Essays/SpaceTime2D/SpaceTime2D08

7 Riemann-Christoffel Tensor

$\begin{gather*} \intertext {\texttt {... \newline ... McCONNELL Chapter XII Section 6 ... \newline ... Sokolnikoff Section 36 ... \newline ... } } \end{gather*}$

7.1 Riemann-Christoffel Tensor of the Second Kind

$\begin{gather*} \intertext {\texttt {... } } \begin{split} B_{\alpha\beta\gamma}^{\epsilon} = & \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\gamma \end{matrix} \Bigr\} } { \partial v^{\beta} } - \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\beta \end{matrix} \Bigr\} } { \partial v^{\gamma} } + \Bigl\{ \begin{matrix} \sigma \\ \alpha\gamma \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\beta \end{matrix} \Bigr\} - \Bigl\{ \begin{matrix} \sigma \\ \alpha\beta \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\gamma \end{matrix} \Bigr\} \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

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

$\begin{gather*} \intertext {\texttt {... } } \begin{split} \frac { \partial B_{\alpha\beta\gamma}^{\epsilon} } { \partial v^{\lambda} } = & \frac { \partial^2 \Bigl\{ \begin{matrix} \epsilon \\ \alpha\gamma \end{matrix} \Bigr\} } { \partial v^{\beta} \partial v^{\lambda} } \\ & - \frac { \partial^2 \Bigl\{ \begin{matrix} \epsilon \\ \alpha\beta \end{matrix} \Bigr\} } { \partial v^{\gamma} \partial v^{\lambda} } \\ & + \Bigl\{ \begin{matrix} \sigma \\ \alpha\gamma \end{matrix} \Bigr\} \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \sigma\beta \end{matrix} \Bigr\} } { \partial v^{\lambda} } \\ & + \frac { \partial \Bigl\{ \begin{matrix} \sigma \\ \alpha\gamma \end{matrix} \Bigr\} } { \partial v^{\lambda} } \Bigl\{ \begin{matrix} \epsilon \\ \sigma\beta \end{matrix} \Bigr\} \\ & - \Bigl\{ \begin{matrix} \sigma \\ \alpha\beta \end{matrix} \Bigr\} \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \sigma\gamma \end{matrix} \Bigr\} } { \partial v^{\lambda} } \\ & - \frac { \partial \Bigl\{ \begin{matrix} \sigma \\ \alpha\beta \end{matrix} \Bigr\} } { \partial v^{\lambda} } \Bigl\{ \begin{matrix} \epsilon \\ \sigma\gamma \end{matrix} \Bigr\} \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

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

$\color{darkblue} \begin{verbatim} v2pts=.0.01 0.646447 10,:0 2p1 33 aRbaseWrite 'P';(0.25;0.25;1;11.62;1);v2pts 1 aRbaseWrite 'Q';(_0.125;0;1;11.62;1);v2pts 1 \end{verbatim}$

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

$\begin{gather*} \intertext {\texttt {... } } \begin{split} B_{\alpha\beta\gamma,\lambda}^{\epsilon} = & \frac { \partial B_{\alpha\beta\gamma}^{\epsilon} } { \partial v^{\lambda} } - \Bigl\{ \begin{matrix} \rho \\ \alpha\lambda \end{matrix} \Bigr\} B_{\rho\beta\gamma}^{\epsilon} - \Bigl\{ \begin{matrix} \rho \\ \beta\lambda \end{matrix} \Bigr\} B_{\alpha\rho\gamma}^{\epsilon} - \Bigl\{ \begin{matrix} \rho \\ \gamma\lambda \end{matrix} \Bigr\} B_{\alpha\beta\rho}^{\epsilon} + \Bigl\{ \begin{matrix} \epsilon \\ \rho\lambda \end{matrix} \Bigr\} B_{\alpha\beta\gamma}^{\rho} \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

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

$\begin{gather*} \intertext {\texttt {... \newline ... Sokolnikoff Section 38 ... \newline ... \newline ... Bianchi Identity ... } } \begin{split} B_{\alpha\beta\gamma,\lambda}^{\epsilon} + B_{\alpha\gamma\lambda,\beta}^{\epsilon} + B_{\alpha\lambda\beta,\gamma}^{\epsilon} = & 0 \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

$\color{darkblue} \begin{verbatim} v2pts=.0.01 0.646447 10,:0 2p1 33 aRbaseWrite 'P';(0.25;0.25;1;11.62;1);v2pts 1 aRbaseWrite 'Q';(_0.125;0;1;11.62;1);v2pts 1 \end{verbatim}$

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

$\begin{gather*} \intertext {\texttt {... \newline ... Sokolnikoff Section 38 ... \newline ... \newline ... equations ... } } \begin{align*} \begin{split} B_{\mu\alpha\delta\gamma,\lambda} = & m_{\mu\epsilon} B_{\alpha\delta\gamma,\lambda}^{\epsilon} \end{split} \\ \begin{split} R_{\alpha\delta,\lambda} = & B_{\alpha\delta\gamma,\lambda}^{\gamma} \end{split} \\ \begin{split} R_{\alpha,\lambda}^{\beta} = & m^{\beta\delta} R_{\alpha\delta,\lambda} \end{split} \\ \begin{split} R_{,\lambda} = & R_{\alpha,\lambda}^{\alpha} \end{split} \\ \end{align*} \intertext {\texttt {... } } \end{gather*}$

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

$\begin{gather*} \intertext {\texttt {... \newline ... lower the superscript $\epsilon$ in the Bianchi Identity ... } } \begin{split} B_{\delta\alpha\beta\gamma,\lambda} + B_{\delta\alpha\gamma\lambda,\beta} + B_{\delta\alpha\lambda\beta,\gamma} = & m_{\delta\epsilon} ( B_{\alpha\beta\gamma,\lambda}^{\epsilon} + B_{\alpha\gamma\lambda,\beta}^{\epsilon} + B_{\alpha\lambda\beta,\gamma}^{\epsilon} ) \\ = & 0 \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

$\color{darkblue} \begin{verbatim} v2pts=.0.01 0.646447 10,:0 2p1 33 aRbaseWrite 'P';(0.25;0.25;1;11.62;1);v2pts 1 aRbaseWrite 'Q';(_0.125;0;1;11.62;1);v2pts 1 \end{verbatim}$

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

   aRsetA''
   (((aRL'P'),<'P');<(aRL'Q'),<'Q')*./@ (*./)@((($$0:)-:(2^_14)&gXtsz@(]+(2 3|:])+2|:]))"5@B1kcv)''
1

$\begin{gather*} \intertext {\texttt {... \newline ... using properties of the Riemann-Christoffel tensor ... } } \begin{split} B_{\beta\gamma\delta\alpha,\lambda} + B_{\lambda\gamma\alpha\delta,\beta} - B_{\lambda\beta\alpha\delta,\gamma} = & 0 \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

$\color{darkblue} \begin{verbatim} v2pts=.0.01 0.646447 10,:0 2p1 33 aRbaseWrite 'P';(0.25;0.25;1;11.62;1);v2pts 1 aRbaseWrite 'Q';(_0.125;0;1;11.62;1);v2pts 1 \end{verbatim}$

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

   aRsetA''
   (((aRL'P'),<'P');<(aRL'Q'),<'Q')*./@ (*./)@((($$0:)-:(2^_8)&gXtsz@(]+(1 3 2 0|:])-3 2 0|:]))"5@B1kcv)''
1

$\begin{gather*} \intertext {\texttt {... \newline ... sums ... } } \begin{split} m^{\lambda\delta} m^{\alpha\beta} B_{\beta\gamma\delta\alpha,\lambda} = & - m^{\lambda\delta} m^{\alpha\beta} \Bigr( B_{\lambda\gamma\alpha\delta,\beta} - B_{\lambda\beta\alpha\delta,\gamma} \Bigl) \end{split} \\ \intertext {\texttt {... \newline ... left side ... } } \begin{split} m^{\lambda\delta} m^{\alpha\beta} B_{\beta\gamma\delta\alpha,\lambda} \end{split} \\ \intertext {\texttt {... sum $\beta$ ... } } \begin{split} m^{\lambda\delta} B_{\gamma\delta\alpha,\lambda}^{\alpha} \end{split} \\ \intertext {\texttt {... sum $\alpha$ ... } } \begin{split} m^{\lambda\delta} R_{\gamma\delta,\lambda} \end{split} \\ \intertext {\texttt {... sum $\delta$ ... } } \begin{split} R_{\gamma,\lambda}^{\lambda} \end{split} \\ \intertext {\texttt {... \newline ... right side ... } } \begin{split} - m^{\lambda\delta} m^{\alpha\beta} \Bigl( B_{\lambda\gamma\alpha\delta,\beta} - B_{\lambda\beta\alpha\delta,\gamma} \Bigr) \end{split} \\ \intertext {\texttt {... sum $\lambda$ ... } } \begin{split} - m^{\alpha\beta} \Bigl( B_{\gamma\alpha\delta,\beta}^{\delta} - B_{\beta\alpha\delta,\gamma}^{\delta} \Bigr) \end{split} \\ \intertext {\texttt {... sum $\delta$ ... } } \begin{split} - m^{\alpha\beta} \Bigl( R_{\gamma\alpha,\beta} - R_{\beta\alpha,\gamma} \Bigr) \end{split} \\ \intertext {\texttt {... sum $\alpha$ ... } } \begin{split} - R_{\gamma,\beta}^{\beta} + R_{\beta,\gamma}^{\beta} \end{split} \\ \intertext {\texttt {... \newline ... sums ... } } \begin{split} R_{\gamma,\lambda}^{\lambda} = & - R_{\gamma,\beta}^{\beta} + R_{\beta,\gamma}^{\beta} \\ = & \frac{1}{2} R_{,\gamma} \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

$\color{darkblue} \begin{verbatim} v2pts=.0.01 0.646447 10,:0 2p1 33 aRbaseWrite 'P';(0.25;0.25;1;11.62;1);v2pts 1 aRbaseWrite 'Q';(_0.125;0;1;11.62;1);v2pts 1 \end{verbatim}$

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

   aRsetA''
   (((aRL'P'),<'P');<(aRL'Q'),<'Q')*./@(*./)@(+/"1@((<1 2)|:])"3@R11icv((2^_26)gXteq[;])(0.5*])"1@Rcv)''
1

$\begin{gather*} \intertext {\texttt {... \newline ... Einstein's tensor $R_{\gamma}^{\lambda}-\frac{1}{2}\delta_{\gamma}^{\lambda}R$ ... \newline ... divergence of Einstein's tensor is zero ... } } \begin{split} R_{\gamma,\lambda}^{\lambda} - \frac{1}{2} \delta_{\gamma}^{\lambda} R_{,\lambda} = & 0 \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

10 Tangents to Coordinate Curves

10.1 v1 Coordinate Curve

$\begin{gather*} \intertext {\texttt {... \newline ... a rank one tensor $\psi^{\alpha}$ ... \newline ... a rank two tensor $\psi_{,\beta}^{\alpha}$ (covariant derivative) ... } } \begin{split} \psi_{,\beta}^{\alpha} = & \frac { \partial\psi^{\alpha} } { \partial v^{\beta} } + \Bigl\{ \begin{matrix} \alpha \\ \gamma\beta \end{matrix} \Bigr\} \psi^{\gamma} \end{split} \\ \intertext {\texttt {... \newline ... a unit vector $\lambda^{\alpha}$ (a tensor) ... \newline ... where $\lambda^1=\frac{dv^1}{ds}$ and $\lambda^2=\frac{dv^2}{ds}$ ... \newline ... the \emph{inner product} of $\lambda_{,\beta}^{\alpha}$ and $\lambda^{\beta}$ (a tensor)... } } \begin{split} \lambda_{,\beta}^{\alpha} \frac { dv^{\beta} } { ds } = & \frac { \partial\lambda^{\alpha} } { \partial v^{\beta} } \frac { dv^{\beta} } { ds } + \Bigl\{ \begin{matrix} \alpha \\ \gamma\beta \end{matrix} \Bigr\} \lambda^{\gamma} \frac { dv^{\beta} } { ds } \\ \frac { \delta\lambda^{\alpha} } { \delta s } = & \frac { d\lambda^{\alpha} } { ds } + \Bigl\{ \begin{matrix} \alpha \\ \gamma\beta \end{matrix} \Bigr\} \lambda^{\gamma} \lambda^{\beta} \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

... consider the tangent to the v1 coordinate curve ...

$\begin{gather*} \intertext {\texttt {... \newline ... where $\lambda^{\alpha}$ is a unit vector ... \newline ... $\lambda^{\alpha}$ is a unit vector tangent to the $v^1$ coordinate curve ... \newline ... so $\lambda^1=\frac{dv^1}{ds}$ and $\lambda^2=0$ } } \begin{align*} \begin{split} m_{11} \frac{dv^1}{ds} \frac{dv^1}{ds} = & 1 \\ \Bigl( \frac{dv^1}{ds} \Bigr)^2 = & (m_{11})^{-1} \\ \frac{dv^1}{ds} = & (m_{11})^{-\frac{1}{2}} \end{split} \\ \intertext {\texttt {... and ... } } \begin{split} \frac{d^2v^1}{ds^2} = & \frac { d \Bigl( \frac{dv^1}{ds} \Bigr) } { dm_{11} } \frac{dm_{11}}{ds} \\ = & \frac { d(m_{11})^{-\frac{1}{2}} } { dm_{11} } \Bigl( \frac { \partial m_{11} } { \partial v^1 } \frac{dv^1}{ds} \Bigr) \\ = & - \frac{1}{2} (m_{11})^{-\frac{3}{2}} \frac { \partial m_{11} } { \partial v^1 } (m_{11})^{-\frac{1}{2}} \\ = & - \frac{1}{2} (m_{11})^{-2} \frac { \partial m_{11} } { \partial v^1 } \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} \frac { \delta\lambda^1 } { \delta s } = & \frac { d^2v^1 } { ds^2 } + \Bigl\{ \begin{matrix} 1 \\ 11 \end{matrix} \Bigr\} \lambda^1 \lambda^1 \\ \frac { \delta\lambda^2 } { \delta s } = & \Bigl\{ \begin{matrix} 2 \\ 11 \end{matrix} \Bigr\} \lambda^1 \lambda^1 \end{split} \\ \end{align*} \intertext {\texttt {... } } \end{gather*}$

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 ...

$\begin{gather*} \intertext {\texttt {... \newline ... where $\lambda^{\alpha}$ is a unit vector ... \newline ... $\lambda^{\alpha}$ is a unit vector tangent to the $v^2$ coordinate curve ... \newline ... so $\lambda^1=0$ and $\lambda^2=\frac{dv^2}{ds}$ } } \begin{align*} \begin{split} m_{22} \frac{dv^2}{ds} \frac{dv^2}{ds} = & 1 \\ \Bigl( \frac{dv^2}{ds} \Bigr)^2 = & (m_{22})^{-1} \\ \frac{dv^2}{ds} = & (m_{22})^{-\frac{1}{2}} \end{split} \\ \intertext {\texttt {... and ... } } \begin{split} \frac{d^2v^2}{ds^2} = & \frac { d \Bigl( \frac{dv^2}{ds} \Bigr) } { dm_{22} } \frac{dm_{22}}{ds} \\ = & \frac { d(m_{22})^{-\frac{1}{2}} } { dm_{22} } \Bigl( \frac { \partial m_{22} } { \partial v^2 } \frac{dv^2}{ds} \Bigr) \\ = & - \frac{1}{2} (m_{22})^{-\frac{3}{2}} \frac { \partial m_{22} } { \partial v^2 } (m_{22})^{-\frac{1}{2}} \\ = & - \frac{1}{2} (m_{22})^{-2} \frac { \partial m_{22} } { \partial v^2 } \end{split} \\ \intertext {\texttt {... so ... } } \begin{split} \frac { \delta\lambda^1 } { \delta s } = & \Bigl\{ \begin{matrix} 1 \\ 22 \end{matrix} \Bigr\} \lambda^2 \lambda^2 \\ \frac { \delta\lambda^2 } { \delta s } = & \frac { d^2v^2 } { ds^2 } + \Bigl\{ \begin{matrix} 2 \\ 22 \end{matrix} \Bigr\} \lambda^2 \lambda^2 \end{split} \\ \end{align*} \intertext {\texttt {... } } \end{gather*}$

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

$\begin{gather*} \intertext {\texttt {... \newline ... $\lambda^{\alpha}$ is orthogonal to $\frac{\delta\lambda^{\alpha}}{\delta s}$ ... } } \begin{split} m_{\alpha\beta} \lambda^{\alpha} \frac { \delta\lambda^{\beta} } { \delta s } = & 0 \end{split} \\ \intertext {\texttt {... } } \end{gather*}$

$\color{darkblue} \begin{verbatim} v2pts=.0.01 0.646447 10,:0 2p1 33 aRbaseWrite 'P';(0.25;0.25;1;11.62;1);v2pts 1 aRbaseWrite 'Q';(_0.125;0;1;11.62;1);v2pts 1 \end{verbatim}$

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

Prev Page: Essays/SpaceTime2D/SpaceTime2D07

Contributed by TomAllen