{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 "limittwo.mws" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 110 "The special case v(theta)=1 , diff(Q[infinify](the ta),theta)=0 for consistency and diff(v(theta),theta) not 0" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 75 "Notation diff(v(thtea),theta$ n)=v[n], diff(Q[infinify](theta),theta$n)=Q[n]" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 "The following sub stitutions for P(t,theta) and Q(t,theta) are made:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "P(t,theta)=P[infinity](theta)-v(theta)*log (t)+V(theta)*t^2*(log(t))^2+V[1](theta)*t^2+V[2](theta)*t^2*(log(t)),Q (t,theta)=Q[infinity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2* (v(theta)))*log(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"PG6$%\"tG% &thetaG,,-&F%6#%)infinityG6#F(\"\"\"*&-%\"vGF.F/-%#lnG6#F'F/!\"\"*(-% \"VGF.F/)F'\"\"#F/)F3F;F/F/*&-&F96#F/F.F/F:F/F/*(-&F96#F;F.F/F:F/F3F/F //-%\"QGF&,(-&FGF,F.F/*&-%\"qGF.F/)F',$*&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$\"#K!\"$" }}}{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 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 63 "Apply consistency relation \+ but keep all derivatives of Q(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 "grmap(_,subs,diff(Q[infini ty](theta),theta$3)=Q[3],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applyin g 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 routine subs to Weyl Sq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grmap(_,subs,diff(Q[i nfinity](theta),theta)=0,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applyin g 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 subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),t heta$3)=v[3],`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$2)=v[2],`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Ap plying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grmap(_,subs,diff(v(theta),theta$1)=v[1],`x`);" }}{PARA 6 "" 1 " " {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "grmap(_,subs,v(theta)=1,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 " " {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine fac tor to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$ \"&g`%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "core:=simplif y(limit(factor(t^1*grcomponent(WeylSq,[])/(exp(gamma(t,theta)))),t=0)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coreG,&*&\"\"%\"\"\")&%\"vG6#F (\"\"#F(!\"\"*(F'F(-%$expG6#,$*&F-F(-&%\"PG6#%)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(the ta),theta$1),Q[2]=diff(Q[infinity](theta),theta$2),core)*exp(gamma(t,t heta))/t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%\"\"\",&*$)-%%d iffG6$-%\"vG6#%&thetaGF0\"\"#F&F&*&-%$expG6#,$*&F1F&-&%\"PG6#%)infinit yGF/F&F&F&)-F+6$-&%\"QGF;F/-%\"$G6$F0F1F1F&!\"\"F&-F46#-%&gammaG6$%\"t GF0F&FLFFFF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kernelopts(c putime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'8^I!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "latex(-4*(diff(v(theta),theta)^2-e xp(2*P[infinity](theta))*diff(Q[infinity](theta),`$`(theta,2))^2)*exp( gamma(t,theta))/t);" }}{PARA 6 "" 1 "" {TEXT -1 63 "-4\\,\{\\frac \{ \+ \\left( \\left( \{\\frac \{d\}\{d\\theta\}\}v \\left( \\theta" }} {PARA 6 "" 1 "" {TEXT -1 68 " \\right) \\right) ^\{2\}-\{e^\{2\\,P_\{ \{\\infty \}\} \\left( \\theta \\right) \}\}" }}{PARA 6 "" 1 "" {TEXT -1 65 " \\left( \{\\frac \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}Q_\{\{\\in fty \}\} \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\right) \\ right) ^\{2\} \\right) \{e^\{\\gamma \\left( t,\\theta \\right) \}\}\} \{" }}{PARA 6 "" 1 "" {TEXT -1 3 "t\}\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "16 0 0" 56 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }