{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {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 "" 0 1 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 0 }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 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "T itle" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 } } {SECT 0 {PARA 18 "" 0 "" {TEXT -1 23 "Catalogue of Spacetimes" }} {PARA 19 "" 0 "" {TEXT -1 27 "Thomas Mueller, Frank Grave" }}{PARA 0 " " 0 "" {TEXT 256 69 "Schwarzschild metric in Kruskal-Szekeres coordina tes (T,X,theta,phi)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 16 "Date: 27.04.2009" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "restart:grtw();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%06~February~2001G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayll~LakeG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~the~authors.G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~available~from:~ http://grtensor.phy.queensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%B /home/muelleta/local/Grii/metricsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "makeg(kruskSzek);" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 46 "Makeg 2. 0: GRTensor metric/basis entry utility" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 41 "To quit makeg, type 'exit' at any pr ompt." }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 44 "Do you wish to enter a 1) metric [g(dn,dn)]," }}{PARA 6 "" 1 "" {TEXT -1 44 " 2) line element [ds]," }}{PARA 6 " " 1 "" {TEXT -1 63 " 3) non-holonomic basis [e(1 )...e(n)], or" }}{PARA 6 "" 1 "" {TEXT -1 49 " 4 ) NP tetrad [l,n,m,mbar]?" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "2:" }}{PARA 6 "" 1 "" {TEXT -1 50 "Enter coordinates as a LIST (eg. [t,r,theta,phi]):" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 16 "[T,X,theta,phi]:" }}{PARA 6 "" 1 "" {TEXT -1 64 "Enter the line element using d[coord] to indicate di fferentials." }}{PARA 6 "" 1 "" {TEXT -1 55 "(for example, r^2*(d[the ta]^2 + sin(theta)^2*d[phi]^2)" }}{PARA 6 "" 1 "" {TEXT -1 27 "[Type ' exit' to quit makeg]" }}{PARA 6 "" 1 "" {TEXT -1 8 " ds^2 = " }}} {EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 91 "4*rs^3/r(T,X)*exp(-r(T,X )/rs)*(-d[T]^2+d[X]^2)+r(T,X)^2*(d[theta]^2+sin(theta)^2*d[phi]^2):" } }{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 67 "If there are any complex valued coordinates, constants or functions" }}{PARA 6 "" 1 "" {TEXT -1 66 "for this spacetime, please enter them as a SET \+ ( eg. \{ z, psi \} )." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 33 "Complex quantities [default=\{\}]: " }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "\{\}" }}{PARA 6 "" 1 "" {TEXT -1 2 "\{ \}" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%AThe~values~you~have~entered~are:G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%,CoordinatesG7&%\"TG%\"XG%&thetaG%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(Metric:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"gG 6#%\"aG\"\"\"&%!G6#%\"bGF)K%'matrixG6#7&7&,$**\"\"%F)%#rsG\"\"$-%\"rG6 $%\"TG%\"XG!\"\"-%$expG6#*&F8F)F6F=F=F=\"\"!FBFB7&FB,$**F5F)F6F7F8F=F> F=F)FBFB7&FBFB*$)F8\"\"#F)FB7&FBFBFB*&FHF))-%$sinG6#%&thetaGFIF)Q(ppri nt26\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "You may choose to 0) Use the me tric WITHOUT saving it," }}{PARA 6 "" 1 "" {TEXT -1 46 " \+ 1) Save the metric as it is," }}{PARA 6 "" 1 "" {TEXT -1 54 " \+ 2) Correct an element of the metric," }}{PARA 6 "" 1 "" {TEXT -1 41 " 3) Re-enter the metric," }}{PARA 6 "" 1 "" {TEXT -1 54 " 4) Add/change constraint equations , " }}{PARA 6 "" 1 "" {TEXT -1 47 " 5) Add a text des cription, or" }}{PARA 6 "" 1 "" {TEXT -1 61 " 6) Aban don this metric and return to Maple." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "0:" }}{PARA 6 "" 1 "" {TEXT -1 43 "Calculated ds for kruskSzek (0.000000 sec.)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%*kruskSzekG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%=For~the~kruskSzek~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aGK%'vector G6#7&%\"TG%\"XG%&thetaG%$phiGQ(pprint36\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$) %$~dsG\"\"#\"\"\",**.\"\"%F(%#rsG\"\"$-%\"rG6$%\"TG%\"XG!\"\"-%$expG6# *&F.F(F,F3F3%#~dGF()F1%#2~GF(F3*.F+F(F,F-F.F3F4F3F8F()F2F:F(F(*()F.F'F (F8F()%&thetaGF:F(F(**F>F()-%$sinG6#F@F'F(F8F()%$phiGF:F(F(" }}{PARA 6 "" 1 "" {TEXT -1 18 "makeg() completed." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "grcalcd(Chr2):" }}{PARA 6 "" 1 "" {TEXT -1 53 "Calc ulated g(dn,dn,pdn) for kruskSzek (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 54 "Calculated Chr(dn,dn,dn) for kruskSzek (0.004000 sec.)" } }{PARA 6 "" 1 "" {TEXT -1 45 "Calculated detg for kruskSzek (0.000000 \+ sec.)" }}{PARA 6 "" 1 "" {TEXT -1 49 "Calculated g(up,up) for kruskSze k (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 54 "Calculated Chr(dn,dn ,up) for kruskSzek (0.004000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%*CPU~Time~G$\"\")!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=For~the~k ruskSzek~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%aoChristoffel ~symbol~of~the~second~kind~(symmetric~in~first~two~indices)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"TG\"\"\"%\"TGF*F*)%!G%\"T GF*,$*&#F*\"\"#F***%#rsG!\"\"-%\"rG6$%\"TG%\"XGF5-%%diffG6$F6F9F*,&F4F *F6F*F*F*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"TG\" \"\"%\"TGF*F*)%!G%\"XGF*,$*&#F*\"\"#F***%#rsG!\"\"-%\"rG6$%\"TG%\"XGF5 -%%diffG6$F6F:F*,&F4F*F6F*F*F*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &&%&GammaG6#*&%\"TG\"\"\"%\"XGF*F*)%!G%\"TGF*,$*&#F*\"\"#F***%#rsG!\" \"-%\"rG6$%\"TG%\"XGF5-%%diffG6$F6F:F*,&F4F*F6F*F*F*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"TG\"\"\"%\"XGF*F*)%!G%\"XGF*,$* &#F*\"\"#F***%#rsG!\"\"-%\"rG6$%\"TG%\"XGF5-%%diffG6$F6F9F*,&F4F*F6F*F *F*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"TG\"\"\"%& thetaGF*F*)%!G%&thetaGF**&-%\"rG6$%\"TG%\"XG!\"\"-%%diffG6$F0F3F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"TG\"\"\"%$phiGF*F*) %!G%$phiGF**&-%\"rG6$%\"TG%\"XG!\"\"-%%diffG6$F0F3F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"XG\"\"\"%\"XGF*F*)%!G%\"TGF*,$*&# F*\"\"#F***%#rsG!\"\"-%\"rG6$%\"TG%\"XGF5-%%diffG6$F6F9F*,&F4F*F6F*F*F *F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"XG\"\"\"%\"X GF*F*)%!G%\"XGF*,$*&#F*\"\"#F***%#rsG!\"\"-%\"rG6$%\"TG%\"XGF5-%%diffG 6$F6F:F*,&F4F*F6F*F*F*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&Gamm aG6#*&%\"XG\"\"\"%&thetaGF*F*)%!G%&thetaGF**&-%\"rG6$%\"TG%\"XG!\"\"-% %diffG6$F0F4F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"X G\"\"\"%$phiGF*F*)%!G%$phiGF**&-%\"rG6$%\"TG%\"XG!\"\"-%%diffG6$F0F4F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%&thetaG\"\"\"%&th etaGF*F*)%!G%\"TGF*,$*&#F*\"\"%F***%#rsG!\"$-%\"rG6$%\"TG%\"XG\"\"#-%$ expG6#*&F6F*F4!\"\"F*-%%diffG6$F6F9F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%&thetaG\"\"\"%&thetaGF*F*)%!G%\"XGF*,$ *&#F*\"\"%F***%#rsG!\"$-%\"rG6$%\"TG%\"XG\"\"#-%$expG6#*&F6F*F4!\"\"F* -%%diffG6$F6F:F*F*F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6# *&%&thetaG\"\"\"%$phiGF*F*)%!G%$phiGF**&-%$sinG6#%&thetaG!\"\"-%$cosGF 2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%$phiG\"\"\"%$p hiGF*F*)%!G%\"TGF*,$*&#F*\"\"%F**,%#rsG!\"$-%\"rG6$%\"TG%\"XG\"\"#-%$e xpG6#*&F6F*F4!\"\"F*-%$sinG6#%&thetaGF;-%%diffG6$F6F9F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%$phiG\"\"\"%$phiGF*F*)%!G% \"XGF*,$*&#F*\"\"%F**,%#rsG!\"$-%\"rG6$%\"TG%\"XG\"\"#-%$expG6#*&F6F*F 4!\"\"F*-%$sinG6#%&thetaGF;-%%diffG6$F6F:F*F*F@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%$phiG\"\"\"%$phiGF*F*)%!G%&thetaGF*,$* &-%$sinG6#%&thetaGF*-%$cosGF3F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "eval(subs(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),gr component(Chr2,[T,T,T])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%#rsG! \"\",&-%)LambertWG6#*&,&*$)%\"XG\"\"#\"\"\"F0*$)%\"TGF/F0F%F0-%$expG6# F0F%F0F0F0!\"#F'F0F+F%F3F0,&F$F0*&F$F0F&F0F0F0" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 44 "subs(LambertW((X^2-T^2)/exp(1))=(r/rs-1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.%#rsG\"\"\"%\"rG!\"#,&*&F&F%F$!\" \"F%F%F*F%,&*$)%\"XG\"\"#F%F%*$)%\"TGF/F%F*F*F2F%,&F$F%F&F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(X^2-T^2=(r/rs-1)*exp(r/ rs),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,%#rsG\"\"\"%\"rG!\"#-%$ex pG6#*&F&F%F$!\"\"F,%\"TGF%,&F$F%F&F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "diff(LambertW((X^2-T^2)/exp(1)),X);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#\"\"\"-%)LambertWG6#*&,&*$)%\"XGF%F&F&*$)% \"TGF%F&!\"\"F&-%$expG6#F&F2F&,&F'F&F&F&F2F+F2F.F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs(LambertW((X^2-T^2)/exp(1))=(r/rs-1), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"\"#\"\"\",&*&%\"rGF&%#rsG !\"\"F&F&F+F&F)F+F*F&,&*$)%\"XGF%F&F&*$)%\"TGF%F&F+F+F/F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(X^2-T^2=(r/rs-1)*exp(r/ rs),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#\"\"\"%\"rG!\"\"%# rsGF&-%$expG6#*&F'F&F)F(F(%\"XGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "diff(LambertW((X^2-T^2)/exp(1)),X,X);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**,\"\"%\"\"\"-%)LambertWG6#*&,&*$)%\"XG\"\"#F&F& *$)%\"TGF/F&!\"\"F&-%$expG6#F&F3F&,&F'F&F&F&!\"#F+F8F.F/F&*,F%F&F'F/F7 !\"$F+F8F.F/F3*,F%F&F'F&F7F3F+F8F.F/F3**F/F&F'F&F7F3F+F3F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "subs(LambertW((X^2-T^2)/exp(1))=(r/ rs-1),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**.\"\"%\"\"\",&*&%\"rGF &%#rsG!\"\"F&F&F+F&F)!\"#F*\"\"#,&*$)%\"XGF-F&F&*$)%\"TGF-F&F+F,F1F-F& *.F%F&F'F-F)!\"$F*\"\"$F.F,F1F-F+*.F%F&F'F&F)F+F*F&F.F,F1F-F+*,F-F&F'F &F)F+F*F&F.F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(X^2 -T^2=(r/rs-1)*exp(r/rs),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**.\" \"%\"\"\",&*&%\"rGF&%#rsG!\"\"F&F&F+F+F)!\"#F*\"\"#-%$expG6#F(F,%\"XGF -F&*,F%F&F)!\"$F*\"\"$F.F,F1F-F+*.F%F&F'F+F)F+F*F&F.F,F1F-F+**F-F&F)F+ F*F&F.F+F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"\"#\"\"\",(**F%F&)%#rsGF%F&- %$expG6#,$*&%\"rGF&F*!\"\"F1F&)%\"XGF%F&F&*,F%F&F+F&F2F&F0F&F*F&F&*$)F 0F%F&F1F&F*F&F+F&F0!\"$F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grcalcd(Riemann);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G $\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=For~the~kruskSzek~spacet ime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Covariant~RiemannG" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#R~G6#**%\"TG\"\"\"%\"XGF)%\"TGF)% \"XGF),$*,\"\"#F)%#rsGF/-%\"rG6$%\"TG%\"XG!\"$-%$expG6#*&F1F)F0!\"\"F; ,.*(F0F)-%%diffG6$F1-%\"$G6$F5F/F)F1F)F)*&F>F))F1F/F)F)*&F0F))-F?6$F1F 5F/F)F;*(F0F)-F?6$F1-FB6$F4F/F)F1F)F;*&FKF)FEF)F;*&F0F))-F?6$F1F4F/F)F )F)F;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#R~G6#**%\"TG\"\"\"%&theta GF)%\"TGF)%&thetaGF),$*&#F)\"\"#F)*&,,**F0F)%#rsGF)-%%diffG6$-%\"rG6$% \"TG%\"XG-%\"$G6$F;F0F)F8F)F)*&F4F))-F66$F8F;F0F)F)*&FAF)F8F)F)*&F4F)) -F66$F8FF)F)*(F@F)FBF)F6F) F)F)F>!\"\"FE" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#R~G6#**%\"XG\"\" \"%&thetaGF)%\"XGF)%&thetaGF),$*&#F)\"\"#F)*&,,**F0F)%#rsGF)-%%diffG6$ -%\"rG6$%\"TG%\"XG-%\"$G6$F " 0 "" {MPLTEXT 1 0 96 "eval(subs(r(T,X) =rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent(Riemann,[theta,phi,thet a,phi]))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "simplify(subs (LambertW((X^2-T^2)/exp(1))=(r/rs-1),%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,,,*(%#rsG\"\"\")%\"XG\"\"#F'%\"rGF'!\"\"*&-%$expG6#*& F+F'F&F,F')F&F*F'F'*(F&F')%\"TGF*F'F+F'F'*&)F+F*F'F.F'F'**F*F'F.F'F&F' F+F'F,F',&F'F,*$)-%$cosG6#%&thetaGF*F'F'F'F+F',&*$F(F'F'*$F4F'F,F,F&F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "simplify(subs(X^2=T^2+( r/rs-1)*exp(r/rs),%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*()-%$sinG6# %&thetaG\"\"#\"\"\"%#rsGF*%\"rGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalcd(Ricci);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% *CPU~Time~G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=For~the~krusk Szek~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Covariant~RicciG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"RG6$%#dnGF&" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#/&%#R~G6#*&%\"TG\"\"\"%\"TGF),$*&#F)\"\"#F)*(,2*(%#rs GF)-%%diffG6$-%\"rG6$%\"TG%\"XG-%\"$G6$F:F.F)F6F)F)*&F3F))F6F.F)F)*(F. F))-F46$F6F:F.F)F6F)F)*(\"\"$F)F2F))-F46$F6F9F.F)F)*(F.F)FFF)F6F)F)**F EF)F2F)-F46$F6-F<6$F9F.F)F6F)F)*&FKF)F?F)!\"\"*&F2F)FAF)F)F)F2FPF6!\"# F)FP" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%#R~G6#*&%\"TG\"\"\"%\"XGF), $**\"\"#F),(*(-%\"rG6$%\"TG%\"XGF)-%%diffG6%F0F3F4F)%#rsGF)F)*(-F66$F0 F3F)-F66$F0F4F)F8F)F)*(F:F)FF)-FE 6$F9-%\"$G6$FF))-FE6$F9F=F4F)FB*(FHF)F>F)-FE6$F9-FL6$F= F4F)FB*&F.F))F5\"\"$F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "eval(subs(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent (Ricci,[phi,phi]))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sim plify(subs(LambertW((X^2-T^2)/exp(1))=r/rs-1,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,**&%#rsG\"\"\")%\"XG\"\"#F(F(*&F'F(-%$expG6#*&%\"r GF(F'!\"\"F(F(*&F'F()%\"TGF+F(F2*&F-F(F1F(F2F(,&F(F2*$)-%$cosG6#%&thet aGF+F(F(F(,&*$F)F(F(*$F4F(F2F2F'F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "simplify(subs(X^2=T^2+(r/rs-1)*exp(r/rs),%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "grcomponent(Ricci,[phi,phi]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"%F&*(-%$sinG6#%&thetaG\"\"#%#rsG!\"$,,*( -%\"rG6$%\"TG%\"XGF&-%$expG6#*&F2F&F.!\"\"F&)-%%diffG6$F2F5F-F&F&*()F2 F-F&F7F&-F>6$F2-%\"$G6$F5F-F&F&*(F2F&F7F&)-F>6$F2F6F-F&F;*(FAF&F7F&-F> 6$F2-FE6$F6F-F&F;*&F'F&)F.\"\"$F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "grcalcd(Ricciscalar);" }}{PARA 6 "" 1 "" {TEXT -1 52 "Calculated Ricciscalar for kruskSzek (0.000000 sec.)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=For~the~kruskSzek~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Ricci~scalarG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%#R ~G,$*&#\"\"\"\"\"%F(*(,0*,\"\"$F(-%$expG6#*&-%\"rG6$%\"TG%\"XGF(%#rsG! \"\"F()F2\"\"#F(F7F(-%%diffG6$F2-%\"$G6$F6F:F(F8*(F.F()F2F-F(F;F(F(*,F -F(F.F(F2F(F7F()-F<6$F2F5F:F(F(*,F-F(F.F(F9F(F7F(-F<6$F2-F?6$F5F:F(F(* (F.F(FBF(FHF(F8*,F-F(F.F(F2F(F7F()-F<6$F2F6F:F(F8*&\"\")F()F7F)F(F(F(F 7!\"%F2!\"#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "eval(sub s(r(T,X)=rs*(LambertW((X^2-T^2)/exp(1))+1),grcomponent(Ricciscalar))): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "simplify(subs(LambertW( (X^2-T^2)/exp(1))=r/rs-1,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*, \"\"#\"\"\",**&%#rsGF&)%\"XGF%F&F&*&-%$expG6#*&%\"rGF&F)!\"\"F&F1F&F2* &F-F&F)F&F&*&F)F&)%\"TGF%F&F2F&F)F2F1!\"#,&*$F*F&F&*$F5F&F2F2F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "simplify(subs(X^2=T^2+(r/rs- 1)*exp(r/rs),%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(RiemSq):" }}{PARA 6 "" 1 "" {TEXT -1 38 "Created definition for R(dn,dn,up,up) " }}{PARA 6 "" 1 " " {TEXT -1 55 "Calculated R(dn,dn,up,up) for kruskSzek (0.004000 sec.) " }}{PARA 6 "" 1 "" {TEXT -1 47 "Calculated RiemSq for kruskSzek (0.00 0000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#C!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "eval(subs(r(T,X)=rs*(Lamb ertW((X^2-T^2)/exp(1))+1),grcomponent(RiemSq))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "simplify(subs(LambertW((X^2-T^2)/exp(1))=r/rs -1,%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,\"\"%\"\"\",>*()%#rsG\" \"#F&)%\"XGF%F&)%\"rGF+F&F&*,F+F&)F-F+F&)F*\"\"$F&-%$expG6#*&F/F&F*!\" \"F&F/F&F8*,F%F&F1F&F)F&F4F&F.F&F&*,F+F&F)F&F1F&F.F&)%\"TGF+F&F8*,F+F& F1F&F*F&F4F&)F/F3F&F8**\"\")F&-F56#,$*(F+F&F/F&F*F8F&F&F2F&F/F&F8**F@F &FAF&F)F&F.F&F&*&FAF&)F/F%F&F&*(F3F&FAF&)F*F%F&F&*,F+F&F*F&F4F&F>F&F;F &F&*,F%F&F)F&F4F&F.F&F;F&F8**F%F&FAF&F*F&F>F&F8*,F+F&F2F&F4F&F/F&F;F&F &*()F " 0 "" {MPLTEXT 1 0 45 "simplify(subs(X^2=T^2+(r/rs-1)*exp( r/rs),%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,\"\"%\"\"\",4*()%#rs G\"\"#F&)%\"XGF%F&)%\"rGF+F&F&**F+F&-%$expG6#,$*(F+F&F/F&F*!\"\"F&F&F) F&F.F&F&**\"\"'F&F1F&)F*\"\"$F&F/F&F6*,F+F&F)F&-F26#*&F/F&F*F6F&F.F&)% \"TGF+F&F&**F+F&F1F&F*F&)F/F:F&F&*()F@F%F&F)F&F.F&F6*,F+F&F*F&F " 0 "" {MPLTEXT 1 0 49 "simp lify(subs(X^4=(T^2+(r/rs-1)*exp(r/rs))^2,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#7\"\"\"%#rsG\"\"#%\"rG!\"'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }