Essays/Schwarzschild/Schwarzschild03
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NB. ... plot (ijs) ... gg=:gs NB. ... set the metric ... NB. ... spherical symmetry ... NB. ... closes to a circle ... load 'plot' NB. ... 'timelike' geodesic (blue/red) ... vS00type =:1"_ vS01config=:1.5,1.5,0,0,0,0"_ vS03xH =:5.7,0.5p1,0,0"_ vS05dxH =:0,0,0.01,0.07"_ vS06ctrl =:1,_,_,_,_,_"_ vS11count =:0,600"_ pd vSgDplot vSgD@vS'' vS00type =:1"_ vS01config=:1.5,1.5,0,0,0,0"_ vS03xH =:5.7,0.5p1,0,0"_ vS05dxH =:0,0,_0.01,0.07"_ vS06ctrl =:1,_,_,_,_,_"_ vS11count =:0,600"_ pd vSgDplot vSgD@vS'' NB. ... Newtonian elliptical orbit (green) ... pd (%@(0.165*1+0.06*cos)(([*cos@]);[*sin@])])@nMsteps@(0,(2p1),80"_)'' pd 'pdf' pd 'show'
NB. ... plot (ijs) ... gg=:gs NB. ... set the metric ... load 'plot' pd 'title geodesics in rotating coordinate system' NB. ... null geodesic (blue) ... NB. ... point electric charge at the origin ... vS00type =:0"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5.7,0.5p1,0,0"_ vS05dxH =:0,0,0.01,0"_ vS06ctrl =:1,0.06,0.08,(2^_30),0,100"_ vS11count =:0,790"_ pd vSgDplot vSgD@vS'' NB. ... 'spacelike' (red) ... NB. ... point electric charge at the origin ... vS00type =:1"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5.7,0.5p1,0,0"_ vS05dxH =:0,0,0.01,0.05"_ vS06ctrl =:1,_,_,_,_,_"_ vS11count =:0,850"_ pd vSgDplot vSgD@vS'' NB. ... 'timelike' (green) ... NB. ... point electric charge at the origin ... vS00type =:1"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5.7,0.5p1,0,0"_ vS05dxH =:0,0,0.01,0.09"_ vS06ctrl =:1,_,_,_,_,_"_ vS11count =:0,820"_ pd vSgDplot vSgD@vS'' NB. ... NB. ... opposite direction ... NB. ... NB. ... null geodesic ... NB. ... point electric charge at the origin ... vS00type =:0"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5,0.5p1,0,0"_ vS05dxH =:0,0,_0.01,0"_ vS06ctrl =:1,0.03,0.05,(2^_30),0,100"_ vS11count =:0,2200"_ pd vSgDplot vSgD@vS'' NB. ... 'spacelike' ... NB. ... point electric charge at the origin ... vS00type =:1"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5,0.5p1,0,0"_ vS05dxH =:0,0,_0.01,0.039"_ vS06ctrl =:1,_,_,_,_,_"_ vS11count =:0,2360"_ pd vSgDplot vSgD@vS'' NB. ... 'timelike' ... NB. ... point electric charge at the origin ... vS00type =:1"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5,0.5p1,0,0"_ vS05dxH =:0,0,_0.01,0.042"_ vS06ctrl =:1,_,_,_,_,_"_ vS11count =:0,2020"_ pd vSgDplot vSgD@vS'' pd 'pdf' pd 'show'
NB. ... execute (ijx) ... gg=:gs NB. ... set the metric ... NB. ... null geodesic ... NB. ... point electric charge at the origin ... vS00type =:0"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5.7,0.5p1,0,0"_ vS05dxH =:0,0,0.01,0"_ vS06ctrl =:1,0.06,0.08,(2^_30),0,100"_ vS11count =:0,790"_ *./ (0=]) ((2^_8)tsz]) (S0600ds{])"2@vSgD@vS'' 1
NB. ... execute (ijx) ... gg=:gs NB. ... set the metric ... NB. ... null geodesic ... NB. ... point electric charge at the origin ... vS00type =:0"_ vS01config=:1.5,1.5,1.2,1.2,0.05,0"_ vS03xH =:5,0.5p1,0,0"_ vS05dxH =:0,0,_0.01,0"_ vS06ctrl =:1,0.03,0.05,(2^_30),0,100"_ vS11count =:0,2200"_ *./ (0=]) ((2^_10)tsz]) (S0600ds{])"2@vSgD@vS'' 1
NB. ... script Schwarzschild.ijs (continued) ... NB. ... point electric charge at the origin ... NB. ... potential for the electromagnetic field of the charge (y coordinates) ... eGptl =:(0,0,0,-@gXe*(gXc^2:)*cy1^_1:)"1 eGptldy =:((3 4$0:),(gXe*(gXc^2:)*cy1^_2:),0,0,0:)"1 eGptldydy=:((3 4 4$0:),((_2*gXe*(gXc^2:)*cy1^_3:),0,0,0:),3 4$0:)"1
NB. ... execute (ijx) ... dTaC=:1.5,1.5,1.2,1.2,0,0"_ dTaP=:100 c4Gen 3.5 10 28,(0.1,(1p1-0.1),28),0 2p1 44,:0 2p1 20"_ dTa0=:eGptldy dTa1=:(0|:[:(dTaC eGptl])D.1])"1 (10 10$]) (dTaC(dTa0((2^_23)teq[;])"2 dTa1)dTaP)'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 dTa2=:eGptldydy dTa3=:(0|:[:(dTaC eGptldy])D.1])"1 (10 10$]) (dTaC(dTa2((2^_22)teq[;])"3 dTa3)dTaP)'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
NB. ... script Schwarzschild.ijs (continued) ... NB. ... covariant derivatives of the potential (y coordinates) ... eGptlcv =:eGptldy+-@(([ch2k gs)smx"3 1 eGptl) eGptlcvdy=:eGptldydy+-@(([ch2k gs)smx"3 2(0|:])"2@eGptldy)+-@((2|:])"4@([ch2kdx gs)smx"4 1 eGptl) eGptlcvcv=:eGptlcvdy+-@(([ch2k gs)(1|:])@:smx"3 2(0|:])"2@eGptlcv)+-@(([ch2k gs)(0 1|:])@:smx"3 2 eGptlcv)
NB. ... execute (ijx) ... dTbC=:1.5,1.5,1.2,1.2,0,0"_ dTbP=:100 c4Gen 3.5 10 28,(0.1,(1p1-0.1),28),0 2p1 44,:0 2p1 20"_ dTb0=:eGptlcvdy dTb1=:(0|:[:(dTbC eGptlcv])D.1])"1 (10 10$]) (dTbC(dTb0(((2^_44);2^_20)tcm[;])"3 dTb1)dTbP)'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
NB. ... script Schwarzschild.ijs (continued) ... NB. ... field tensor ... eGF20=:(0|:"2 ycdx)([smx"2 smx"2)((0|:])-])"2@([eGptlcv yfromx) eGF02y=:gXw0y@[(([gcn gs)([smx"2 smx"2)((0|:])-])"2@eGptlcv)yfromx eGF02 =:([xcdy yfromx)([smx"2 smx"2)eGF02y
NB. ... script Schwarzschild.ijs (continued) ... NB. ... covariant derivatives of the field tensor ... eGF20cv=:(0|:"2 ycdx)([smx"2 3[smx"2 3 smx"2 3)((0 2|:])-])"3@([eGptlcvcv yfromx) eGF02cvy=:gXw0y@[(([gcn gs)([([smx 1|:])"2 3([smx 1|:])"2 3)((0 2|:])-])"3@eGptlcvcv)yfromx eGF02cv =:([xcdy yfromx)([smx"2 3 smx"2 3)(0|:"2 ycdx)smx"2 3 eGF02cvy
NB. ... script Schwarzschild.ijs (continued) ... NB. ... Maxwell's equations ... eGMx1=:+/"1@((<1 2)|:])"3@eGF02cv eGMx2=:(]+(0 1|:])+0|:])"3@eGF20cv
NB. ... execute (ijx) ... dTcC=:1.5,1.5,1.2,1.2,1.5,0"_ dTcP=:100 c4Gen 3.5 10 28,(0.1,(1p1-0.1),28),0 2p1 44,:0 2p1 20"_ (10 10$]) *./"1 (0=]) ((2^_44)tsz]) (dTcC eGMx1 dTcP)'' NB. ... for this example J=0 ... 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (10 10$]) *./^:3:"3 (0=]) ((2^_43)tsz]) (dTcC eGMx2 dTcP)'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
NB. ... script Schwarzschild.ijs (continued) ... NB. ... Einstein's tensor ... eGE=:(]++/@((<0 1)|:])*=@i.@$)"2@(gcn(0|:])@:smx"2 R20i)
NB. ... script Schwarzschild.ijs (continued) ... NB. ... mixed energy-momentum tensor ... k =:((<0 0 0){])@(([eGE gs)%eGT) eGT=:((]%(4p1)"_)+-@(+/@((<0 1)|:])*=@i.@$)"2%16p1"_)@(eGF20 smx"2 eGF02)
NB. ... execute (ijx) ... NB. ... Einstein's Law of Gravitation ... dTdC=:1.5,1.5,1.2,1.2,0,0"_ dTdP=:100 c4Gen 3.5 10 28,(0.1,(1p1-0.1),28),0 2p1 44,:0 2p1 20"_ (10 10$]) ((2^_36)teq])"1 ((2^_44)tsz])&.> (dTdC(([eGE gs);"2 k*eGT)dTdP)'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
NB. ... execute (ijx) ... NB. ... Einstein's Law of Gravitation ... NB. ... in a rotating reference frame ... dTeC=:1.5,1.5,1.2,1.2,1.5,0"_ dTeP=:100 c4Gen 3.5 10 28,(0.1,(1p1-0.1),28),0 2p1 44,:0 2p1 20"_ (10 10$]) ((2^_31)teq])"1 ((2^_44)tsz])&.> (dTeC(([eGE gs);"2 k*eGT)dTeP)'' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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Contributed by Tom Allen