{VERSION 5 0 "DEC ALPHA UNIX" "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 "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {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 1 }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 1 }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 1 }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 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT 256 69 "Calculating the Dirac e quation on a Schwarzschild metric with torsion" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 257 35 "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,sig ma(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),pr(sigma(2),sigma(1))=-pr(sigma(1),sigma(2)),pr(sigma(1),si gma(2))=I*sigma(3),pr(sigma(3),sigma(1))=-pr(sigma(1),sigma(3)),pr(sig ma(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::algebra ic),(d::integer)*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::algebraic,0)=0,pd(1,1)=1,pd(I*a::algebraic,b::algebraic)=I*p d(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)=-p d(a,b));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1407 "define(`&p`,` &p`(-a::algebraic,-b::algebraic)=`&p`(a,b),(-pd(sigma(a::algebraic),si gma(b::algebraic))) &p (-pd(sigma(c::algebraic),sigma(d::algebraic)))= pd(sigma(a),sigma(b)) &p pd(sigma(c),sigma(d)),(-pd(a::algebraic,b::al gebraic)) &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::algebr aic,d::algebraic))=-m*(pd(a,b) &p pd(c,d)),(-pd(a::algebraic,b::algebr aic)) &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))=pd(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::alge braic)) &p pd(1,sigma(c::algebraic))=pd(pr(sigma(a),1),pr(sigma(b),sig ma(c))),pd(sigma(a::algebraic),1) &p pd(sigma(b::algebraic),sigma(c::a lgebraic))=pd(pr(sigma(a),sigma(b)),pr(1,sigma(c))),pd(sigma(a::algebr aic),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::a lgebraic))=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::algeb raic)=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 "definemore(`&p`,`&p`(c::algebraic*pd(sigma(a::algeb raic),sigma(b::algebraic)),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),sigma(f::algebraic)))=c*d*`&p`(pd(a,sigma(b)),pd(s igma(e),sigma(f))),`&p`(c::algebraic*pd(sigma(a::algebraic),b::algebra ic),d::algebraic*pd(sigma(e::algebraic),sigma(f::algebraic)))=c*d*`&p` (pd(sigma(a),b),pd(sigma(e),sigma(f))),`&p`(c::algebraic*pd(sigma(a::a lgebraic),sigma(b::algebraic)),d::algebraic*pd(e::algebraic,sigma(f::a lgebraic)))=c*d*`&p`(pd(sigma(a),sigma(b)),pd(e,sigma(f))),`&p`(c::alg ebraic*pd(sigma(a::algebraic),sigma(b::algebraic)),d::algebraic*pd(sig ma(e::algebraic),f::algebraic))=c*d*`&p`(pd(sigma(a),sigma(b)),pd(sigm a(e),f)),`&p`(pd(sigma(a::algebraic),1),pd(1,sigma(b::algebraic)))=pd( pr(sigma(a),1),pr(1,sigma(b))),`&p`(c::algebraic*pd(sigma(a::algebraic ),e::algebraic),d::algebraic*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 "" {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 "" {MPLTEXT 1 0 150 "define(gam,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::algebraic,b::algebraic)=a &p b - \+ b &p a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "comu(gam(1),gam (2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&^#!\"#\"\"\"-%#pdG6$F&-%&si gmaG6#\"\"$F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "I*comu(gam (2),gam(1))/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#pdG6$\"\"\"-%&s igmaG6#\"\"$#!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " qload(sscw_2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetime G%'sscw_2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spaceti me:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ )%#x~G%\"aG-%'vectorG6#7&%\"rG%&thetaG%$phiG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Basis~inner~productG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$etaG6$%$bupGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$etaG%$(a )G\"\"\")%!G%$(b)GF(-%'matrixG6#7&7&!\"\"\"\"!F2F27&F2F1F2F27&F2F2F1F2 7&F2F2F2F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=Basis~(covariant~compo nents)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w1G6#%#dnG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%'omega1G6#%\"aG-%'vectorG6#7&-%$expG6#,$-%'la mbdaG6#%\"rG#\"\"\"\"\"#\"\"!F7F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %#w2G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega2G6#%\"aG-%'v ectorG6#7&\"\"!%\"rGF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w3G6#%# dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega3G6#%\"aG-%'vectorG6#7 &\"\"!F,*&%\"rG\"\"\"-%$sinG6#%&thetaGF/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w4G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'om ega4G6#%\"aG-%'vectorG6#7&\"\"!F,F,-%$expG6#,$-%#nuG6#%\"rG#\"\"\"\"\" #" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(metric);" }} {PARA 6 "" 1 "" {TEXT -1 46 "Calculated e(bup,dn) for sscw_2 (.005000 \+ sec.)" }}{PARA 6 "" 1 "" {TEXT -1 45 "Calculated g(dn,dn) for sscw_2 ( .011000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#8!\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "gralter(metric,simplif y);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTe nsorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine simplify to object g(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$0\"!\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 18 "grdisplay(metric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Covariant~metric~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&& %#g~G6#%\"aG\"\"\"&%!G6#%\"bGF)-%'matrixG6#7&7&,$-%$expG6#-%'lambdaG6# %\"rG!\"\"\"\"!F " 0 " " {MPLTEXT 1 0 11 "grcalc(ds);" }}{PARA 6 "" 1 "" {TEXT -1 39 "Calcula ted ds for sscw_2 (.002000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% *CPU~Time~G$\"\"%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gr display(ds);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~space time:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",**(-%$expG6#-%'lambdaG6#% \"rGF(%#~dGF()F1%#2~GF(!\"\"*()F1F'F(F2F()%&thetaGF4F(F5**F7F(,&F5F(*$ )-%$cosG6#F9F'F(F(F(F2F()%$phiGF4F(F(*(-F,6#-%#nuGF0F(F2F()%\"tGF4F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gralter(ds,trigsin);" } }{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 49 "Applying routine `simplify[trigsin]` 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# %:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line ~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",**( -%$expG6#-%'lambdaG6#%\"rGF(%#~dGF()F1%#2~GF(!\"\"*()F1F'F(F2F()%&thet aGF4F(F5**F7F(F2F()%$phiGF4F()-%$sinG6#F9F'F(F5*(-F,6#-%#nuGF0F(F2F()% \"tGF4F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "grcalc(w1(dn, pdn));grcalc(w2(dn,pdn));grcalc(w3(dn,pdn));grcalc(w4(dn,pdn));" }} {PARA 6 "" 1 "" {TEXT -1 35 "Created a definition for w1(dn,pdn)" }} {PARA 6 "" 1 "" {TEXT -1 47 "Calculated w1(dn,pdn) for sscw_2 (.004000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$l\"!\"$" }} {PARA 6 "" 1 "" {TEXT -1 35 "Created a definition for w2(dn,pdn)" }} {PARA 6 "" 1 "" {TEXT -1 47 "Calculated w2(dn,pdn) for sscw_2 (.004000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#I!\"$" }} {PARA 6 "" 1 "" {TEXT -1 35 "Created a definition for w3(dn,pdn)" }} {PARA 6 "" 1 "" {TEXT -1 47 "Calculated w3(dn,pdn) for sscw_2 (.003000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#D!\"$" }} {PARA 6 "" 1 "" {TEXT -1 35 "Created a definition for w4(dn,pdn)" }} {PARA 6 "" 1 "" {TEXT -1 47 "Calculated w4(dn,pdn) for sscw_2 (.003000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#I!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grdisplay(w1(dn,pdn));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+w1(dn,pdn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w1G6$%#dnG%$pdnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%'ome ga1G6#%\"aG\"\"\"&%!G6#%#,bGF)-%'matrixG6#7&7&,$*&-%%diffG6$-%'lambdaG 6#%\"rGF;F)-%$expG6#,$F8#F)\"\"#F)F@\"\"!FBFB7&FBFBFBFBFCFC" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grdisplay(w2(dn,pdn));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+w2(dn,pdn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w2G6$%#dnG%$pdnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%'ome ga2G6#%\"aG\"\"\"&%!G6#%#,bGF)-%'matrixG6#7&7&\"\"!F3F3F37&F)F3F3F3F2F 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grdisplay(w3(dn,pdn)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%+w3(dn,pdn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w3G6$%#dnG%$pdnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&&%'omega3G6#%\"aG\"\"\"&%!G6#%#,bGF)-%'matrixG6#7&7&\"\"!F3F3F3F27& -%$sinG6#%&thetaG*&%\"rGF)-%$cosGF7F)F3F3F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grdisplay(w4(dn,pdn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+w4(dn,pdn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w4G 6$%#dnG%$pdnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%'omega4G6#%\"aG \"\"\"&%!G6#%#,bGF)-%'matrixG6#7&7&\"\"!F3F3F3F2F27&,$*&-%%diffG6$-%#n uG6#%\"rGF=F)-%$expG6#,$F:#F)\"\"#F)FBF3F3F3" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 88 "Now the definition of the torsion components. Observ e how we used the previous results !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "grdef(`tor\{ ^a b c \}:=w1\{ b ,c \}*kdelta\{ ^a $r \+ \} + w2\{ b ,c \}*kdelta\{ ^a $theta \} + w3\{ b ,c \}*kdelta\{ ^a $ph i \} + w4\{ b ,c \}*kdelta\{ ^a $t \}`);" }}{PARA 6 "" 1 "" {TEXT -1 36 "Created definition for tor(up,dn,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grcalc(tor(up,dn,dn));" }}{PARA 6 "" 1 "" {TEXT -1 51 "Calculated kdelta(up,dn) for sscw_2 (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 50 "Calculated tor(up,dn,dn) for sscw_2 (.022000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#A!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grdisplay(tor(up,dn,dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.tor(up,dn,dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&)%$torG%\"rG\"\"\"&%!G6#*&%\"rGF(%\"rGF(F(,$*&-%%diffG6$-%'lambda G6#%\"rGF7F(-%$expG6#,$F4#F(\"\"#F(F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%&thetaG\"\"\"&%!G6#*&%&thetaGF(%\"rGF(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/*&)%$torG%$phiG\"\"\"&%!G6#*&%$phiGF(%\"rGF(F(- %$sinG6#%&thetaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%\"tG\" \"\"&%!G6#*&%\"tGF(%\"rGF(F(,$*&-%%diffG6$-%#nuG6#%\"rGF7F(-%$expG6#,$ F4#F(\"\"#F(F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%$phiG\"\" \"&%!G6#*&%$phiGF(%&thetaGF(F(*&%\"rGF(-%$cosG6#%&thetaGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "grmap(tor(up,dn,dn),subs,lambda(r)= f1(r),nu(r)=f2(r),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 38 "Applying routi ne subs to tor(up,dn,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grdisplay(tor(up,dn,dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~th e~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.tor(up,dn,dn )G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%\"rG\"\"\"&%!G6#*&%\" rGF(%\"rGF(F(,$*&-%%diffG6$-%#f1G6#%\"rGF7F(-%$expG6#,$F4#F(\"\"#F(F< " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%&thetaG\"\"\"&%!G6#*&%& thetaGF(%\"rGF(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%$phi G\"\"\"&%!G6#*&%$phiGF(%\"rGF(F(-%$sinG6#%&thetaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%\"tG\"\"\"&%!G6#*&%\"tGF(%\"rGF(F(,$*&-%%dif fG6$-%#f2G6#%\"rGF7F(-%$expG6#,$F4#F(\"\"#F(F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG%$phiG\"\"\"&%!G6#*&%$phiGF(%&thetaGF(F(*&%\" rGF(-%$cosG6#%&thetaGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " grcalc(tor(bup,bdn,bdn));" }}{PARA 6 "" 1 "" {TEXT -1 40 "Created defi nition for tor(bup,bdn,bdn) " }}{PARA 6 "" 1 "" {TEXT -1 42 "Calculate d detg for sscw_2 (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 45 "Calc ulated g(up,up) for sscw_2 (.002000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 46 "Calculated e(bdn,up) for sscw_2 (.004000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 53 "Calculated tor(bup,bdn,bdn) for sscw_2 (.263000 sec.)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$)Q!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "gralter(tor(bup,bdn,bdn),trigsin,po wer,simplify);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplificatio n of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 63 "Applying routine `simplify[trigsin]` to object tor (bup,bdn,bdn)" }}{PARA 6 "" 1 "" {TEXT -1 61 "Applying routine `simpli fy[power]` to object tor(bup,bdn,bdn)" }}{PARA 6 "" 1 "" {TEXT -1 52 " Applying routine simplify to object tor(bup,bdn,bdn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$/\"!\"$" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "grdisplay(tor(bup,bdn,bdn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1tor(bup,bdn,bdn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /*&)%$torG-%!G6#%\"1G\"\"\"&F(6#*&-F(6#%\"1GF+-F(6#%\"1GF+F+,$*&-%$exp G6#,&-%'lambdaG6#%\"rG#!\"\"\"\"#*&#F+FAF+-%#f1GF=F+F+F+-%%diffG6$FDF> F+F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG-%!G6#%\"2G\"\"\"&F( 6#*&-F(6#%\"2GF+-F(6#%\"1GF+F+,$-%$expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\" #F?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG-%!G6#%\"3G\"\"\"&F(6 #*&-F(6#%\"3GF+-F(6#%\"1GF+F+,$*&-%$sinG6#%&thetaGF+-%$expG6#,$-%'lamb daG6#%\"rG#!\"\"\"\"#F+FD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tor G-%!G6#%\"4G\"\"\"&F(6#*&-F(6#%\"4GF+-F(6#%\"1GF+F+,$*&-%$expG6#,&-%'l ambdaG6#%\"rG#!\"\"\"\"#*&#F+FAF+-%#f2GF=F+F+F+-%%diffG6$FDF>F+F?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$torG-%!G6#%\"3G\"\"\"&F(6#*&-F(6 #%\"3GF+-F(6#%\"2GF+F+,$-%$cosG6#%&thetaG!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "grdef(`SS\{ ^a ^b \}`);" }}{PARA 6 "" 1 "" {TEXT -1 32 "Created definition for SS(up,up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grcalc(SS(up,up)):" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 37 "Enter components for object SS(up, up)" }}{PARA 6 "" 1 "" {TEXT -1 56 " If you wish to quit at any point and leave this object" }}{PARA 6 "" 1 "" {TEXT -1 39 " uninitialized , enter the string exit." }}{PARA 6 "" 1 "" {TEXT -1 51 " REMEMBER to complete each entry with a semicolon." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"rG\"\"\")%!G%\"rGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(1),ga m(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%&thetaG\"\"\")%!G% \"rGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(2),gam(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%$phiG\"\"\")%!G%\"rGF'" }}{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%\"rGF'" }}{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%\"rG\"\"\")%!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#*&)%#SS G%&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%$phiG\"\"\")%!G%&theta GF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(3),gam(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"tG\"\"\")%!G%&thetaGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)* comu(gam(0),gam(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SSG%\"rG \"\"\")%!G%$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#*&)%#SSG%&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%\"rG\"\"\")%!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%&thet aG\"\"\")%!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#*&)%#SSG%$phiG\"\"\")%!G%\"tGF'" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(3),gam(0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%#SS G%\"tG\"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 26 "(I/4)*comu(gam(0),gam(0));" }} {PARA 6 "" 1 "" {TEXT -1 46 "Calculated SS(up,up) for sscw_2 (.652000 \+ sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$`'!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "grdisplay(SS(up,up));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*SS(up,up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#SSG6$%#upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#SSG%\"a G\"\"\")%!G%\"bGF(-%'matrixG6#7&7&\"\"!,$-%#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 "" {MPLTEXT 1 0 22 "g rcalc(Chr(dn,dn,dn));" }}{PARA 6 "" 1 "" {TEXT -1 49 "Calculated g(dn, dn,pdn) for sscw_2 (.009000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 50 "Calc ulated Chr(dn,dn,dn) for sscw_2 (.008000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#?!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "gralter(Chr(dn,dn,dn),simplify);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 49 "Applying \+ routine simplify to object Chr(dn,dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#\\!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grdisplay(Chr(dn,dn,dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:F or~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%`oChrist offel~symbol~of~the~first~kind~(symmetric~in~first~two~indices)G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"rG\"\"\"%\"rGF)%\"rGF ),$*&-%%diffG6$-%'lambdaG6#%\"rGF4F)-%$expG6#F1F)#!\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"rG\"\"\"%&thetaGF)%&thetaGF ),$%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"rG\" \"\"%$phiGF)%$phiGF),&%\"rG!\"\"*&F-F))-%$cosG6#%&thetaG\"\"#F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%\"rG\"\"\"%\"tGF)%\"tGF ),$*&-%%diffG6$-%#nuG6#%\"rGF4F)-%$expG6#F1F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%&thetaG\"\"\"%&thetaGF)%\"rGF)%\"rG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%&thetaG\"\"\"%$phiG F)%$phiGF),$*()%\"rG\"\"#F)-%$cosG6#%&thetaGF)-%$sinGF3F)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%$phiG\"\"\"%$phiGF)%\"r GF),&%\"rGF)*&F-F))-%$cosG6#%&thetaG\"\"#F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&GammaG6#*(%$phiG\"\"\"%$phiGF)%&thetaGF)*()%\"rG\" \"#F)-%$cosG6#%&thetaGF)-%$sinGF2F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%&GammaG6#*(%\"tG\"\"\"%\"tGF)%\"rGF),$*&-%%diffG6$-%#nuG6#%\"rGF4F )-%$expG6#F1F)#!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grcalc(Chr(bdn,bdn,bdn));" }}{PARA 6 "" 1 "" {TEXT -1 40 "Created \+ definition for Chr(bdn,bdn,bdn) " }}{PARA 6 "" 1 "" {TEXT -1 53 "Calcu lated Chr(bdn,bdn,bdn) for sscw_2 (.211000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$=#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "gralter(Chr(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 `simplify[trigsin]` to object Chr(bdn,bdn,bdn)" } }{PARA 6 "" 1 "" {TEXT -1 50 "Applying routine expand to object Chr(bd n,bdn,bdn)" }}{PARA 6 "" 1 "" {TEXT -1 52 "Applying routine simplify t o object Chr(bdn,bdn,bdn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Ti me~G$\"$3\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "grdispla y(Chr(bdn,bdn,bdn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw _2~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#%\"1GF,,$*&-%$expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F,-%%d iffG6$F9F " 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 "" {TEXT 258 59 "Here we have a new special term with torsion contribution !" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "grdef(`det\{ i \}:=(I/2)* SS\{ ^a ^b \}*(Chr\{ (a) (i) (b) \}+(tor\{ (a) (i) (b) \}-tor\{ (i) (b ) (a) \} + tor\{ (b) (a) (i) \}))`);" }}{PARA 6 "" 1 "" {TEXT -1 40 "C reated definition for tor(bdn,bdn,bdn) " }}{PARA 6 "" 1 "" {TEXT -1 30 "Created definition for det(dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "grcalc(de(dn));grcalc(det(dn));" }}{PARA 6 "" 1 "" {TEXT -1 43 "Calculated de(dn) for sscw_2 (.016000 sec.)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"# " 0 "" {MPLTEXT 1 0 77 "gralter(de(dn),trigsin,factor,expan d);gralter(det(dn),trigsin,factor,expand);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 53 "Applying routine `si mplify[trigsin]` to object de(dn)" }}{PARA 6 "" 1 "" {TEXT -1 40 "Appl ying routine factor to object de(dn)" }}{PARA 6 "" 1 "" {TEXT -1 40 "A pplying routine expand to object de(dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#;!\"$" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 54 "Applying routine `simplify[trigsin]` to object det(dn)" }}{PARA 6 "" 1 "" {TEXT -1 41 "Applying routine fa ctor to object det(dn)" }}{PARA 6 "" 1 "" {TEXT -1 41 "Applying routin e expand to object det(dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~T ime~G$\"#K!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grdisplay (de(dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetim e:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'de(dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#deG6#%#dnG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#de G6#%\"aG-%'vectorG6#7&\"\"!**^##\"\"\"\"\"#F0-%#pdG6$F0-%&sigmaG6#\"\" $F0%\"rG!\"\"-%$expG6#,$-%'lambdaG6#F9#F:F1F0,&**^#FBF0F9F:-F36$F0-F66 #F1F0F;F0F0*,F.F0-%$sinG6#%&thetaGF:F9F:-F36$F0-F66#F0F0-%$cosGFMF0F0, $*(-F36$F5FQF0F;F0-%%diffG6$-%#nuGFAF9F0#F0\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "grdisplay(det(dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(det(dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$detG6# %#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%$detG6#%&thetaG,&**^##\"\" \"\"\"#F,-%#pdG6$F,-%&sigmaG6#\"\"$F,%\"rG!\"\"-%$expG6#,$-%'lambdaG6# F5#F6F-F,F,*(^#F>F,F.F,F7F,F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%$d etG6#%$phiG,***^##!\"\"\"\"#\"\"\"%\"rGF,-%#pdG6$F.-%&sigmaG6#F-F.-%$e xpG6#,$-%'lambdaG6#F/F+F.F.**^##F.F-F.-%$sinG6#%&thetaGF.F0F.F6F.F.*,F >F.F@F,F/F,-F16$F.-F46#F.F.-%$cosGFBF.F.*(F*F.FEF.FIF.F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%$detG6#%\"tG,&*(-%#pdG6$-%&sigmaG6#\"\"$-F.6# \"\"\"F3-%$expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F3-%%diffG6$-%#nuGF:F;F 3#F3\"\"%*&#F3FEF3**F*F3F4F3-F56#,$-%#f2GF:#F3F>F3-F@6$FLF;F3F3F=" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "grdef(`ga\{ ^a \}:=[gam(1),g am(2),gam(3),gam(0)]`);" }}{PARA 6 "" 1 "" {TEXT -1 38 "Components ass igned for metric: sscw_2" }}{PARA 6 "" 1 "" {TEXT -1 29 "Created defin ition for ga(up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grdispl ay(ga(up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacet ime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'ga(up)G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%#gaG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%# gaG%\"aG-%'vectorG6#7&*&^#\"\"\"F--%#pdG6$-%&sigmaG6#\"\"#-F26#F-F-*&F ,F--F/6$F1F1F-*&F,F--F/6$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#*&)%$verG%\"rG\"\"\")%!G%\"rGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grcom ponent(ga(up),[r]) &p grcomponent(ga(up),[r]) +grcomponent(ga(up),[r]) &p grcomponent(ga(up),[r])+2*grcomponent(eta(bup,bup),[r,r]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%&thetaG\"\"\")%!G%\"rGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 150 "grcomponent(ga(up),[theta]) &p grcomponent(ga(up),[r ]) +grcomponent(ga(up),[r]) &p grcomponent(ga(up),[theta])+2*grcompone nt(eta(bup,bup),[theta,r]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$ verG%$phiG\"\"\")%!G%\"rGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 144 "grcomponent(ga(up),[ph i]) &p grcomponent(ga(up),[r]) +grcomponent(ga(up),[r]) &p grcomponent (ga(up),[phi])+2*grcomponent(eta(bup,bup),[phi,r]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%\"tG\"\"\")%!G%\"rGF'" }}{PARA 6 "" 1 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grc omponent(ga(up),[t]) &p grcomponent(ga(up),[r]) +grcomponent(ga(up),[r ]) &p grcomponent(ga(up),[t])+2*grcomponent(eta(bup,bup),[t,r]);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$verG%\"rG\"\"\")%!G%&thetaGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 150 "grcomponent(ga(up),[r]) &p grcomponent(ga(up),[theta ]) +grcomponent(ga(up),[theta]) &p grcomponent(ga(up),[r])+2*grcompone nt(eta(bup,bup),[r,theta]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$ 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%\"rG\"\"\")%!G%$phiGF'" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 144 "grcomponent(ga(up),[r]) &p grcomponent(ga(up),[phi]) +grcomponent(ga(up),[phi]) &p grcomponent(ga(up),[r])+2*grcomponent(e ta(bup,bup),[r,phi]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$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# *&)%$verG%\"rG\"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grcomponent(ga(up),[r] ) &p grcomponent(ga(up),[t]) +grcomponent(ga(up),[t]) &p grcomponent(g a(up),[r])+2*grcomponent(eta(bup,bup),[r,t]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%$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#*&)%$verG %\"tG\"\"\")%!G%\"tGF'" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 138 "grcomponent(ga(up),[t]) &p gr component(ga(up),[t]) +grcomponent(ga(up),[t]) &p grcomponent(ga(up),[ t])+2*grcomponent(eta(bup,bup),[t,t]);\n" }}{PARA 6 "" 1 "" {TEXT -1 47 "Calculated ver(up,up) for sscw_2 (.539000 sec.)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%*CPU~Time~G$\"$S&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "grdisplay(ver(up,up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~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#/*&)%$verG%\"aG\"\"\")%!G%\"bGF(-%'mat rixG6#7&7&!\"%\"\"!F2F27&F2F1F2F27&F2F2F1F27&F2F2F2\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "grdef(`Psid\{ a \}:=[diff(psi(r),r) ,diff(psi(r),theta),diff(psi(r),phi),diff(psi(r),t)]`);" }}{PARA 6 "" 1 "" {TEXT -1 38 "Components assigned for metric: sscw_2" }}{PARA 6 " " 1 "" {TEXT -1 31 "Created definition for Psid(dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grcalc(Psid(dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\"\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "grdisplay(Psid(dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)P sid(dn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%PsidG6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%PsidG6#%\"aG-%'vectorG6#7&-%%diffG6$-%$p siG6#%\"rGF2\"\"!F3F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "gr calc(Psid(bdn));" }}{PARA 6 "" 1 "" {TEXT -1 33 "Created definition fo r Psid(bdn) " }}{PARA 6 "" 1 "" {TEXT -1 46 "Calculated Psid(bdn) for \+ sscw_2 (.002000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G $\"\"*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gralter(Psid( bdn),simplify);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplificati on of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 45 "Applying routine simplify to object Psid(bdn)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#7!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "grdisplay(Psid(bdn));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%*Psid(bdn)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Ps idG6#%$bdnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%PsidG6#%$(a)G-%'vec torG6#7&,$*&-%%diffG6$-%$psiG6#%\"rGF4\"\"\"-%$expG6#,$-%'lambdaGF3#! \"\"\"\"#F5F=\"\"!F?F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a 0:=expand(grcomponent(de(dn),[t]));a00:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a0G,$*(-%#pdG6$-%&sigmaG6#\"\"$-F+6#\"\"\"F0-%$expG6 #,$-%'lambdaG6#%\"rG#!\"\"\"\"#F0-%%diffG6$-%#nuGF7F8F0#F0\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a00G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u0:=whattype(a0);u0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G%\"*G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"*G" } }}{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 11 "" 1 "" {XPPMATH 20 "6#>%$a00G,$**%\"hG\"\"\"-%$expG6 #,$-%'lambdaG6#%\"rG#!\"\"\"\"#F(-%%diffG6$-%#nuGF/F0F(-%#pdG6$-%&sigm aG6#F3-F=6#F(F(#F(\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a 00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"hG\"\"\"-%$expG6#,$-%'la mbdaG6#%\"rG#!\"\"\"\"#F&-%%diffG6$-%#nuGF-F.F&-%#pdG6$-%&sigmaG6#F1-F ;6#F&F&#F&\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a1:=expa nd(grcomponent(de(dn),[r]));a11:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#a1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a11G\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "u1:=whattype(a1);u1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u1G%(integerG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(integerG" }}}{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),[r]) &p op(i,a1) od else a11:=I*h* grcomponent(ga(up),[r]) &p a1 fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$a11G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a11;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a2:=expand(grcomponent(de(dn),[theta]));a22:=0;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a2G**^##\"\"\"\"\"#F(-%#pdG6$F(-%&s igmaG6#\"\"$F(%\"rG!\"\"-%$expG6#,$-%'lambdaG6#F1#F2F)F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$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#>%$a22G,$**%\"hG\"\"\"-%$ expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F(F0F2-%#pdG6$-%&sigmaG6#F3-F86#F( F(#F(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a22;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**%\"hG\"\"\"-%$expG6#,$-%'lambdaG6#%\"rG#!\" \"\"\"#F&F.F0-%#pdG6$-%&sigmaG6#F1-F66#F&F&#F&F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a3:=expand(grcomponent(de(dn),[phi]));a33:=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a3G,&**^##!\"\"\"\"#\"\"\"%\"rG F)-%#pdG6$F+-%&sigmaG6#F*F+-%$expG6#,$-%'lambdaG6#F,F(F+F+*,^##F+F*F+- %$sinG6#%&thetaGF)F,F)-F.6$F+-F16#F+F+-%$cosGF?F+F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$a33G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "u3:=whattype(a3);u3;\n " }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#u3G%\"+G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"+G " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(a3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "if u3=`+` then for i from 1 to nops(a3) do a33:=a33+I*h*grcompon ent(ga(up),[phi]) &p op(i,a3) od else a33:=I*h*grcomponent(ga(up),[phi ]) &p a3 fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a33;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&**%\"hG\"\"\"-%$expG6#,$-%'lambdaG6# %\"rG#!\"\"\"\"#F&F.F0-%#pdG6$-%&sigmaG6#F1-F66#F&F&#F&F1*.F:F&F%F&-%$ sinG6#%&thetaGF0-%$cosGF>F&F.F0-F36$F5F5F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "grdef(`dd`);" }}{PARA 6 "" 1 "" {TEXT -1 25 "Cre ated definition for dd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g rcalc(dd);" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 30 "Enter components for object dd" }}{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 semico lon." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#ddG" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 16 "a00+a1 1+a22+a33;" }}{PARA 6 "" 1 "" {TEXT -1 39 "Calculated dd for sscw_2 (. 007000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\")!\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "grdisplay(dd);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#ddG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%# ddG,$**%\"hG\"\"\",(*,-%$expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F(-%%diff G6$-%#nuGF1F2F(-%#pdG6$-%&sigmaG6#F5-F?6#F(F(-%$sinG6#%&thetaGF(F2F(F( **\"\"%F(F+F(F;F(FCF(F(*(F5F(-%$cosGFEF(-F<6$F>F>F(F(F(FCF4F2F4#F(FH" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gralter(dd,factor,expand) ;" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTens orII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying routine factor to object dd" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying routine expand to object dd" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\"'!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "grdisplay(dd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:F or~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#ddG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#ddG,(**%\"hG\"\"\"-%$expG6#,$-%'lam bdaG6#%\"rG#!\"\"\"\"#F(-%%diffG6$-%#nuGF/F0F(-%#pdG6$-%&sigmaG6#F3-F= 6#F(F(#F(\"\"%**F'F(F)F(F0F2F9F(F(*.#F(F3F(F'F(-%$sinG6#%&thetaGF2-%$c osGFHF(F0F2-F:6$F " 0 "" {MPLTEXT 1 0 46 "a0t:=expand(grcomponent(det(dn),[t]));a00t:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a0tG,&*(-%#pdG6$-%&sigmaG6#\"\"$-F+6#\"\"\"F0-%$expG 6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F0-%%diffG6$-%#nuGF7F8F0#F0\"\"%*&#F0F BF0**F'F0F1F0-F26#,$-%#f2GF7#F0F;F0-F=6$FIF8F0F0F:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%a00tG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u0t:=whattype(a0t);u0t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ u0tG%\"+G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"+G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nops(a0t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "if u0t=`+` the n for i from 1 to nops(a0t) do a00t:=a00t+I*h*grcomponent(ga(up),[t]) &p op(i,a0t) od else a00t:=I*h*grcomponent(ga(up),[t]) &p a0t fi;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a00t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**%\"hG\"\"\"-%$expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F &-%%diffG6$-%#nuGF-F.F&-%#pdG6$-%&sigmaG6#F1-F;6#F&F&#F&\"\"%*&#F&F@F& *,F%F&F'F&-F36$-%#f2GF-F.F&-F(6#,$FF#F&F1F&F7F&F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "a1t:=expand(grcomponent(det(dn),[r]));a11 t:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a1tG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%a11tG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u1t:=whattype(a1t);u1t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$u1tG%(integerG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(integerG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nops(a1t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "if u1t=`+` then for i from 1 to nops(a1t) do a11t:=a11t+I*h*gr component(ga(up),[r]) &p op(i,a1t) od else a11t:=I*h*grcomponent(ga(up ),[r]) &p a1t fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%a11tG\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a11t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a2t :=expand(grcomponent(det(dn),[theta]));a22t:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$a2tG,&**^##\"\"\"\"\"#F)-%#pdG6$F)-%&sigmaG6#\"\"$F) %\"rG!\"\"-%$expG6#,$-%'lambdaG6#F2#F3F*F)F)*(^#F;F)F+F)F4F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%a22tG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u2t:=whattype(a2t);u2t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$u2tG%\"+G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"+G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nops(a2t);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "if u2t=`+` then for i from 1 to nops(a2t) do a22t:=a22t+I*h*grco mponent(ga(up),[theta]) &p op(i,a2t) od else a22t:=I*h*grcomponent(ga( up),[theta]) &p a2t fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a 22t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**%\"hG\"\"\"-%$expG6#,$-%'l ambdaG6#%\"rG#!\"\"\"\"#F&F.F0-%#pdG6$-%&sigmaG6#F1-F66#F&F&#F&F1*&#F& F1F&*(F%F&F'F&F2F&F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "a 3t:=expand(grcomponent(det(dn),[phi]));a33t:=0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$a3tG,***^##!\"\"\"\"#\"\"\"%\"rGF)-%#pdG6$F+-%&sigma G6#F*F+-%$expG6#,$-%'lambdaG6#F,F(F+F+**^##F+F*F+-%$sinG6#%&thetaGF+F- F+F3F+F+*,F;F+F=F)F,F)-F.6$F+-F16#F+F+-%$cosGF?F+F+*(F'F+FBF+FFF+F+" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%a33tG\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 46 "u3t:=whattype(a3t);u3t;\n \+ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$u3tG%\"+G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"+G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nop s(a3t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "if u3t=`+` then for i from 1 to nops(a3t) d o a33t:=a33t+I*h*grcomponent(ga(up),[phi]) &p op(i,a3t) od else a33t:= I*h*grcomponent(ga(up),[phi]) &p a3t fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a33t;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,***%\"hG\"\" \"-%$expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F&F.F0-%#pdG6$-%&sigmaG6#F1-F 66#F&F&#F&F1*&#F&F1F&**F%F&F'F&F2F&-%$sinG6#%&thetaGF&F&F0*.F:F&F%F&F> F0-%$cosGF@F&F.F0-F36$F5F5F&F&*&#F&F1F&*(F%F&FCF&FEF&F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdef(`ddt`);" }}{PARA 6 "" 1 "" {TEXT -1 26 "Created definition for ddt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "grcalc(ddt);" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 31 "Enter components for object ddt" }}{PARA 6 "" 1 "" {TEXT -1 56 " If you wish to quit at any point and leave th is object" }}{PARA 6 "" 1 "" {TEXT -1 39 " uninitialized, enter the s tring exit." }}{PARA 6 "" 1 "" {TEXT -1 51 " REMEMBER to complete eac h entry with a semicolon." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$ddtG" } }{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "grcalc>" 0 "" {MPLTEXT 1 0 20 "a00t+a11t+a22t+a33t;" }}{PARA 6 "" 1 "" {TEXT -1 40 " Calculated ddt for sscw_2 (.010000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#6!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grdisplay(ddt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~ssc w_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$ddtG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$ddtG,$**%\"hG\"\"\",0*,-%$expG6#,$-%'lambdaG 6#%\"rG#!\"\"\"\"#F(-%%diffG6$-%#nuGF1F2F(-%#pdG6$-%&sigmaG6#F5-F?6#F( F(-%$sinG6#%&thetaGF(F2F(F4*.F+F(-F76$-%#f2GF1F2F(-F,6#,$FJ#F(F5F(F;F( FCF(F2F(F(**\"\"%F(F+F(F;F(FCF(F4*,F5F(F+F(F;F(FCF(F2F(F(*,F5F(F+F(F;F ()FCF5F(F2F(F(*(F5F(-%$cosGFEF(-F<6$F>F>F(F4*,F5F(FVF(FXF(FCF(F2F(F(F( FCF4F2F4#F4FQ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gralter(dd t,trigsin,factor,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 50 "Applying routine `simplify[trigsin]` to object ddt" }}{PARA 6 "" 1 "" {TEXT -1 37 "Applying routine factor to object ddt" }}{PARA 6 "" 1 "" {TEXT -1 37 "Applying routine expand to object ddt" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#B!\"$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grdisplay(ddt);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$ddtG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/% $ddtG,0**%\"hG\"\"\"-%$expG6#,$-%'lambdaG6#%\"rG#!\"\"\"\"#F(-%%diffG6 $-%#nuGF/F0F(-%#pdG6$-%&sigmaG6#F3-F=6#F(F(#F(\"\"%*&#F(FBF(*,F'F(F)F( -F56$-%#f2GF/F0F(-F*6#,$FH#F(F3F(F9F(F(F2**F'F(F)F(F0F2F9F(F(*&#F(F3F( *(F'F(F)F(F9F(F(F2*&#F(F3F(**F'F(F)F(F9F(-%$sinG6#%&thetaGF(F(F2*.FMF( F'F(FUF2-%$cosGFWF(F0F2-F:6$F " 0 "" {MPLTEXT 1 0 66 "grdef(`dirac:=I*h*ga\{ ^l \} *Psid\{ (l) \} + dd*psi(r) - m*c*psi(r)`);" }}{PARA 6 "" 1 "" {TEXT -1 28 "Created definition for dirac" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "grdef(`diract:=I*h*ga\{ ^l \}*Psid\{ (l) \} + ddt*psi (r) - m*c*psi(r)`);" }}{PARA 6 "" 1 "" {TEXT -1 29 "Created definition for diract" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "grcalc(dirac t);grcalc(dirac);" }}{PARA 6 "" 1 "" {TEXT -1 43 "Calculated diract fo r sscw_2 (.009000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time ~G$\"#7!\"$" }}{PARA 6 "" 1 "" {TEXT -1 42 "Calculated dirac for sscw_ 2 (.007000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\" )!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "gralter(dirac,fact or,power,normal,expand);gralter(diract,factor,power,normal,expand);" } }{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 "Applying routine factor to object dirac" }}{PARA 6 "" 1 "" {TEXT -1 50 "Applying routine `simplify[power]` to object dirac" }}{PARA 6 " " 1 "" {TEXT -1 39 "Applying routine normal to object dirac" }}{PARA 6 "" 1 "" {TEXT -1 39 "Applying routine expand to object dirac" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#B!\"$" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying \+ routine factor to object diract" }}{PARA 6 "" 1 "" {TEXT -1 51 "Applyi ng routine `simplify[power]` to object diract" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine normal to object diract" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routine expand to object diract" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#K!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grdisplay(dirac);" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%&diracG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%&diracG,,*,%\"hG\"\"\"-%$psiG6#%\"rGF(-%$expG6#,$-%'l ambdaGF+#!\"\"\"\"#F(-%%diffG6$-%#nuGF+F,F(-%#pdG6$-%&sigmaG6#F5-F?6#F (F(#F(\"\"%*,F,F4F'F(F)F(F-F(F;F(F(*0#F(F5F(-%$sinG6#%&thetaGF4F,F4F'F (F)F(-%$cosGFJF(-F<6$F>F>F(F(*(%\"mGF(%\"cGF(F)F(F4**F'F(F;F(-F76$F)F, F(F-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grdisplay(dirac t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%'diractG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%'diractG,4*,%\"hG\"\"\"-%$psiG6#%\"rGF(-%$expG6#,$-%' lambdaGF+#!\"\"\"\"#F(-%%diffG6$-%#nuGF+F,F(-%#pdG6$-%&sigmaG6#F5-F?6# F(F(#F(\"\"%*&#F(FDF(*.F'F(F)F(F-F(-F76$-%#f2GF+F,F(-F.6#,$FJ#F(F5F(F; F(F(F4*,F,F4F'F(F)F(F-F(F;F(F(*&#F(F5F(**F'F(F)F(F-F(F;F(F(F4*&#F(F5F( *,F'F(F)F(F-F(F;F(-%$sinG6#%&thetaGF(F(F4*0FOF(FWF4F,F4F'F(F)F(-%$cosG FYF(-F<6$F>F>F(F(*&#F(F5F(**F'F(F)F(FfnF(FhnF(F(F4*(%\"mGF(%\"cGF(F)F( F4**F'F(F;F(-F76$F)F,F(F-F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "define(`gen`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "def ine(`gama`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 499 "grmap(dira c,subs,pd(sigma(1),1)=gama(0),pd(sigma(2),1)=-I*gama(0)*gama(5),pd(sig ma(3),1)=-gama(5),pd(1,sigma(1))=2*gen(2,3),pd(1,sigma(2))=2*gen(3,1), pd(1,sigma(3))=2*gen(1,2),pd(sigma(2),sigma(1))=-I*gama(1),pd(sigma(2) ,sigma(2))=-I*gama(2),pd(sigma(2),sigma(3))=-I*gama(3),pd(sigma(1),sig ma(1))=gama(1)*gama(5),pd(sigma(1),sigma(2))=gama(2)*gama(5),pd(sigma( 1),sigma(3))=gama(3)*gama(5),pd(sigma(3),sigma(1))=2*I*gen(0,1),pd(sig ma(3),sigma(2))=2*I*gen(0,2),pd(sigma(3),sigma(3))=2*I*gen(0,3),`x`); " }}{PARA 6 "" 1 "" {TEXT -1 30 "Applying routine subs to dirac" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "grmap(diract,subs,pd(sigma( 1),1)=gama(0),pd(sigma(2),1)=-I*gama(0)*gama(5),pd(sigma(3),1)=-gama(5 ),pd(1,sigma(1))=2*gen(2,3),pd(1,sigma(2))=2*gen(3,1),pd(1,sigma(3))=2 *gen(1,2),pd(sigma(2),sigma(1))=-I*gama(1),pd(sigma(2),sigma(2))=-I*ga ma(2),pd(sigma(2),sigma(3))=-I*gama(3),pd(sigma(1),sigma(1))=gama(1)*g ama(5),pd(sigma(1),sigma(2))=gama(2)*gama(5),pd(sigma(1),sigma(3))=gam a(3)*gama(5),pd(sigma(3),sigma(1))=2*I*gen(0,1),pd(sigma(3),sigma(2))= 2*I*gen(0,2),pd(sigma(3),sigma(3))=2*I*gen(0,3),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 31 "Applying routine subs to diract" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "grdisplay(dirac);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&diracG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%&diracG,, *.^##!\"\"\"\"%\"\"\"%\"hGF+-%$psiG6#%\"rGF+-%$expG6#,$-%'lambdaGF/#F) \"\"#F+-%%diffG6$-%#nuGF/F0F+-%%gamaG6#F+F+F+*.^#F)F+F0F)F,F+F-F+F1F+F >F+F+*0^#F7F+-%$sinG6#%&thetaGF)F0F)F,F+F-F+-%$cosGFGF+-F?6#F8F+F+*(% \"mGF+%\"cGF+F-F+F)*,FBF+F,F+F>F+-F:6$F-F0F+F1F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grdisplay(diract);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~sscw_2~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'diractG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%'diractG ,4*.^##!\"\"\"\"%\"\"\"%\"hGF+-%$psiG6#%\"rGF+-%$expG6#,$-%'lambdaGF/# F)\"\"#F+-%%diffG6$-%#nuGF/F0F+-%%gamaG6#F+F+F+*0^##F+F*F+F,F+F-F+F1F+ -F:6$-%#f2GF/F0F+-F26#,$FF#F+F8F+F>F+F+*.^#F)F+F0F)F,F+F-F+F1F+F>F+F+* ,^#FKF+F,F+F-F+F1F+F>F+F+*.FOF+F,F+F-F+F1F+F>F+-%$sinG6#%&thetaGF+F+*0 ^#F7F+FQF)F0F)F,F+F-F+-%$cosGFSF+-F?6#F8F+F+*,FOF+F,F+F-F+FWF+FYF+F+*( 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