{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 13 "limitfour.mws" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 108 "The special case v(theta)=1 , diff(Q[infinify](the ta),theta)=0 for consistency and diff(v(theta),theta) = 0" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 64 "Notation 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 f or P(t,theta) and Q(t,theta) are made:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*e xp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*l n(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q [infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta, 2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infini ty](theta)+psi[Q](theta)*t^(2*v(theta))+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#%)infinityG6#F(\"\"\"*&-%\"vGF.F/-%#lnG6# F'F/!\"\"*&#F/\"\"%F/**)-%$expG6#F*\"\"#F/)-%%diffG6$-&%\"QGF,F.-%\"$G 6$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/F 6F/FJF/F3F/F//-FFF&,(FDF/*&FXF/)F',$*&F?F/F1F/F/F/F/*&#F/F?F/*(FAF/Fao F/F3F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "grtw();" }}{PARA 6 "" 1 "" {TEXT -1 25 "Scalar invar iant library." }}{PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 1 997." }}{PARA 6 "" 1 "" {TEXT -1 25 "`Differential Invariants`" }} {PARA 6 "" 1 "" {TEXT -1 29 "`Last modified Jan. 20, 1995`" }}{PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad related object definitions`" }} {PARA 6 "" 1 "" {TEXT -1 31 "`Last modified 23 January 2001`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~b y~Peter~Musgrave,~Denis~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~available~from:~http://grtensor. phy.queensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(6)/Metr icsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(gowdy);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&gowdyG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~gowdy~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%' vectorG6#7&%\"tG%&thetaG%#x1G%#x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\" \",,**-%$expG6#,$*&#F(F'F(-%&gammaG6$%\"tG%&thetaGF(!\"\"F(F4#F6F'%#~d GF()F4%#2~GF(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(*(,&*(F4 F(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-\"\"\")-%$expG6#-%\"PGF,\"\"#F1)-F(6$-%\"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$FF>*(F8F1FAF1F:F1F>*(F8F1FIF1FEF1F 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(WeylSq);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#%*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gralter(_,13,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "Applying \+ routine `Apply constraints repeatedly` to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to object WeylSq" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#J!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "grmap(_,subs,P(t,theta) = P[infini ty](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinit y](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinit y](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*dif f(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))* t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta)) +1/2*diff(Q[infinity](theta),`$`(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 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 57 "Apply consistency relation but keep all other deriva tives" }}}{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 56 "grmap(_,subs,diff(Q[infinity](theta),theta$3)=Q[3],`x `);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "grmap(_,subs,diff(Q[infini ty](theta),theta$2)=Q[2],`x`);" }{MPLTEXT 0 21 0 "" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs 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 deri vatives of v(theta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grma p(_,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 43 "grmap(_,sub s,diff(v(theta),theta$1)=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Apply ing routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "grmap(_,subs,v(theta)=1,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Appl ying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplifi cation of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq " }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to object Wey lSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"'%)**=!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "core:=simplify(limit(factor( t*grcomponent(WeylSq,[])/(exp(gamma(t,theta)))),t=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coreG,$*(\"\"%\"\"\"-%$expG6#,$*&\"\"#F(-&%\"PG 6#%)infinityG6#%&thetaGF(F(F()&%\"QG6#F.F.F(F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "factor(subs(v[1]=diff(v(theta),theta$1),Q[2]=diff(Q[infinity](theta), theta$2),core)*exp(gamma(t,theta))/t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"%\"\"\"-%$expG6#,$*&\"\"#F&-&%\"PG6#%)infinityG6#%&theta GF&F&F&-%%diffG6$-&%\"QGF0F2-%\"$G6$F3F,F,-F(6#-%&gammaG6$%\"tGF3F&FB! \"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kernelopts(cputim e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'ys[!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "latex(4*exp(2*P[infinity](theta))*diff(Q[in finity](theta),`$`(theta,2))^2*exp(gamma(t,theta))/t);" }}{PARA 6 "" 1 "" {TEXT -1 65 "4\\,\{\\frac \{\{e^\{2\\,P_\{\{\\infty \}\} \\left( \+ \\theta \\right) \}\} \\left( \{" }}{PARA 6 "" 1 "" {TEXT -1 65 "\\fra c \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}Q_\{\{\\infty \}\} \\left( \\thet a \\right) " }}{PARA 6 "" 1 "" {TEXT -1 55 " \\right) ^\{2\}\{e^\{\\ga mma \\left( t,\\theta \\right) \}\}\}\{t\}\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }