Essays/SpaceTime2D/SpaceTime2D05
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4 Metric Tensor
... McCONNELL Chapter XIV Section 3 ...
... Sokolnikoff Sections 29 and 30 ...
... Sokolnikoff Section 41 ...
4.1 Covariant Metric Tensor
NB. ... script SpaceTime2D.ijs (continued) ... mXsetV=:3 : 0 mXv02=:xcdv;0;0;0;tdv;0;0;0: mXv04=:xcdv;xcdvdv;0;0;tdv;tdvdv;0;0: mXv06=:xcdv;xcdvdv;xcdvdvdv;0;tdv;tdvdv;tdvdvdv;0: mXv08=:xcdv;xcdvdv;xcdvdvdv;xcdvdvdvdv;tdv;tdvdv;tdvdvdv;tdvdvdvdv 0 0$0 ) mXm1v =:>@(0{]) mXm2v =:>@(1{]) mXm3v =:>@(2{]) mXm4v =:>@(3{]) mXt1v =:>@(4{]) mXt2v =:>@(5{]) mXt3v =:>@(6{]) mXt4v =:>@(7{]) mXmcvI=:+/@:(*/~"1)"2@mXm1v+-@(gXc^2:)**/~"1@mXt1v mXmcvIdvt1 =:mXm1v+/@:((]+0 2|:])@(*/)"1 2)"2 3 mXm2v mXmcvIdvt2 =:-@(gXc^2:)*mXt1v(]+0 2|:])@(*/)"1 2 mXt2v mXmcvIdv =:mXmcvIdvt1+mXmcvIdvt2 mXmcvIdvdvM1 =:]+0 2 3|:] mXmcvIdvdvM2 =:(1|:])+1 3|:] mXmcvIdvdvt1 =:mXm1v+/@:(mXmcvIdvdvM1@(*/)"1 3)"2 4 mXm3v mXmcvIdvdvt2 =:mXm2v+/@:(mXmcvIdvdvM2@(*/)"2 2)"3 3 mXm2v mXmcvIdvdvt3 =:-@(gXc^2:)*mXt1v mXmcvIdvdvM1@(*/)"1 3 mXt3v mXmcvIdvdvt4 =:-@(gXc^2:)*mXt2v mXmcvIdvdvM2@(*/)"2 2 mXt2v mXmcvIdvdv =:mXmcvIdvdvt1+mXmcvIdvdvt2+mXmcvIdvdvt3+mXmcvIdvdvt4 mXmcvIdvdvdvM1=:]+0 2 3 4|:] mXmcvIdvdvdvM2=:(1|:])+(1 4|:])+(0 1 3 4|:])+(1 3 4|:])+(0 3 1 4|:])+0 3 4 1|:] mXmcvIdvdvdvt1=:mXm1v+/@:(mXmcvIdvdvdvM1@(*/)"1 4)"2 5 mXm4v mXmcvIdvdvdvt2=:mXm2v+/@:(mXmcvIdvdvdvM2@(*/)"2 3)"3 4 mXm3v mXmcvIdvdvdvt3=:-@(gXc^2:)*mXt1v mXmcvIdvdvdvM1@(*/)"1 4 mXt4v mXmcvIdvdvdvt4=:-@(gXc^2:)*mXt2v mXmcvIdvdvdvM2@(*/)"2 3 mXt3v mXmcvIdvdvdv =:mXmcvIdvdvdvt1+mXmcvIdvdvdvt2+mXmcvIdvdvdvt3+mXmcvIdvdvdvt4 mcv =:[((gXL0 mXmcvI gXR0)+gXL1 mXmcvI gXR1)(gXL0 mXv02]);<@(gXL1 mXv02]) mcvdv =:[((gXL0 mXmcvIdv gXR0)+gXL1 mXmcvIdv gXR1)(gXL0 mXv04]);<@(gXL1 mXv04]) mcvdvdv =:[((gXL0 mXmcvIdvdv gXR0)+gXL1 mXmcvIdvdv gXR1)(gXL0 mXv06]);<@(gXL1 mXv06]) mcvdvdvdv=:[((gXL0 mXmcvIdvdvdv gXR0)+gXL1 mXmcvIdvdvdv gXR1)(gXL0 mXv08]);<@(gXL1 mXv08])
NB. ... execute (ijx) ... mXsetV'' p5aXd1=.(0.25;0.25;1;11.62;1);<_0.125;0;1;11.62;1 p5aXd2=.0.01 0.646447 10,:0 2p1 33 p5a1d1=.p5aXd1 mcvdv vGen(p5aXd2) p5a1d2=.p5aXd1(0|:[:(p5aXd1 mcv])D.1])"1 vGen(p5aXd2) p5a1d1((2^_10)gXteq[;])p5a1d2 1 (p5a1d1=.0),p5a1d2=.0 0 0 p5a2d1=.p5aXd1 mcvdvdv vGen(p5aXd2) p5a2d2=.p5aXd1(0|:[:(p5aXd1 mcvdv])D.1])"1 vGen(p5aXd2) p5a2d1((2^_6)gXteq[;])p5a2d2 1 (p5a2d1=.0),p5a2d2=.0 0 0 p5a3d1=.p5aXd1 mcvdvdvdv vGen(p5aXd2) p5a3d2=.p5aXd1(0|:[:(p5aXd1 mcvdvdv])D.1])"1 vGen(p5aXd2) p5a3d1((2^_6)gXteq[;])p5a3d2 1 (p5a3d1=.0),p5a3d2=.0 0 0
4.2 Cofactors of Covariant Metric Tensor
NB. ... script SpaceTime2D.ijs (continued) ... mXsign =:*/~@({.@$$1,_1:) mXdet =:-/ . * NB. see Vocabulary 'Determinant' mXminors=:1&(|:\.)"2^:2 NB. see Vocabulary 'Outfix' mXMcf =:(mXsign*mXdet@mXminors)"2
NB. ... script SpaceTime2D.ijs (continued) ... M =:mXMcf@mcv Mdv =:mXminors^:1"2@(0 1|:])"3@mcvdv(0|:(mXsign*])@:(+/@(+/)@([*mXMcf@])"2)"4)"5 4 mXminors^:1"2@mcv Mdvdv=:mXminors^:1"2@(0 1|:])"4@mcvdvdv(0 1|:(mXsign*])@:(+/@(+/)@([*mXMcf@])"2)"4)"6 4 mXminors^:1"2@mcv
4.3 Determinant of Covariant Metric Tensor
NB. ... script SpaceTime2D.ijs (continued) ... m =:mXdet@mcv mdv =:(0 1|:])"3@mcvdv+/@(+/)@([*mXMcf@])"2"3 2 mcv mdvdvt1=:(0 1|:])"3@mcvdv+/@(+/)@(*"2)"2 _"3(0 1|:])"3@Mdv mdvdvt2=:(0 1|:])"4@mcvdvdv+/@(+/)@(*"2)"4 2 M mdvdv =:mdvdvt1+mdvdvt2
NB. ... execute (ijx) ... mXsetV'' p5bXd1=.(0.25;0.25;1;11.62;1);<_0.125;0;1;11.62;1 p5bXd2=.0.01 0.646447 10,:0 2p1 33 p5b1d1=.p5bXd1 Mdv vGen(p5bXd2) p5b1d2=.p5bXd1(0|:[:(p5bXd1 M])D.1])"1 vGen(p5bXd2) p5b1d1((2^_10)gXteq[;])p5b1d2 1 (p5b1d1=.0),p5b1d2=.0 0 0 p5b2d1=.p5bXd1 Mdvdv vGen(p5bXd2) p5b2d2=.p5bXd1(0|:[:(p5bXd1 Mdv])D.1])"1 vGen(p5bXd2) p5b2d1((2^_6)gXteq[;])p5b2d2 1 (p5b2d1=.0),p5b2d2=.0 0 0 p5b3d1=.p5bXd1 mdv vGen(p5bXd2) p5b3d2=.p5bXd1(0|:[:(p5bXd1 m])D.1])"1 vGen(p5bXd2) p5b3d1((2^_9)gXteq[;])p5b3d2 1 (p5b3d1=.0),p5b3d2=.0 0 0 p5b4d1=.p5bXd1 mdvdv vGen(p5bXd2) p5b4d2=.p5bXd1(0|:[:(p5bXd1 mdv])D.1])"1 vGen(p5bXd2) p5b4d1((2^_8)gXteq[;])p5b4d2 1 (p5b4d1=.0),p5b4d2=.0 0 0
4.4 Contravariant Metric Tensor
NB. ... script SpaceTime2D.ijs (continued) ... mcn=:(m^_1:)*"0 2 M mcndvt1=:(m^_1:)*"0 3 Mdv mcndvt2=:(-@(m^_2:)*"0 1 mdv)(0|:])"3@(*/"1 2)M mcndv =:mcndvt1+mcndvt2 mcndvdvt1=:(m^_1:)*"0 4 Mdvdv mcndvdvt2=:(-@(m^_2:)*"0 1 mdv)(0|:])"4@:(*/"1 3)Mdv mcndvdvt3=:(-@(m^_2:)*"0 1 mdv)(0 3|:])"4@:(*/"1 3)Mdv mcndvdvt4=:(-@(m^_2:)*"0 2 mdvdv)(0 1|:])"4@:(*/"2)M mcndvdvt5=:((2*m^_3:)*"0 2 mdv*/"1 mdv)(0 1|:])"4@:(*/"2)M mcndvdv =:mcndvdvt1+mcndvdvt2+mcndvdvt3+mcndvdvt4+mcndvdvt5
NB. ... execute (ijx) ... mXsetV'' p5cXd1=.(0.25;0.25;1;11.62;1);<_0.125;0;1;11.62;1 p5cXd2=.0.01 0.646447 10,:0 2p1 33 p5c1d1=.p5cXd1 mcndv vGen(p5cXd2) p5c1d2=.p5cXd1(0|:[:(p5cXd1 mcn])D.1])"1 vGen(p5cXd2) p5c1d1((2^_7)gXteq[;])p5c1d2 1 (p5c1d1=.0),p5c1d2=.0 0 0 p5c2d1=.p5cXd1 mcndvdv vGen(p5cXd2) p5c2d2=.p5cXd1(0|:[:(p5cXd1 mcndv])D.1])"1 vGen(p5cXd2) p5c2d1((2^_8)gXteq[;])p5c2d2 1 (p5c2d1=.0),p5c2d2=.0 0 0
NB. ... execute (ijx) ... mXsetV'' p5dXd1=.(0.25;0.25;1;11.62;1);<_0.125;0;1;11.62;1 p5dXd2=.0.01 0.646447 10,:0 2p1 33 p5dXd1*./@(*./)@((2 2$1 0 0 1"_)((2^_41)gXteq[;])"2(2^_39)&gXtsz@(mcv gXsmx"2 mcn))vGen(p5dXd2) 1
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Contributed by Tom Allen