{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 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 0 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 "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 255 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Elementary rudiments of GR TensorII: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "This is a brief introduction only. For more interesting exampl es consult the demonstrations page on the Web." }}{PARA 0 "" 0 "" {TEXT -1 71 "For more detailed information consult the help pages and \+ Release Notes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The following (optional) line allows the worksheet to be \+ re-executed." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "We now load the GRTensorII package:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "grtw():" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1.79~(R6)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" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "T he current environment can be checked as follows:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "groptions();" }}{PARA 6 "" 1 "" {TEXT -1 29 "grOpti onAlterSize = false" }}{PARA 6 "" 1 "" {TEXT -1 28 "grOptionCoordN ames = true" }}{PARA 6 "" 1 "" {TEXT -1 25 "grOptionDefaultSimp = 8" }}{PARA 6 "" 1 "" {TEXT -1 28 "grOptionDisplayLimit = 5000" }} {PARA 6 "" 1 "" {TEXT -1 28 "grOptionLLSC = true" }}{PARA 6 " " 1 "" {TEXT -1 45 "grOptionMetricPath = `e:/Grtii(6)/Metrics`" }} {PARA 6 "" 1 "" {TEXT -1 38 "grOptionqloadPath = (not assigned)" } }{PARA 6 "" 1 "" {TEXT -1 27 "grOptionTermSize = 100" }}{PARA 6 " " 1 "" {TEXT -1 29 "grOptionTrace = false" }}{PARA 6 "" 1 "" {TEXT -1 28 "grOptionTimeStamp = true" }}{PARA 6 "" 1 "" {TEXT -1 29 "grOptionVerbose = false" }}{PARA 6 "" 1 "" {TEXT -1 28 "grOp tionWindows = true" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 65 "grOptionDefaultSimp values: 0=None, 1=simplify, 2= simplify[trig]," }}{PARA 6 "" 1 "" {TEXT -1 63 " 3=simplify[power] 4= simplify[hypergeom], 5=simplify[radical]," }}{PARA 6 "" 1 "" {TEXT -1 57 " 6=expand, 7=factor, 8=normal, 9=sort, 10=simplify[sqrt]" }} {PARA 6 "" 1 "" {TEXT -1 22 " 11=simplify[trigsin]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "We now use makeg to create a file (schwarz) wh ich is the Schwarzschild exterior metric in curvature coordinates. Her e we choose to enter the line" }}{PARA 0 "" 0 "" {TEXT -1 17 "element \+ directly:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "makeg(schwarz);" }} {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 ut ility" }}{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) metr ic [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 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}} {EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 16 "[r,theta,phi,t];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&%\"rG%&thetaG%$phiG%\"tG" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 73 "d[r]^2/(1-2*m/r)+r^2*(d[theta]^ 2+sin(theta)^2*d[phi]^2)-(1-2*m/r)*d[t]^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&*$)&%\"dG6#%\"rG\"\"#\"\"\"F,,&F,F,*&*&F+F,%\"mGF,F ,F*!\"\"F1F1F,*&)F*F+F,,&*$)&F(6#%&thetaGF+F,F,*&)-%$sinGF8F+F,)&F(6#% $phiGF+F,F,F,F,*&F-F,)&F(6#%\"tGF+F,F1" }}}{EXCHG {PARA 0 "makeg>" 0 " " {MPLTEXT 1 0 3 "\{\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<\"" }}} {EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "The Schwarzschild exterior metric in curvature coordinates;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%AThe~values~you~have~entered~are:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,CoordinatesG7&%\"rG%&thetaG%$phiG%\"tG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%(Metric:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"gG6#%\"aG\"\"\"&%!G6#%\"bGF)-%'matrixG6#7&7&*&F) F),&F)F)*&*&\"\"#F)%\"mGF)F)%\"rG!\"\"F:F:\"\"!F;F;7&F;*$)F9F7F)F;F;7& F;F;*&F>F))-%$sinG6#%&thetaGF7F)F;7&F;F;F;,&F:F)*&*&F7F)F8F)F)F9F:F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#QfnThe~Schwarzschild~exterior~metric ~in~curvature~coordinates;6\"" }}{PARA 6 "" 1 "" {TEXT -1 54 "You may \+ choose to 0) Use the metric 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/c hange 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 return to Maple." }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "We choose not to load it so as to \+ demonstrate the loading procedure." }}{PARA 0 "makeg>" 0 "" {MPLTEXT 1 0 2 "0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We now load the metric we have created:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "qload(schwarz);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%(schwarzG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~schwarz~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& %\"rG%&thetaG%$phiG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~ele mentG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",**&*&%# ~dGF()%\"rG%#2~GF(F(,&F(F(*&*&F'F(%\"mGF(F(F.!\"\"F4F4F(*()F.F'F(F,F() %&thetaGF/F(F(**F6F()-%$sinG6#F8F'F(F,F()%$phiGF/F(F(*(,&F4F(*&*&F'F(F 3F(F(F.F4F(F(F,F()%\"tGF/F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%fnTh e~Schwarzschild~exterior~metric~in~curvature~coordinates;G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 283 "To see what objects are predefined, load the help library (?grtensor; for information on the help system, read lib(griihelp); to load the help library) and do ?grt_objects;. Here we calculate and display (grcalcd) the covariant Ricci tensor (R(dn,dn)) and Kretschmann scalar (RiemSq)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grcalcd(R(dn,dn),RiemSq);" }}{PARA 6 "" 1 "" {TEXT -1 38 "Created \+ definition for R(dn,dn,up,up) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*C PU~Time~G$\"$,#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~schwa rz~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Covariant~RicciG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"RG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%\"aG\"\"\"&%!G6#%\"bGF)%8All~components~are ~zeroG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 45 "grdef(`E\{a b\}:=R\{a b\}-Ricciscalar*g\{ a b\}/2`);" }}{PARA 6 "" 1 "" {TEXT -1 68 "This object is already defi ned. The new definition has been ignored." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "Notice that the attempt was ignored because the 2-tensor \"E\" is predefined. (It is the\"electric part\" of the Weyl tensor, \+ described in ?grt_operators;). Let's use \"En\":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "grdef(`En\{a b\}:=R\{a b\}-Ricciscalar*g\{a b\}/2`); " }}{PARA 6 "" 1 "" {TEXT -1 32 "Created definition for En(dn,dn)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Now let us ask for the mixed compo nents. The definition for these is created automatically:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "grcalcd(En(dn,up));" }}{PARA 6 "" 1 "" {TEXT -1 33 "Created definition for En(dn,up) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#q!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~schwarz~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%* En(dn,up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#EnG6$%#dnG%#upG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/)&%#EnG6#%\"aG%\"bG%8All~components~a re~zeroG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Spacetimes which are implicitly defined are handled by the application of constraints. Con sider the Kruskal-Szekeres metric:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "qload(krn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetim eG%$krnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7For~the~krn~spacetime:G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%# x~G%\"aG-%'vectorG6#7&%\"uG%\"vG%&thetaG%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$) %$~dsG\"\"#\"\"\",(*&*.)%\"mGF'F(,&F-F'-%\"rG6$%\"uG%\"vG!\"\"F(%#~dGF ()F2%\"~GF(%#d~GF()F3F7F(F(*(F2F(F3F(F/F(F4!#;*()F/F'F(F5F()%&thetaG%# 2~GF(F(**F=F()-%$sinG6#F?F'F(F5F()%$phiGF@F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,ConstraintsG7$/-%%diffG6$-%\"rG6$%\"uG%\"vGF-,$*&*&% \"mG\"\"\",&F2\"\"#F*!\"\"F3F3*&F*F3F-F3F6!\"#/-F(6$F*F.,$*&*&F2F3F4F3 F3*&F*F3F.F3F6F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D~~~~Null~form~of ~Kruskal~metric~~~~G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "Since ob jects calculated for this metric must be simplified (at least by the a pplication of constraints) we calculate some objects without displayin g them (grcalc):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "grcalc(R(dn,dn) ,RiemSq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$,\"!\"$ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 228 "To simplify the objects we a pply gralter. If no routine is given, a menu of options is given as sh own below. Refer to the help on gralter. The argument _ is shorthand \+ for the previous objects (in this case R(dn,dn) and RiemSq)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "gralter(_);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 " " {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 59 "(use ?name for help on \+ a particular simplification routine)" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 30 "Choose which routine to apply:" }} {PARA 6 "" 1 "" {TEXT -1 9 " 0) none" }}{PARA 6 "" 1 "" {TEXT -1 58 " 1) simplify() try all simplification techniques" }}{PARA 6 "" 1 "" {TEXT -1 50 " 2) simplify[trig] apply trig simplificatio n" }}{PARA 6 "" 1 "" {TEXT -1 52 " 3) simplify[power] simplify po wers, exp and ln" }}{PARA 6 "" 1 "" {TEXT -1 58 " 4) simplify[hyperge om] simplify hypergeometric functions" }}{PARA 6 "" 1 "" {TEXT -1 67 " 5) simplify[radical] convert radicals,log,exp to canonical form" } }{PARA 6 "" 1 "" {TEXT -1 13 " 6) expand()" }}{PARA 6 "" 1 "" {TEXT -1 13 " 7) factor()" }}{PARA 6 "" 1 "" {TEXT -1 13 " 8) normal()" }} {PARA 6 "" 1 "" {TEXT -1 11 " 9) sort()" }}{PARA 6 "" 1 "" {TEXT -1 49 "10) simplify[sqrt,symbolic] allows sqrt(r^2) = r" }}{PARA 6 "" 1 "" {TEXT -1 52 "11) simplify[trigsin] trig simp biased to sin" }}{PARA 6 "" 1 "" {TEXT -1 31 "12) Apply constraint equations" }} {PARA 6 "" 1 "" {TEXT -1 33 "13) Apply constraints repeatedly" }} {PARA 6 "" 1 "" {TEXT -1 33 "14) other user specified routine" }} {PARA 6 "" 1 "" {TEXT -1 44 "Number of routine to apply (followed by ; ) >" }}}{EXCHG {PARA 0 "gralter>" 0 "" {MPLTEXT 1 0 3 "13;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "We now display the objects:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grd isplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7For~the~krn~spacetime: G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Covariant~RicciG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%\"RG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#R~G6#%\"aG\"\"\"&%!G6#%\"bGF)%8All~components~are~zeroG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 13 "qload(schwb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%&schwbG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~schwb~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"x G6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7& %\"rG%&thetaG%$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&*&*$-%%sqrtG6#%\"rG\"\"\"F2*$-F/6#,&F1F 2*&\"\"#F2%\"mGF2!\"\"F2F:\"\"!F;F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%#w2G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega2G6#%\"aG-% 'vectorG6#7&\"\"!%\"rGF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w3G6# %#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'omega3G6#%\"aG-%'vectorG6 #7&\"\"!F,*&%\"rG\"\"\"-%$sinG6#%&thetaGF/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#w4G6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%'om ega4G6#%\"aG-%'vectorG6#7&\"\"!F,F,*&*$-%%sqrtG6#,&%\"rG\"\"\"*&\"\"#F 4%\"mGF4!\"\"F4F4*$-F06#F3F4F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;~~ ~~Schwarzschild~basis~~~G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "The basis components are distinguished by the prefix b. For example, let' s calculate the mixed basis components of the Weyl tensor:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "grcalc(C(bdn,bdn,bup,bup));" }}{PARA 6 "" 1 "" {TEXT -1 42 "Created definition for C(bdn,bdn,bup,bup) " }}{PARA 6 "" 1 "" {TEXT -1 40 "Created definition for rot(bdn,bup,bdn) " }} {PARA 6 "" 1 "" {TEXT -1 38 "Created a definition for e(bdn,dn,pdn)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$+#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9For~the~schwb~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3C(bdn,bdn,bup,bup)G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/*&&%\"CG6#%((1)~(2)G\"\"\")%!GF(F)*&%\"mGF)*$)%\"rG\"\"$F)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"CG6#%((1)~(3)G\"\"\")%!GF(F)*&% \"mGF)*$)%\"rG\"\"$F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"C G6#%((1)~(4)G\"\"\")%!GF(F),$*&%\"mGF)*$)%\"rG\"\"$F)!\"\"!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"CG6#%((2)~(3)G\"\"\")%!GF(F),$* &%\"mGF)*$)%\"rG\"\"$F)!\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/* &&%\"CG6#%((2)~(4)G\"\"\")%!GF(F)*&%\"mGF)*$)%\"rG\"\"$F)!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%\"CG6#%((3)~(4)G\"\"\")%!GF(F)*&% \"mGF)*$)%\"rG\"\"$F)!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Th e Newman-Penrose formalism can be used from a contravariant or a covar iant null tetrad. We start with the contravariant tetrad:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "qload(npschw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%'npschwG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%:For~the~npschw~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Co ordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~G%\"aG-%'vectorG6#7&%\"rG%&thetaG%$phi G%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Basis~inner~productG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$etaG6$%$bupGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$etaG%$(a)G\"\"\")%!G%$(b)GF(-%'matrixG6#7&7&\"\"! F(F1F17&F(F1F1F17&F1F1F1!\"\"7&F1F1F4F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%GNull~tetrad~(contravaraint~components)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$NPlG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\" lG%\"aG-%'vectorG6#7&,$*&*$-%%sqrtG6#\"\"#\"\"\"F2*$-F/6#*&%\"rGF2,&F7 F2*&F1F2%\"mGF2!\"\"F;F2F;#F2F1\"\"!F=,$*&*$F.F2F2*$-F/6#*&F8F2F7F;F2F ;F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$NPnG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"nG%\"aG-%'vectorG6#7&,$*&*$-%%sqrtG6#\"\"#\" \"\"F2*$-F/6#*&%\"rGF2,&F7F2*&F1F2%\"mGF2!\"\"F;F2F;#F;F1\"\"!F=,$*&*$ F.F2F2*$-F/6#*&F8F2F7F;F2F;#F2F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $NPmG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"mG%\"aG-%'vectorG 6#7&\"\"!,$*&*$-%%sqrtG6#\"\"#\"\"\"F3%\"rG!\"\"#F3F2*&*&^#F6F3F/F3F3* &F4F3-%$sinG6#%&thetaGF3F5F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'NPm barG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%%mbarG%\"aG-%'vector G6#7&\"\"!,$*&*$-%%sqrtG6#\"\"#\"\"\"F3%\"rG!\"\"#F3F2*&*&^##F5F2F3F/F 3F3*&F4F3-%$sinG6#%&thetaGF3F5F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%_ p~~~~Contravariant~NPtetrad~for~the~Schwarzschild~metric~in~curvature~ coordinates~~~~G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Let's calcula te the Ricci scalars:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grcalcd(Ri cciSc);" }}{PARA 6 "" 1 "" {TEXT -1 41 "`Basis/tetrad related object d efinitions`" }}{PARA 6 "" 1 "" {TEXT -1 31 "`Last modified 23 January \+ 2001`" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$,%!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~npschw~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Ricci~Scalar,~Phi00G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Phi00G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Ric ci~Scalar,~Phi01G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Phi01G\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Ricci~Scalar,~Phi02G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%&Phi02G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%4Ricci~Scalar,~Phi11G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Phi11G \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Ricci~Scalar,~Phi12G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Phi12G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Ricci~Scalar,~Phi22G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&Phi22G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 15 "qload(n pcschw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%(npc schwG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~npcschw~spacetime:G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%# x~G%\"aG-%'vectorG6#7&%\"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%$(b)GF(-%'matrixG6#7&7&\"\"!F(F1F17&F(F1F1F17&F1F1F1!\"\" 7&F1F1F4F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%CNull~tetrad~(covariant ~components)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$NPlG6#%#dnG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"lG6#%\"aG-%'vectorG6#7&,$*&*&%\"r G\"\"\"-%%sqrtG6#\"\"#F0F0*&,&F/F0*&F4F0%\"mGF0!\"\"F0-F26#*&F/F0F6F9F 0F9#F9F4\"\"!F>,$*&*&F6F0F1F0F0*&F/F0-F26#*&F6F0F/F9F0F9#F0F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$NPnG6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"nG6#%\"aG-%'vectorG6#7&,$*&*&%\"rG\"\"\"-%%sqrtG6# \"\"#F0F0*&,&F/F0*&F4F0%\"mGF0!\"\"F0-F26#*&F/F0F6F9F0F9#F0F4\"\"!F>,$ *&*&F6F0F1F0F0*&F/F0-F26#*&F6F0F/F9F0F9F=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$NPmG6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" mG6#%\"aG-%'vectorG6#7&\"\"!,$*&%\"rG\"\"\"-%%sqrtG6#\"\"#F0#!\"\"F4** ^#F5F0F/F0-%$sinG6#%&thetaGF0F1F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #-%'NPmbarG6#%#dnG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%mbarG6#%\"aG -%'vectorG6#7&\"\"!,$*&%\"rG\"\"\"-%%sqrtG6#\"\"#F0#!\"\"F4**^##F0F4F0 F/F0-%$sinG6#%&thetaGF0F1F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%jo~~ ~Covariant~NPtetrad~for~the~Schwarzschild~metric~in~curvature~coordina tes~~~~G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "grcalcd(Petrov) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#!*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;For~the~npcschw~spacetime:G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%,Petrov~TypeG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%-Petrov~Type~G%/D~(or~simpler)G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Since we were not careful to ensure that the Weyl scalars were in fully simplifed form, we ask for a report:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "PetrovReport();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%N The~conclusion~'Petrov~type~=~D~(or~simpler)'G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7for~the~npcschw~metricG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dwas~based~on~the~following~results:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1Weyl~scalar~Psi0G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1Weyl~scalar~Psi1G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%QW eyl~scalar~Psi2~could~not~be~evaluated~to~zero.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1Weyl~scalar~Psi3G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1Weyl~scalar~Psi4G\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%T- -->~Therefore~the~metric~is~Petrov~D~(or~simpler).G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%X--------------------------------------------------- ----G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%YThe~quantities~that~could~n ot~be~evaluated~to~zero~are:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%1W eyl~scalar~Psi2G,$*&%\"mG\"\"\"*$)%\"rG\"\"$F(!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "Notice that we have 5 spacetimes active \+ in this session (schwarz, krn, schwb, npschw, npcschw). The current de fault is npcschw (the last one loaded). We can calculate an object for any one of the spacetimes (e.g. the Weyl invariant W1R for krn)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grcalc(W1R[krn]);" }}{PARA 6 "" 1 " " {TEXT -1 25 "Scalar invariant library." }}{PARA 6 "" 1 "" {TEXT -1 28 "Last modified 25 March 1997." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% *CPU~Time~G$\"$!>!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gr alter(_,13);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification \+ of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 61 "Applying routine `Apply constraints repeatedly` to o bject W1R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#?!\"$" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7For~the~krn~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4CM~invariant~Re(W1)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%W1R~G,$*&*$)%\"mG\"\"#\"\"\"F+*$)-%\"rG6$%\"uG%\"vG\"\"'F+!\"\"F 3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Let us load another implicit form of the metric" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "qload(israel );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%'israelG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:For~the~israel~spacetime:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%#x~ G%\"aG-%'vectorG6#7&%\"uG%\"wG%&thetaG%$phiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$) %$~dsG\"\"#\"\"\",**&*()%\"wGF'F(%#~dGF()%\"uG%#2~GF(F(*&%\"mGF(-%\"rG 6$F0F-F(!\"\"#F(F'*,F'F(F.F()F0%\"~GF(%#d~GF()F-F;F(F(*()F4F'F(F.F()%& thetaGF1F(F(**F?F()-%$sinG6#FAF'F(F.F()%$phiGF1F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%,ConstraintsG7#/-%\"rG6$%\"uG%\"wG,&%\"mG\"\"#*&*(# \"\"\"\"\"%F2F*F2F+F2F2F-!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%I Israel~coordinates~(Phys.~Rev.~143,1016)G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "We define a differential invariant:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "grdef(`DiRiem:=R\{a b ^c ^d ; e\}*R\{c d ^a ^b ; ^e\} `);" }}{PARA 6 "" 1 "" {TEXT -1 43 "Created a definition for R(dn,dn,u p,up,cdn)" }}{PARA 6 "" 1 "" {TEXT -1 43 "Created a definition for R(d n,dn,up,up,cdn)" }}{PARA 6 "" 1 "" {TEXT -1 42 "Created definition for R(dn,dn,up,up,cup) " }}{PARA 6 "" 1 "" {TEXT -1 29 "Created definitio n for DiRiem" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "grcalc(DiRi em);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$]#!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "gralter(_,13);" }}{PARA 6 " " 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 64 "Applyin g routine `Apply constraints repeatedly` to object DiRiem" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#?!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%:For~the~israel~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'DiR iemG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'DiRiemG,$*&*()%\"mG\"#5\"\" \"%\"uGF+%\"wGF+F+*$),&*$)F)\"\"#F+\"\")*&F,F+F-F+F+\"\"*F+!\"\"\")?f= Z" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Notice that the invariant v anishes at u=0 (r(0,w)=2m), a property of this object that holds more \+ generally. " }}{PARA 0 "" 0 "" {TEXT -1 176 "Whereas we have interacti vely defined this invariant here, it is also predefined in the differe ntial invariants library dinvar as diRiem. Do grlib(dinvar); to load t he library." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "We can also change the default spacetime with grmetric. Let's \+ go back to schwarz and CREATE a null tetrad:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grmetric(schwarz);" }}{PARA 6 "" 1 "" {TEXT -1 30 "De fault metric is now schwarz." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nptetrad([r,t]);" }}{PARA 6 "" 1 "" {TEXT -1 52 "The metric sign ature of the schwarz spacetime is +2." }}{PARA 6 "" 1 "" {TEXT -1 81 " In order to create an NP-tetrad, the signature of g(dn,dn) will be cha nged to -2." }}{PARA 6 "" 1 "" {TEXT -1 39 "Continue? (1=yes [default] , other=no) :" }}}{EXCHG {PARA 0 "nptetrad>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "The situation here (in S chwarzschild) is unusually simple. The basis vectors would normally re quire further simplification for efficient use." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "42 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }