{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " Vessiot_Text" -1 256 "Intrepid" 1 12 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 1 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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 8 2 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 }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 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_Title2" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 128 0 64 1 2 2 0 0 0 2 0 0 0 }1 0 0 0 4 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Vess_IO" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 0 0 0 0 0 0 1 0 0 }1 0 0 -1 -1 -1 3 30 0 0 0 0 -1 3 }{PSTYLE "Vess_Title1" -1 258 1 {CSTYLE "" -1 -1 "Helvetica " 1 18 128 0 64 1 0 0 0 0 0 0 3 0 0 }2 1 0 0 10 10 3 6 3 30 0 0 -1 0 } {PSTYLE "Example" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 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 "" 256 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 80 " \+ Vessiot Tutorial: Isometries of Metrics " }}{PARA 260 "" 0 "" {TEXT 258 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 168 "In thi s tutorial we compute the Lie algebras of infinitesimal isometries of a variety of metrics. We then classify these Lie algebras using the package Chevalley" }{TEXT 257 0 "" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 343 "If X is a Killing vector of a metric g, then L_X g = 0 and L_X R^k = 0 where R^k is the k-th covariant derivative of \+ the curvature tensor R. A fixed point these equations become algebr aic equations for X and it its first derivative which can be used to d etermine the dimension and structure equations of the Lie algebra of \+ isometries. " }}{PARA 257 "" 0 "" {TEXT -1 188 "As k increases the ran k of the system of algebraic equations for X and its first derivative s increases. The algorithm terminates if this rank is constant for \+ 2 consecutive values of k." }}{PARA 257 "" 0 "" {TEXT -1 36 "See A. A shetar and A. M. Ashtekar, " }{TEXT 261 55 "A technique for analysing the structure of isometries," }{TEXT 262 0 "" }{TEXT -1 31 " J.Math \+ Phys. 19(2) July 1978." }}{PARA 256 "" 0 "" {TEXT 259 22 "Procedures I llustrated" }}{PARA 257 "" 0 "" {HYPERLNK 17 "Killing_data" 2 "Killing _data" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "Killing_data_to_Lie_algebra " 2 "Killing_data_to_Lie_algebra" "" }{TEXT -1 3 ", " }{HYPERLNK 17 " Lie_alg_init" 2 "Lie_alg_init" "" }{TEXT -1 3 ", " }{HYPERLNK 17 "Lev i_decomposition" 2 "Levi_decomposition" "" }{TEXT -1 2 ", " } {HYPERLNK 17 "decompose_Lie_algebra" 2 "decompose_Lie_algebra" "" } {TEXT -1 1 "," }{HYPERLNK 17 "classify_Lie_alebra" 2 "classify_Lie_ale bra" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "reductive_complement" 2 "reduc tive_complement" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "check_symmetric_pa ir" 2 "check_symmetric_pair" "" }{TEXT -1 1 "." }}}{PARA 256 "" 0 "" {TEXT 260 8 "Examples" }}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 8 "r estart:" }}}{EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 74 "with(Vess iot):with(Koszul):with(tensors):with(Chevalley):with(isometries):" }}} {EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 21 "Th e de Sitter metric." }}{PARA 257 "" 0 "" {TEXT -1 54 "the metric is g iven in Hawking and Ellis, page 125." }}{PARA 257 "" 0 "" {TEXT -1 89 "the metric has constant curvature so the isometry group with be o f maximal dimension 10." }}{PARA 257 "" 0 "" {TEXT -1 121 "the isomet ry group is o(1,4) although we do not yet have the software to ident ify the Lie algebra we compute as such. " }}{PARA 257 "" 0 "" {TEXT -1 60 "anti-de Sitter space is the symmetric space o(1,4)/o(1,3) ??" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "coord_init([t,chi,theta,phi ],[],deSitter):" }}}{EXCHG {PARA 0 "deSitter>" 0 "" {MPLTEXT 1 0 27 "l :=alpha^2*cosh(t/alpha)^2:" }}}{EXCHG {PARA 0 "deSitter>" 0 "" {MPLTEXT 1 0 64 "gm:=linalg[diag](-1, l, l*sin(chi)^2,l*sin(chi)^2*sin (theta)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gmG-%'matrixG6#7&7&! \"\"\"\"!F+F+7&F+*&)%&alphaG\"\"#\"\"\")-%%coshG6#*&%\"tGF1F/!\"\"F0F1 F+F+7&F+F+*(F.F1F2F1)-%$sinG6#%$chiGF0F1F+7&F+F+F+**F.F1F2F1F;F1)-F=6# %&thetaGF0F1" }}}{EXCHG {PARA 0 "deSitter>" 0 "" {MPLTEXT 1 0 39 "g:=a rray_to_tens(gm,[cov_hor,cov_hor]):" }}}{EXCHG {PARA 0 "deSitter > " 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }}}{EXCHG {PARA 0 "deSitter > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C):" }}}{EXCHG {PARA 0 "deSi tter > " 0 "" {MPLTEXT 1 0 20 "pt:=[0,Pi/2,Pi/2,0];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#ptG7&\"\"!,$%#PiG#\"\"\"\"\"#F'F&" }}}{EXCHG {PARA 0 "deSitter > " 0 "" {MPLTEXT 1 0 27 "KD:=Killing_data(g,C,R,pt) :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curvature~derivative G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry ~alg:G\"#5" }}}{EXCHG {PARA 0 "deSitter > " 0 "" {MPLTEXT 1 0 44 "L:=K illing_data_to_Lie_algebra(KD,R,pt,deS):" }}}{EXCHG {PARA 0 "deSitter \+ > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L):" }}}{EXCHG {PARA 0 "deS > \+ " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7.7.%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F (&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(&%#e7G6#F(&%#e8G6#F(&%#e9G6#F(&%$e10G6# F(7.F(%$---G%%----GFJFJFJFJFJFJFJFJFJ7.&F+6#F(F)\"\"!,$&F76#F(!\"#,$&F :6#F(FR,$&F=6#F(FR*&&,$*&\"\"\"Fgn*$)%&alphaG\"\"#Fgn!\"\"#!\"\"\"\"#6 #F(\"\"\"&F.6#F(Fao*&FZFgn&F16#F(Fao*&FZFgn&F46#F(FaoFNFNFN7.&F.6#F(F) ,$FPF_oFN,$*&)FjnF_oFgn&F@6#F(FaoF_o,$*&F`pFgn&FC6#F(FaoF_o*&&F]oF`oFa o&F+6#F(FaoFNFN*&FZFgn&F16#F(Fao*&FZFgn&F46#F(FaoFN7.&F16#F(F),$FTF_o, $F_pFRFN,$*&F`pFgn&FF6#F(FaoF_oFN*&FhpFgn&F+6#F(FaoFN*&&,$Ffn#FaoF_oF` oFao&F.6#F(FaoFN*&FZFgn&F46#F(Fao7.&F46#F(F),$FWF_o,$FdpFR,$FgqFRFNFNF N*&FhpFgn&F+6#F(FaoFN*&F^rFgn&F.6#F(Fao*&F^rFgn&F16#F(Fao7.&F76#F(F),$ FYF^o,$FgpF^oFNFNFN*&FhpFgn&F@6#F(Fao*&FhpFgn&FC6#F(Fao*&FZFgn&F:6#F(F ao*&FZFgn&F=6#F(FaoFN7.&F:6#F(F),$FdoF^oFN,$FjqF^oFN,$FjsF^oFN*&FhpFgn &FF6#F(Fao*&F^rFgn&F76#F(FaoFN*&FZFgn&F=6#F(Fao7.&F=6#F(F),$FgoF^oFNFN ,$F\\sF^o,$F]tF^o,$F\\uF^oFNFN*&F^rFgn&F76#F(Fao*&F^rFgn&F:6#F(Fao7.&F @6#F(F)FN,$F[qF^o,$F]rF^oFN,$F`tF^o,$F_uF^oFNFN*&F^rFgn&FF6#F(Fao*&FZF gn&FC6#F(Fao7.&FC6#F(F)FN,$F^qF^oFN,$F_sF^o,$FctF^oFN,$F\\vF^o,$FivF^o FN*&F^rFgn&F@6#F(Fao7.&FF6#F(F)FNFN,$FcrF^o,$FbsF^oFN,$FbuF^o,$F_vF^o, $F\\wF^o,$FgwF^oFN" }}}{EXCHG {PARA 0 "deS>" 0 "" {MPLTEXT 1 0 26 "che ck_indecomposable(deS);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }} }{EXCHG {PARA 0 "deS>" 0 "" {MPLTEXT 1 0 20 "check_semi_simple();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "deS > " 0 "" {MPLTEXT 1 0 52 "h:=Killing_data_to_isotropy_subalgebra(KD,deS,R,pt ):" }}}{EXCHG {PARA 0 "deS > " 0 "" {MPLTEXT 1 0 8 "Show(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(&%#e5G6#%!G&%#e6G6#F'&%#e7G6#F'&%#e8G6#F'& %#e9G6#F'&%$e10G6#F'" }}}{EXCHG {PARA 0 "deS>" 0 "" {MPLTEXT 1 0 27 "M :=reductive_complement(h):" }}}{EXCHG {PARA 0 "deS > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%#e1G6#%!G&%#e2G 6#F'&%#e3G6#F'&%#e4G6#F'" }}}{EXCHG {PARA 0 "deS>" 0 "" {MPLTEXT 1 0 26 "check_symmetric_pair(h,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr ueG" }}}{EXCHG {PARA 0 "deS > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 25 "The anti-de Sitter metric" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 55 "the metric is g iven in Hawking and Ellis, eq (5.9). " }}{PARA 257 "" 0 "" {TEXT -1 89 "the metric has constant curvature so the isometry group with be o f maximal dimension 10." }}{PARA 257 "" 0 "" {TEXT -1 121 "the isomet ry group is o(2,3) although we do not yet have the software to ident ify the Lie algebra we compute as such. " }}{PARA 257 "" 0 "" {TEXT -1 60 "anti-de Sitter space is the symmetric space o(2,3)/o(1,3) ??" } }{EXCHG {PARA 0 "deS>" 0 "" {MPLTEXT 1 0 49 "coord_init([t,chi,theta,p hi],[],`anti-deSitter`):" }}}{EXCHG {PARA 0 "anti-deSitter>" 0 "" {MPLTEXT 1 0 12 "l:=cos(t)^2:" }}}{EXCHG {PARA 0 "anti-deSitter>" 0 " " {MPLTEXT 1 0 65 "gm:=linalg[diag](-1,l,l*sinh(chi)^2, l*sinh(chi)^2* sin(theta)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gmG-%'matrixG6#7& 7&!\"\"\"\"!F+F+7&F+*$)-%$cosG6#%\"tG\"\"#\"\"\"F+F+7&F+F+*&F.F4)-%%si nhG6#%$chiGF3F4F+7&F+F+F+*(F.F4F7F4)-%$sinG6#%&thetaGF3F4" }}}{EXCHG {PARA 0 "anti-deSitter>" 0 "" {MPLTEXT 1 0 40 "g:=array_to_tens(gm, [c ov_hor,cov_hor]):" }}}{EXCHG {PARA 0 "anti-deSitter>" 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }}}{EXCHG {PARA 0 "anti-deSitter > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C):" }}}{EXCHG {PARA 0 "anti-deSi tter > " 0 "" {MPLTEXT 1 0 26 "pt:=[0,arcsinh(1),Pi/2,0];" }}{PARA 0 " anti-deSitter > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7&\"\"!-%(arcsinhG6#\"\"\",$%#PiG#F*\"\"#F&" }}}{EXCHG {PARA 0 "anti-deSitter > " 0 "" {MPLTEXT 1 0 27 "KD:=Killing_data(g,C, R,pt):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curvature~deriv ativeG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~sym metry~alg:G\"#5" }}}{EXCHG {PARA 0 "anti-deSitter > " 0 "" {MPLTEXT 1 0 51 "L:=Killing_data_to_Lie_algebra(KD,R,pt,`anti-deS`):" }}}{EXCHG {PARA 0 "anti-deSitter > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%6Lie~algebra:~anti-deSG" }}}{EXCHG {PARA 0 "anti-deS > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table(); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7.7.%!G%\"|grG&%#e1G6# F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(&%#e7G6#F(&%#e8G6# F(&%#e9G6#F(&%$e10G6#F(7.F(%$---G%%----GFJFJFJFJFJFJFJFJFJ7.&F+6#F(F) \"\"!,$&F76#F(\"\"#,$&F:6#F(FR,$&F=6#F(FR*&&#!\"\"FR6#F(\"\"\"&F.6#F(F hn*&FZ\"\"\"&F16#F(Fhn*&FZF\\o&F46#F(FhnFNFNFN7.&F.6#F(F),$FP!\"#FN,$& F@6#F(Ffo,$&FC6#F(Ffo*&FZF\\o&F+6#F(FhnFNFN*&FZF\\o&F16#F(Fhn*&FZF\\o& F46#F(FhnFN7.&F16#F(F),$FTFfo,$FhoFRFN,$&FF6#F(FfoFN*&FZF\\o&F+6#F(Fhn FN*&&#FhnFRFgnFhn&F.6#F(FhnFN*&FZF\\o&F46#F(Fhn7.&F46#F(F),$FWFfo,$F[p FR,$F\\qFRFNFNFN*&FZF\\o&F+6#F(FhnFN*&FbqF\\o&F.6#F(Fhn*&FbqF\\o&F16#F (Fhn7.&F76#F(F),$FYFfn,$F]pFfnFNFNFN*&FZF\\o&F@6#F(Fhn*&FZF\\o&FC6#F(F hn*&FZF\\o&F:6#F(Fhn*&FZF\\o&F=6#F(FhnFN7.&F:6#F(F),$F[oFfnFN,$F^qFfnF N,$F]sFfnFN*&FZF\\o&FF6#F(Fhn*&FbqF\\o&F76#F(FhnFN*&FZF\\o&F=6#F(Fhn7. &F=6#F(F),$F_oFfnFNFN,$F_rFfn,$F`sFfn,$F_tFfnFNFN*&FbqF\\o&F76#F(Fhn*& FbqF\\o&F:6#F(Fhn7.&F@6#F(F)FN,$F`pFfn,$FaqFfnFN,$FcsFfn,$FbtFfnFNFN*& FbqF\\o&FF6#F(Fhn*&FZF\\o&FC6#F(Fhn7.&FC6#F(F)FN,$FcpFfnFN,$FbrFfn,$Ff sFfnFN,$F_uFfn,$F\\vFfnFN*&FbqF\\o&F@6#F(Fhn7.&FF6#F(F)FNFN,$FfqFfn,$F erFfnFN,$FetFfn,$FbuFfn,$F_vFfn,$FjvFfnFN" }}}{EXCHG {PARA 0 "anti-deS > " 0 "" {MPLTEXT 1 0 20 "check_semi_simple();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "anti-deS > " 0 "" {MPLTEXT 1 0 33 "check_indecomposable(`anti-deS`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "anti-deS > " 0 "" {MPLTEXT 1 0 59 "h:=Killing_data_to_isotropy_subalgebra(KD,`anti-deS`,R,pt);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG7(7$7%%%vectG%)anti-deSG7\"7#7$ 7#\"\"&\"\"\"7$F'7#7$7#\"\"'F/7$F'7#7$7#\"\"(F/7$F'7#7$7#\"\")F/7$F'7# 7$7#\"\"*F/7$F'7#7$7#\"#5F/" }}}{EXCHG {PARA 0 "anti-deS > " 0 "" {MPLTEXT 1 0 8 "Show(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(&%#e5G6# %!G&%#e6G6#F'&%#e7G6#F'&%#e8G6#F'&%#e9G6#F'&%$e10G6#F'" }}}{EXCHG {PARA 0 "anti-deS>" 0 "" {MPLTEXT 1 0 27 "M:=reductive_complement(h): " }}}{EXCHG {PARA 0 "anti-deS > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%#e1G6#%!G&%#e2G6#F'&%#e3G6#F'&%#e4 G6#F'" }}}{EXCHG {PARA 0 "anti-deS>" 0 "" {MPLTEXT 1 0 26 "check_symme tric_pair(h,M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 24 "The Schwarzschild metric" }}{PARA 257 "" 1 "" {TEXT -1 46 "of course, we are obliged to do this example ." }}{PARA 257 "" 0 "" {TEXT -1 36 "the isometry algebra is so(3) + \+ R." }}{PARA 257 "" 0 "" {TEXT -1 59 "the isometry, isotropy pair de termines a symmetry space." }}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 35 " coord_frame([t, r ,theta, phi],[]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 97 "gm:=matrix([[-(1- 2*m/r), 0,0, 0], [0, 1/(1-2*m/r),0,0], [0,0, r^2,0], [0,0,0,r^2*sin(theta)^2]]);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gmG-%'matrixG6#7&7&,&!\"\"\"\"\"*& %\"mG\"\"\"%\"rG!\"\"\"\"#\"\"!F3F37&F3*&F/F/,&F,F,F-!\"#F1F3F37&F3F3* $)F0F2F/F37&F3F3F3*&F:F/)-%$sinG6#%&thetaGF2F/" }}}{EXCHG {PARA 0 "euc lid>" 0 "" {MPLTEXT 1 0 40 "g:=array_to_tens(gm, [cov_hor,cov_hor]):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }}} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C ):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 24 "basept:=[0,3*m, Pi/2,0]:" }} }{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 33 "KD:=Killing_data(g,C ,R,basept,4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curvatur e~derivativeG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size ~of~symmetry~alg:G\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing ~curvature~derivativeG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maxim al~~size~of~symmetry~alg:G\"\"%" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 44 "L:=Killing_data_to_Lie_algebra(KD,R,basept):" }}} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L):" }}} {EXCHG {PARA 0 "Killing > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_tab le();" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#-%'matrixG6#7( 7(%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(7(F(%$---G%%----GF 8F8F87(&F+6#F(F)\"\"!F" 0 "" {MPLTEXT 1 0 60 "h:=Killing_data_to_is otropy_subalgebra(KD,Killing,R,basept):" }}}{EXCHG {PARA 0 "Killing > \+ " 0 "" {MPLTEXT 1 0 8 "Show(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#& %#e4G6#%!G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "WE MAKE A CHANGE O F BASIS IN THE LIE ALGEBRA TO ELIMENIATE THE METRIC PARAMETER m." }} }{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 34 "A:=linalg[diag](1,1,1 ,1/(18*m^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7 &\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+F*F+7&F+F+F+,$*&\"\"\"F1*$)%\"mG\"\"#F 1!\"\"#F*\"#=" }}}{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 42 "newL :=change_Lie_algebra_basis(A,Sch_iso):" }}}{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 5 "newL;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%(Li e_algG%(Sch_isoG7#\"\"%7%7$7%\"\"#\"\"$F(\"\"\"7$7%F,F(F-!\"\"7$7%F-F( F,F." }}}{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 19 "Lie_alg_init( newL);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5Lie~algebra:~Sch_isoG" }}} {EXCHG {PARA 0 "Sch_iso > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_tab le();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(%!G%\"|grG&%# e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(7(F(%$---G%%----GF8F8F87(&F+6#F(F )\"\"!F" 0 "" {MPLTEXT 1 0 23 "classify_Lie_algebra();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7&7&\"\"!F)!\"\"F)7&F)\"\"\"F)F)7&F)F)F)F,7&F,F)F) F)7$7$%+winternitzG7$\"\"$\"\"'7$F17$F,F," }}}{EXCHG {PARA 0 "Sch_iso \+ > " 0 "" {MPLTEXT 1 0 34 "M:=reductive_complement([e4],[a]):" }}{PARA 0 "Sch_iso > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&&%#e1G6#%!G\"\"\"*&%\"aGF)&%#e4G6#F(F)F)&%#e2G6#F(& %#e3G6#F(" }}}{EXCHG {PARA 0 "Sch_iso>" 0 "" {MPLTEXT 1 0 33 "check_sy mmetric_pair([e4],M,\{a\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$,$%\" aG!\"\"F%<#/F%\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "SO a=0 \+ MAKES (g,h) A SYMMETRIC PAIR." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}} {SECT 1 {PARA 256 "" 0 "" {TEXT -1 27 "The Robertson Walker metric" }} {PARA 257 "" 0 "" {TEXT -1 57 "We take this metric from Hawking and \+ Ellis, page 135. " }}{PARA 257 "" 0 "" {TEXT -1 184 "This metric in volves an arbitrary function S(t). In order to aid in the evaluatio n of the curvature tensor and its derivatives, it is useful to use \+ the optional arguments of " }{TEXT 267 14 "Killing_data. " }{TEXT -1 392 "The fifth argument is an upper bound on the dimension of th e isometry algebra. We set this to 10(which is the largest possible d imension ona 4 manifold). The sixth argument is a list of the arb itrary fuctions and their coordinate variables. This argument is use d to creat an auxilary jet space whose variables are used to in the evaluation of the curvature and its derivatives." }}{PARA 257 "" 0 " " {TEXT -1 42 "The isometry algebra is 6 dimensional." }}{PARA 257 "" 0 "" {TEXT -1 44 "The isometry , isotropy pair is symmetry. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "coord_init([t,chi,theta, phi],[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "CONSTRUCT OTHOGONAL COFRAME" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 12 "omega1:= dt:" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 25 "omega2:= S(t) &mult dchi:" }}}{EXCHG {PARA 0 "euclid \+ > " 0 "" {MPLTEXT 1 0 40 "omega3:= (S(t)*sinh(chi)) &mult dtheta:" }} }{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 48 "omega4:=(S(t)*sinh(c hi)*sin(theta)) &mult dphi :" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 34 "cF:=[omega1,omega2,omega3,omega4]:" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 9 "Show(cF);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&&%#dtG6#%!G*&-%\"SG6#%\"tG\"\"\"&%%dchiG6#F'F-*(F)\" \"\"-%%sinhG6#%$chiGF-&%'dthetaG6#F'F-**F)F2F3F2-%$sinG6#%&thetaGF-&%% dphiG6#F'F-" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 45 "Cdata:=coframe_data([t, chi,theta,phi],cF,RW):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 30 "frame_init(Cdata,[E],[omega]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%*frame:~RWG" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 59 "g:=array_to_tens(linalg[diag](-1,1,1,1),[cov_hor,cov_ hor]):" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 8 "show(g);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%'omega1G6#%!G\"\"\"&F&6#F(F)!\" \"*&&%'omega2G6#F(F)&F/6#F(F)F)*&&%'omega3G6#F(F)&F56#F(F)F)*&&%'omega 4G6#F(F)&F;6#F(F)F)" }}}{EXCHG {PARA 0 "RW>" 0 "" {MPLTEXT 1 0 13 "C:= offel2(g):" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 23 "R:=curvatu re_tensor(C):" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 26 "pt:=[0,arcsinh(2),Pi/2,0] ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7&\"\"!-%(arcsinhG6#\"\"#,$ %#PiG#\"\"\"F*F&" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 40 "KD:=Killing_data(g,C,R,pt ,10,[[t],[S]]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curvat ure~derivativeG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~si ze~of~symmetry~alg:G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computi ng~curvature~derivativeG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Max imal~~size~of~symmetry~alg:G\"\"'" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 62 "L:=Killing_data_to_Lie_algebra(KD,R,pt,RW_isometry,[[ t],[S]]):" }}}{EXCHG {PARA 0 "RW > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_ini t(L):" }}}{EXCHG {PARA 0 "RW_isometry > " 0 "" {MPLTEXT 1 0 25 "Lie_br acket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7*7 *%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&%#e6G6#F (7*F(%$---G%%----GF>F>F>F>F>7*&F+6#F(F)\"\"!*&&,$*&\"\"\"FG*$)&%\"SG6# FB\"\"#FG!\"\"!\"#6#F(\"\"\"&F46#F(FQ*&FDFG&F76#F(FQ*&&#!\"\"\"\"#FPFQ &F.6#F(FQ*&FXFG&F16#F(FQFB7*&F.6#F(F),$FCFZFB*&FDFG&F:6#F(FQ*&&#FQFenF PFQ&F+6#F(FQFB*&FXFG&F16#F(FQ7*&F16#F(F),$FTFZ,$F_oFZFBFB*&FcoFG&F+6#F (FQ*&FcoFG&F.6#F(FQ7*&F46#F(F),$FWFZ,$FboFZFBFB*&FcoFG&F:6#F(FQ*&FXFG& F76#F(FQ7*&F76#F(F),$FhnFZFB,$F_pFZ,$FjpFZFB*&FcoFG&F46#F(FQ7*&F:6#F(F )FB,$FgoFZ,$FbpFZ,$F]qFZ,$FfqFZFB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "SCALE THE LIE ALGEBRA BA SIS TO ELIMENIATE THE PARAMETER S[0]." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "RW_isometry>" 0 "" {MPLTEXT 1 0 45 "A:=linalg[dia g](1/S[0],1/S[0], 1/S[0],1,1,1):" }}}{EXCHG {PARA 0 "RW_isometry>" 0 " " {MPLTEXT 1 0 45 "nL:=change_Lie_algebra_basis(A,nRW_isometry):" }} {PARA 0 "RW_isometry>" 0 "" {MPLTEXT 1 0 17 "Lie_alg_init(nL);" }} {PARA 0 "RW_isometry>" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Lie~algebra:~nRW_isometryG" }}}{EXCHG {PARA 0 "nRW_is ometry > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7*7*%!G%\"|grG&%#e1G6#F(&%#e2G6# F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(7*F(%$---G%%----GF>F>F>F>F>7 *&F+6#F(F)\"\"!,$&F46#F(!\"#,$&F76#F(FF*&&#!\"\"\"\"#6#F(\"\"\"&F.6#F( FP*&FK\"\"\"&F16#F(FPFB7*&F.6#F(F),$FDFNFB,$&F:6#F(FF*&&#FPFNFOFP&F+6# F(FPFB*&FKFT&F16#F(FP7*&F16#F(F),$FHFN,$FfnFNFBFB*&FinFT&F+6#F(FP*&Fin FT&F.6#F(FP7*&F46#F(F),$FJFM,$FhnFMFBFB*&FinFT&F:6#F(FP*&FKFT&F76#F(FP 7*&F76#F(F),$FSFMFB,$FeoFM,$F`pFMFB*&FinFT&F46#F(FP7*&F:6#F(F)FB,$F]oF M,$FhoFM,$FcpFM,$F\\qFMFB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "COMP UTE THE ISOTROPY" }}}{EXCHG {PARA 0 "nRW_isometry>" 0 "" {MPLTEXT 1 0 29 "change_frame_to(RW_isometry);" }}}{EXCHG {PARA 0 "RW_isometry > \+ " 0 "" {MPLTEXT 1 0 60 "h:=Killing_data_to_isotropy_subalgebra(KD,RW_i sometry,R,pt):" }}}{EXCHG {PARA 0 "RW_isometry > " 0 "" {MPLTEXT 1 0 8 "Show(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e4G6#%!G&%#e5G6#F' &%#e6G6#F'" }}}{EXCHG {PARA 0 "RW_isometry>" 0 "" {MPLTEXT 1 0 31 "M:= reductive_complement(h,[a]):" }}}{EXCHG {PARA 0 "RW_isometry > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&&%#e1G 6#%!G\"\"\"*&%\"aGF)&%#e6G6#F(F)F),&&%#e2G6#F(F)*&F+\"\"\"&%#e5G6#F(F) !\"\",&&%#e3G6#F(F)*&F+F4&%#e4G6#F(F)F)" }}}{EXCHG {PARA 0 "RW_isometr y>" 0 "" {MPLTEXT 1 0 30 "check_symmetric_pair(h,M,\{a\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$%\"aG,$F$!\"\"<#/F$\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "SO a=0 MAKES (g,h) A SYMMETRIC PAIR." }}}{EXCHG {PARA 0 "RW_isometry > \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 16 "Th e Godel metric" }}{PARA 257 "" 0 "" {TEXT -1 44 "we take the form of t he Goedel metric from " }{TEXT 263 48 "Exact Solutions of the Einstei n Field Equations " }{TEXT 264 0 "" }{TEXT -1 7 " 10.25." }}{PARA 257 "" 0 "" {TEXT -1 66 "the isometry group is 5 dim and we classify it as sl(2) +R +R. " }}{PARA 257 "" 0 "" {TEXT -1 82 "the action is tr ansitive but the homogeneous space is not a symmetric space." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "RW_isometry > " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 30 "coord_init([t,x,y,z],[],Kurt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1frame~name:~KurtG" }}}{EXCHG {PARA 0 "Kurt>" 0 "" {MPLTEXT 1 0 96 "g:=array([[-a^2,0,0,-a^2*exp(x)],[0,a^2,0,0],[0,0,a^2,0],[-a^2*e xp(x),0,0,\n(a^2/2)*exp(2*x)]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"gG-%'matrixG6#7&7&,$*$)%\"aG\"\"#\"\"\"!\"\"\"\"!F1,$*&F,F/-%$expG 6#%\"xG\"\"\"F07&F1F+F1F17&F1F1F+F17&F2F1F1,$*&F,F/-F56#,$F7F.F8#F8F. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "WE INTRODUCE A MOVING FRAME " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "Kurt>" 0 "" {MPLTEXT 1 0 44 "omega1:= v_zip([a, a*exp(x)],[dt,dz],plus):" }}} {EXCHG {PARA 0 "Kurt > " 0 "" {MPLTEXT 1 0 20 "omega2:= a &mult dx:" } }}{EXCHG {PARA 0 "Kurt > " 0 "" {MPLTEXT 1 0 20 "omega3:= a&mult dy: " }}}{EXCHG {PARA 0 "Kurt > " 0 "" {MPLTEXT 1 0 39 "omega4:= (a*exp(x) *sqrt(3/2)) &mult dz:" }}}{EXCHG {PARA 0 "Kurt > " 0 "" {MPLTEXT 1 0 34 "cF:=[omega1,omega2,omega3,omega4]:" }}}{EXCHG {PARA 0 "Kurt > " 0 "" {MPLTEXT 1 0 18 "F:=dual_frame(cF):" }}}{EXCHG {PARA 0 "Kurt > " 0 "" {MPLTEXT 1 0 9 "Show(cF);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&*& %\"aG\"\"\"&%#dtG6#%!GF'F'*(F&\"\"\"-%$expG6#%\"xGF'&%#dzG6#F+F'F'*&F& F-&%#dxG6#F+F'*&F&F-&%#dyG6#F+F',$**F&F-F.F--%%sqrtG6#\"\"'F-&F36#F+F' #F'\"\"#" }}}{EXCHG {PARA 0 "Kurt>" 0 "" {MPLTEXT 1 0 42 "Cdata:=cofra me_data([t,x,y,z],cF, Goedel):" }}}{EXCHG {PARA 0 "Kurt > " 0 "" {MPLTEXT 1 0 18 "frame_init(Cdata);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%.frame:~GoedelG" }}}{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 59 "g:=array_to_tens(linalg[diag](-1,1,1,1),[cov_hor,cov_hor]):" }}} {EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 17 "Gamma:=offel2(g):" }} }{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 27 "R:=curvature_tensor( Gamma):" }}}{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 38 "KD:=Killing_data(g,Ga mma,R,[0,0,0,0]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curv ature~derivativeG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~ size~of~symmetry~alg:G\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Compu ting~curvature~derivativeG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?M aximal~~size~of~symmetry~alg:G\"\"&" }}}{EXCHG {PARA 0 "Goedel > " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 51 "L:=Killing_data_to_Lie_algebra(KD,R, [0,0,0,0],kg):" }}}{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0Lie~algebra:~kgG" }}}{EXCHG {PARA 0 "kg > \+ " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F (&%#e4G6#F(&%#e5G6#F(7)F(%$---G%%----GF;F;F;F;7)&F+6#F(F)\"\"!*&&,$*&* $-%%sqrtG6#\"\"'\"\"\"FI%\"aG!\"\"#!\"\"FH6#F(\"\"\"&F46#F(FOF?*&&,$FC #FOFHFNFO&F.6#F(FOF?7)&F.6#F(F),$F@FMF?F?,&*&&,$FC#FO\"\"$FNFO&F+6#F(F OFO*&&,$*&FIFI*$)FJ\"\"#FIFK#!\"%F[oFNFO&F76#F(FOFO*&&#FM\"\"#FNFO&F46 #F(FO7)&F16#F(F)F?F?F?F?F?7)&F46#F(F),$FRFM,&FgnFMF^oFMF?F?*&&#FOF\\pF NFO&F.6#F(FO7)&F76#F(F)F?,$FioFMF?,$FgpFMF?" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 69 "WE SCALE THE LIE ALGEBRA BASIS TO SIMPLIFY THE STRUCT URE CONSTANTS." }}}{EXCHG {PARA 0 "kg>" 0 "" {MPLTEXT 1 0 42 "A:=linal g[diag](a^q1,a^q2,a^q3,a^q4,a^q5):" }}}{EXCHG {PARA 0 "kg>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "kg>" 0 "" {MPLTEXT 1 0 32 "change _Lie_algebra_basis(A,nkg);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$7%%(Li e_algG%$nkgG7#\"\"&7(7$7%\"\"\"\"\"#\"\"%,$*&-%%sqrtG6#\"\"'\"\"\")%\" aG,*%#q1G!\"\"%#q2GF:%#q4GF,F:F,F,#F:F47$7%F,F.F-,$*&F1F5)F7,*F9F:F" 0 "" {MPLTEXT 1 0 99 "solve(\{-q1-q2+q4 -1, -q1-q4+q2 -1, -q2-q4+q1-1,-q2-q 4+q5-2,-q2-q5+q4, -q4-q5+q2\},\{q1,q2,q3,q4,q5\});" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<'/%#q5G\"\"!/%#q3GF(/%#q1G!\"\"/%#q2GF+/%#q4GF+" }}} {EXCHG {PARA 0 "kg>" 0 "" {MPLTEXT 1 0 33 "A:=linalg[diag](1/a,1/a,1,1 /a,1):" }}}{EXCHG {PARA 0 "kg>" 0 "" {MPLTEXT 1 0 39 "change_Lie_algeb ra_basis(A,Goedel_iso);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7%%(Lie_a lgG%+Goedel_isoG7#\"\"&7(7$7%\"\"\"\"\"#\"\"%,$*$-%%sqrtG6#\"\"'\"\"\" #!\"\"F47$7%F,F.F-,$F0#F,F47$7%F-F.F,,$F0#F,\"\"$7$7%F-F.F(#!\"%F@7$7% F-F(F.#F7F-7$7%F.F(F-#F,F-" }}}{EXCHG {PARA 0 "kg>" 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Lie~algebra: ~Goedel_isoG" }}}{EXCHG {PARA 0 "Goedel_iso > " 0 "" {MPLTEXT 1 0 25 " Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrix G6#7)7)%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(7)F (%$---G%%----GF;F;F;F;7)&F+6#F(F)\"\"!*&&,$*$-%%sqrtG6#\"\"'\"\"\"#!\" \"FG6#F(\"\"\"&F46#F(FLF?*&&,$FC#FLFGFKFL&F.6#F(FLF?7)&F.6#F(F),$F@FJF ?F?,&*&&,$FC#FL\"\"$FKFL&F+6#F(FLFL*&&#!\"%FhnFKFL&F76#F(FLFL*&&#FJ\" \"#FKFL&F46#F(FL7)&F16#F(F)F?F?F?F?F?7)&F46#F(F),$FOFJ,&FZFJF[oFJF?F?* &&#FLFdoFKFL&F.6#F(FL7)&F76#F(F)F?,$FaoFJF?,$F_pFJF?" }}}{EXCHG {PARA 0 "Goedel_iso>" 0 "" {MPLTEXT 1 0 23 "classify_Lie_algebra();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7'7'#!\"\"\"\"#,$*$-%%sq rtG6#\"\"'\"\"\"#\"\"\"F+\"\"!F5,$F-#F4\"\"%7'F5F5F5F*F57'#F4\"#7,$F-F ;F5F5,$F-#F*\"#C7'#F+\"\"$F5F5F5,$F-#F4F17'F5F5F4F5F57$7$%+winternitzG 7$FC\"\"&7$FI7$F+F5" }}}{EXCHG {PARA 0 "Goedel_iso > " 0 "" {MPLTEXT 1 0 59 "h:=Killing_data_to_isotropy_subalgebra(KD, kg,R,[0,0,0,0]):" } }{PARA 0 "Goedel_iso > " 0 "" {MPLTEXT 1 0 8 "Show(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e5G6#%!G" }}}{EXCHG {PARA 0 "kg>" 0 "" {MPLTEXT 1 0 28 "change_frame_to(Goedel_iso);" }}}{EXCHG {PARA 0 "Goed el_iso > " 0 "" {MPLTEXT 1 0 36 "M:=reductive_complement([e5],[a,b]): " }}}{EXCHG {PARA 0 "Goedel_iso > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&&%#e1G6#%!G\"\"\"*&%\"aGF)&%#e5G6# F(F)F)&%#e2G6#F(,&&%#e3G6#F(F)*&%\"bGF)&F-6#F(F)F)&%#e4G6#F(" }}} {EXCHG {PARA 0 "Goedel_iso>" 0 "" {MPLTEXT 1 0 43 "sym_eq:=check_symme tric_pair([e5],M,\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sym_e qG<%,&%\"aG#!\"\"\"\"#*$-%%sqrtG6#\"\"'\"\"\"#\"\"\"F/,$F+#F2\"\"$,&F' #F2F*F+#F)F/" }}}{EXCHG {PARA 0 "Goedel_iso > " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "WE SEE THAT THERE IS NO SOLU TION FOR \{a,b\} AND SO THIS IS NOT A SYMMETRIC SPACE." }}}{EXCHG {PARA 0 "Goedel_iso > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 " " 0 "" {TEXT -1 14 "The NUT metric" }}{PARA 257 "" 0 "" {TEXT -1 38 "w e use the form of the metric in the " }{TEXT 265 48 "Exact Solutions \+ of the Einstein Field Equations " }{TEXT 266 0 "" }{TEXT -1 7 " 11.43. " }}{PARA 257 "" 0 "" {TEXT -1 32 "the isometry algebra is so(3)+R." } }{PARA 257 "" 0 "" {TEXT -1 109 "in contrast to the Schwarzschild metr ic, the isometry,isotropy pair is does not determine a symmetric spac e." }}{EXCHG {PARA 0 "NUT_ISO > " 0 "" {MPLTEXT 1 0 39 "coord_frame([t ,r,theta,phi],[],M11_43):" }}}{EXCHG {PARA 0 "M11_43>" 0 "" {MPLTEXT 1 0 36 "f:=(r^2 - 2*m*r - l^2)/(r^2 + l^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,(*$)%\"rG\"\"#\"\"\"\"\"\"*&%\"mGF,F)F,!\"#*$) %\"lGF*F+!\"\"F+,&F'F,F0F,!\"\"" }}}{EXCHG {PARA 0 "M11_43>" 0 "" {MPLTEXT 1 0 144 "M:=array([[-f,0,0,-2*f*l*cos(theta)],[0,1/f,0,0],[0, 0,r^2 +\nl^2,0],[-2*f*l*cos(theta),0,0,(r^2 + l^2)*sin(theta)^2 -\nf*4 *l^2*cos(theta)^2]]);\n\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG- %'matrixG6#7&7&,$*&,(*$)%\"rG\"\"#\"\"\"\"\"\"*&%\"mGF2F/F2!\"#*$)%\"l GF0F1!\"\"F1,&F-F2F6F2!\"\"F9\"\"!F<,$*&*(F,F2F8F2-%$cosG6#%&thetaGF2F 1F:F;F57&F<*&F:F1F,F;F" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "WE INTRODUCE A MOVING \+ FRAME. " }}}{EXCHG {PARA 0 "M11_43>" 0 "" {MPLTEXT 1 0 58 "omega1:= 1 \+ &mult(dt &plus ( (2*l*cos(theta)) &mult dphi)):" }}}{EXCHG {PARA 0 "M1 1_43 > " 0 "" {MPLTEXT 1 0 23 "omega2:= (1) &mult dr:" }}}{EXCHG {PARA 0 "M11_43 > " 0 "" {MPLTEXT 1 0 37 "omega3:= sqrt(r^2 +l^2) &mul t dtheta:" }}}{EXCHG {PARA 0 "M11_43 > " 0 "" {MPLTEXT 1 0 47 "omega4: =(sqrt(r^2 +l^2)*sin(theta)) &mult dphi:" }}}{EXCHG {PARA 0 "M11_43 > \+ " 0 "" {MPLTEXT 1 0 38 "NcF:=[omega1, omega2, omega3, omega4]:" }}} {EXCHG {PARA 0 "M11_43 > " 0 "" {MPLTEXT 1 0 10 "Show(NcF);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&&%#dtG6#%!G\"\"\"*(%\"lGF)-%$cosG6#%&the taGF)&%%dphiG6#F(F)\"\"#&%#drG6#F(*&-%%sqrtG6#,&*$)%\"rGF3\"\"\"F)*$)F +F3F?F)F?&%'dthetaG6#F(F)*(F8F?-%$sinGF.F)&F16#F(F)" }}}{EXCHG {PARA 0 "M11_43>" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "M11_43>" 0 "" {MPLTEXT 1 0 20 "NF:=dual_frame(NcF):" }}}{EXCHG {PARA 0 "M11_43 > " 0 "" {MPLTEXT 1 0 42 "CN:=coframe_data([t,r,theta,phi],NcF,NUT):" }}} {EXCHG {PARA 0 "M11_43 > " 0 "" {MPLTEXT 1 0 15 "frame_init(CN):" }}} {EXCHG {PARA 0 "NUT > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "NUT \+ > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "NUT > " 0 "" {MPLTEXT 1 0 31 "MM:=linalg[diag](-f, 1/f, 1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MMG-%'matrixG6#7&7&,$*&,(*$)%\"rG\"\"#\"\"\"\"\"\"*&%\"mGF2F/F2! \"#*$)%\"lGF0F1!\"\"F1,&F-F2F6F2!\"\"F9\"\"!F " 0 "" {MPLTEXT 1 0 39 "g:= array_to_tens(MM,[cov_hor,cov_hor]):" }}}{EXCHG {PARA 0 "NUT > " 0 "" {MPLTEXT 1 0 25 "Gamma:=factor(offel2(g)):" }}}{EXCHG {PARA 0 "NUT > \+ " 0 "" {MPLTEXT 1 0 27 "R:=curvature_tensor(Gamma):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "COMPUTE THE KILLING DATA (It takes a couple \+ of minutes)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "Goede l>" 0 "" {MPLTEXT 1 0 41 "KD:=Killing_data(g,Gamma,R,[0,0,Pi/2,0]):" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curvature~derivativeG\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry~al g:G\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing~curvature~deri vativeG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~sym metry~alg:G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing~curvat ure~derivativeG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~siz e~of~symmetry~alg:G\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "FOR \+ CONVENIENCE, WE STORE THE RESULT." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "NUT > " 0 " " {MPLTEXT 1 0 498 "storedKD:=[[[[tensor, NUT, [con_hor, []]], [[[1], \+ 1]]], [[tensor, NUT, [con_hor, cov_hor, []]], [[[1, 2], m/(l^2)], [[2, 1], m/(l^2)]]]], [[[tensor, NUT, [con_hor, []]], [[[3], 1]]], [[tenso r, NUT, [con_hor, cov_hor, []]], [[[1, 4], 1/l], [[4, 1], -1/l]]]], [[ [tensor, NUT, [con_hor, []]], [[[4], 1]]], [[tensor, NUT, [con_hor, co v_hor, []]], [[[1, 3], -1/l], [[3, 1], 1/l]]]], [[[tensor, NUT, [con_h or, []]], [[[1], 0]]], [[tensor, NUT, [con_hor, cov_hor, []]], [[[3, 4 ], 1/2], [[4, 3], -1/2]]]]]:" }}}{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 13 "KD:=storedKD:" }}}}{EXCHG {PARA 0 "Goedel > " 0 "" {MPLTEXT 1 0 50 "L:=Killing_data_to_Lie_algebra(KD,R,[0,0,Pi/2,0]):" } }}{EXCHG {PARA 0 "NUT > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%5Lie~algebra:~KillingG" }}}{EXCHG {PARA 0 "Killing > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(%!G%\"|grG&%#e1G6#F( &%#e2G6#F(&%#e3G6#F(&%#e4G6#F(7(F(%$---G%%----GF8F8F87(&F+6#F(F)\"\"!* &&,$*&\"\"\"FA%\"lG!\"\"!\"\"6#F(\"\"\"&F16#F(FF*&&F@FEFF&F.6#F(FFF<7( &F.6#F(F),$F=FDF<,&*&&,$F@!\"#FEFF&F+6#F(FFFF*&&,$*&FAFA*$)FB\"\"#FAFC FUFEFF&F46#F(FFFF*&&#FD\"\"#FEFF&F16#F(FF7(&F16#F(F),$FIFD,&FRFDFXFDF< *&&#FFF^oFEFF&F.6#F(FF7(&F46#F(F)F<,$F[oFD,$FfoFDF<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "MANY LIE ALGEBRAS . WILL HAVE METRIC PARAMET ERS IN THEM. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "THESE SHOULD BE ELIMENIATED BY A CHANGE OF BASIS BEFORE USING LIE ALG CLASSIFY. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "OFTEN A SIMPLE SCALING WILL DO." }}}{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 33 "A:=linalg[diag](l^a,l^b,l^c,l ^d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&)%\"lG% \"aG\"\"!F-F-7&F-)F+%\"bGF-F-7&F-F-)F+%\"cGF-7&F-F-F-)F+%\"dG" }}} {EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 139 "change_Lie_algebra_ba sis(A); \+ " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$7%%(Lie_algG%'KoszulG7#\"\"%7(7$7%\"\"\"\"\"#\"\"$,$) %\"lG,*%\"aG!\"\"%\"bGF4%\"cGF,F4F,F47$7%F,F.F-)F1,*F3F4F6F4F5F,F4F,7$ 7%F-F.F,,$)F1,*F5F4F6F4F3F,F4F,!\"#7$7%F-F.F(,$)F1,*F5F4F6F4%\"dGF,F@F ,F@7$7%F-F(F.,$)F1,(F5F4FFF4F6F,#F4F-7$7%F.F(F-,$)F1,(F6F4FFF4F5F,#F,F -" }}}{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 70 "solve(\{ -a -c + b -1, -b-c+a -1, -b-c+d -2, -b-d+c, -c-d+b\},\{a,b,c,d\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"dG\"\"!/%\"bG!\"\"/%\"aGF)/%\"cGF)" }} }{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 33 "A:=linalg[diag](1/l, \+ 1/l, 1/l,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7& *&\"\"\"F+%\"lG!\"\"\"\"!F.F.7&F.F*F.F.7&F.F.F*F.7&F.F.F.\"\"\"" }}} {EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 43 "newL:=change_Lie_algeb ra_basis(A,NUT_ISO): " }}}{EXCHG {PARA 0 "Killing>" 0 "" {MPLTEXT 1 0 19 "Lie_alg_init(newL):" }}}{EXCHG {PARA 0 "NUT_ISO > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'matrixG6#7(7(%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F(&%#e4G6#F(7(F(% $---G%%----GF8F8F87(&F+6#F(F)\"\"!,$&F16#F(!\"\"&F.6#F(F<7(&F.6#F(F)F> F<,&&F+6#F(!\"#&F46#F(FI*&&#F@\"\"#6#F(\"\"\"&F16#F(FQ7(&F16#F(F),$FAF @,&FGFOFJFOF<*&&#FQFOFPFQ&F.6#F(FQ7(&F46#F(F)F<,$FLF@,$FYF@F<" }}} {EXCHG {PARA 0 "NUT_ISO>" 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "NUT_ISO > \+ " 0 "" {MPLTEXT 1 0 23 "classify_Lie_algebra();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7&7&\"\"!\"\"\"F)F)7&!\"\"F)F)#F*\"\"#7&F )F)F,F)7&F,F)F)F*7$7$%+winternitzG7$\"\"$\"\"'7$F37$F*F*" }}}{EXCHG {PARA 0 "NUT_ISO > " 0 "" {MPLTEXT 1 0 63 "Killing_data_to_isotropy_su balgebra(KD,Killing,R,[0,0,Pi/2,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7#7$7%%%vectG%(KillingG7\"7#7$7#\"\"%\"\"\"" }}}{EXCHG {PARA 0 "Kill ing > " 0 "" {MPLTEXT 1 0 25 "change_frame_to(NUT_ISO);" }}}{EXCHG {PARA 0 "NUT_ISO > " 0 "" {MPLTEXT 1 0 8 "h:=[e4];" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"hG7#7$7%%%vectG%(NUT_ISOG7\"7#7$7#\"\"%\"\"\"" }} }{EXCHG {PARA 0 "NUT_ISO > " 0 "" {MPLTEXT 1 0 31 "M:=reductive_comple ment(h,[a]):" }}}{EXCHG {PARA 0 "NUT_ISO > " 0 "" {MPLTEXT 1 0 8 "Show (M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&&%#e1G6#%!G\"\"\"*&%\"aGF) &%#e4G6#F(F)F)&%#e2G6#F(&%#e3G6#F(" }}}{EXCHG {PARA 0 "NUT_ISO>" 0 "" {MPLTEXT 1 0 30 "check_symmetric_pair(h,M,\{a\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%!\"#,&%\"aG#\"\"\"\"\"#!\"\"F(,&F&#F*F)F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "THERE IS NO SYMMETRIC COMPLEMENT. " }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 10 "Plane Wave" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 63 "The plane wave metric is given in H awking and Ellis, p178." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "RW_isometry > " 0 "" {MPLTEXT 1 0 28 "coord_init([y,z,u,v],[] ,pw):" }}}{EXCHG {PARA 0 "pw>" 0 "" {MPLTEXT 1 0 31 "H:=(y^2 -z^2)*F(u ) -2*y*z*G(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG,&*&,&*$)%\"yG \"\"#\"\"\"\"\"\"*$)%\"zGF+F,!\"\"F--%\"FG6#%\"uGF-F-*(F*F-F0F--%\"GGF 4F-!\"#" }}}{EXCHG {PARA 0 "pw>" 0 "" {MPLTEXT 1 0 117 "g:=(du &tensor dv) &plus (dv &tensor du) &plus (dy &tensor dy) &plus (dz &tensor dz) &plus ( H &mult du &tensor du):" }}}{EXCHG {PARA 0 "pw > " 0 "" {MPLTEXT 1 0 8 "show(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%#dyG 6#%!G\"\"\"&F&6#F(F)F)*&&%#dzG6#F(F)&F.6#F(F)F)*(,(*&-%\"FG6#%\"uGF))% \"yG\"\"#\"\"\"F)*&F5F<)%\"zGF;F" 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }}}{EXCHG {PARA 0 "pw > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C):" }}} {EXCHG {PARA 0 "pw > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pw > \+ " 0 "" {MPLTEXT 1 0 18 "pt:=[y0,z0,u0,v0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7&%#y0G%#z0G%#u0G%#v0G" }}}{EXCHG {PARA 0 "pw > \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pw > " 0 "" {MPLTEXT 1 0 44 "KD:=Killing_data(g,C,R, pt,10,[[u], [F,G]]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curvature~derivativeG\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry~alg:G\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing~curvature~derivativeG\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry~alg:G\"\"& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing~curvature~derivativeG \"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry~a lg:G\"\"&" }}}{EXCHG {PARA 0 "pw > " 0 "" {MPLTEXT 1 0 65 "L:=Killing_ data_to_Lie_algebra(KD,R,pt,pw_isometry, [[u],[F,G]]):" }}}{EXCHG {PARA 0 "pw > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9Lie~algebra:~pw_isometryG" }}}{EXCHG {PARA 0 "pw _isometry > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7)7)%!G%\"|grG&%#e1G6#F(&% #e2G6#F(&%#e3G6#F(&%#e4G6#F(&%#e5G6#F(7)F(%$---G%%----GF;F;F;F;7)&F+6# F(F)\"\"!F?F?*&&#!\"\"\"\"#6#F(\"\"\"&F16#F(FFF?7)&F.6#F(F)F?F?F?F?*&F A\"\"\"&F16#F(FF7)&F16#F(F)F?F?F?F?F?7)&F46#F(F),$F@FCF?F?F?F?7)&F76#F (F)F?,$FLFCF?F?F?" }}}{EXCHG {PARA 0 "pw_isometry > " 0 "" {MPLTEXT 1 0 17 "check_solvable();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }} }{EXCHG {PARA 0 "pw_isometry > " 0 "" {MPLTEXT 1 0 34 "check_indecompo sable(pw_isometry);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "pw_isometry > " 0 "" {MPLTEXT 1 0 55 "h:=Killing_data_ to_isotropy_subalgebra(KD,pw_isometry):" }}{PARA 0 "pw_isometry > " 0 "" {MPLTEXT 1 0 8 "Show(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e4 G6#%!G&%#e5G6#F'" }}}{EXCHG {PARA 0 "pw_isometry>" 0 "" {MPLTEXT 1 0 37 "M:=reductive_complement(h,[a,b,c,d]):" }}}{EXCHG {PARA 0 "pw_isome try > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,(&%#e1G6#%!G\"\"\"*&%\"cGF)&%#e4G6#F(F)F)*&%\"aGF)&%#e5G6#F(F)F) ,(&%#e2G6#F(F)*&%\"dGF)&F-6#F(F)F)*&%\"bGF)&F26#F(F)F)&%#e3G6#F(" }}} {EXCHG {PARA 0 "pw_isometry>" 0 "" {MPLTEXT 1 0 36 "check_symmetric_pa ir(h,M,\{a,b,c,d\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#,&%\"aG#\"\" \"\"\"#%\"dG#!\"\"F(<&/F)F)/%\"bGF//%\"cGF1/F%F)" }}}{EXCHG {PARA 0 "p w_isometry > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 25 "Plane Wave (Special Case)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "pw > " 0 "" {MPLTEXT 1 0 29 "coord_init([y,z,u,v], [],spw):" }}}{EXCHG {PARA 0 "spw>" 0 "" {MPLTEXT 1 0 39 "H:=(y^2 -z^2) *cos(2*u) -2*y*z*sin(2*u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG,& *&,&*$)%\"yG\"\"#\"\"\"\"\"\"*$)%\"zGF+F,!\"\"F--%$cosG6#,$%\"uGF+F-F- *(F*F-F0F--%$sinGF4F-!\"#" }}}{EXCHG {PARA 0 "spw>" 0 "" {MPLTEXT 1 0 117 "g:=(du &tensor dv) &plus (dv &tensor du) &plus (dy &tensor dy) &p lus (dz &tensor dz) &plus ( H &mult du &tensor du):" }}}{EXCHG {PARA 0 "spw > " 0 "" {MPLTEXT 1 0 8 "show(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&&%#dyG6#%!G\"\"\"&F&6#F(F)F)*&&%#dzG6#F(F)&F.6#F(F)F)*(,(*&- %$cosG6#,$%\"uG\"\"#F))%\"yGF:\"\"\"F)*&F5F=)%\"zGF:F=!\"\"*(F" 0 "" {MPLTEXT 1 0 13 "C:=of fel2(g):" }}}{EXCHG {PARA 0 "spw > " 0 "" {MPLTEXT 1 0 23 "R:=curvatur e_tensor(C):" }}}{EXCHG {PARA 0 "spw > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "spw > " 0 "" {MPLTEXT 1 0 14 "pt:=[0,0,0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7&\"\"!F&F&F&" }}}{EXCHG {PARA 0 "pw_i sometry > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "pw_isometry > " 0 "" {MPLTEXT 1 0 28 "KD:=Killing_data(g,C,R, pt):" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%@Computing~~curvature~derivativeG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry~alg:G\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing~curvature~derivativeG\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry~alg:G \"\"'" }}}{EXCHG {PARA 0 "spw > " 0 "" {MPLTEXT 1 0 53 "L:=Killing_dat a_to_Lie_algebra(KD,R,pt,spw_isometry):" }}}{EXCHG {PARA 0 "spw > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%:Lie~algebra:~spw_isometryG" }}}{EXCHG {PARA 0 "spw_isometry > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7*7*%!G%\"|grG&%#e1G6#F(&%#e2G6#F(&%#e3G6#F (&%#e4G6#F(&%#e5G6#F(&%#e6G6#F(7*F(%$---G%%----GF>F>F>F>F>7*&F+6#F(F) \"\"!FB,&&F.6#F(!\"\"&F76#F(!\"#FB*&&#FF\"\"#6#F(\"\"\"&F46#F(FOFB7*&F .6#F(F)FBFB,&&F+6#F(FO&F:6#F(FMFBFB*&FK\"\"\"&F46#F(FO7*&F16#F(F),&FDF OFGFM,&FVFFFXFIFBFB,&*&&#FOFMFNFO&F+6#F(FOFO&F:6#F(FO,&*&F_oFen&F.6#F( FOFO&F76#F(FF7*&F46#F(F)FBFBFBFBFBFB7*&F76#F(F),$FJFFFB,&F^oFFFcoFFFBF BFB7*&F:6#F(F)FB,$FZFF,&FfoFFFioFOFBFBFB" }}}{EXCHG {PARA 0 "spw_isome try > " 0 "" {MPLTEXT 1 0 17 "check_solvable();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "spw_isometry > " 0 "" {MPLTEXT 1 0 35 "check_indecomposable(spw_isometry);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "pw_isometry > " 0 "" {MPLTEXT 1 0 56 "h:=Killing_data_to_isotropy_subalgebra(KD,spw_isometr y):" }}{PARA 0 "spw_isometry > " 0 "" {MPLTEXT 1 0 8 "Show(h);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$&%#e5G6#%!G&%#e6G6#F'" }}}{EXCHG {PARA 0 "spw_isometry>" 0 "" {MPLTEXT 1 0 33 "M:=reductive_complement( h,[a,b]):" }}}{EXCHG {PARA 0 "spw_isometry > " 0 "" {MPLTEXT 1 0 8 "Sh ow(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&&%#e1G6#%!G\"\"\"&%#e6G6 #F(\"\"#,&&%#e2G6#F(F)&%#e5G6#F(!\"#,(&%#e3G6#F(F)*&%\"aGF)&F36#F(F)F) *&%\"bGF)&F+6#F(F)F)&%#e4G6#F(" }}}{EXCHG {PARA 0 "spw_isometry>" 0 " " {MPLTEXT 1 0 32 "check_symmetric_pair(h,M,\{a,b\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&!\"#\"\"#,$%\"aG#!\"\"F%,$%\"bGF(" }}}{EXCHG {PARA 0 "spw_isometry > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 13 "MacCallum A2c" }}{PARA 257 "" 0 "" {TEXT -1 43 " We analyze the metric A2c in the paper " }{TEXT 268 68 " Local ly Isotropic SpaceTimes with Nonnull Homogeneous Hypersurfaces" } {TEXT -1 22 " by M.A. H. MacCallum." }}{PARA 257 "" 0 "" {TEXT -1 129 " We find the isometry algebra to be sl(2) + R, the isotropy type to \+ be a rotation and the reductive complement to be an ideal." }}{PARA 257 "" 0 "" {TEXT -1 66 "Using PetrovOnLine, we identity this metric in Petrov as 32.24- " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "coord_init([t,x,y,z],[]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}}{EXCHG {PARA 0 "euclid> " 0 "" {MPLTEXT 1 0 9 "sig0:=dt:" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 35 "sig1:= dx &plus (cosh(y) &mult dz):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 59 "sig2:=(cos(x) &mult dy) &plus ( (s inh(y)*sin(x)) &mult dz):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 62 "sig3:=((-sin(x)) &mult dy) &plus ( (sinh(y)*cos(x)) & mult dz):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 57 "CD:=coframe_data([t,x ,y,z],[sig0,sig1,sig2,sig3],MM_A2c):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 49 "frame_init(CD,[E],[sigma0,sigma1,sigma2,sigma3]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.frame:~MM_A2cG" }}}{EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 70 "gm:=matrix([[-1,0,0,0],[0,E(t)^2,0 ,0],[0,0,F(t)^2,0],[0,0,0,F(t)^2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#gmG-%'matrixG6#7&7&!\"\"\"\"!F+F+7&F+*$)-%\"EG6#%\"tG\"\"#\"\"\"F +F+7&F+F+*$)-%\"FGF1F3F4F+7&F+F+F+F6" }}}{EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 39 "g:=array_to_tens(gm,[cov_hor,cov_hor]):" }}} {EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 8 "show(g);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,**&&%'sigma0G6#%!G\"\"\"&F&6#%!GF)!\"\"*()-%\"E G6#%\"tG\"\"#\"\"\"&%'sigma1G6#%!GF)&F76#%!GF)F)*()-%\"FGF2F4F5&%'sigm a2G6#%!GF)&FB6#%!GF)F)*(F>F5&%'sigma3G6#%!GF)&FJ6#%!GF)F)" }}}{EXCHG {PARA 0 "MM_A2c>" 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }}}{EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C):" }}} {EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 23 "pt:=[0,0,arcsinh(2),0 ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7&\"\"!F&-%(arcsinhG6#\"\" #F&" }}}{EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 42 "KD:=Killing_d ata(g,C,R,pt,10,[[t],[E,F]]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Com puting~~curvature~derivativeG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%?Maximal~~size~of~symmetry~alg:G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing~curvature~derivativeG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symmetry~alg:G\"\"%" }}}{EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 65 "L:=Killing_data_to_Lie_algeb ra(KD,R,pt,A3c_isometry,[[t],[E,F]]):" }}}{EXCHG {PARA 0 "MM_A2c > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Lie~algebra:~A3c_isometryG" }}}{EXCHG {PARA 0 "A3c_isometry > \+ " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(%!G%\"|grG&%#e1G6#%!G&%#e2G6#%!G&%#e3G6 #%!G&%#e4G6#%!G7(F(%$---G%%----GF" 0 "" {MPLTEXT 1 0 59 "h:=Killing_data_to_isotropy_subalgebra(KD,A3c_isometr y,pt):" }}}{EXCHG {PARA 0 "A3c_isometry > " 0 "" {MPLTEXT 1 0 8 "Show( h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e4G6#%!G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "WE NOW MAKE A CHANGE OF BASIS TO ELIMINATE THE CONSTANTS." }}{PARA 0 "" 0 "" {TEXT -1 47 "FIRST WE STRAIGHTEN OUT TH E CENTER. THEN SCALE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "A3c_isometry>" 0 "" {MPLTEXT 1 0 15 "Show(center());" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7#,&&%#e1G6#%!G\"\"\"*&)&%\"EG6#\"\"! \"\"#\"\"\"&%#e4G6#%!GF)!\"\"" }}}{EXCHG {PARA 0 "A3c_isometry>" 0 "" {MPLTEXT 1 0 60 "A1:=matrix([[1,0,0,1],[0,1,0,0],[0,0,1,0],[0,0,0,-E[0 ]^2]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G-%'matrixG6#7&7&\"\" \"\"\"!F+F*7&F+F*F+F+7&F+F+F*F+7&F+F+F+,$*$)&%\"EG6#F+\"\"#\"\"\"!\"\" " }}}{EXCHG {PARA 0 "A3c_isometry>" 0 "" {MPLTEXT 1 0 55 "L2:=change_L ie_algebra_basis(inverse(A),A3c_isometry1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L2G7$7%%(Lie_algG%.A3c_isometry1G7#\"\"%7&7$7%\"\"\" \"\"#\"\"$,$*&*$)&%\"EG6#\"\"!F/\"\"\"F9*$)&%\"FGF7\"\"#F9!\"\"#F.F/7$ 7%F.F0F/,$F2#!\"\"F/7$7%F/F0F.,$*&*$)F" 0 "" {MPLTEXT 1 0 38 "A2:=linalg[diag](E[0]^2/F[0]^2,1,1,1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G-%'matrixG6#7&7&*&*$)&%\"EG6#\" \"!\"\"#\"\"\"F2*$)&%\"FGF/\"\"#F2!\"\"F0F0F07&F0\"\"\"F0F07&F0F0F:F07 &F0F0F0F:" }}}{EXCHG {PARA 0 "A3c_isometry>" 0 "" {MPLTEXT 1 0 58 "L3: =expand(change_Lie_algebra_basis(A2,A3c_isometry2,L2));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#L3G7$7%%(Lie_algG%.A3c_isometry2G7#\"\"%7&7$7 %\"\"\"\"\"#\"\"$#F.F/7$7%F.F0F/#!\"\"F/7$7%F/F0F.!\"#7$7%F/F0F*,&F.F. *&*$)&%\"FG6#\"\"!F/\"\"\"FC*$)&%\"EGFA\"\"#FC!\"\"F/" }}}{EXCHG {PARA 0 "A3c_isometry>" 0 "" {MPLTEXT 1 0 77 "A3:=matrix([[1,0,0,0],[0 ,1,0,0],[0,0,1,0],[1/2*(1 +2*F[0]^2/E[0]^2),0,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F +F+F*F+7&,&#F*\"\"#F**&*$)&%\"FG6#F+F1\"\"\"F8*$)&%\"EGF7\"\"#F8!\"\"F *F+F+F*" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "A 3c_isometry>" 0 "" {MPLTEXT 1 0 50 "L4:=change_Lie_algebra_basis(A3,A3 c_isometry3,L3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L4G7$7%%(Lie_al gG%.A3c_isometry3G7#\"\"%7%7$7%\"\"\"\"\"#\"\"$#F.F/7$7%F.F0F/#!\"\"F/ 7$7%F/F0F.!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "THIS IS OUR FIN AL FORM OF THE ALEBRA. " }}{PARA 0 "" 0 "" {TEXT -1 52 "NOW WE HAVE \+ TO TRANSFORM THE ISOTROPY SUBALGEBRA. " }}{PARA 0 "" 0 "" {TEXT -1 62 "CREATE THE ISOMORPHISM FROM A3c_isometry TO A3c_isometry3." }} }{EXCHG {PARA 0 "A3c_isometry3>" 0 "" {MPLTEXT 1 0 17 "Lie_alg_init(L4 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%;Lie~algebra:~A3c_isometry3G" } }}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "A3c_isometr y3 > " 0 "" {MPLTEXT 1 0 103 "Phi:=simplify(matrix_to_Lie_algebra_tran sform(A3c_isometry,A3c_isometry3,evalm(A3&* A2&*inverse(A1)))):" }}} {EXCHG {PARA 0 "A3c_isometry > " 0 "" {MPLTEXT 1 0 39 "h3:=map2(Lie_al gebra_transform, Phi,h):" }}}{EXCHG {PARA 0 "A3c_isometry3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "A3c_isometry3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "A3c_isometry3 > " 0 "" {MPLTEXT 1 0 25 "Li e_bracket_mult_table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6 #7(7(%!G%\"|grG&%#e1G6#%!G&%#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G7(F(%$---G% %----GF " 0 "" {MPLTEXT 1 0 32 "restricted_ad(h3[1],[e1,e2,e3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7%7%\"\"!F(F(7%F(F(,$*&\"\"\"F,*$)&%\"FG6#F (\"\"#F,!\"\"#!\"\"\"\"#7%F(,$F+#\"\"\"F6F(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 30 "FIND REDUCTIVE COMPLEMENT TYPE" }}}{EXCHG {PARA 0 "A3c_ isometry3 > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "A3c_isometry3 \+ > " 0 "" {MPLTEXT 1 0 32 "M:=reductive_complement(h3,[a]):" }}}{EXCHG {PARA 0 "A3c_isometry3 > " 0 "" {MPLTEXT 1 0 8 "Show(M);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7%,&*&&*&,&%\"aG\"\"\"*$)&%\"FG6#\"\"!\"\"#\"\" \"F*F2*$)F-\"\"#F2!\"\"6#%!GF*&%#e1G6#%!GF*F**&&,$*&F)F2*$)F-\"\"#F2F6 #F*F1F7F*&%#e4G6#%!GF*F*&%#e2G6#%!G&%#e3G6#%!G" }}}{EXCHG {PARA 0 "A3c _isometry3>" 0 "" {MPLTEXT 1 0 30 "M0:=map(helmsimp,subs(a=0,M)):" }}} {EXCHG {PARA 0 "A3c_isometry3 > " 0 "" {MPLTEXT 1 0 9 "Show(M0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%&%#e1G6#%!G&%#e2G6#%!G&%#e3G6#%!G" } }}{EXCHG {PARA 0 "A3c_isometry3>" 0 "" {MPLTEXT 1 0 16 "check_ideal(M0 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "A3c_ isometry3 > " 0 "" {MPLTEXT 1 0 31 "check_symmetric_pair(h3,M,\{a\}); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%!\"#,$*&,&%\"aG\"\"\"*$)&%\"FG6# \"\"!\"\"#\"\"\"F)F1*$)F,\"\"#F1!\"\"#F)F0,$F&#!\"\"F0" }}}{EXCHG {PARA 0 "A3c_isometry3 > " 0 "" {MPLTEXT 1 0 23 "classify_Lie_algebra( );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7&7&#!\"\"\"\"%#F* \"\"#\"\"!F.7&F.F.F*F.7&#\"\"\"F+F,F.F.7&F.F.F.F27$7$%+winternitzG7$\" \"$\"\"&7$F67$F2F2" }}}}{SECT 1 {PARA 256 "" 0 "" {TEXT -1 13 "MacCull um B2b" }}{PARA 257 "" 0 "" {TEXT -1 42 "We analyze the metric B2b \+ in the paper " }{TEXT 269 68 " Locally Isotropic SpaceTimes with Nonnu ll Homogeneous Hypersurfaces" }{TEXT -1 22 " by M.A. H. MacCallum." }} {PARA 257 "" 0 "" {TEXT -1 128 " We find the isometry algebra to be s l(2) + R, the isotropy type to be a boost and the reductive complem ent to be symmetric." }}{PARA 257 "" 0 "" {TEXT -1 65 "Using PetrovOn Line, we identity this metric in Petrov as 32.07 " }}{EXCHG {PARA 0 " A3c_isometry3 > " 0 "" {MPLTEXT 1 0 25 "coord_init([t,x,u,v],[]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 9 "sig0:=dt:" }}}{EXCHG {PARA 0 "e uclid>" 0 "" {MPLTEXT 1 0 9 "sig1:=dx:" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 56 "sig2:= du &plus( (Gamma/beta *exp(-beta*u) ) &mul t dv):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 30 "sig3:=(exp (-beta*u)) &mult dv:" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 54 "CD:=coframe_data([t,x,u,v],[sig0,sig1,sig2,sig3],B2b):" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 49 "frame_init(CD,[E],[sigma0,si gma1,sigma2,sigma3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+frame:~B2bG " }}}{EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 73 "gm:=matrix([[1, 0,0 ,0],[0,E(t)^2,0,0],[0,0,0,-F(t)^2], [0,0,-F(t)^2,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gmG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+*$)-%\"EG 6#%\"tG\"\"#\"\"\"F+F+7&F+F+F+,$*$)-%\"FGF1F3F4!\"\"7&F+F+F6F+" }}} {EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 39 "g:=array_to_tens(gm,[cov _hor,cov_hor]):" }}}{EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 8 "show( g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&&%'sigma0G6#%!G\"\"\"&F&6#% !GF)F)*()-%\"EG6#%\"tG\"\"#\"\"\"&%'sigma1G6#%!GF)&F66#%!GF)F)*()-%\"F GF1F3F4&%'sigma2G6#%!GF)&%'sigma3G6#%!GF)!\"\"*(F=F4&FE6#%!GF)&FA6#%!G F)FH" }}}{EXCHG {PARA 0 "B2b>" 0 "" {MPLTEXT 1 0 13 "C:=offel2(g):" }} }{EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 23 "R:=curvature_tensor(C): " }}}{EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 14 "pt:=[0,0,0,0];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7&\"\"!F&F&F&" }}}{EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 42 "KD:=Killing_data(g,C,R,pt,10,[[t],[E, F]]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%@Computing~~curvature~deriva tiveG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size~of~symm etry~alg:G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Computing~curvatu re~derivativeG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%?Maximal~~size ~of~symmetry~alg:G\"\"%" }}}{EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 65 "L:=Killing_data_to_Lie_algebra(KD,R,pt,B2b_isometry,[[t],[E,F]]): " }}}{EXCHG {PARA 0 "B2b > " 0 "" {MPLTEXT 1 0 16 "Lie_alg_init(L);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Lie~algebra:~B2b_isometryG" }}} {EXCHG {PARA 0 "B2b_isometry > " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "B2b _isometry > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table();" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(%!G%\"|grG&%#e1G6#%!G& %#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G7(F(%$---G%%----GF" 0 "" {MPLTEXT 1 0 59 "h:=Killing_data_to_isotropy_subalgebra(KD,A3c_isometry,pt):" }}{PARA 0 "A3c_isometry > " 0 "" {MPLTEXT 1 0 8 "Show(h);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#&%#e4G6#%!G" }}}{EXCHG {PARA 0 "A3c_isometry>" 0 "" {MPLTEXT 1 0 43 "A:=linalg[diag](1,1,4*beta*Gamma,1/F[0]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\"\"\"\"!F+F+7&F+F*F +F+7&F+F+,$*&%%betaGF*%&GammaGF*\"\"%F+7&F+F+F+*&\"\"\"F5*$)&%\"FG6#F+ \"\"#F5!\"\"" }}}{EXCHG {PARA 0 "A3c_isometry>" 0 "" {MPLTEXT 1 0 49 " L1:=change_Lie_algebra_basis(A,AB2b_isometry2,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1G7$7%%(Lie_algG%/AB2b_isometry2G7#\"\"%7%7$7%\"\"# \"\"$F*\"\"\"7$7%F.F*F.#F0F.7$7%F/F*F/#!\"\"F." }}}{EXCHG {PARA 0 "A3c _isometry>" 0 "" {MPLTEXT 1 0 17 "Lie_alg_init(L1):" }}}{EXCHG {PARA 0 "AB2b_isometry2 > " 0 "" {MPLTEXT 1 0 25 "Lie_bracket_mult_table(); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7(7(%!G%\"|grG&%#e1G6# %!G&%#e2G6#%!G&%#e3G6#%!G&%#e4G6#%!G7(F(%$---G%%----GF" 0 "" {MPLTEXT 1 0 29 "restrict ed_ad(e4,[e1,e2,e3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7 %7%\"\"!F(F(7%F(#!\"\"\"\"#F(7%F(F(#\"\"\"F," }}}{EXCHG {PARA 0 "AB2b_ isometry2 > " 0 "" {MPLTEXT 1 0 34 "M:=reductive_complement([e4],[a]): " }}}{EXCHG {PARA 0 "AB2b_isometry2 > " 0 "" {MPLTEXT 1 0 8 "Show(M); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&&%#e1G6#%!G\"\"\"*&%\"aGF)&%#e 4G6#%!GF)F)&%#e2G6#%!G&%#e3G6#%!G" }}}{EXCHG {PARA 0 "AB2b_isometry2> " 0 "" {MPLTEXT 1 0 33 "check_symmetric_pair([e4],M,\{a\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$,$%\"aG#!\"\"\"\"#,$F%#\"\"\"F(<#/F%\"\"! " }}}{EXCHG {PARA 0 "AB2b_isometry2 > " 0 "" {MPLTEXT 1 0 23 "classify _Lie_algebra();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$-%'matrixG6#7&7& \"\"!F)!\"#F)7&F)F)F)#!\"\"\"\"#7&F)#\"\"\"\"\")F)F)7&F1F)F)F)7$7$%+wi nternitzG7$\"\"$\"\"&7$F67$F1F1" }}}{EXCHG {PARA 0 "AB2b_isometry2 > \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "14" 0 }{VIEWOPTS 1 1 0 3 4 1802 }