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'


SCSchwarzschild0300.jpg 


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'


SCSchwarzschild0301.jpg 


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


GSC0300C.jpg 


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


GSC0301C.jpg 


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


GSC0302C.jpg 


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


GSC0303C.jpg 


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


GSC0304C.jpg 


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


GSC0305C.jpg 


NB. ... script Schwarzschild.ijs (continued) ...

NB. ... Einstein's tensor ...

eGE=:(]++/@((<0 1)|:])*=@i.@$)"2@(gcn(0|:])@:smx"2 R20i)


GSC0306C.jpg 


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)


GSC0307C.jpg 


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


Download: File:LSC0300C.txt
Download: File:LSC0301C.txt
Download: File:LSC0302C.txt
Download: File:LSC0303C.txt
Download: File:LSC0304C.txt
Download: File:LSC0305C.txt
Download: File:LSC0306C.txt
Download: File:LSC0307C.txt


Download MoinMoin source: File:Schwarzschild03.ijs


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Contributed by Tom Allen

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