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From Ulrich.Bruchholz@t-online.de Tue~Jul~24~18:25:04~2001\\
Message-ID: \textless9j71vm\$lp8mr\$1@ID-28113.news.dfncis.de\textgreater\\
From: Ulrich Bruchholz \textless{}Ulrich.Bruchholz@t-online.de\textgreater\\
Newsgroups: sci.physics.research\\
Subject: Ricci Main Directions in an EM Vacuum\\
Date: 19 Jul 2001 16:30:46 GMT\\
Approved: spr@rosencrantz.stcloudstate.edu (sci.physics.research)\\
\\
The known source-free Einstein-Maxwell equations
{\large
\begin{displaymath}
R_{ik} = \kappa \cdot ( \frac{1}{4}~g_{ik} F_{ab} F^{ab} - F_{ia} F_k{}^a )
\qquad ,
\end{displaymath}
\begin{displaymath}
F^{ia}{}_{;a} = 0 \qquad ,
\end{displaymath}
\begin{displaymath}
F_{ik} = A_{i,k} - A_{k,i}
\end{displaymath}}
\normalsize
involve a special kind of Riemannian geometry, what is explained
as follows.
The Ricci main directions (written in terms according to Eisenhart)
follow from
{\large
\begin{displaymath}
\mathsf{det} | R_{ik} + \rho g_{ik} | = 0
\end{displaymath}}
\normalsize
with the solutions
{\large
\begin{displaymath}
\rho_{|1} = \rho_{|4} = +\rho_o \qquad ,
\qquad \qquad \rho_{|2} = \rho_{|3} = -\rho_o
\end{displaymath}}
\normalsize
with
{\large
\begin{displaymath}
\rho_o{}^2 = R_1{}^a R^1{}_a = R_2{}^a R^2{}_a = R_3{}^a R^3{}_a = R_4{}^a R^4{}_a
\qquad .
\end{displaymath}}
\normalsize
Characteristical are the two double-roots, that means: There are
two dual surfaces of the congruences $ e_{|1}{}^i e_{|4}{}^k - e_{|1}{}^k e_{|4}{}^i $
and $ e_{|2}{}^i e_{|3}{}^k - e_{|2}{}^k e_{|3}{}^i $ with extreme mean Riemannian
curvature. ($e_{|1} ... e_{|4}$ are the vectors of an orthogonal quadrupel
in those "main surfaces". At single roots we had 4 main directions.)
With it we get
{\large
\begin{displaymath}
g_{ik} = e_{|1\_i} e_{|1\_k} + e_{|2\_i} e_{|2\_k} + e_{|3\_i} e_{|3\_k} - e_{|4\_i} e_{|4\_k}
\qquad ,
\end{displaymath}
\begin{displaymath}
\frac{R_{ik}}{\rho_o}
= - e_{|1\_i} e_{|1\_k} + e_{|2\_i} e_{|2\_k} + e_{|3\_i} e_{|3\_k} + e_{|4\_i} e_{|4\_k}
\qquad .
\end{displaymath}}
\normalsize
If we set
{\large
\begin{displaymath}
c_{|ik} = -c_{|ki} = F_{ab} e_{|i}{}^a e_{|k}{}^b
\end{displaymath}}
\normalsize
follows [elementary calculations snipped]
{\large
\begin{displaymath}
-\kappa \Bigl( (c_{|23})^2 + (c_{|14})^2 \Bigr) = 2 \rho_o \qquad ,
\end{displaymath}
\begin{displaymath}
c_{|12} = c_{|34} = c_{|13} = c_{|24} = 0 \qquad .
\end{displaymath}}
\normalsize
With it, the field tensor
{\large
\begin{displaymath}
F_{ik} = - c_{|14} (e_{|1\_i} e_{|4\_k} - e_{|1\_k} e_{|4\_i})
+ c_{|23} (e_{|2\_i} e_{|3\_k} - e_{|2\_k} e_{|3\_i})
\end{displaymath}}
\normalsize
is performed from the main surfaces !
That works of course solely without sources (in an "electrovacuum"),
but it is not the author's problem, because relevant quantities like
charge, mass, which one can measure, are integration constants of the
above mentioned equations. The Bianchi identities are met always here.
This brief summary was a sample from the works in
http://www.markt-2000.de/patent/section2/sc\_works.html\footnote{This
site has been moved to http://bruchholz.psf.net/index.html~.}~.
It was first published 1980, see literary index in both works.
Details about the derivation in the German-language work, pages 33 and 34,
about the Ricci main directions in general pages 22 to 24. The latter
was drawn from Eisenhart: Riemannian Geometry, Princeton university press.
See also solutions relevant for the microcosmos, achieved from
computations.
Please take no offence at the author's claim about the sources, because
it is well founded with the fact that known values of particles let
recognize themselves from the got "electrovacua". These solutions exist
independently on that claim !\\
\\
Ulrich Bruchholz\footnote{This document has been later composed with \LaTeX{}
from a plain text. For it, the text has been negligibly changed}
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