{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 12 "limitten.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-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$FF>*(F8F1FAF1F:F1F>*(F8F1FIF1FEF1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(WeylSq);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%*CPU~Time~G$\"#%*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gralter(_,13,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Co mponent simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "Applying routine `Apply c onstraints repeatedly` to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#J!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "grmap(_,subs,P(t,theta) = P[infinity](theta)-v(theta )*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(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$4)=Q[4],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying \+ routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "g rmap(_,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 54 "grmap(_,subs,diff(Q[infinity](theta),theta)= Q[1],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to We ylSq" }}}{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 WeylSq" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 32 "Keep all derivatives 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 s ubs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,su bs,diff(v(theta),theta$3)=v[3],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "A pplying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),theta$2)=v[2],`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$1)=v[1],`x` ):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }} }{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 37 "Keep all derivatives of psi[ Q](theta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "grmap(_,subs,d iff(psi[Q](theta),`$`(theta,2))=psi[Q2],`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),`$`(theta,3))=ps i[Q3],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to W eylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "grmap(_,subs,diff( psi[Q](theta),`$`(theta,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 c onstraints" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs, diff(psi[Q](theta),theta)=0,`x`):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Appl ying 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 GRTens orII 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 WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$d'!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "denom(grcomponent(WeylSq,[]));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*(\"&%Q;\"\"\")-%$expG6#,$*&#F&\"\"#F&-%&gammaG6$% \"tG%&thetaGF&!\"\"F.F&)F2\"\"$F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "factor(subs(t=0,numer(grcomponent(WeylSq,[]))));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"#k\"\"\")-%$expG6#-&%\"PG6#%)inf inityG6#%&thetaG\"\"#F&)-F)6#\"\"!F2F&)&%\"QG6#\"\"$F2F&,.**)F8\"\"%F& )F4F?F&)F(F?F&)-%#lnGF5\"\"'F&F&*&\"$c#F&)FCF2F&F&*&FGF&FCF&!\"\"FGF&* ,\"#KF&F7F&F3F&F'F&)FCF;F&FJ*,\"#'*F&F7F&F3F&F'F&)FCF?F&FJF&FJ" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "core:=limit(factor(t^3*grcomponent(WeylSq,[])/(log(t) ^6*exp(gamma(t,theta)))),t=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%c oreG,$*&#\"\"\"\"$c#F(*&)-%$expG6#-&%\"PG6#%)infinityG6#%&thetaG\"\"'F ()&%\"QG6#\"\"$F6F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Ww:=core*log(t)^6*exp(gamma(t,theta))/t^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#WwG,$*&#\"\"\"\"$c#F(*,-%$expG6#-&%\"PG6#%)infinityG 6#%&thetaG\"\"'&%\"QG6#\"\"$F5-%#lnG6#%\"tGF5-F,6#-%&gammaG6$F=F4F(F=! \"$F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "subs(Q[3]=dif f(Q[infinity](theta),theta$3),Ww);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&#\"\"\"\"$c#F&*,-%$expG6#-&%\"PG6#%)infinityG6#%&thetaG\"\"'-%%dif fG6$-&%\"QGF/F1-%\"$G6$F2\"\"$F3-%#lnG6#%\"tGF3-F*6#-%&gammaG6$FAF2F&F A!\"$F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" } }{PARA 6 "" 1 "" {TEXT -1 66 "-\{\\frac \{1\}\{256\}\}\\,\{\\frac \{ \+ \\left( \{e^\{P_\{\{\\infty \}\} \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 66 " \\right) \}\} \\right) ^\{6\} \\left( \{\\frac \{d^\{3\} \}\{d\{\\theta\}^\{3\}\}\}Q_\{\{" }}{PARA 6 "" 1 "" {TEXT -1 66 "\\inf ty \}\} \\left( \\theta \\right) \\right) ^\{6\} \\left( \\ln \\left ( t" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\right) \\right) ^\{6\}\{e^\{\\ gamma \\left( t,\\theta \\right) \}\}\}\{\{t\}^\{3\}\}\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 8 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }