\begin{gather*} \intertext {\texttt {... in this case ... \newline ... the covariant derivative ... } } \begin{split} ( F_{ij} F_{km} )_{,n} = & F_{ij} F_{km,n} + F_{ij,n} F_{km} \end{split} \\ \intertext {\texttt {... does not satisfy the Bianchi Identity ... \newline ... \newline ... the first term on the RHS does ... } } \end{gather*}

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

   gTaC=:1.5,1.5,1.2,1.2,1.5,0"_
   gTaP=:16 c4Gen 3.5 10 28,(0.1,(1p1-0.1),28),0 2p1 44,:0 2p1 20"_

   (4 4$]) *./^:5"5 (0=]) ((2^_44)tsz]) (gTaC(]+(2|:])+2 3|:])"5@(eGF20*/"2 3 eGF20cv)gTaP)''
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1

\begin{gather*} \intertext {\texttt {... but the second term on the RHS doesn't ... \newline ... for example ... } } \end{gather*}

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

   gTbC=:1.5,1.5,1.2,1.2,1.5,0"_
   gTbP=:16 c4Gen 3.5 10 28,(0.1,(1p1-0.1),28),0 2p1 44,:0 2p1 20"_

   nzmax ((2^_44)tsz]) (gTbC(]+(2|:])+2 3|:])@(2|:])"5@(eGF20cv*/"3 2 eGF20)gTbP)''
0.0349497

\begin{gather*} \intertext {\texttt {... so (as already shown) ... \newline ... the divergence of Einstein's tensor is 0 ... } } \begin{align*} \begin{split} 0 = & \Bigl( R_i^j - \frac{1}{2} \delta_i^j R \Bigr)_{,\,j} \end{split} \\ \intertext {\texttt {... but ... } } \begin{split} 0 \neq & \biggl( F_{i\,. }^{\,\,\,k} F_{k\,. }^{\,\,\,\,j} - \frac{1}{2} \delta_i^j \Bigl( F_{m\,. }^{\,\,\,\,\,n} F_{n\,. }^{\,\,\,m} \Bigr) \biggr)_{,\,j} \\ \neq & + F_{i\,. }^{\,\,\,k} F_{k\,.\,j}^{\,\,\,\,j} \\ & + F_{i\,.\,j}^{\,\,\,k} F_{k\,. }^{\,\,\,\,j} \\ & - \frac{1}{2} \delta_i^j \Bigl( F_{m\,. }^{\,\,\,\,\,n} F_{n\,.\,j}^{\,\,\,m} \Bigr) \\ & - \frac{1}{2} \delta_i^j \Bigl( F_{m\,.\,j}^{\,\,\,\,\,n} F_{n\,. }^{\,\,\,m} \Bigr) \\ \neq & + F_{i\,. }^{\,\,\,k} F_{k\,.\,j}^{\,\,\,\,j} \\ & + F_{i\,.\,j}^{\,\,\,k} F_{k\,. }^{\,\,\,\,j} \\ & - \delta_i^j \Bigl( F_{m\,. }^{\,\,\,\,\,n} F_{n\,.\,j}^{\,\,\,m} \Bigr) \\ \neq & + F_{i\,. }^{\,\,\,k} F_{k\,.\,j}^{\,\,\,\,j} \\ & + F_{k\,. }^{\,\,\,\,j} F_{i\,.\,j}^{\,\,\,k} \\ & - F_{m\,. }^{\,\,\,\,\,n} F_{n\,.\,i}^{\,\,\,m} \end{split} \\ \end{align*} \end{gather*}

\begin{gather*} \intertext {\texttt {... \newline ... if $(F_{ij}F_{km})$ were equal to $B_{ijkm}$ ... \newline ... then ... } } \begin{align*} \begin{split} 0 = & ( F_{ij} F_{km} )_{,n} + ( F_{ij} F_{mn} )_{,k} + ( F_{ij} F_{nk} )_{,m} \end{split} \\ \intertext {\texttt {... and ... } } \begin{split} 0 = & \biggl( F_{i\,.}^{\,\,\,k} F_{k\,.}^{\,\,\,\,j} - \frac{1}{2} \delta_i^j \Bigl( F_{m\,.}^{\,\,\,\,\,n} F_{n\,.}^{\,\,\,m} \Bigr) \biggr)_{,\,j} \end{split} \\ \end{align*} \intertext {\texttt {... which might be ... \newline ... the equation for the conservation of mass and energy ... } } \end{gather*}

\begin{gather*} \intertext {\texttt {... \newline ... also ... } } \begin{split} 0 = & g^{ni} g^{jk} \Bigl( ( F_{ij} F_{km} )_{,n} + ( F_{ij} F_{mn} )_{,k} + ( F_{ij} F_{nk} )_{,m} \Bigr) \\ = & g^{ni} g^{jk} \Bigl( ( F_{mk} F_{ji} )_{,n} + ( F_{mn} F_{ij} )_{,k} - ( F_{kn} F_{ij} )_{,m} \Bigr) \end{split} \\ \intertext {\texttt {... term 1 on RHS ... } } \begin{align*} \begin{split} g^{ni} g^{jk} & ( F_{mk} F_{ji} )_{,n} \\ g^{ni} g^{jk} & ( F_{mk} F_{ji,n} + F_{mk,n} F_{ji} ) \\ g^{ni} & ( F_{m\,. }^{\,\,\,\,\,\,j} F_{ji,n} + F_{m\,.\,n}^{\,\,\,\,\,\,j} F_{ji} ) \\ & ( F_{m\,. }^{\,\,\,\,\,\,j} F_{j\,.\,n}^{\,\,n} + F_{j\,. }^{\,\,n} F_{m\,.\,n}^{\,\,\,\,\,\,j} ) \end{split} \\ \intertext {\texttt {... term 2 on RHS ... } } \begin{split} g^{ni} g^{jk} & ( F_{mn} F_{ij} )_{,k} \\ g^{ni} g^{jk} & ( F_{mn} F_{ij,k} + F_{mn,k} F_{ij} ) \\ g^{jk} & ( F_{m\,. }^{\,\,\,\,\,\,i} F_{ij,k} + F_{m\,.\,k}^{\,\,\,\,\,\,i} F_{ij} ) \\ & ( F_{m\,. }^{\,\,\,\,\,\,i} F_{i\,.\,k}^{\,\,k} + F_{i\,. }^{\,\,k} F_{m\,.\,k}^{\,\,\,\,\,\,i} ) \end{split} \\ \intertext {\texttt {... term 3 on RHS ... } } \begin{split} - g^{ni} g^{jk} & ( F_{kn} F_{ij} )_{,m} \\ - g^{ni} g^{jk} & ( F_{kn} F_{ij,m} + F_{kn,m} F_{ij} ) \\ - g^{jk} & ( F_{k\,. }^{\,\,\,\,i} F_{ij,m} + F_{k\,.\,m}^{\,\,\,\,i} F_{ij} ) \\ - & ( F_{k\,. }^{\,\,\,\,i} F_{i\,.\,m}^{\,\,k} + F_{i\,. }^{\,\,k} F_{k\,.\,m}^{\,\,\,\,i} ) \end{split} \\ \end{align*} \end{gather*}


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Essays/Schwarzschild/Schwarzschild06 (last edited 2010-05-14 21:11:54 by TomAllen)