Essays/Torsion Tensor/TorsionTensor05

... ADDENDUM ...

$\begin{gather*} \intertext {\texttt {Script 'torsiontensorE.ijs' is for use with the addendum. \newline Verbs in 'torsiontensor.ijs' are replaced to use 'k1', and \newline not 'k0' together with 'k1', to construct the torsion tensor. } } \end{gather*}$

NB. ... script torsiontensorE.ijs ...

NB. ... use only k1 ...

hkAy=:hycv;hycvdy;hycvdydy;hycvdydydy;Oymg;Oymgdy;Oymgdydy;Oymg;Oymgdy;Oymgdydy;Oymgdydydy
hkAx=:hxcv;hxcvdx;hxcvdxdx;hxcvdxdxdx;Oxmg;Oxmgdx;Oxmgdxdx;Oxmg;Oxmgdx;Oxmgdxdx;Oxmgdxdxdx

NB. ... plot (ijs) ...

gXcme =:(  1.5  1.5   0 )"_
gXpAll=:(  0    0     0 )"_
gXqAll=:( _8    0.5  _1 )"_
gXwAll=:(  0    _     _ )"_

yCpts=:5 15 50,(0.1,(1p1-0.1),28),0 2p1 50,:0 20 50"_

load 'plot'
pd (gXT((cy0*cos@cy2);(cy0*sin@cy2);k1)]) (51 51$]) (~:@:((0 2{])"1)#]) yCtotal@yCpts''
pd 'pdf'
pd 'show'

... k1 ...

... contractions of the curvature tensor and it's covariant derivative ...

$\begin{gather*} \intertext {\texttt {... \newline ... component of the field tensor ... } } \begin{split} F_{np} = & B_{\,.\,knp}^k \end{split} \\ \intertext {\texttt {... Ricci tensor ... } } \begin{split} R_{mn} = & B_{\,.\,mnk}^k \end{split} \\ \intertext {\texttt {... $R$ ... } } \begin{split} R = & B_{\,.\,m\,\,.\,\,k}^{k\,\,\,\,\,m} \end{split} \\ \intertext {\texttt {... covariant derivative of component of the field tensor ... } } \begin{split} F_{np\,,\,r} = & B_{\,.\,knp\,,\,r}^k \end{split} \\ \intertext {\texttt {... divergence of component of the field tensor ... } } \begin{split} F_n = & B_{\,.\,kn\,.\,,\,r}^{k\,\,\,\,\,\,\,\,r} \end{split} \\ \intertext {\texttt {... covariant derivative of Ricci tensor ... } } \begin{split} R_{mn\,,\,r} = & B_{\,.\,mnk\,,\,r}^k \end{split} \\ \intertext {\texttt {... divergence of Ricci tensor ... } } \begin{split} & B_{\,.\,\,\,.\,\,nk\,,\,r}^{k\,\,r} \\ & B_{\,.\,m\,.\,k\,,\,r}^{k\,\,\,\,\,\,r} \end{split} \\ \intertext {\texttt {... covariant derivative of $R$ ... } } \begin{split} R_{\,,\,r} = & B_{\,.\,m\,\,.\,\,k\,,\,r}^{k\,\,\,\,\,m} \end{split} \\ \end{gather*}$

NB. ... script torsiontensorE.ijs ...

R20i=:+/"1@((<0 3)|:])"4@B2k

RI0=:hcn+/@((<0 1)|:])@(   ])@([smx 0|:])"2 2 R20i
RI1=:hcn+/@((<0 1)|:])@(0|:])@([smx    ])"2 2 R20i
R  =:RI1


R20icv=:+/"1@((<0 3)|:])"5@B2kcv

R20iI0cv=:hcn         ([smx 0|:])"2 3 R20icv
R20iI1cv=:hcn(0 2|:])@([smx 1|:])"2 3 R20icv

R20iI0dv=:+/"1@((<0 2)|:])"3@R20iI0cv
R20iI1dv=:+/"1@((<1 2)|:])"3@R20iI1cv

RI0cv=:+/"1@((<0 1)|:])"3@R20iI0cv
RI1cv=:+/"1@((<0 1)|:])"3@R20iI1cv
Rcv  =:RI1cv

NB. ... execute (ijx) ...

NB. ... checks ...

   gXcme =:(  1.5  1.5   0 )"_
   gXpAll=:(  0    0     0 )"_
   gXqAll=:( _8    0.5  _1 )"_
   gXwAll=:(  0    _     _ )"_

   yCpts=:5 15 7,(0.1,(1p1-0.1),7),0 2p1 7,:0 20 7"_

   (gXT([(RI0-:RI1)hkAx)]) (50 yCrandom yCpts)''
1
   (gXT([(R20iI0dv(((2^_43);2^_42)qteq[;])R20iI1dv)hkAx)]) (50 yCrandom yCpts)''
1
   (gXT([(RI0cv(((2^_44);2^_44)qteq[;])RI1cv)hkAx)]) (10 yCrandom yCpts)''
1

   # RArray=:(gXT([R hkAx)]) yCtotal@yCpts''
4096
   *./ (]<0:) RArray
1

$\begin{gather*} \intertext {\texttt {... Einstein's tensor ... } } \begin{align*} \begin{split} E_{ij} = & R_{ij} - \frac{1}{2} h_{ij} R \end{split} \\ \intertext {\texttt {... divergence of Einstein's tensor ... } } \begin{split} E_i = & R_{i\,.\,,\,j}^{\,\,\,j} - \frac{1}{2} R_{\,,\,i} \end{split} \\ \end{align*} \end{gather*}$

NB. ... script torsiontensorE.ijs ...

Edv=:R20iI1dv+-@(0.5*Rcv)

NB. ... script torsiontensorE.ijs ...

S00xL       =:00 00;00 01;00 02;00 03"_
S01dxL      =:01 00;01 01;01 02;01 03"_
S02d1L      =:02 00;02 01;02 02;02 03"_
S03xH       =:03 00;03 01;03 02;03 03"_
S04dxH      =:04 00;04 01;04 02;04 03"_
S05d1H      =:05 00;05 01;05 02;05 03"_
S06ds       =:(<06 00)"_
S07count    =:07 00;07 01"_
   S0700nmbr=:(<07 00)"_
   S0701max =:(<07 01)"_

vS00=:''vS00xL   `S00xL   `]}]
vS01=:''vS01dxL  `S01dxL  `]}]
vS02=:''vS02d1L  `S02d1L  `]}]
vS06=:''vS06ds   `S06ds   `]}]
vS07=:''vS07count`S07count`]}]
vS  =:vS07@vS06@vS02@vS01@vS00@(8 4$_"_)

vSgDd2 =:gXT(-@(([([ch2k hkAx)S03xH{])+/"1@((<0 2)|:])@((0|:[)smx])(]*/])@(S02d1L{]))+vSgDd2M)]

vSgDxH    =:''((S00xL{])+(S02d1L{])*S06ds{])`S03xH`]}]
vSgDdxH   =:''((S02d1L{])*S06ds{])`S04dxH`]}]
vSgDd1H   =:''((S02d1L{])+vSgDd2*S06ds{])`S05d1H`]}]
vSgDxHtoL =:''(S03xH{]) `S00xL `]}]
vSgDdxHtoL=:''(S04dxH{])`S01dxL`]}]
vSgDd1HtoL=:''(S05d1H{])`S02d1L`]}]
vSgDnmbr  =:''(1+S0700nmbr{])`S0700nmbr`]}]
vSgD      =:(vSgDnmbr@vSgDd1HtoL@vSgDdxHtoL@vSgDxHtoL@vSgDd1H@vSgDdxH@vSgDxH)`]@.((S0700nmbr{])=S0701max{])^:a:

vSgDplot=:<"1@(0|:])@:(((cx0*cos@cx2),cx0*sin@cx2)@(S00xL{])"2)

$\begin{gather*} \intertext {\texttt {... an invariant ... } } \begin{split} ds^2 = & h_{ij} dx^i dx^j \end{split} \\ \end{gather*}$

NB. ... script torsiontensorE.ijs ...

Initds=:%:@((gXT([hcv hkAx)Initx)+/@(+/)@:*(]*/])@Initdx)
Initd1=:Initdx%|@Initds

$\begin{gather*} \intertext {\texttt {... a tensor equation ... } } \begin{split} \frac{d^2x^i}{ds^2} + \Bigl\{ \begin{matrix} i \\ jk \end{matrix} \Bigr\} \frac{dx^j}{ds} \frac{dx^k}{ds} - E^i = & 0 \end{split} \\ \end{gather*}$

NB. ... plot (ijs) ...

vSgDd2M=:[([(hcn smx"2 1 Edv)hkAx)S03xH{]

load 'plot'

NB. ... mass only (blue) ...

gXcme =:(  1.5   1.5   0 )"_
gXpAll=:(  0     0     0 )"_
gXqAll=:(  0     0     0 )"_
gXwAll=:(  0     _     _ )"_

Initx =: 7 , 0.5p1 ,  1p1  , 0        "_
Initdx=: 0 , 0     , _0.01 , 0.1008115"_

vS00xL    =:Initx
vS01dxL   =:0,0,0,0:
vS02d1L   =:Initd1
vS06ds    =:|@Initds
vS07count =:0,628"_
pd vSgDplot pdDa=:vSgD@vS''

NB. ... increase mass (red) ...

gXcme =:(  1.5   1.8   0 )"_
gXpAll=:(  0     0     0 )"_
gXqAll=:(  0     0     0 )"_
gXwAll=:(  0     _     _ )"_

Initx =: 7 , 0.5p1 ,  1p1  , 0        "_
Initdx=: 0 , 0     , _0.01 , 0.1008115"_

vS00xL    =:Initx
vS01dxL   =:0,0,0,0:
vS02d1L   =:Initd1
vS06ds    =:|@Initds
vS07count =:0,150"_
pd vSgDplot pdDb=:vSgD@vS''

NB. ... increase (negative) potential (green) ...

gXcme =:(  1.5         1.5   0 )"_
gXpAll=:(  0           0     0 )"_
gXqAll=:( _18.4638663  0.5  _1 )"_
gXwAll=:(  0           _     _ )"_

Initx =: 7 , 0.5p1 ,  1p1  , 0        "_
Initdx=: 0 , 0     , _0.01 , 0.1008115"_

vS00xL    =:Initx
vS01dxL   =:0,0,0,0:
vS02d1L   =:Initd1
vS06ds    =:|@Initds
vS07count =:0,200"_
pd vSgDplot pdDc=:vSgD@vS''

NB. ... increase mass and (negative) potential (purple overwrites blue) ...

gXcme =:(  1.5         1.8   0 )"_
gXpAll=:(  0           0     0 )"_
gXqAll=:( _18.4638663  0.5  _1 )"_
gXwAll=:(  0           _     _ )"_

Initx =: 7 , 0.5p1 ,  1p1  , 0        "_
Initdx=: 0 , 0     , _0.01 , 0.1008115"_

vS00xL    =:Initx
vS01dxL   =:0,0,0,0:
vS02d1L   =:Initd1
vS06ds    =:|@Initds
vS07count =:0,628"_
pd vSgDplot pdDd=:vSgD@vS''

pd 'pdf'
pd 'show'

... geodesics ...

NB. ... execute (ijx) ...

   (S00xL{]) {: pdDa
7.00008 1.5708 _3.13837 63.3095
   
   (S00xL{]) {: pdDd
7 1.5708 _3.13841 63.3096

...

With 'm=0' the potential generated by the Schwarzschild metric is not used.

...

NB. ... plot (ijs) ...

vSgDd2M=:[([-@(hcn smx"2 1 Edv)hkAx)S03xH{]

load 'plot'

NB. ... (negative) potential (blue) ...

gXcme =:(  1.5           0     0 )"_
gXpAll=:(  0             0     0 )"_
gXqAll=:( _020.62867905  0.5  _1 )"_
gXwAll=:(  0             _     _ )"_

Initx =:  7 , 0.5p1 ,  1p1  , 0     "_
Initdx=: (0 , 0     , _0.02 , 0.1200"_)%2:

vS00xL    =:Initx
vS01dxL   =:0,0,0,0:
vS02d1L   =:Initd1
vS06ds    =:|@Initds
vS07count =:(0,314"_)*2:
pd vSgDplot pdDe=:vSgD@vS''

NB. ... increase (negative) potential (red overwrites blue) ...

gXcme =:(  1.5           0     0 )"_
gXpAll=:(  0             0     0 )"_
gXqAll=:( _100.75958575  0.5  _1 )"_
gXwAll=:(  0             _     _ )"_

Initx =:  7 , 0.5p1 ,  1p1  , 0     "_
Initdx=: (0 , 0     , _0.02 , 0.0934"_)%2:

vS00xL    =:Initx
vS01dxL   =:0,0,0,0:
vS02d1L   =:Initd1
vS06ds    =:|@Initds
vS07count =:(0,314"_)*2:
pd vSgDplot pdDf=:vSgD@vS''

pd 'pdf'
pd 'show'

... geodesics ...

NB. ... execute (ijx) ...

   (S00xL{]) {: pdDe
7.00009 1.5708 _3.13839 37.68
   
   (S00xL{]) {: pdDf
7.00001 1.5708 _3.13841 29.3276

NB. ... time difference (blue - red) ...   
   
   (cx3 (S00xL{]) {: pdDe)-(cx3 (S00xL{]) {: pdDf)
8.3524

$\begin{gather*} \intertext {\texttt {... for this potential ... \newline ... \newline ... component of the field tensor ... } } \begin{split} F_{np} = & B_{\,.\,knp}^k \\ = & 0 \end{split} \\ \intertext {\texttt {... covariant derivative of component of the field tensor ... } } \begin{split} F_{np\,,\,r} = & B_{\,.\,knp\,,\,r}^k \\ = & 0 \end{split} \\ \end{gather*}$

NB. ... execute (ijx) ...

NB. ... for this potential ...

   gXcme =:(  1.5  0     0 )"_
   gXpAll=:(  0    0     0 )"_
   gXqAll=:( _8    0.5  _1 )"_
   gXwAll=:(  0    _     _ )"_

   yCpts=:5 15 7,(0.1,(1p1-0.1),7),0 2p1 7,:0 20 7"_

   *./^:2"2 (0=]) ((2^_44)qtsz]) (gXT([+/"1@((<0 1)|:])"4@B2k   hkAx)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
   
   *./^:3"3 (0=]) ((2^_44)qtsz]) (gXT([+/"1@((<0 1)|:])"5@B2kcv hkAx)]) (50 yCrandom yCpts)''
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 TomAllen

Essays/Torsion Tensor/TorsionTensor05 (last edited 2010-12-22 04:39:58 by TomAllen)