{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 -1 185 "Demonstration 1 (kruskalo ):Transformation to original Kruskal coordinates; reduction of Ricci t o zero, calculation and simplification of the Kretschmann scalar in th ese new coordinates." }}{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~(R6)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%02~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#%4e:/G rtii(6)/MetricsG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "First we load in the original Scwarzschild metric" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "qload(schw);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~space timeG%%schwG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8For~the~schw~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%$phiG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$) %$~dsG\"\"#\"\"\",**&*&%#~dGF()%\"rG%#2~GF(F(,&F(F(*&*&F'F(%\"mGF(F(F. !\"\"F4F4F(*()F.F'F(F,F()%&thetaGF/F(F(**F6F()-%$sinG6#F8F'F(F,F()%$ph iGF/F(F(*(,&F4F(*&*&F'F(F3F(F(F.F4F(F(F,F()%\"tGF/F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RThe~Schwarzschild~metric~in~curvature~coordinate sG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "The following transformatio ns are given by Kruskal:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "xform: =[u(r,t)=sqrt(r/(2*m)-1)*exp(r/(4*m))*cosh(t/(4*m)),v(r,t)=sqrt(r/(2*m )-1)*exp(r/(4*m))*sinh(t/(4*m)),Theta(theta)=theta,Phi(phi)=phi];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xformG7&/-%\"uG6$%\"rG%\"tG,$*(-%%s qrtG6#,&*&F*\"\"\"%\"mG!\"\"\"\"#\"\"%F5F3-%$expG6#,$F2#F3F7F3-%%coshG 6#,$*&F+F3F4F5F " 0 "" {MPLTEXT 1 0 33 "grtransform(schw,kruskalo,xform):" }}{PARA 6 "" 1 "" {TEXT -1 35 " The new default metric is: kruskalo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "We simplify and display the new form of the metric:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gralter(g(dn,dn),1);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 44 "Applying \+ routine simplify to object g(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%*CPU~Time~G$\"$+\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%F?\"#K \"\"!FCFC7&FC,$F4!#KFCFC7&FCFC*$)F>FAF)FC7&FCFCFC,$*&FIF),&F?F)*$)-%$c osG6#%&thetaGFAF)F)F)F?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Note \+ that in the new coordinates $[u,v,\\Theta,\\Phi]$ $r=r(u,v)$. We fini sh the transformation as follows:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "grmap(g(dn,dn),subs,r=r(u,v),theta=Theta,phi=Phi,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 33 "Applying routine subs to g(dn,dn)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We force MapleV to use sine as follows:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gralter(g(dn,dn),11);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object: " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 55 "Apply ing routine `simplify[trigsin]` to object g(dn,dn)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%*CPU~Time~G$\"#5!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "grdisplay(g(dn,dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%FC\"#K\"\"!FGFG7&FG,$F4!#KFGFG7&FGFG*$)F>FE F)FG7&FGFGFG*&FMF))-%$sinG6#%&ThetaGFEF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "This is the original Kruskal metric. " }}{PARA 0 "" 0 "" {TEXT -1 216 "We now wish to actually do some calculations with it. Be cause of the implicit nature of the metric, it is subject to the const raints which define r(u,v). We chose to eliminate the exponential func tion in what folows." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "P(u,v):=u^ 2-v^2-simplify((sqrt(r(u,v)/(2*m)-1)*exp(r(u,v)/(4*m))*cosh(t/(4*m)))^ 2-(sqrt(r(u,v)/(2*m)-1)*exp(r(u,v)/(4*m))*sinh(t/(4*m)))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"PG6$%\"uG%\"vG,(*$)F'\"\"#\"\"\"F-*$)F( F,F-!\"\"*&#F-F,F-*&*&,&-%\"rGF&F-*&F,F-%\"mGF-F0F--%$expG6#,$*&F6F-F9 F0#F-F,F-F-F9F0F-F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "simp lify(solve(P(u,v)=0,exp(r(u,v)/(2*m))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"mG\"\"\",&*$)%\"uG\"\"#F'!\"\"*$)%\"vGF,F'F'F'F',&-%\"r G6$F+F0F'*&F,F'F&F'F-F-!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P(u,v)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"uG\"\"#\"\"\" F)*$)%\"vGF(F)!\"\"*&#F)F(F)*&*&,&-%\"rG6$F'F,F)*&F(F)%\"mGF)F-F)-%$ex pG6#,$*&F3F)F7F-#F)F(F)F)F7F-F)F-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(diff(%,v),diff(r(u,v),v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"vG\"\"\")%\"mG\"\"#F'F'*&-%$expG6#,$*&-%\"rG6$ %\"uGF&F'F)!\"\"#F'F*F'F1F'F5!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "B(u,v):=subs(exp(r(u,v)/(2*m))=2*(-u^2+v^2)*m/(-r(u,v )+2*m),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"BG6$%\"uG%\"vG,$*&* (F(\"\"\"%\"mGF,,&-%\"rGF&!\"\"*&\"\"#F,F-F,F,F,F,*&,&*$)F'F3F,F1*$)F( F3F,F,F,F/F,F1!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "P(u,v) =0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"uG\"\"#\"\"\"F)*$)%\"v GF(F)!\"\"*&#F)F(F)*&*&,&-%\"rG6$F'F,F)*&F(F)%\"mGF)F-F)-%$expG6#,$*&F 3F)F7F-#F)F(F)F)F7F-F)F-\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve(diff(%,u),diff(r(u,v),u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%\"uG\"\"\")%\"mG\"\"#F'F'*&-%$expG6#,$*&-%\"rG6$F&%\"vGF' F)!\"\"#F'F*F'F1F'F5\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A(u,v):=subs(exp(r(u,v)/(2*m))=2*(-u^2+v^2)*m/(-r(u,v)+2*m),%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"AG6$%\"uG%\"vG,$*&*(F'\"\"\"%\"m GF,,&-%\"rGF&!\"\"*&\"\"#F,F-F,F,F,F,*&,&*$)F'F3F,F1*$)F(F3F,F,F,F/F,F 1\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "The functions A(u,v) a nd B(u,v) constrain the derivatives of r(u,v) wrt u and v. We now writ e the Kruskal metric without the exponent and attach these constraints ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "grmap(g(dn,dn),subs,exp(-r(u,v )/(2*m))=(r(u,v)-2*m)/(2*m*(u^2-v^2)),`x`);" }}{PARA 6 "" 1 "" {TEXT -1 33 "Applying routine subs to g(dn,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%F8F)F@F)F:F)F@\"#;\"\"!FHFH7&FH,$F4!#;FHFH 7&FHFH*$)F:F8F)FH7&FHFHFH*&FNF))-%$sinG6#%&ThetaGF8F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grconstraint(kruskalo);" }}{PARA 6 "" 1 "" {TEXT -1 41 "Constraint specification and manipulation" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 14 "Do you wi sh to" }}{PARA 6 "" 1 "" {TEXT -1 33 "1) Add a constraint to the metri c" }}{PARA 6 "" 1 "" {TEXT -1 38 "2) Remove a constraint from the metr ic" }}{PARA 6 "" 1 "" {TEXT -1 29 "3) Modify a metric constraint" }} {PARA 6 "" 1 "" {TEXT -1 35 "4) Display the existing constraints" }} {PARA 6 "" 1 "" {TEXT -1 7 "5) Exit" }}{PARA 6 "" 1 "" {TEXT -1 11 "En ter 1-5 >" }}}{EXCHG {PARA 0 "grconstraint>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "grconstra int>" 0 "" {MPLTEXT 1 0 22 "diff(r(u,v),u)=A(u,v);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%%diffG6$-%\"rG6$%\"uG%\"vGF*,$*&*(F*\"\"\"%\"mGF/, &F'!\"\"*&\"\"#F/F0F/F/F/F/*&,&*$)F*F4F/F2*$)F+F4F/F/F/F'F/F2\"\"%" }} }{EXCHG {PARA 0 "grconstraint>" 0 "" {MPLTEXT 1 0 2 "1;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "grconstraint>" 0 "" {MPLTEXT 1 0 22 "diff(r(u,v),v)=B(u,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"rG6$%\"uG%\"vGF+,$*&*(F+\"\"\"%\"mGF/,&F'!\"\"*& \"\"#F/F0F/F/F/F/*&,&*$)F*F4F/F2*$)F+F4F/F/F/F'F/F2!\"%" }}}{EXCHG {PARA 0 "grconstraint>" 0 "" {MPLTEXT 1 0 2 "4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "grconstraint>" 0 "" {MPLTEXT 1 0 2 "5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We now save the Kruskal metric with the constrain ts." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grsaveg(kruskalo);" }}{PARA 6 "" 1 "" {TEXT -1 58 "Information written to: `e:/Grtii(6)/Metrics/kr uskalo.mpl`" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Note that there is now NO reference to the original coordinates. " }}{PARA 0 "" 0 "" {TEXT -1 85 "We go on now to show that the metric is vacuum, and calcu late the Kretschmann scalar." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "grc alc(R(dn,dn));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#!)! \"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "gralter(_,13,7);" }} {PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII \+ object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 66 "Applying routine `Apply constraints repeatedly` to object R(dn,dn) " }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine factor to object R(d n,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#I!\"$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 15 "grcalc(RiemSq);" }}{PARA 6 "" 1 "" {TEXT -1 38 " Created definition for R(dn,dn,up,up) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$S#!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "gralter(_,13,7);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simpl ification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 64 "Applying routine `Apply constraints repea tedly` to object RiemSq" }}{PARA 6 "" 1 "" {TEXT -1 40 "Applying routi ne factor to object RiemSq" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~T ime~G$\"#g!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay (_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "33 3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }