{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 19 "limittwospecial.mws" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 33 "v(theta)=1,dv(theta)/dthet a not 0" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 86 "(dv(theta)/dtheta )^2=exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 "The following substitutions for P(t, theta) and Q(t,theta) are made:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "P(t,theta)=P[infinity](theta)-v(theta)*log(t)+V(theta)*t^2*(l og(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t)),Q(t,theta)=Q[infinit y](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v(theta)))*log(t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"PG6$%\"tG%&thetaG,,-&F%6#%)in finityG6#F(\"\"\"*&-%\"vGF.F/-%#lnG6#F'F/!\"\"*(-%\"VGF.F/)F'\"\"#F/)F 3F;F/F/*&-&F96#F/F.F/F:F/F/*(-&F96#F;F.F/F:F/F3F/F//-%\"QGF&,(-&FGF,F. F/*&-%\"qGF.F/)F',$*&F;F/F1F/F/F/F/*(-%#qqGF.F/FNF/F3F/F/" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 4 "with" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "qq(theta):=diff(Q[infinity](theta),`$`(theta,2))/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#qqG6#%&thetaG,$*&#\"\"\"\"\"#F+- %%diffG6$-&%\"QG6#%)infinityGF&-%\"$G6$F'F,F+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 78 "V(theta):=exp(P[infinity](theta))^2*diff(Q[inf inity](theta),`$`(theta,2))^2/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>- %\"VG6#%&thetaG,$*&#\"\"\"\"\"%F+*&)-%$expG6#-&%\"PG6#%)infinityGF&\" \"#F+)-%%diffG6$-&%\"QGF5F&-%\"$G6$F'F7F7F+F+F+" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{PARA 6 "" 1 "" {TEXT -1 25 "Scalar invariant librar y." }}{PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 1997." }} {PARA 6 "" 1 "" {TEXT -1 25 "`Differential Invariants`" }}{PARA 6 "" 1 "" {TEXT -1 29 "`Last modified Jan. 20, 1995`" }}{PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad related object definitions`" }}{PARA 6 "" 1 "" {TEXT -1 31 "`Last modified 23 January 2001`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTens orII~Version~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~Februar y~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgra ve,~Denis~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% DCopyright~1994-2001~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~available~from:~http://grtensor.phy.queensu.ca/G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(gowdy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&gowdyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~gowdy~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"x G6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7& %\"tG%&thetaG%#x1G%#x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elem entG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",,**-%$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 0 219 "grmap(_,subs,P(t,theta)=P[infinity](theta)-v(theta)* log(t)+V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t) ),Q(t,theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^ (2*(v(theta)))*log(t),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying r outine subs to WeylSq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 51 "App ly consistency relation but keep all 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 W eylSq" }}}{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[infinity](theta),theta$2)=Q[2],`x `);" }{MPLTEXT 0 21 0 "" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routi ne 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 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$1)=v[1],`x` );" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "grmap(_,subs,v(theta)=1,`x` );" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }} }{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 96 "To do the special case we al so insert the following (it turns out that the sign does not matter)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "grmap(_,subs,v[1]=Q[2]*ex p(P[infinity](theta)),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying r outine subs to WeylSq" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 53 "grm ap(_,subs,v[1]=-Q[2]*exp(P[infinity](theta)),`x`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "gralter(_,6);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 " " {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$ Q%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "expand(grcomponen t(WeylSq,[])/(t*log(t)^4*exp(gamma(t,theta)))):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "limit(%,t=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"#;\"\"\")-%\"VG6#%&thetaG\"\"#F&F&*(F%F&)-%$expG6#-&%\"PG6#%) infinityGF*\"\"%F&)-%#qqGF*F7F&F&**\"#KF&F(F&)F9F,F&)F/F,F&!\"\"*(\"\" $F&)&%\"QG6#F,F7F&F.F&F&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 16 " Put in qq(theta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "subs(qq (theta)=diff(Q[infinity](theta),`$`(theta,2))/2,%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,**&\"#;\"\"\")-%\"VG6#%&thetaG\"\"#F&F&*&)-%$expG6#- &%\"PG6#%)infinityGF*\"\"%F&)-%%diffG6$-&%\"QGF5F*-%\"$G6$F+F,F7F&F&** \"\")F&F(F&)F9F,F&)F/F,F&!\"\"*(\"\"$F&)&F>6#F,F7F&F.F&F&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 11 "Put in Q[2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "subs(Q[2]=diff(Q[infinity](theta),theta$2),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"#;\"\"\")-%\"VG6#%&thetaG\"\"# F&F&*(\"\"%F&)-%$expG6#-&%\"PG6#%)infinityGF*F.F&)-%%diffG6$-&%\"QGF6F *-%\"$G6$F+F,F.F&F&**\"\")F&F(F&)F9F,F&)F0F,F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Ww:=t*log(t)^4*exp(gamma(t,theta))*%;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#WwG**%\"tG\"\"\")-%#lnG6#F&\"\"%F'- %$expG6#-%&gammaG6$F&%&thetaGF',(*&\"#;F')-%\"VG6#F3\"\"#F'F'*(F,F')-F .6#-&%\"PG6#%)infinityGF:F,F')-%%diffG6$-&%\"QGFCF:-%\"$G6$F3F;F,F'F'* *\"\")F'F8F')FFF;F')F>F;F'!\"\"F'" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 48 "Note that the form of V(theta) must be inserted." }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 15 "Put in V(theta)" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 78 "V(theta):=exp(P[infinity](theta))^2*diff(Q[i nfinity](theta),`$`(theta,2))^2/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >-%\"VG6#%&thetaG,$*&#\"\"\"\"\"%F+*&)-%$expG6#-&%\"PG6#%)infinityGF& \"\"#F+)-%%diffG6$-&%\"QGF5F&-%\"$G6$F'F7F7F+F+F+" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 11 "factor(Ww);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"\"$\"\"\"%\"tGF&)-%#lnG6#F'\"\"%F&-%$expG6#-%&gammaG6$F'%&the taGF&)-F.6#-&%\"PG6#%)infinityG6#F3F,F&)-%%diffG6$-&%\"QGF:F<-%\"$G6$F 3\"\"#F,F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" } }{PARA 6 "" 1 "" {TEXT -1 67 "3\\,t \\left( \\ln \\left( t \\right) \+ \\right) ^\{4\}\{e^\{\\gamma \\left( t," }}{PARA 6 "" 1 "" {TEXT -1 67 "\\theta \\right) \}\} \\left( \{e^\{P_\{\{\\infty \}\} \\left( \\t heta \\right) \}\}" }}{PARA 6 "" 1 "" {TEXT -1 64 " \\right) ^\{4\} \\ left( \{\\frac \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}Q_\{\{\\infty \}\}" }}{PARA 6 "" 1 "" {TEXT -1 36 " \\left( \\theta \\right) \\right) ^\{ 4\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kernelopts(cputime); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'[q7!\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "43 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }