{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 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 14 "limitthree.mws" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 80 "The special case v(theta)=1 , diff(Q[infinify](thet a),theta)=0 for consistency " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 36 "Notation diff(v(thtea),theta$n)=v[n]" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 23 "Here dv(theta)/dtheta=0" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 "The following substit utions for P(t,theta) and Q(t,theta) are made:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 201 "P(t,theta)=P[infinity](theta)-v(theta)*log(t) +V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t)),Q(t, theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v( theta)))*log(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"PG6$%\"tG%&th etaG,,-&F%6#%)infinityG6#F(\"\"\"*&-%\"vGF.F/-%#lnG6#F'F/!\"\"*(-%\"VG F.F/)F'\"\"#F/)F3F;F/F/*&-&F96#F/F.F/F:F/F/*(-&F96#F;F.F/F:F/F3F/F//-% \"QGF&,(-&FGF,F.F/*&-%\"qGF.F/)F',$*&F;F/F1F/F/F/F/*(-%#qqGF.F/FNF/F3F /F/" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 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 invaria nt library." }}{PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 199 7." }}{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#%=GRT ensorII~Version~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~Febr uary~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Mus grave,~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 26 "App ly consistency relation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " grmap(_,subs,diff(Q[infinity](theta),theta)=0,`x`);" }{MPLTEXT 0 21 0 "" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 46 "Make sure you keep all deriva tives of v(theta)" }}}{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 43 "grmap(_,subs,diff(v(theta),theta$1)=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "grmap(_,subs,v(theta)=1,`x`);" }} {PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);" }}{PARA 6 " " 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applyin g routine expand to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Appl ying routine factor to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%*CPU~Time~G$\"%>K!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "core1:=expand(grcomponent(WeylSq,[])/(t*log(t)^4*exp(gamma(t,theta )))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "core2:=factor(limi t(core1,t=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&core2G,$*&\"#;\" \"\"),&-%\"VG6#%&thetaG!\"\"*&)-%$expG6#-&%\"PG6#%)infinityGF-\"\"#F() -%#qqGF-F:F(F(F:F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "t*l og(t)^4*exp(gamma(t,theta))*core2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*,\"#;\"\"\"%\"tGF&)-%#lnG6#F'\"\"%F&-%$expG6#-%&gammaG6$F'%&thetaGF &),&-%\"VG6#F3!\"\"*&)-F.6#-&%\"PG6#%)infinityGF8\"\"#F&)-%#qqGF8FCF&F &FCF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kernelopts(cputi me);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&E`$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "27 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }