinity ](theta),theta)^2)*exp(P[infinity](theta)),theta)+_C1)*exp(-2*P[infini ty](theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%$psiG6#%\"QG6#%&th etaG*&,&-%$IntG6$*&,&*&#\"\"\"\"\"#F4-%%diffG6$-&%\"PG6#%)infinityGF)- %\"$G6$F*F5F4F4*&F3F4*$)-F76$F9F*F5F4F4F4F4-%$expG6#F9F4F*F4%$_C1GF4F4 -FG6#,$*&F5F4F9F4!\"\"F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(WeylSq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$ \"G!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gralter(_,13,6,7 );" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTen sorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "Applying routine `Apply constraints repeatedly` to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to obj ect WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to \+ object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$a%! \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "grmap(_,subs,P(t,th eta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2 *diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+( exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(th eta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta), `$`(theta,2)))*t^2*ln(t)+t^4*Z(t,thea),Q(t,theta) = Q[infinity](theta) +psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2) )*t^(2*v(theta))*ln(t)+t^4*W(t,theta),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 57 "Apply co nsistency relation but keep all other derivatives" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 56 "grmap(_,subs,diff(Q[infinity](theta),theta$4 )=Q[4],`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 "A pplying 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 sub s to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grmap(_,subs ,diff(Q[infinity](theta),theta)=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 46 "Make sure you keep all derivatives of v(theta)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),t heta$4)=v[4],`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$3)=v[3],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Ap plying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "grmap(_,subs,diff(v(theta),theta$2)=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "grmap(_,subs,diff(v(theta),theta$1)=0,`x`);" }} {PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "grmap(_,subs,v(theta)=1,`x`) ;" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "grmap(_,subs,diff(P[infinit y](theta),`$`(theta,2))=-diff(P[infinity](theta),theta)^2+4*diff(P[inf inity](theta),theta)*exp(P[infinity](theta))*psi[Q](theta)+2*exp(P[inf inity](theta))*diff(psi[Q](theta),theta),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gr alter(_,6);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification o f a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$%f!\"$" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 209 "V[1](theta):= (exp(-2*P[infinity](theta))*( Int((1/2*diff(P[infinity](theta),`$`(theta,2))+1/2*diff(P[infinity](th eta),theta)^2)*exp(P[infinity](theta)),theta)+_C1)^2+1/4*diff(P[infini ty](theta),`$`(theta,2)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"VG6#\"\"\"6#%&thetaG,&*&-%$expG6 #,$*&\"\"#F(-&%\"PG6#%)infinityGF)F(!\"\"F(),&-%$IntG6$*&,&*&#F(F2F(-% %diffG6$F3-%\"$G6$F*F2F(F(*&FAF(*$)-FC6$F3F*F2F(F(F(F(-F.6#F3F(F*F(%$_ C1GF(F2F(F(*&#F(\"\"%F(FBF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "core2:=(subs(t=0,expand(grcomponent(WeylSq,[])/(t*exp(gamma(t,th eta)))))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(core2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ker nelopts(cputime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&\\e\"!\"$" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 10 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }