{VERSION 5 0 "IBM INTEL LINUX" "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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 33 "Fun with the Schwarzschil d metric" }}{PARA 0 "" 0 "" {TEXT -1 15 "\nDirk Puetzfeld" }}{PARA 0 " " 0 "" {TEXT -1 31 "\nlast update: 2004-12-09 by dp\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "If you do not have grtensor in your path name you should the run the following procedure with the correct path" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "grtw := proc () \n global libname , grOptionMetricPath, grOptionqloadPath:\n\n libname := `/home/dpuetz /additional/grtensor179/lib`, libname:\n\n readlib (grii):\n grtenso r():\n\n grOptionqloadPath := `/home/dpuetz/additional/grtensor179/me trics`:\n\nend:\n" }}}{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~February~2001G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~Denis~Pollney~and~Kayl l~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-2001~by~th e~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%inLatest~version~avail able~from:~http://grtensor.phy.queensu.ca/G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%L/home/dpuetz/additional/grtensor179/metricsG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "makeg(sschild);" }}{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 prompt." }}{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 ele ment [ds]," }}{PARA 6 "" 1 "" {TEXT -1 63 " 3) n on-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 68 "Invalid input. Coordinates must be in the form of a list. Try again." }}{PARA 6 "" 1 "" {TEXT -1 50 "Enter coor dinates as a LIST (eg. [t,r,theta,phi]):" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 16 "[t,r,theta,phi];" }}{PARA 6 "" 1 "" {TEXT -1 64 "Enter the line element using d[coord] to indicate differentials." }} {PARA 6 "" 1 "" {TEXT -1 55 "(for example, r^2*(d[theta]^2 + sin(thet a)^2*d[phi]^2)" }}{PARA 6 "" 1 "" {TEXT -1 27 "[Type 'exit' to quit ma keg]" }}{PARA 6 "" 1 "" {TEXT -1 8 " ds^2 = " }}}{EXCHG {PARA 0 "makeg >" 0 "" {MPLTEXT 1 0 78 "(1-2*m/r)*d[t]^2-(1-2*m/r)^(-1)*d[r]^2-r^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 coordinat es, constants or functions" }}{PARA 6 "" 1 "" {TEXT -1 66 "for this sp acetime, 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 1 ";" }}{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%\"rG%&thetaG% $phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(Metric:G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&&%\"gG6#%\"aG\"\"\"&%!G6#%\"bGF)-%'matrixG6#7&7&,& F)F)*(\"\"#F)%\"mGF)%\"rG!\"\"F8\"\"!F9F97&F9,$*&F)F)F3F8F8F9F97&F9F9, $*$)F7F5F)F8F97&F9F9F9,$*&F@F))-%$sinG6#%&thetaGF5F)F8" }}{PARA 6 "" 1 "" {TEXT -1 54 "You may choose to 0) Use the metric WITHOUT saving i t," }}{PARA 6 "" 1 "" {TEXT -1 46 " 1) Save the metri c as it is," }}{PARA 6 "" 1 "" {TEXT -1 54 " 2) Corre ct 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 description, or" }}{PARA 6 "" 1 "" {TEXT -1 61 " 6) Abandon this metric and re turn to Maple." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "mak eg>" 0 "" {MPLTEXT 1 0 2 "0;" }}{PARA 6 "" 1 "" {TEXT -1 41 "Calculate d ds for sschild (0.000000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% 2Default~spacetimeG%(sschildG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For ~the~sschild~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Coordina tesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7&%\"tG%\"rG%&thetaG%$phiG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",**(,&F(F(*(F'F(%\"mGF(%\"rG!\" \"F/F(%#~dGF()%\"tG%#2~GF(F(*(F+F/F0F()F.F3F(F/*()F.F'F(F0F()%&thetaGF 3F(F/**F7F()-%$sinG6#F9F'F(F0F()%$phiGF3F(F/" }}{PARA 6 "" 1 "" {TEXT -1 18 "makeg() completed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "grcalc(g(up,up));grdisplay(g(up,up));" }}{PARA 6 "" 1 "" {TEXT -1 43 "Calculated detg for sschild (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 46 "Calculated g(up,up) for sschild (.011000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#6!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~sschild~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 47 "grcalc(Chr(dn,dn,up));grdisplay(Chr(dn,dn,up ));" }}{PARA 6 "" 1 "" {TEXT -1 51 "Calculated g(dn,dn,pdn) for sschil d (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 52 "Calculated Chr(dn,dn ,dn) for sschild (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 52 "Calcu lated Chr(dn,dn,up) for sschild (0.000000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%;For~the~sschild~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ao Christoffel~symbol~of~the~second~kind~(symmetric~in~first~two~indices) G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"tG\"\"\"%\"tGF *F*)%!G%\"rGF**(,&%\"rGF**&\"\"#F*%\"mGF*!\"\"F*F1!\"$F4F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"tG\"\"\"%\"rGF*F*)%!G%\"t GF**(%\"rG!\"\",&F0F**&\"\"#F*%\"mGF*F1F1F5F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"rG\"\"\"%\"rGF*F*)%!G%\"rGF*,$*(%\"r G!\"\",&F1F**&\"\"#F*%\"mGF*F2F2F6F*F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"rG\"\"\"%&thetaGF*F*)%!G%&thetaGF**&F*F*%\"rG !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%\"rG\"\"\"%$ phiGF*F*)%!G%$phiGF**&F*F*%\"rG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&&%&GammaG6#*&%&thetaG\"\"\"%&thetaGF*F*)%!G%\"rGF*,&%\"rG!\"\"*& \"\"#F*%\"mGF*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%& thetaG\"\"\"%$phiGF*F*)%!G%$phiGF**&-%$sinG6#%&thetaG!\"\"-%$cosGF2F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%$phiG\"\"\"%$phiG F*F*)%!G%\"rGF*,$*&,&%\"rGF**&\"\"#F*%\"mGF*!\"\"F*)-%$sinG6#%&thetaGF 4F*F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%&GammaG6#*&%$phiG\"\"\"% $phiGF*F*)%!G%&thetaGF*,$*&-%$sinG6#%&thetaGF*-%$cosGF3F*!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "grcalc(R(dn,dn));grdisplay(R (dn,dn));" }}{PARA 6 "" 1 "" {TEXT -1 46 "Calculated R(dn,dn) for ssch ild (.011000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\" #?!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~sschild~spacetime: G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Covariant~RicciG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%\"RG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%\"aG\"\"\"&%!G6#%\"bGF)%8All~components~are~zeroG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "grcalc(G(dn,dn));grdisplay(G (dn,dn));" }}{PARA 6 "" 1 "" {TEXT -1 50 "Calculated Ricciscalar for s schild (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Calculated G(dn ,dn) for sschild (0.000000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% *CPU~Time~G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~sschi ld~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3Covariant~Einstein G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"GG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#G~G6#%\"aG\"\"\"&%!G6#%\"bGF)%8All~components~ are~zeroG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "grcalc(Riccisc alar);grdisplay(Ricciscalar);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CP U~Time~G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~sschild~ spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Ricci~scalarG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#R~G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "grcalc(R(up,dn,dn,dn));grdisplay(R(up,dn,dn,dn)); " }}{PARA 6 "" 1 "" {TEXT -1 38 "Created definition for R(up,dn,dn,dn) " }}{PARA 6 "" 1 "" {TEXT -1 53 "Calculated R(dn,dn,dn,dn) for sschil d (0.000000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 53 "Calculated R(up,dn,d n,dn) for sschild (0.000000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %*CPU~Time~G$\"#@!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~ssc hild~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/R(up,dn,dn,dn)G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%\"tG\"\"\"&%!G6#*(%\"rGF (%\"tGF(%\"rGF(F(,$**\"\"#F(%\"rG!\"#,&F3F(*&F2F(%\"mGF(!\"\"F8F7F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%\"tG\"\"\"&%!G6#*(%&thet aGF(%\"tGF(%&thetaGF(F(,$*&%\"mGF(%\"rG!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%\"tG\"\"\"&%!G6#*(%$phiGF(%\"tGF(%$phiGF(F(,$ *(%\"mGF(%\"rG!\"\"-%$sinG6#%&thetaG\"\"#F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%\"rG\"\"\"&%!G6#*(%\"tGF(%\"tGF(%\"rGF(F(,$** \"\"#F(,&%\"rGF(*&F2F(%\"mGF(!\"\"F(F4!\"%F6F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%\"rG\"\"\"&%!G6#*(%&thetaGF(%\"rGF(%&thetaGF( F(,$*&%\"mGF(%\"rG!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G %\"rG\"\"\"&%!G6#*(%$phiGF(%\"rGF(%$phiGF(F(,$*(%\"mGF(%\"rG!\"\"-%$si nG6#%&thetaG\"\"#F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%&thet aG\"\"\"&%!G6#*(%\"tGF(%\"tGF(%&thetaGF(F(,$*(,&%\"rGF(*&\"\"#F(%\"mGF (!\"\"F(F3!\"%F6F(F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%&the taG\"\"\"&%!G6#*(%\"rGF(%\"rGF(%&thetaGF(F(*(%\"rG!\"#,&F1F(*&\"\"#F(% \"mGF(!\"\"F7F6F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%&thetaG \"\"\"&%!G6#*(%$phiGF(%&thetaGF(%$phiGF(F(,$**\"\"#F(%\"mGF(%\"rG!\"\" -%$sinG6#%&thetaGF2F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%$ph iG\"\"\"&%!G6#*(%\"tGF(%\"tGF(%$phiGF(F(,$*(,&%\"rGF(*&\"\"#F(%\"mGF(! \"\"F(F3!\"%F6F(F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%$phiG \"\"\"&%!G6#*(%\"rGF(%\"rGF(%$phiGF(F(*(%\"rG!\"#,&F1F(*&\"\"#F(%\"mGF (!\"\"F7F6F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#R~G%$phiG\"\"\"& %!G6#*(%&thetaGF(%&thetaGF(%$phiGF(F(,$*(\"\"#F(%\"mGF(%\"rG!\"\"F5" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "grdef(`curvinvariant := R \{^a ^b ^c ^d\} * R\{a b c d\}`);" }}{PARA 6 "" 1 "" {TEXT -1 38 "Crea ted definition for R(up,up,up,up) " }}{PARA 6 "" 1 "" {TEXT -1 36 "Cre ated definition for curvinvariant" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "grcalc(curvinvariant);grdisplay(curvinvariant);" }} {PARA 6 "" 1 "" {TEXT -1 52 "Calculated R(up,up,dn,dn) for sschild (.0 10000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 52 "Calculated R(up,up,up,up) \+ for sschild (.021000 sec.)" }}{PARA 6 "" 1 "" {TEXT -1 52 "Calculated \+ curvinvariant for sschild (0.000000 sec.)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#J!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~sschild~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%. curvinvariantG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%.curvinvariantG,$* (\"#[\"\"\"%\"mG\"\"#%\"rG!\"'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 13 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }