{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 "limitonerevised.mws" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 98 "This worksheet examines We ylSq (=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 f or P(t,theta) and Q(t,theta) are made:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*e xp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*l n(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q [infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta, 2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infini ty](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$ `(theta,2))*t^(2*v(theta))*ln(t);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/-%\"PG6$%\"tG%&thetaG,,-&F%6#%)infini tyG6#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/*&-&%\"VG 6#F/F.F/FJF/F/*(,(*(-F=6#,$*&F?F/F*F/F/F/-&%$psiG6#FFF.F/FAF/F/*&#F/F9 F/*$F@F/F/F6*&#F/F9F/-FB6$F1FGF/F6F/FJF/F3F/F//-FFF&,(FDF/*&FXF/)F',$* &F?F/F1F/F/F/F/*&#F/F?F/*(FAF/FaoF/F3F/F/F/" }}}{EXCHG {PARA 0 "" 0 " " {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{PARA 6 " " 1 "" {TEXT -1 25 "Scalar invariant library." }}{PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 1997." }}{PARA 6 "" 1 "" {TEXT -1 25 "`Differential Invariants`" }}{PARA 6 "" 1 "" {TEXT -1 29 "`Last mo dified Jan. 20, 1995`" }}{PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad re lated object definitions`" }}{PARA 6 "" 1 "" {TEXT -1 31 "`Last modifi ed 23 January 2001`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May , 1999`" }}{PARA 6 "" 1 "" {TEXT -1 25 "`Last built 27 May, 1999`" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R4)G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayl l~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~th e~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~avail able~from:~http://grtensor.phy.queensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(6)/MetricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(gowdy);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2D efault~spacetimeG%&gowdyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the ~gowdy~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7&%\"tG%&thetaG%#x1G%#x2G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",,**-%$expG6#,$*&#F(F'F(-%&gammaG6 $%\"tG%&thetaGF(!\"\"F(F4#F6F'%#~dGF()F4%#2~GF(F6**F+F(F4F7F8F()F5F:F( F(**F4F(-F,6#-%\"PGF3F(F8F()%#x1GF:F(F(*2F'F(F4F(F>F(-%\"QGF3F(F8F()FC %\"~GF(%#d~GF()%#x2GFHF(F(*(,&*(F4F(F>F()FEF'F(F(*&F4F(F>F6F(F(F8F()FK F:F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%,ConstraintsG7&/-%%diffG6$ -%&gammaG6$%\"tG%&thetaGF-,**(F-\"\"\")-%$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$F*F.,&**F8F1F-F1FIF1FAF1F>*,F8F1F-F1F2F1F:F1FE F1F>/-F(6$F6-%\"$G6$F-F8,(-F(6$F6-FU6$F.F8F1*&FAF1F-F>F>*&-F46#,$*&F8F 1F6F1F1F1,&*$F9F1F1*$FDF1F>F1F1/-F(6$FF>*(F8 F1FAF1F:F1F>*(F8F1FIF1FEF1F1" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 49 "Note that the predefined 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 `Apply constraints repeatedly` to object WeylSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object WeylS q" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to object We ylSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#K!\"$" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{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),`$`(theta,2))*t^(2*v(theta))*ln(t),`x`);" }}{PARA 6 "" 1 " " {TEXT -1 31 "Applying routine subs to WeylSq" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 92 "h:=factor(simplify(subs(t=0,factor((4*t^3*grco mponent(WeylSq,[])/(exp(gamma(t,theta))))))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG*(,&*$)-%\"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&*,-%$expG6#-%&gammaG6$%\"tG%&thetaGF& ,&*$)-%\"vG6#F0\"\"#F&F&\"\"$F&F&,&F4F&F&!\"\"F7,&F4F&F&F&F7F/!\"$F&F& " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 51 "The following shows the \+ execution time (in seconds)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "kernelopts(cputime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(8+\\#! \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "20 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }