{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 20 "limitfivespecial.mws " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 144 "The special case v(theta)=1 , diff(Q[infinify]( theta),theta)=0 for consistency and diff(v(theta),theta) = 0, diff(Q[i nfinity](theta),theta$2)=0" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 107 "The subcase diff(v(theta),`$`(theta,2))^2-diff(Q[infinity](theta) ,`$`(theta,3))^2*exp(2*P[infinity](theta))" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 64 "Notation diff(v(thtea),theta$n)=v[n] diff(Q(thtea),t heta$n)=Q[n]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 67 "The following substitutions for P(t,theta) and Q(t,t heta) are made:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "P(t,the ta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2* diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(e xp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(the ta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),` $`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)* t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(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.-%\"$G6$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/F AF/F/*&#F/F9F/*$F@F/F/F6*&#F/F9F/-FB6$F1FGF/F6F/FJF/F3F/F//-FFF&,(FDF/ *&FXF/)F',$*&F?F/F1F/F/F/F/*&#F/F?F/*(FAF/FaoF/F3F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " grtw();" }}{PARA 6 "" 1 "" {TEXT -1 25 "Scalar invariant library." }} {PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 1997." }}{PARA 6 " " 1 "" {TEXT -1 25 "`Differential Invariants`" }}{PARA 6 "" 1 "" {TEXT -1 29 "`Last modified Jan. 20, 1995`" }}{PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad related object definitions`" }}{PARA 6 "" 1 "" {TEXT -1 31 "`Last modified 23 January 2001`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~V ersion~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001 G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~De nis~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyr ight~1994-2001~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%in Latest~version~available~from:~http://grtensor.phy.queensu.ca/G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(gowdy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&gowdyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~gowdy~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"x G6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7& %\"tG%&thetaG%#x1G%#x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elem entG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",,**-%$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 57 "Apply consistency relation but keep all other d erivatives" }}}{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 98 "grmap(_,subs,diff(Q[infinity](theta),theta$3)=diff(v( theta),theta$2)/exp(P[infinity](theta)),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}}{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 rou tine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "grma p(_,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 43 "grmap(_,subs,diff(v(theta),theta$1)=0,`x`); " }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "grmap(_,subs,v(theta)=1,`x`) ;" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to WeylSq" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "gralter(_,6,7);" }}{PARA 6 " " 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applyin g routine expand to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Appl ying routine factor to object WeylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%*CPU~Time~G$\"$f)!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "core:=simplify(limit(grcomponent(WeylSq,[])/(t*log(t)*exp(gamma(t, theta))),t=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coreG,$*(\"\"$\" \"\"&%\"vG6#\"\"#F(,,*&F,F()-%%diffG6$-&%\"PG6#%)infinityG6#%&thetaGF9 F,F(F(F)F(*(\"\"%F(-%$expG6#F3F(-F16$-&%$psiG6#%\"QGF8F9F(F(*&F,F(-F16 $F3-%\"$G6$F9F,F(F(**\"\")F(F " 0 "" {MPLTEXT 1 0 76 "factor(subs(v[2]=diff(v(theta),theta$2),core )*t*log(t)*exp(gamma(t,theta)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *.\"\"$\"\"\"-%%diffG6$-%\"vG6#%&thetaG-%\"$G6$F-\"\"#F&,,*&F1F&)-F(6$ -&%\"PG6#%)infinityGF,F-F1F&F&F'F&*(\"\"%F&-%$expG6#F7F&-F(6$-&%$psiG6 #%\"QGF,F-F&F&*&F1F&-F(6$F7F.F&F&**\"\")F&F>F&FCF&F5F&F&F&%\"tGF&-%#ln G6#FMF&-F?6#-%&gammaG6$FMF-F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "latex(%);" }}{PARA 6 "" 1 "" {TEXT -1 66 "-3\\, \\left ( \{\\frac \{d^\{2\}\}\{d\{\\theta\}^\{2\}\}\}v \\left( \\theta \\righ t) " }}{PARA 6 "" 1 "" {TEXT -1 69 " \\right) \\left( 2\\, \\left( \{ \\frac \{d\}\{d\\theta\}\}P_\{\{\\infty \}\} \\left( " }}{PARA 6 "" 1 "" {TEXT -1 68 "\\theta \\right) \\right) ^\{2\}+\{\\frac \{d^\{2\}\} \{d\{\\theta\}^\{2\}\}\}v \\left( " }}{PARA 6 "" 1 "" {TEXT -1 70 "\\t heta \\right) +4\\,\{e^\{P_\{\{\\infty \}\} \\left( \\theta \\right) \+ \}\}\{\\frac \{d" }}{PARA 6 "" 1 "" {TEXT -1 65 "\}\{d\\theta\}\}\\psi _\{\{Q\}\} \\left( \\theta \\right) +2\\,\{\\frac \{d^\{2\}\}\{d\{" }} {PARA 6 "" 1 "" {TEXT -1 70 "\\theta\}^\{2\}\}\}P_\{\{\\infty \}\} \\l eft( \\theta \\right) +8\\,\{e^\{P_\{\{\\infty \}\}" }}{PARA 6 "" 1 " " {TEXT -1 70 " \\left( \\theta \\right) \}\}\\psi_\{\{Q\}\} \\left( \+ \\theta \\right) \{\\frac \{d\}\{d" }}{PARA 6 "" 1 "" {TEXT -1 67 "\\t heta\}\}P_\{\{\\infty \}\} \\left( \\theta \\right) \\right) t\\ln \+ \\left( t" }}{PARA 6 "" 1 "" {TEXT -1 46 " \\right) \{e^\{\\gamma \\le ft( t,\\theta \\right) \}\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kernelopts(cputime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%ST!\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "32 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }