Essays/SpaceTime2D/SpaceTime2D08
7 Riemann-Christoffel Tensor
... McCONNELL Chapter XII Section 6 ...
... Sokolnikoff Section 36 ...
7.1 Riemann-Christoffel Tensor of the Second Kind
NB. ... script SpaceTime2D.ijs (continued) ... B2k=:((1 2|:])-2|:])"4@ch2kdv+ch2k((1 3|:[gXsmx 0|:])-[gXsmx 0|:])"3 3 ch2k
7.2 Derivative of the Riemann-Christoffel Tensor of the Second Kind
7.2.1 Derivative
NB. ... script SpaceTime2D.ijs (continued) ... B2kdvt1=:((1 2 4|:])-2 4|:])"5@ch2kdvdv B2kdvt2=:ch2k(((0 3 1 4|:])-0 1 4|:])@((0|:[)gXsmx 2|:])+((1 3 4|:])-])@([gXsmx 0|:]))"3 4 ch2kdv B2kdv =:B2kdvt1+B2kdvt2
7.2.2 Verify Derivative
NB. ... execute (ijx) ... p8aXd1=:((aRL'P'),<'P');<(aRL'Q'),<'Q' aRsetA'' p8a1d1=.p8aXd1 B2kdv'' mXsetV'' p8a1d2=.p8aXd1(0|:[:(p8aXd1 B2k])D.1])"1(vGen aRR'P') p8a1d1((2^_6)gXteq[;])p8a1d2 1 (p8a1d1=.0),p8a1d2=.0 0 0
7.3 Covariant Derivative of the Riemann-Christoffel Tensor of the Second Kind
NB. ... script SpaceTime2D.ijs (continued) ... B2kcvt1=:-@(1|:[gXsmx 0|:]) B2kcvt2=:-@(0 3 4 1|:[gXsmx 1|:]) B2kcvt3=:-@(0 4 1|:[gXsmx 2|:]) B2kcvt4=:1 0|:(0|:[)gXsmx] B2kcv =:B2kdv+ch2k(B2kcvt1+B2kcvt2+B2kcvt3+B2kcvt4)"3 4 B2k
8 Bianchi Identity
... Sokolnikoff Section 38 ...
NB. ... execute (ijx) ... aRsetA'' (((aRL'P'),<'P');<(aRL'Q'),<'Q')*./@ (*./)@((($$0:)-:(2^_19)&gXtsz@(]+(1 3 2|:])+3 1|:]))"5@B2kcv)'' 1
9 Einstein Tensor
... Sokolnikoff Section 38 ...
NB. ... script SpaceTime2D.ijs (continued) ... B1kcv =:mcv([gXsmx 3|:])"2 5 B2kcv R20icv=:+/"1@((<2 3)|:])"5@B2kcv R11icv=:mcn(0 2|:[gXsmx 1|:])"2 3 R20icv Rcv =:+/"1@((<0 1)|:])"3@R11icv
NB. ... execute (ijx) ... aRsetA'' (((aRL'P'),<'P');<(aRL'Q'),<'Q')*./@ (*./)@((($$0:)-:(2^_14)&gXtsz@(]+(2 3|:])+2|:]))"5@B1kcv)'' 1
NB. ... execute (ijx) ... aRsetA'' (((aRL'P'),<'P');<(aRL'Q'),<'Q')*./@ (*./)@((($$0:)-:(2^_8)&gXtsz@(]+(1 3 2 0|:])-3 2 0|:]))"5@B1kcv)'' 1
NB. ... execute (ijx) ... aRsetA'' (((aRL'P'),<'P');<(aRL'Q'),<'Q')*./@(*./)@(+/"1@((<1 2)|:])"3@R11icv((2^_26)gXteq[;])(0.5*])"1@Rcv)'' 1
10 Tangents to Coordinate Curves
10.1 v1 Coordinate Curve
... consider the tangent to the v1 coordinate curve ...
NB. ... script SpaceTime2D.ijs (continued) ... tGntT1 =:(((0{0{])^_0.5"_),0:)"2@mcv tGntI11=:-@(1r2*((0{0{])^_2:)"2@mcv*(0{0{0{])"3@mcvdv)+(0{0{0{])"3@ch2k%(0{0{])"2@mcv tGntI12=:(1{0{0{])"3@ch2k%(0{0{])"2@mcv tGntI1 =:tGntI11,"0 tGntI12
10.2 v2 Coordinate Curve
... consider the tangent to the v2 coordinate curve ...
NB. ... script SpaceTime2D.ijs (continued) ... tGntT2 =:(0,(1{1{])^_0.5"_)"2@mcv tGntI21=:(0{1{1{])"3@ch2k%(1{1{])"2@mcv tGntI22=:-@(1r2*((1{1{])^_2:)"2@mcv*(1{1{1{])"3@mcvdv)+(1{1{1{])"3@ch2k%(1{1{])"2@mcv tGntI2 =:tGntI21,"0 tGntI22
10.3 Orthogonality
NB. ... execute (ijx) ... p8bXd1=:((aRL'P'),<'P');<(aRL'Q'),<'Q' aRsetA'' p8bXd1*./@(*./)@(0=])@((2^_41)&gXtsz)@:(+/"1)@((<0 1)|:"2 mcv gXsmx"2 tGntT1*/"1 tGntI1)'' 1 p8bXd1*./@(*./)@(0=])@((2^_42)&gXtsz)@:(+/"1)@((<0 1)|:"2 mcv gXsmx"2 tGntT2*/"1 tGntI2)'' 1
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Contributed by Tom Allen