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{EXCHG {PARA 257 "" 0 "" {TEXT 269 64 "Program for calculating the Dirac equation on curved spacetimes." }}{PARA 258 "" 0 "" {TEXT -1 53 "Here the program is adapted to a Taub-NUT spacetime !" }}{PARA 259 "" 0 "" {TEXT -1 37 "See gr-qc/0209096 and gr-qc/0010085" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "grtw();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%= GRTensorII~Version~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~F ebruary~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.q ueensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 25 "define(sigma,sigma(0)=1);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 566 "define(pr,pr(1,1)=1,pr(1, sigma(1))=sigma(1),pr(1,sigma(2))=sigma(2),pr(1,sigma(3))=sigma(3),pr( sigma(1),1)=sigma(1),pr(sigma(2),1)=sigma(2),pr(sigma(3),1)=sigma(3),p r(sigma(2),sigma(1))=-pr(sigma(1),sigma(2)),pr(sigma(1),sigma(2))=I*si gma(3),pr(sigma(3),sigma(1))=-pr(sigma(1),sigma(3)),pr(sigma(1),sigma( 3))=-I*sigma(2),pr(sigma(3),sigma(2))=-pr(sigma(2),sigma(3)),pr(sigma( 2),sigma(3))=I*sigma(1),pr(sigma(1),sigma(1))=1,pr(sigma(2),sigma(2))= 1,pr(sigma(3),sigma(3))=1,pr((a::integer)*sigma(b::algebraic),(d::inte ger)*sigma(c::integer))=a*d*pr(sigma(b),sigma(c)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 315 "define(pd,pd(0,a::algebraic)=0,pd(a::alg ebraic,0)=0,pd(1,1)=1,pd(I*a::algebraic,b::algebraic)=I*pd(a,b),pd(-I* a::algebraic,b::algebraic)=-I*pd(a,b),pd(a::algebraic,I*b::algebraic)= I*pd(a,b),pd(a::algebraic,-I*b::algebraic)=-I*pd(a,b),pd(-a::algebraic ,b::algebraic)=-pd(a,b),pd(a::algebraic,-b::algebraic)=-pd(a,b));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1407 "define(`&p`,`&p`(-a::algeb raic,-b::algebraic)=`&p`(a,b),(-pd(sigma(a::algebraic),sigma(b::algebr aic))) &p (-pd(sigma(c::algebraic),sigma(d::algebraic)))=pd(sigma(a),s igma(b)) &p pd(sigma(c),sigma(d)),(-pd(a::algebraic,b::algebraic)) &p \+ (-m::function*pd(c::algebraic,d::algebraic))=m*(pd(a,b) &p pd(c,d)),pd (a::algebraic,b::algebraic) &p (-m::function*pd(c::algebraic,d::algebr aic))=-m*(pd(a,b) &p pd(c,d)),(-pd(a::algebraic,b::algebraic)) &p pd(c ::algebraic,d::algebraic)=-(pd(a,b) &p pd(c,d)),pd(sigma(a::algebraic) ,sigma(b::algebraic)) &p pd(sigma(c::algebraic),sigma(d::algebraic))=p d(pr(sigma(a),sigma(c)),pr(sigma(b),sigma(d))),pd(1,sigma(a::algebraic )) &p pd(sigma(b::algebraic),sigma(c::algebraic))=pd(pr(1,sigma(b)),pr (sigma(a),sigma(c))),pd(sigma(a::algebraic),sigma(b::algebraic)) &p pd (1,sigma(c::algebraic))=pd(pr(sigma(a),1),pr(sigma(b),sigma(c))),pd(si gma(a::algebraic),1) &p pd(sigma(b::algebraic),sigma(c::algebraic))=pd (pr(sigma(a),sigma(b)),pr(1,sigma(c))),pd(sigma(a::algebraic),sigma(b: :algebraic)) &p pd(sigma(c::algebraic),1)=pd(pr(sigma(a),sigma(c)),pr( sigma(b),1)),pd(1,sigma(a::algebraic)) &p pd(1,sigma(b::algebraic))=pd (pr(1,1),pr(sigma(a),sigma(b))),pd(sigma(a::algebraic),1) &p pd(sigma( b::algebraic),1)=pd(pr(sigma(a),sigma(b)),pr(1,1)),I*(a::algebraic) &p (I*b::algebraic)=-(a &p b),(I*a::algebraic) &p (b::algebraic)=I*(a &p b),(a::algebraic) &p (I*b::algebraic)=I*(a &p b));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1093 "de finemore(`&p`,`&p`(c::algebraic*pd(sigma(a::algebraic),sigma(b::algebr aic)),d::algebraic*pd(sigma(e::algebraic),sigma(f::algebraic)))=c*d*`& p`(pd(sigma(a),sigma(b)),pd(sigma(e),sigma(f))),`&p`(c::algebraic*pd(a ::algebraic,sigma(b::algebraic)),d::algebraic*pd(sigma(e::algebraic),s igma(f::algebraic)))=c*d*`&p`(pd(a,sigma(b)),pd(sigma(e),sigma(f))),`& p`(c::algebraic*pd(sigma(a::algebraic),b::algebraic),d::algebraic*pd(s igma(e::algebraic),sigma(f::algebraic)))=c*d*`&p`(pd(sigma(a),b),pd(si gma(e),sigma(f))),`&p`(c::algebraic*pd(sigma(a::algebraic),sigma(b::al gebraic)),d::algebraic*pd(e::algebraic,sigma(f::algebraic)))=c*d*`&p`( pd(sigma(a),sigma(b)),pd(e,sigma(f))),`&p`(c::algebraic*pd(sigma(a::al gebraic),sigma(b::algebraic)),d::algebraic*pd(sigma(e::algebraic),f::a lgebraic))=c*d*`&p`(pd(sigma(a),sigma(b)),pd(sigma(e),f)),`&p`(pd(sigm a(a::algebraic),1),pd(1,sigma(b::algebraic)))=pd(pr(sigma(a),1),pr(1,s igma(b))),`&p`(c::algebraic*pd(sigma(a::algebraic),e::algebraic),d::al gebraic*pd(f::algebraic,sigma(b::algebraic)))=c*d*`&p`(pd(sigma(a),e), pd(f,sigma(b))),`&p`(a::algebraic,0)=0 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 20 "Just some checkings " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "(-3*x*pd(sigma(1),sigma(2))) &p (-r*pd(sigma(0),sigma (3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**^#\"\"$\"\"\"%\"xGF&%\"rGF &-%#pdG6$-%&sigmaG6#F&F,F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 65 "De fine now the Dirac gamma matrices and their commutator with \"&p" } {TEXT -1 1 "\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "define(g am,gam(1)=I*pd(sigma(2),sigma(1)),gam(2)=I*pd(sigma(2),sigma(2)),gam(3 )=I*pd(sigma(2),sigma(3)),gam(0)=pd(sigma(1),1),gam(5)=-pd(sigma(3),1) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "define(comu,comu(a::a lgebraic,b::algebraic)=a &p b - b &p a);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 19 "Again a small test " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "comu(gam(1),gam(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&^#!\"#\"\"\"-%#pdG6$F&-%&sigmaG6#\"\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "I*comu(gam(2),gam(1))/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#pdG6$\"\"\"-%&sigmaG6#\"\"$#!\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 25 "Download now the metric !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qload(tnb_1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&tnb_1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"x G6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7& %\"yG%&thetaG%$phiG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Basis~in ner~productG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$etaG6$%$bupGF&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$etaG%$(a)G\"\"\")%!G%$(b)GF(-%'m atrixG6#7&7&!\"\"\"\"!F2F27&F2F1F2F27&F2F2F1F27&F2F2F2F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%=Basis~(covariant~components)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w1G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%'omega1G6#%\"aG-%'vectorG6#7&,$*&%\"lG\"\"\"-%%sqrtG6#-%\"UG6#%\"tGF /\"\"#\"\"!,$*(F.F/F0F/-%$cosG6#%&thetaGF/F7F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w2G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'om ega2G6#%\"aG-%'vectorG6#7&\"\"!*$-%%sqrtG6#,&*$)%\"tG\"\"#\"\"\"F6*$)% \"lGF5F6F6F6F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w3G6#%#dnG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega3G6#%\"aG-%'vectorG6#7&\"\"!F ,*$-%%sqrtG6#*&,&*$)%\"tG\"\"#\"\"\"F7*$)%\"lGF6F7F7F7)-%$sinG6#%&thet aGF6F7F7F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w4G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega4G6#%\"aG-%'vectorG6#7&\"\"!F,F,,$* &\"\"\"F/*$-%%sqrtG6#-%\"UG6#%\"tGF/!\"\"F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7~~~~Taub-NUT~space~~~~G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(metric);" }}{PARA 6 "" 1 "" {TEXT -1 45 "Calcu lated e(bup,dn) for tnb_1 (.009000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 44 "Calculated g(dn,dn) for tnb_1 (.012000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#9!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gralter(metric,simplify);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine sim plify to object g(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Tim e~G$\"#s!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grdisplay(m etric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Covariant~metric~tensorG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"aG\"\"\"&%!G6#%\"bGF)-%'matrixG6#7&7&,$*& )%\"lG\"\"#F)-%\"UG6#%\"tGF)!\"%\"\"!,$*(F5F)F8F)-%$cosG6#%&thetaGF)F< F=7&F=,&*$)F;F7F)!\"\"*$F5F)FHF=F=7&F>F=,,*(F5F)F8F))F@F7F)F " 0 "" {MPLTEXT 1 0 11 "grcalc(ds);" }}{PARA 6 "" 1 "" {TEXT -1 38 "Calcul ated ds for tnb_1 (.002000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% *CPU~Time~G$\"\"$!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "gr alter(ds,trigsin,expand,normal);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Compo nent simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 49 "Applying routine `simplify[trig sin]` to object ds" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying routine ex pand to object ds" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying routine nor mal to object ds" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#? !\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "grdisplay(ds);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\"*(%#~dGF(,4*()%\"lGF'F()-%\"UG6#% \"tGF'F()%\"yG%#2~GF(!\"%*0\"\")F(F-F(F/F(-%$cosG6#%&thetaGF()F5%\"~GF (%#d~GF()%$phiGF?F(!\"\"*()F=F6F(F0F()F3F'F(FC*(FEF(F0F(F-F(FC**\"\"%F ()FBF6F(F/F(F-F(FC*,FIF(FJF(F/F(F-F()-%$sinGF " 0 "" {MPLTEXT 1 0 21 "grdef(`SS\{ ^a ^b \}`);" }}{PARA 6 "" 1 "" {TEXT -1 32 "Created definition for SS(up ,up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grcalc(SS(up,up)): " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 37 "Enter components for object SS(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 56 " If \+ you wish to quit at any point and leave this object" }}{PARA 6 "" 1 " " {TEXT -1 39 " uninitialized, enter the string exit." }}{PARA 6 "" 1 "" {TEXT -1 51 " REMEMBER to complete each entry with a semicolon. " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"yG\"\"\")%!G%\"yGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(1),gam(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%&thetaG\"\"\")%!G%\"yGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)* comu(gam(2),gam(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%$phiG \"\"\")%!G%\"yGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(3),gam(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"tG\"\"\")%!G%\"yGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/ 4)*comu(gam(0),gam(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\" yG\"\"\")%!G%&thetaGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(1),gam(2));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%&thetaG\"\"\")%!G%&thetaGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(2),gam(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%$phiG\"\"\")%!G%&thetaGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)* comu(gam(3),gam(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"tG \"\"\")%!G%&thetaGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(0),gam(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"yG\"\"\")%!G%$phiGF'" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(1),gam(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SS G%&thetaG\"\"\")%!G%$phiGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(2),gam(3 ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%$phiG\"\"\")%!G%$phiGF '" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(3),gam(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"tG\"\"\")%!G%$phiGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)* comu(gam(0),gam(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"yG \"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(1),gam(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%&thetaG\"\"\")%!G%\"tGF'" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(2),gam(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SS G%$phiG\"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(3),gam(0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"tG\"\"\")%!G%\"tGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(0),gam(0));" }}{PARA 6 "" 1 "" {TEXT -1 45 "Calculated SS(up,up) for tnb_1 (.579000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$\"e!\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "grdisplay(SS(up,up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% *SS(up,up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#SSG6$%#upGF&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#SSG%\"aG\"\"\")%!G%\"bGF(-%'matr ixG6#7&7&\"\"!,$-%#pdG6$F(-%&sigmaG6#\"\"$#F(\"\"#,$-F46$F(-F76#F;#!\" \"F;*&^#F:F(-F46$F6-F76#F(F(7&,$F3FAF1,$-F46$F(FGF:*&FDF(-F46$F6F?F(7& ,$F=F:,$FLFAF1*&FDF(-F46$F6F6F(7&*&^#FAF(FEF(*&FYF(FOF(*&FYF(FUF(F1" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 24 "The Christoffell symbols" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grcalc(Chr(dn,dn,dn));" }} {PARA 6 "" 1 "" {TEXT -1 48 "Calculated g(dn,dn,pdn) for tnb_1 (.00500 0 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 49 "Calculated Chr(dn,dn,dn) for t nb_1 (.005000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$ \"#7!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gralter(Chr(dn, dn,dn),trigsin);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplificat ion of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 60 "Applying routine `simplify[trigsin]` to object \+ Chr(dn,dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#J!\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grdisplay(Chr(dn,dn,dn ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%`oChristoffel~symbol~of~the~first~kin d~(symmetric~in~first~two~indices)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%&GammaG6#*(%\"yG\"\"\"%\"yGF)%\"tGF),$*&)%\"lG\"\"#F)-%%diffG6$-% \"UG6#%\"tGF7F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\" yG\"\"\"%&thetaGF)%$phiGF),$*()%\"lG\"\"#F)-%\"UG6#%\"tGF)-%$sinG6#%&t hetaGF)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"yG\"\"\" %$phiGF)%&thetaGF),$*()%\"lG\"\"#F)-%\"UG6#%\"tGF)-%$sinG6#%&thetaGF)! \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"yG\"\"\"%$phiG F)%\"tGF),$*()%\"lG\"\"#F)-%%diffG6$-%\"UG6#%\"tGF7F)-%$cosG6#%&thetaG F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"yG\"\"\"%\"tG F)%\"yGF),$*&)%\"lG\"\"#F)-%%diffG6$-%\"UG6#%\"tGF7F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"yG\"\"\"%\"tGF)%$phiGF),$*()% \"lG\"\"#F)-%%diffG6$-%\"UG6#%\"tGF7F)-%$cosG6#%&thetaGF)!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%&thetaG\"\"\"%&thetaGF) %\"tGF)%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%&thetaG \"\"\"%$phiGF)%\"yGF),$*()%\"lG\"\"#F)-%\"UG6#%\"tGF)-%$sinG6#%&thetaG F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%&thetaG\"\"\"%$ phiGF)%$phiGF),(**)%\"lG\"\"#F)-%\"UG6#%\"tGF)-%$cosG6#%&thetaGF)-%$si nGF7F)\"\"%*()F4F0F)F5F)F9F)!\"\"*(F.F)F5F)F9F)F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%&thetaG\"\"\"%\"tGF)%&thetaGF),$%\"tG!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%$phiG\"\"\"%$phiG F)%&thetaGF),(**)%\"lG\"\"#F)-%\"UG6#%\"tGF)-%$cosG6#%&thetaGF)-%$sinG F7F)!\"%*()F4F0F)F5F)F9F)F)*(F.F)F5F)F9F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%$phiG\"\"\"%$phiGF)%\"tGF),(*&)%\"lG\"\" #F)-%%diffG6$-%\"UG6#%\"tGF7F)F0**F0F)F.F)F1F))-%$sinG6#%&thetaGF0F)! \"\"*&F7F)F9F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%$ph iG\"\"\"%\"tGF)%\"yGF),$*()%\"lG\"\"#F)-%%diffG6$-%\"UG6#%\"tGF7F)-%$c osG6#%&thetaGF)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(% $phiG\"\"\"%\"tGF)%$phiGF),(*&)%\"lG\"\"#F)-%%diffG6$-%\"UG6#%\"tGF7F) !\"#**F0F)F.F)F1F))-%$sinG6#%&thetaGF0F)F)*&F7F)F:F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"tG\"\"\"%\"tGF)%\"tGF),$*&-%\" UG6#%\"tG!\"#-%%diffG6$F.F1F)#!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grcalc(Chr(bdn,bdn,bdn));" }}{PARA 6 "" 1 "" {TEXT -1 40 "Created definition for Chr(bdn,bdn,bdn) " }}{PARA 6 "" 1 "" {TEXT -1 40 "Calculated detg for tnb_1 (.003000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 44 "Calculated g(up,up) for tnb_1 (.024000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 45 "Calculated e(bdn,up) for tnb_1 (.008000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 49 "Calculated Chr(dn,dn,up) for tnb_1 (.02 5000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 52 "Calculated Chr(bdn,bdn,bdn) for tnb_1 (.234000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Ti me~G$\"$B%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gralter(C hr(bdn,bdn,bdn),trigsin,expand,simplify);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 " " {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 63 "Applying routine `simpl ify[trigsin]` to object Chr(bdn,bdn,bdn)" }}{PARA 6 "" 1 "" {TEXT -1 50 "Applying routine expand to object Chr(bdn,bdn,bdn)" }}{PARA 6 "" 1 "" {TEXT -1 52 "Applying routine simplify to object Chr(bdn,bdn,bdn) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$.&!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "grdisplay(Chr(bdn,bdn,bdn)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%1Chr(bdn,bdn,bdn)G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"1G\"\"\"-F)6#%\"1GF,-F)6#%\"4G F,,$*&-%\"UG6#%\"tG#!\"\"\"\"#-%%diffG6$F5F8F,F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"1G\"\"\"-F)6#%\"2GF,-F)6#%\"3GF, ,$*,%\"lGF,-%\"UG6#%\"tG#F,\"\"#,$*&,&*$)F9F;F,F,*$)F5F;F,F,F,,&!\"\"F ,*$)-%$cosG6#%&thetaGF;F,F,F,FDF:-%$sinGFIFDF>#!\"$F;FD" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"1G\"\"\"-F)6#%\"3GF,-F)6# %\"2GF,*,%\"lGF,-%\"UG6#%\"tG#F,\"\"#,$*&,&*$)F8F:F,F,*$)F4F:F,F,F,,&! \"\"F,*$)-%$cosG6#%&thetaGF:F,F,F,FCF9-%$sinGFHFCF=#!\"$F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"1G\"\"\"-F)6#%\"4GF,- F)6#%\"1GF,,$*&-%\"UG6#%\"tG#!\"\"\"\"#-%%diffG6$F5F8F,#F,F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"2G\"\"\"-F)6#%\"2GF,- F)6#%\"4GF,,$*(,&*$)%\"tG\"\"#F,F,*$)%\"lGF9F,F,!\"\"-%\"UG6#F8#F,F9F8 F,F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"2G\"\" \"-F)6#%\"3GF,-F)6#%\"1GF,,$*,%\"lGF,-%\"UG6#%\"tG#F,\"\"#,$*&,&*$)F9F ;F,F,*$)F5F;F,F,F,,&!\"\"F,*$)-%$cosG6#%&thetaGF;F,F,F,FDF:-%$sinGFIFD F>#!\"$F;FD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\" 2G\"\"\"-F)6#%\"3GF,-F)6#%\"3GF,*(,&*$)%\"tG\"\"#F,F,*$)%\"lGF8F,F,#! \"\"F8-%$cosG6#%&thetaGF,-%$sinGF@F=" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%&GammaG6#*(-%!G6#%\"2G\"\"\"-F)6#%\"4GF,-F)6#%\"2GF,*(,&*$)%\"tG \"\"#F,F,*$)%\"lGF8F,F,!\"\"-%\"UG6#F7#F,F8F7F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"3G\"\"\"-F)6#%\"3GF,-F)6#%\"2GF, ,$*(,&*$)%\"tG\"\"#F,F,*$)%\"lGF9F,F,#!\"\"F9-%$cosG6#%&thetaGF,-%$sin GFAF>F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(-%!G6#%\"3G\" \"\"-F)6#%\"3GF,-F)6#%\"4GF,,$*(,&*$)%\"tG\"\"#F,F,*$)%\"lGF9F,F,!\"\" -%\"UG6#F8#F,F9F8F,F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#* (-%!G6#%\"3G\"\"\"-F)6#%\"4GF,-F)6#%\"3GF,*(,&*$)%\"tG\"\"#F,F,*$)%\"l GF8F,F,!\"\"-%\"UG6#F7#F,F8F7F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% &GammaG6#*(-%!G6#%\"4G\"\"\"-F)6#%\"4GF,-F)6#%\"4GF,,$*&-%\"UG6#%\"tG# !\"\"\"\"#-%%diffG6$F5F8F,#F,F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 46 "The special term from eq. (9) of gr-qc/0209096" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "grdef(`de\{ i \}:=(I/2)*SS\{ ^a ^b \}*Chr\{ (a) (i) (b) \}`); " }}{PARA 6 "" 1 "" {TEXT -1 29 "Created definition for de(dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(de(dn ));" }}{PARA 6 "" 1 "" {TEXT -1 42 "Calculated de(dn) for tnb_1 (.0140 00 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#:!\"$" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gralter(de(dn),factor,trigs in,expand);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification o f a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine factor to object de(dn)" }}{PARA 6 "" 1 "" {TEXT -1 53 "Applying routine `simplify[trigsin]` to object de (dn)" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object de(dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#V!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grdisplay(de(dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'de(dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#de G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#deG6#%\"yG,&*(-%\"UG6# %\"tG#!\"\"\"\"#-%#pdG6$-%&sigmaG6#\"\"$-F56#\"\"\"F:-%%diffG6$F*F-F:# F:\"\"%*0^#F.F:F*#F:F0-%$sinG6#%&thetaGF/,&*$)F-F0F:F:*$)%\"lGF0F:F:#! \"$F0-F26$F:F8F:FLF:,&*&)FCF0F:FIF:F:*&FSF:FKF:F:FBF:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#deG6#%&thetaG,$**-%#pdG6$-%&sigmaG6#\"\"$-F.6# \"\"#\"\"\",&*$)%\"tGF3F4F4*$)%\"lGF3F4F4!\"\"-%\"UG6#F8#F4F3F8F4F@" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#deG6#%$phiG,.*2^##\"\"\"\"\"#F,-% $sinG6#%&thetaG!\"\",&*$)%\"tGF-F,F,*$)%\"lGF-F,F,#!\"&F--%#pdG6$F,-%& sigmaG6#\"\"$F,F9F,-%\"UG6#F6F+,&*&)F.F-F,F5F,F,*&FHF,F8F,F,F+F6F-F,*0 F*F,F.F2F3F:F " 0 "" {MPLTEXT 1 0 49 "grdef(`ga\{ ^a \}:=[gam(1), gam(2),gam(3),gam(0)]`);" }}{PARA 6 "" 1 "" {TEXT -1 37 "Components as signed for metric: tnb_1" }}{PARA 6 "" 1 "" {TEXT -1 29 "Created defin ition for ga(up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grdispl ay(ga(up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spaceti me:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'ga(up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#gaG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#ga G%\"aG-%'vectorG6#7&*&^#\"\"\"F--%#pdG6$-%&sigmaG6#\"\"#-F26#F-F-*&F,F --F/6$F1F1F-*&F,F--F/6$F1-F26#\"\"$F--F/6$F5F-" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "grdef(`ver\{ ^a ^b \}`);" }}{PARA 6 "" 1 "" {TEXT -1 33 "Created definition for ver(up,up)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "grcalc(ver(up,up));" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 38 "Enter components for object ver (up,up)" }}{PARA 6 "" 1 "" {TEXT -1 56 " If you wish to quit at any p oint and leave this object" }}{PARA 6 "" 1 "" {TEXT -1 39 " uninitial ized, enter the string exit." }}{PARA 6 "" 1 "" {TEXT -1 51 " REMEMBE R to complete each entry with a semicolon." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%\"yG\"\"\")%!G%\"yGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grcom ponent(ga(up),[y]) &p grcomponent(ga(up),[y]) +grcomponent(ga(up),[y]) &p grcomponent(ga(up),[y])-2*grcomponent(eta(bup,bup),[y,y]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%&thetaG\"\"\")%!G%\"yGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 150 "grcomponent(ga(up),[theta]) &p grcomponent(ga(up),[y ]) +grcomponent(ga(up),[y]) &p grcomponent(ga(up),[theta])-2*grcompone nt(eta(bup,bup),[theta,y]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$ verG%$phiG\"\"\")%!G%\"yGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 144 "grcomponent(ga(up),[ph i]) &p grcomponent(ga(up),[y]) +grcomponent(ga(up),[y]) &p grcomponent (ga(up),[phi])-2*grcomponent(eta(bup,bup),[phi,y]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%\"tG\"\"\")%!G%\"yGF'" }}{PARA 6 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grc omponent(ga(up),[t]) &p grcomponent(ga(up),[y]) +grcomponent(ga(up),[y ]) &p grcomponent(ga(up),[t])-2*grcomponent(eta(bup,bup),[t,y]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%\"yG\"\"\")%!G%&thetaGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 150 "grcomponent(ga(up),[y]) &p grcomponent(ga(up),[theta ]) +grcomponent(ga(up),[theta]) &p grcomponent(ga(up),[y])-2*grcompone nt(eta(bup,bup),[y,theta]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$ verG%&thetaG\"\"\")%!G%&thetaGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 162 "grcomponent(ga(up),[th eta]) &p grcomponent(ga(up),[theta]) +grcomponent(ga(up),[theta]) &p g rcomponent(ga(up),[theta])-2*grcomponent(eta(bup,bup),[theta,theta]); \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%$phiG\"\"\")%!G%&theta GF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 156 "grcomponent(ga(up),[phi]) &p grcomponent(ga(up),[the ta]) +grcomponent(ga(up),[theta]) &p grcomponent(ga(up),[phi])-2*grcom ponent(eta(bup,bup),[phi,theta]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&)%$verG%\"tG\"\"\")%!G%&thetaGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 150 "grcomponent(ga(up),[ t]) &p grcomponent(ga(up),[theta]) +grcomponent(ga(up),[theta]) &p grc omponent(ga(up),[t])-2*grcomponent(eta(bup,bup),[t,theta]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%\"yG\"\"\")%!G%$phiGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 144 "grcomponent(ga(up),[y]) &p grcomponent(ga(up),[phi]) +grcomponent(ga(up),[phi]) &p grcomponent(ga(up),[y])-2*grcomponent(e ta(bup,bup),[y,phi]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%& thetaG\"\"\")%!G%$phiGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 156 "grcomponent(ga(up),[theta]) & p grcomponent(ga(up),[phi]) +grcomponent(ga(up),[phi]) &p grcomponent( ga(up),[theta])-2*grcomponent(eta(bup,bup),[theta,phi]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%$phiG\"\"\")%!G%$phiGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 150 "grcomponent(ga(up),[phi]) &p grcomponent(ga(up),[phi]) +grcompone nt(ga(up),[phi]) &p grcomponent(ga(up),[phi])-2*grcomponent(eta(bup,bu p),[phi,phi]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%\"tG\"\" \")%!G%$phiGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grc alc>" 0 "" {MPLTEXT 1 0 144 "grcomponent(ga(up),[t]) &p grcomponent(ga (up),[phi]) +grcomponent(ga(up),[phi]) &p grcomponent(ga(up),[t])-2*gr component(eta(bup,bup),[t,phi]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&)%$verG%\"yG\"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grcomponent(ga(up),[y] ) &p grcomponent(ga(up),[t]) +grcomponent(ga(up),[t]) &p grcomponent(g a(up),[y])-2*grcomponent(eta(bup,bup),[y,t]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%&thetaG\"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 150 "grcom ponent(ga(up),[theta]) &p grcomponent(ga(up),[t]) +grcomponent(ga(up), [t]) &p grcomponent(ga(up),[theta])-2*grcomponent(eta(bup,bup),[theta, t]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%$phiG\"\"\")%!G%\" tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 " " {MPLTEXT 1 0 144 "grcomponent(ga(up),[phi]) &p grcomponent(ga(up),[t ]) +grcomponent(ga(up),[t]) &p grcomponent(ga(up),[phi])-2*grcomponent (eta(bup,bup),[phi,t]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG %\"tG\"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grcomponent(ga(up),[t]) &p gr component(ga(up),[t]) +grcomponent(ga(up),[t]) &p grcomponent(ga(up),[ t])-2*grcomponent(eta(bup,bup),[t,t]);\n" }}{PARA 6 "" 1 "" {TEXT -1 46 "Calculated ver(up,up) for tnb_1 (.545000 sec.)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%*CPU~Time~G$\"$Y&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grdisplay(ver(up,up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% +ver(up,up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$verG6$%#upGF&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$verG%\"aG\"\"\")%!G%\"bGF(%8All~ components~are~zeroG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 39 "The wave function as a covariant vector" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "grdef(`Psid\{ a \}:=[diff(psi(t),y),diff(psi(t),theta),diff(ps i(t),phi),diff(psi(t),t)]`);" }}{PARA 6 "" 1 "" {TEXT -1 37 "Component s assigned for metric: tnb_1" }}{PARA 6 "" 1 "" {TEXT -1 31 "Created d efinition for Psid(dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "g rcalc(Psid(dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\" !F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "grdisplay(Psid(dn)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%)Psid(dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%PsidG6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%% PsidG6#%\"aG-%'vectorG6#7&\"\"!F,F,-%%diffG6$-%$psiG6#%\"tGF3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grcalc(Psid(bdn));" }}{PARA 6 "" 1 "" {TEXT -1 33 "Created definition for Psid(bdn) " }}{PARA 6 " " 1 "" {TEXT -1 45 "Calculated Psid(bdn) for tnb_1 (.003000 sec.)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$Q\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "gralter(Psid(bdn),trigsin,simplify) ;" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTens orII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 56 "Applying routine `simplify[trigsin]` to object Psid(bdn)" }} {PARA 6 "" 1 "" {TEXT -1 45 "Applying routine simplify to object Psid( bdn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#?!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "grdisplay(Psid(bdn));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*Psid(bdn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%PsidG6#%$bdnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%PsidG6#% $(a)G-%'vectorG6#7&\"\"!F,F,,$*&-%%sqrtG6#-%\"UG6#%\"tG\"\"\"-%%diffG6 $-%$psiGF4F5F6!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 82 "A series \+ of tricks to obtain the Dirac operator (\"ddd\" see below) in monomial form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a0:=expand(grcompo nent(de(dn),[t]));a00:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a00G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u0:=whattype(a0);u0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G%(integerG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(i ntegerG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(a0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "if u0=`+` then for i from 1 to nops(a0) do a00:=a00 +I*h*grcomponent(ga(up),[t]) &p op(i,a0) od else a00:=I*h*grcomponent( ga(up),[t]) &p a0 fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a00G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a1:=expand(grcomponent(de(dn ),[y]));a11:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G,&*(-%\"UG6#% \"tG#!\"\"\"\"#-%#pdG6$-%&sigmaG6#\"\"$-F26#\"\"\"F7-%%diffG6$F'F*F7#F 7\"\"%*0^#F+F7F'#F7F--%$sinG6#%&thetaGF,,&*$)F*F-F7F7*$)%\"lGF-F7F7#! \"$F--F/6$F7F5F7FIF7,&*&)F@F-F7FFF7F7*&FPF7FHF7F7F?F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a11G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u1:=whattype(a1);u1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G%\"+G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"+G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(a1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "if u1=`+` then for i from 1 to nops(a1) do a11:=a11+I*h*grcomponent(ga( up),[y]) &p op(i,a1) od else a11:=I*h*grcomponent(ga(up),[y]) &p a1 fi ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a11;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*,^##!\"\"\"\"%\"\"\"%\"hGF)-%\"UG6#%\"tG#F'\"\"#-% %diffG6$F+F.F)-%#pdG6$-%&sigmaG6#F)F)F)F)*2^##F)F0F)F*F)F+F<,&*&)-%$si nG6#%&thetaGF0F))F.F0F)F)*&F?F))%\"lGF0F)F)F<,&*$FDF)F)*$FFF)F)#!\"$F0 F@F'FGF)-F56$-F86#F0F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a2:=expand(grcomponent(de(dn),[theta]));a22:=0;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#a2G,$**-%#pdG6$-%&sigmaG6#\"\"$-F+6#\"\"#\"\"\",&* $)%\"tGF0F1F1*$)%\"lGF0F1F1!\"\"-%\"UG6#F5#F1F0F5F1F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a22G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u2:=whattype(a2);u2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u2G%\"*G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"*G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(a2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "if u2=`+` then for i from 1 to nops(a2) do a22:=a22+I*h*grcomponent(ga( up),[theta]) &p op(i,a2) od else a22:=I*h*grcomponent(ga(up),[theta]) \+ &p a2 fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a22G*.^##!\"\"\"\"#\" \"\"%\"hGF*,&*$)%\"tGF)F*F**$)%\"lGF)F*F*F(-%\"UG6#F/#F*F)F/F*-%#pdG6$ -%&sigmaG6#F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a22;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*.^##!\"\"\"\"#\"\"\"%\"hGF(,&*$)%\"t GF'F(F(*$)%\"lGF'F(F(F&-%\"UG6#F-#F(F'F-F(-%#pdG6$-%&sigmaG6#F(F(F(" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a3:=expand(grcomponent(de( dn),[phi]));a33:=0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#a3G,.*2^##\" \"\"\"\"#F)-%$sinG6#%&thetaG!\"\",&*$)%\"tGF*F)F)*$)%\"lGF*F)F)#!\"&F* -%#pdG6$F)-%&sigmaG6#\"\"$F)F6F)-%\"UG6#F3F(,&*&)F+F*F)F2F)F)*&FEF)F5F )F)F(F3F*F)*0F'F)F+F/F0F7F9F)F6F?F@F(FCF(F)*.F'F)F+F/F0F7-F:6$F)-F=6#F )F)-%$cosGF-F)F3\"\"%F)*0^#F)F)F+F/F0F7FIF)FMF)F3F*F6F*F)*.F'F)F+F/F0F 7FIF)FMF)F6FOF)*,F(F)F0F/-F:6$F%$a33G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "u3:=whattype(a3);u3;\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u3G%\"+G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"+G" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(a3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "if u3=`+` then for i from 1 to nops(a3) do a33:=a33+I*h*grcomponent( ga(up),[phi]) &p op(i,a3) od else a33:=I*h*grcomponent(ga(up),[phi]) & p a3 fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a33;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*4^##!\"\"\"\"#\"\"\"%\"hGF)-%\"UG6#%\"tG# F)F(F.F(,&*$)F.F(F)F)*$)%\"lGF(F)F)#!\"&F(,&*&)-%$sinG6#%&thetaGF(F)F2 F)F)*&F:F)F4F)F)F/F;F'F5F)-%#pdG6$-%&sigmaG6#F(F)F)F)*2F%F)F*F)F+F/F0F 6F5\"\"$F8F/F;F'F@F)F)*0F/F)F*F)-%$cosGF=F)F0F6F;F'F.\"\"%-FA6$FCFCF)F )*0F*F)F.F(FIF)F5F(F0F6F;F'FLF)F)*0F/F)F*F)FIF)F0F6F5FKF;F'FLF)F)*.F%F )F*F)F0F'F+F/F.F)-FA6$-FD6#F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdef(`ddd`);" }}{PARA 6 "" 1 "" {TEXT -1 26 "Created definition for ddd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "grca lc(ddd);" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 31 "Enter components for object ddd" }}{PARA 6 "" 1 "" {TEXT -1 56 " \+ If you wish to quit at any point and leave this object" }}{PARA 6 "" 1 "" {TEXT -1 39 " uninitialized, enter the string exit." }}{PARA 6 " " 1 "" {TEXT -1 51 " REMEMBER to complete each entry with a semicolon ." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$dddG" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 16 "a00+a11+a22+ a33;" }}{PARA 6 "" 1 "" {TEXT -1 39 "Calculated ddd for tnb_1 (.022000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#D!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grdisplay(ddd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%$dddG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$dddG,$*, %\"hG\"\"\",0*.^#!\"\"F(-%%diffG6$-%\"UG6#%\"tGF3F(-%#pdG6$-%&sigmaG6# F(F(F(),&*$)F3\"\"#F(F(*$)%\"lGF>F(F(#\"\"&F>F(-%$sinG6#%&thetaGF(F=F( F(*.F+F(F-F(F4F(F:F(FDF(F@F(F(*.^#!\"%F(F0F(F3F(F4F(F:F(FDF(F(*,F>F(-% $cosGFFF()F3\"\"'F(-F56$-F86#F>FSF(-%%sqrtG6#F0F(F(*.FPF(FMF()F3\"\"%F (FQF(FUF(F@F(F(*.FPF(F=F(FMF()FAFZF(FQF(FUF(F(*,F>F(FMF()FAFPF(FQF(FUF (F(F(F0#F,F>F;#!\"(F>FDF,#F(FZ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gralter(ddd,trigsin,factor,expand);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 50 "Applying routine `si mplify[trigsin]` to object ddd" }}{PARA 6 "" 1 "" {TEXT -1 37 "Applyin g routine factor to object ddd" }}{PARA 6 "" 1 "" {TEXT -1 37 "Applyin g routine expand to object ddd" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C PU~Time~G$\"#H!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grdis play(ddd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetim e:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$dddG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$dddG,0*0^##!\"\"\"\"%\"\"\"%\"hGF+-%\"UG6#%\"tG#F)\" \"#,&*$)F0F2F+F+*$)%\"lGF2F+F+F)-%%diffG6$F-F0F+-%#pdG6$-%&sigmaG6#F+F +F+F0F2F+*0F'F+F,F+F-F1F3F)F9F+F " 0 "" {MPLTEXT 1 0 18 "grdisplay(ga(up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'ga(up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#gaG6#%# upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#gaG%\"aG-%'vectorG6#7&*&^# \"\"\"F--%#pdG6$-%&sigmaG6#\"\"#-F26#F-F-*&F,F--F/6$F1F1F-*&F,F--F/6$F 1-F26#\"\"$F--F/6$F5F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 34 "And no w finally the Dirac equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "grdef(`dirac:=I*h*ga\{ ^l \}*Psid\{ (l) \}+ddd*psi(t)-m*c*psi(t) `);" }}{PARA 6 "" 1 "" {TEXT -1 28 "Created definition for dirac" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grdisplay(ddd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%$dddG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$dddG,0*0 ^##!\"\"\"\"%\"\"\"%\"hGF+-%\"UG6#%\"tG#F)\"\"#,&*$)F0F2F+F+*$)%\"lGF2 F+F+F)-%%diffG6$F-F0F+-%#pdG6$-%&sigmaG6#F+F+F+F0F2F+*0F'F+F,F+F-F1F3F )F9F+F " 0 "" {MPLTEXT 1 0 14 "g rcalc(dirac);" }}{PARA 6 "" 1 "" {TEXT -1 41 "Calculated dirac for tnb _1 (.021000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"# B!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "gralter(dirac,trig sin,factor,power,expand);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component si mplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 52 "Applying routine `simplify[trigsin]` to object dirac" }}{PARA 6 "" 1 "" {TEXT -1 39 "Applying routine factor \+ to object dirac" }}{PARA 6 "" 1 "" {TEXT -1 50 "Applying routine `simp lify[power]` to object dirac" }}{PARA 6 "" 1 "" {TEXT -1 39 "Applying \+ routine expand to object dirac" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C PU~Time~G$\"$T\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grd isplay(dirac);" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&diracG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%&diracG,: *2^##!\"\"\"\"%\"\"\"-%\"UG6#%\"tG#F)\"\"#,&*$)F/F1F+F+*$)%\"lGF1F+F+F )%\"hGF+-%$psiGF.F+-%%diffG6$F,F/F+-%#pdG6$-%&sigmaG6#F+F+F+F/F1F+*2F' F+F,F0F2F)F8F+F9F+F;F+F>F+F7F1F+*0^#F)F+F,#F+F1F2F)F8F+F9F+F/F+F>F+F+* 2FGF+F2#!\"*F1-%$sinG6#%&thetaGF)F8F+F9F+-%$cosGFMF+F/\"\")-F?6$-FB6#F 1FTF+F+*4F1F+F2FIFKF)F8F+F9F+FOF+F/\"\"'FRF+F7F1F+*4\"\"$F+F2FIFKF)F8F +F9F+FOF+F/F*FRF+F7F*F+*4F1F+F2FIFKF)F8F+F9F+F/F1FOF+F7FWFRF+F+*2FGF+F 2FIFKF)F8F+F9F+FOF+F7FQFRF+F+*,F2F)%\"mGF+%\"cGF+F9F+F/F1F)*,F2F)FgnF+ FhnF+F9F+F7F1F)*0FFF+F,FGF2F)F8F+F>F+-F<6$F9F/F+F/F1F+*0FFF+F,FGF2F)F8 F+F>F+F[oF+F7F1F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 78 "New definit ions for providing the Dirac equation in a more comprehensible form" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "define(`gen`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "define(`gama`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "grmap(dirac,subs,pd(sigma(1),1)=gama(0),pd (sigma(2),1)=-I*gama(0)*gama(5),pd(sigma(3),1)=-gama(5),pd(1,sigma(1)) =2*gen(2,3),pd(1,sigma(2))=2*gen(3,1),pd(1,sigma(3))=2*gen(1,2),pd(sig ma(2),sigma(1))=-I*gama(1),pd(sigma(2),sigma(2))=-I*gama(2),pd(sigma(2 ),sigma(3))=-I*gama(3),pd(sigma(1),sigma(1))=gama(1)*gama(5),pd(sigma( 1),sigma(2))=gama(2)*gama(5),pd(sigma(1),sigma(3))=gama(3)*gama(5),pd( sigma(3),sigma(1))=2*I*gen(0,1),pd(sigma(3),sigma(2))=2*I*gen(0,2),pd( sigma(3),sigma(3))=2*I*gen(0,3),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 30 " Applying routine subs to dirac" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grdisplay(dirac);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the ~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&diracG" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/%&diracG,:*2^##!\"\"\"\"%\"\"\"-%\"UG 6#%\"tG#F)\"\"#,&*$)F/F1F+F+*$)%\"lGF1F+F+F)%\"hGF+-%$psiGF.F+-%%diffG 6$F,F/F+-%%gamaG6#\"\"!F+F/F1F+*2F'F+F,F0F2F)F8F+F9F+F;F+F>F+F7F1F+*0^ #F)F+F,#F+F1F2F)F8F+F9F+F/F+F>F+F+*2^#F0F+F2#!\"*F1-%$sinG6#%&thetaGF) F8F+F9F+-%$cosGFLF+F/\"\")-F?6#F1F+F+*4^#!\"#F+F2FHFJF)F8F+F9F+FNF+F/ \"\"'FQF+F7F1F+*4^#!\"$F+F2FHFJF)F8F+F9F+FNF+F/F*FQF+F7F*F+*4FTF+F2FHF JF)F8F+F9F+F/F1FNF+F7FVFQF+F+*2FGF+F2FHFJF)F8F+F9F+FNF+F7FPFQF+F+*,F2F )%\"mGF+%\"cGF+F9F+F/F1F)*,F2F)FgnF+FhnF+F9F+F7F1F)*0FDF+F,FEF2F)F8F+F >F+-F<6$F9F/F+F/F1F+*0FDF+F,FEF2F)F8F+F>F+F[oF+F7F1F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "nops(grcomponent(dirac));" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gralter(dirac,normal,simplif y);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTe nsorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 "Applying routine normal to object dirac" }}{PARA 6 "" 1 " " {TEXT -1 41 "Applying routine simplify to object dirac" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$%R!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grdisplay(dirac);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~tnb_1~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% &diracG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%&diracG,$**,4*0%\"mG\"\" \"%\"cGF*-%$psiG6#%\"tGF*)F/\"\"#F*-%%sqrtG6#-%\"UGF.F*-F36#,&*$F0F*F* *$)%\"lGF1F*F*F*-%$sinG6#%&thetaGF*\"\"%*2^#F*F*%\"hGF*F,F*-%%diffG6$F 5F/F*-%%gamaG6#\"\"!F*F0F*F7F*F>F*F**2^#FBF*F5F*FEF*FIF*-FG6$F,F/F*F0F *F7F*F>F*F**0^#F1F*F0F*FEF*F,F*-%$cosGF@F*-FJ6#F1F*F2F*F**2FNF*F5F*FEF *F,F*F/F*FIF*F7F*F>F*F**2FNF*F5F*FEF*FIF*FOF*FF*F**2FDF*FEF*F ,F*FFF*FIF*FF*F**2FBF*F)F*F+F*F,F*FF*F**0FRF*FEF *F,F*FSF*FUF*F2F*F!\"\"F9#!\"$F1F5#FfnF1#FfnFB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "76 0 0" 105 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }