{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 "limitone.mws" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 98 "This worksheet examines WeylS q (=C_\{a b c d\}*C^\{a b c d\}) and distinguishes the case v(theta)^2 =1." }}}{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 0 "" 0 "" {TEXT -1 0 "" }}{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',$*&F;F/F1F/F/F/F/*(- %#qqGF.F/FNF/F3F/F/" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{PARA 6 "" 1 "" {TEXT -1 25 "Scala r invariant library." }}{PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 \+ March 1997." }}{PARA 6 "" 1 "" {TEXT -1 25 "`Differential Invariants` " }}{PARA 6 "" 1 "" {TEXT -1 29 "`Last modified Jan. 20, 1995`" }} {PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad related object definitions` " }}{PARA 6 "" 1 "" {TEXT -1 31 "`Last modified 23 January 2001`" }} {PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~the~authors.G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~available~from:~http ://grtensor.phy.queensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/G rtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(g owdy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&gowdy G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~gowdy~spacetime:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~ G%\"aG-%'vectorG6#7&%\"tG%&thetaG%#x1G%#x2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$) %$~dsG\"\"#\"\"\",,**-%$expG6#,$*&#F(F'F(-%&gammaG6$%\"tG%&thetaGF(!\" \"F(F4#F6F'%#~dGF()F4%#2~GF(F6**F+F(F4F7F8F()F5F:F(F(**F4F(-F,6#-%\"PG F3F(F8F()%#x1GF:F(F(*2F'F(F4F(F>F(-%\"QGF3F(F8F()FC%\"~GF(%#d~GF()%#x2 GFHF(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%&t hetaGF-,**(F-\"\"\")-%$expG6#-%\"PGF,\"\"#F1)-F(6$-%\"QGF,F-F8F1!\"\"* &F-F1)-F(6$F6F-F8F1F>*(F-F1F2F1)-F(6$F*&F-F1)-F(6$F6F.F8F1F>/ -F(6$F*F.,&**F8F1F-F1FIF1FAF1F>*,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$FF>*(F8F1FAF1F:F1F>*(F8F1F IF1FEF1F1" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 49 "Note that the p redefined invariant WeylSq is used" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(WeylSq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% *CPU~Time~G$\"#$*!\"$" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 49 "The constraints are applied and WeylSq simplified" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "gralter(_,13,6,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "Applying routine `Ap ply constraints 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 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "grmap (_,subs,P(t,theta)=P[infinity](theta)-v(theta)*log(t)+V(theta)*t^2*(lo g(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 routine subs to WeylSq" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "h:=factor(simplify(subs(t =0,factor((4*t^3*grcomponent(WeylSq,[])/(exp(gamma(t,theta))))))));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG*(,&*$)-%\"vG6#%&thetaG\"\"#\" \"\"F.\"\"$F.F.),&F)F.F.!\"\"F-F.),&F)F.F.F.F-F." }}}{EXCHG {PARA 0 " " 0 "" {MPLTEXT 0 21 24 "and so as t->0, WeylSq->" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "simplify(exp(gamma(t,theta))*h/(4*t^3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&*,-%$expG6#-%&gammaG 6$%\"tG%&thetaGF&,&*$)-%\"vG6#F0\"\"#F&F&\"\"$F&F&,&F4F&F&!\"\"F7,&F4F &F&F&F7F/!\"$F&F&" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 51 "The fol lowing shows the execution time (in seconds)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kernelopts(cputime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&am&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 32 "The following just generates tex" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "lat ex(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)));" }}{PARA 6 "" 1 "" {TEXT -1 65 "P \\left( t,\\theta \\right) =P_\{\{\\infty \}\} \\left( \+ \\theta \\right) -v" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\left( \\theta \+ \\right) \\ln \\left( t \\right) +V \\left( \\theta \\right) " }} {PARA 6 "" 1 "" {TEXT -1 66 "\{t\}^\{2\} \\left( \\ln \\left( t \\rig ht) \\right) ^\{2\}+V_\{\{1\}\} \\left( " }}{PARA 6 "" 1 "" {TEXT -1 64 "\\theta \\right) \{t\}^\{2\}+V_\{\{2\}\} \\left( \\theta \\right) \+ \{t\}^\{2\}\\ln " }}{PARA 6 "" 1 "" {TEXT -1 18 " \\left( t \\right) \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "latex(Q(t,theta)=Q[infi nity](theta)+q(theta)*t^(2*(v(theta)))+qq(theta)*t^(2*(v(theta)))*log( t));" }}{PARA 6 "" 1 "" {TEXT -1 65 "Q \\left( t,\\theta \\right) =Q_ \{\{\\infty \}\} \\left( \\theta \\right) +q" }}{PARA 6 "" 1 "" {TEXT -1 65 " \\left( \\theta \\right) \{t\}^\{2\\,v \\left( \\theta \\right ) \}+\{\\it qq\}" }}{PARA 6 "" 1 "" {TEXT -1 69 " \\left( \\theta \\ri ght) \{t\}^\{2\\,v \\left( \\theta \\right) \}\\ln \\left( t" }} {PARA 6 "" 1 "" {TEXT -1 9 " \\right) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "latex(1/4*exp(gamma(t,theta))*(v(theta)^2+3)*(v(theta )-1)^2*(v(theta)+1)^2/t^3);" }}{PARA 6 "" 1 "" {TEXT -1 67 "1/4\\,\{\\ frac \{\{e^\{\\gamma \\left( t,\\theta \\right) \}\} \\left( \\left( \+ v" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\left( \\theta \\right) \\right) \+ ^\{2\}+3 \\right) \\left( v \\left( \\theta" }}{PARA 6 "" 1 "" {TEXT -1 70 " \\right) -1 \\right) ^\{2\} \\left( v \\left( \\theta \\right) +1 \\right) ^\{2" }}{PARA 6 "" 1 "" {TEXT -1 12 "\}\}\{\{t\}^\{3\}\} \}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "25 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }