Essays/Schwarzschild/Schwarzschild07

Reference

'Geometry of Einstein's Unified Field Theory' by Vaclav Hlavaty

An objective is to attempt to use J to implement tensor equations derived using the torsion tensor (VH II (3)) and to calculate using numerical methods the field tensor having been given an initial value.

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

$\begin{gather*} \intertext {\texttt {... use the torsion tensor ... } } \begin{split} S_{\beta\gamma}^{\rho} B_{\alpha\lambda\rho}^{\epsilon} = & + \Bigl\{ \begin{matrix} \rho \\ \beta\gamma \end{matrix} \Bigr\} \Biggl( \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\rho \end{matrix} \Bigr\} } { \partial v^{\lambda} } - \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\lambda \end{matrix} \Bigr\} } { \partial v^{\rho} } + \Bigl\{ \begin{matrix} \sigma \\ \alpha\rho \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\lambda \end{matrix} \Bigr\} - \Bigl\{ \begin{matrix} \sigma \\ \alpha\lambda \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\rho \end{matrix} \Bigr\} \Biggr) \\ & + \Bigl\{ \begin{matrix} \rho \\ \gamma\beta \end{matrix} \Bigr\} \Biggl( \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\lambda \end{matrix} \Bigr\} } { \partial v^{\rho} } - \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\rho \end{matrix} \Bigr\} } { \partial v^{\lambda} } + \Bigl\{ \begin{matrix} \sigma \\ \alpha\lambda \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\rho \end{matrix} \Bigr\} - \Bigl\{ \begin{matrix} \sigma \\ \alpha\rho \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\lambda \end{matrix} \Bigr\} \Biggr) \end{split} \\ \begin{split} S_{\gamma\lambda}^{\rho} B_{\alpha\beta\rho}^{\epsilon} = & + \Bigl\{ \begin{matrix} \rho \\ \gamma\lambda \end{matrix} \Bigr\} \Biggl( \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\rho \end{matrix} \Bigr\} } { \partial v^{\beta} } - \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\beta \end{matrix} \Bigr\} } { \partial v^{\rho} } + \Bigl\{ \begin{matrix} \sigma \\ \alpha\rho \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\rho \end{matrix} \Bigr\} \Biggr) \\ & + \Bigl\{ \begin{matrix} \rho \\ \lambda\gamma \end{matrix} \Bigr\} \Biggl( \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\beta \end{matrix} \Bigr\} } { \partial v^{\rho} } - \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\rho \end{matrix} \Bigr\} } { \partial v^{\beta} } + \Bigl\{ \begin{matrix} \sigma \\ \alpha\beta \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\rho \end{matrix} \Bigr\} - \Bigl\{ \begin{matrix} \sigma \\ \alpha\rho \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\beta \end{matrix} \Bigr\} \Biggr) \end{split} \\ \begin{split} S_{\lambda\beta}^{\rho} B_{\alpha\gamma\rho}^{\epsilon} = & + \Bigl\{ \begin{matrix} \rho \\ \lambda\beta \end{matrix} \Bigr\} \Biggl( \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\rho \end{matrix} \Bigr\} } { \partial v^{\gamma} } - \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\gamma \end{matrix} \Bigr\} } { \partial v^{\rho} } + \Bigl\{ \begin{matrix} \sigma \\ \alpha\rho \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\gamma \end{matrix} \Bigr\} - \Bigl\{ \begin{matrix} \sigma \\ \alpha\gamma \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\rho \end{matrix} \Bigr\} \Biggr) \\ & + \Bigl\{ \begin{matrix} \rho \\ \beta\lambda \end{matrix} \Bigr\} \Biggl( \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\gamma \end{matrix} \Bigr\} } { \partial v^{\rho} } - \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \alpha\rho \end{matrix} \Bigr\} } { \partial v^{\gamma} } + \Bigl\{ \begin{matrix} \sigma \\ \alpha\gamma \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\rho \end{matrix} \Bigr\} - \Bigl\{ \begin{matrix} \sigma \\ \alpha\rho \end{matrix} \Bigr\} \Bigl\{ \begin{matrix} \epsilon \\ \sigma\gamma \end{matrix} \Bigr\} \Biggr) \end{split} \\ \end{gather*}$

$\begin{gather*} \intertext {\texttt {... covariant derivative of the Riemann-Christoffel tensor ... } } \begin{split} B_{\alpha\beta\gamma,\lambda}^{\epsilon} = & + \frac { \partial^2 \Bigl\{ \begin{matrix} \epsilon \\ \alpha\gamma \end{matrix} \Bigr\} } { \partial v^{\beta} \partial v^{\lambda} } \\ & - \frac { \partial^2 \Bigl\{ \begin{matrix} \epsilon \\ \alpha\beta \end{matrix} \Bigr\} } { \partial v^{\gamma} \partial v^{\lambda} } \\ & + \Bigl\{ \begin{matrix} \sigma \\ \alpha\gamma \end{matrix} \Bigr\} \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \sigma\beta \end{matrix} \Bigr\} } { \partial v^{\lambda} } \\ & + \frac { \partial \Bigl\{ \begin{matrix} \sigma \\ \alpha\gamma \end{matrix} \Bigr\} } { \partial v^{\lambda} } \Bigl\{ \begin{matrix} \epsilon \\ \sigma\beta \end{matrix} \Bigr\} \\ & - \Bigl\{ \begin{matrix} \sigma \\ \alpha\beta \end{matrix} \Bigr\} \frac { \partial \Bigl\{ \begin{matrix} \epsilon \\ \sigma\gamma \end{matrix} \Bigr\} } { \partial v^{\lambda} } \\ & - \frac { \partial \Bigl\{ \begin{matrix} \sigma \\ \alpha\beta \end{matrix} \Bigr\} } { \partial v^{\lambda} } \Bigl\{ \begin{matrix} \epsilon \\ \sigma\gamma \end{matrix} \Bigr\} \\ & - \Bigl\{ \begin{matrix} \rho \\ \alpha\lambda \end{matrix} \Bigr\} B_{\rho\beta\gamma}^{\epsilon} \\ & - \Bigl\{ \begin{matrix} \rho \\ \beta\lambda \end{matrix} \Bigr\} B_{\alpha\rho\gamma}^{\epsilon} \\ & - \Bigl\{ \begin{matrix} \rho \\ \gamma\lambda \end{matrix} \Bigr\} B_{\alpha\beta\rho}^{\epsilon} \\ & + \Bigl\{ \begin{matrix} \epsilon \\ \rho\lambda \end{matrix} \Bigr\} B^{\rho}_{\alpha\beta\gamma} \end{split} \\ \end{gather*}$

$\begin{gather*} \intertext {\texttt {... Maxwell's equation ... } } \begin{split} ( B_{\alpha\beta\gamma,\lambda}^{\alpha} + S_{\beta\gamma}^{\rho} B_{\alpha\lambda\rho}^{\alpha} ) + ( B_{\alpha\gamma\lambda,\beta}^{\alpha} + S_{\gamma\lambda}^{\rho} B_{\alpha\beta\rho}^{\alpha} ) + ( B_{\alpha\lambda\beta,\gamma}^{\alpha} + S_{\lambda\beta}^{\rho} B_{\alpha\gamma\rho}^{\alpha} ) = 0 \end{split} \\ \end{gather*}$

$\begin{gather*} \intertext {\texttt {... covariant derivative of the field tensor ... } } \begin{split} B_{\alpha\beta\gamma,\lambda}^{\alpha} + S_{\beta\gamma}^{\rho} B_{\alpha\lambda\rho}^{\alpha} = \frac { \partial F_{\beta\gamma} } { \partial v^{\lambda} } - \Bigl\{ \begin{matrix} \sigma \\ \beta\lambda \end{matrix} \Bigr\} F_{\sigma\gamma} - \Bigl\{ \begin{matrix} \sigma \\ \gamma\lambda \end{matrix} \Bigr\} F_{\beta\sigma} \end{split} \\ \end{gather*}$

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Contributed by TomAllen

Essays/Schwarzschild/Schwarzschild07 (last edited 2010-05-14 21:53:26 by TomAllen)