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\date{16 December 2003}
\author{Ulrich~\textsc{Bruchholz}\footnote{Dipl.-Ing.
\textsl{Ulrich Bruchholz},~http://www.bruchholz-acoustics.de}}
\title{Numerical~Simulations~with~Strange~Results}
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\maketitle
People know two kinds of field which are not tied to a medium but
mean special states of space and time: Electromagnetism and gravitation.
Here is the great question: What is the origin of these fields~? And
people cannot file the observed quantification in any (classical) theory
of fields. As well known, quantum theories (with the standard model
as most accepted) have their way in physics for long time.
In order to find the origin of the quantification in the fields
(as impossible that seems), one must ask two innocent questions:\\
1) What quantities are conserved ?\\
2) What quantities have discrete values ?\\
The answer to both questions is:\\
Integration constants of partial differential equations meet these
conditions. At this place, the unusual way is gone to search the fitting
equations in a continuum theory.\\
\section*{1. Choice of the field equations}
Experimentally well confirmed are the Maxwell equations for electro-
magnetism and Einstein's gravitation equations [1] for gravitation. Both
use sources, namely distributed charges and masses, to explain the origin
of the fields.
The attempt to unify these systems of equations succeeds only in
going without sources. That leads to the (source-free) Einstein-Maxwell
equations [1,3,4,5]. These are tensor equations, and contain Lorentz'
electromagnetic energy and impulse components (known as energy-momentum
stress tensor). The first integration constants are just the wanted
mass, spin, charge, magnetical momentum, and further are higher momenta
of them. Both gravitation and electromagnetism have geometric nature
[2,3,5,7].
How can one see particles in the tensor equations~? In order to avoid
any speculations, the most suitable step seems to be to calculate these
equations.\\
\section*{2. What quantities can one calculate how~?}
The favoured Einstein-Maxwell equations are a system of non-linear
partial differential equations [1,3,4,6,7]. The entire system of equations
involves a special Riemannian geometry [3,5]. These are to solve in the
several components of the electromagnetic vector potential and the metric
tensor. With the Bianchi identities [2], we get 10 independent equations
for 14 to determining components.
What happens if we introduce distributed charges and currents~?
The divergences of the energy-stress tensor do not more vanish, so this
tensor cannot equal the Einstein tensor, because the Einstein tensor
holds the Bianchi identities. Physically does that mean forces and powers
which cannot be compensated. - Allegedly do additional terms to the
energy-stress tensor help, but the author saw no evidence for the
possibility of any compensation of mentioned forces and powers up to now.
In general, it is impossible to solve these equations with conventional
methods. The special case of central symmetry allows solutions this way.
Known are for example the Schwarzschild vacuum or the Weyl-Rei\ss{}ner-Nordstr\o{}m
electrovacuum.
Keep numerical simulations. 22 years ago, the author proposed such
simulations [3], and could do them himself 10 years ago. A detailed report
is to find in [6], more understandable explanations in the
4\textsuperscript{th} chapter of [7].
Calculated were the courses of the several components. As well,
the differential equations become difference equations, because one
can computate only in finite steps over finite differences. - Moreover,
solely integration constants different from zero lead to non-zero solutions .
The first integration constants for stationary solutions are mass, spin,
charge and magnetical momentum, further are dipols, quadrupols etc.
If we assume a particle on a place, the initial conditions are set in
an area outside the particle, where the selected field equations are
evident. The computation is done step by step nearer and nearer to
the particle. Those partial differential equations have as well an
evolution dependent on the integration constants similarly to the chaos.\\
\section*{3. The results}
A lot of computations with several values of the integration constants
was done. During each computation the steps were counted until the physical
components took on an amount of 1 . (That could be a kind of event
horizon). As well, the number of steps has periodic behaviour in relation
to the integration constants. Maximal numbers of steps appeared when the
values of the integration constants were identical with the values of spin,
charge, and magnetical momentum from literature [8,9] (of course normalized
for the computer). Needs the clue that the latter are empirical values~? -
Could not the mentioned evolution be the reason for all quantification~?\\
\section*{4. Consequences}
The most important consequence consists in it that we can recognize
seen particles in solutions of tensor equations. The author did it with
the free electron, proton, deuteron, and Helium nucleus. It should
presently be possible to do it with all nuclei, but one needs to test
with higher momenta, and with higher precision (more than 80 bit) too.
We get concrete evidence about the geometric appearance of particles.
That provides an alternative view of particles, and a valuable
addition to the quantum theories.\\
\\
\section*{Acknowledgement}
The author thanks\\
Prof. \emph{Ernst Schmutzer} for patient hearings in the past,\\
Prof. \emph{EL Sharony Hemet} for valuable hints and discussions,\\
\emph{Sabbir A. Rahman}, Ph.D., for kind encouragement.\\
The author owes recent discussions to the gentlemen \emph{Boris Unrau}
(www.einsteins-erben.de) who gave the valuable hint to reference [9],
and \emph{Ilja Schmelzer} (www.ilja-schmelzer.de) who named highest precision
libraries and pointed at misunderstandable statements.\\
%\newpage
\section*{Literary index}
{}[1] \textsc{Einstein}, A.: Grundz\"uge der Relativit\"atstheorie.\\
A back-translation from the Four Lectures on Theory of Relativity.\\
Berlin: Akademie-Verlag, Oxford: Pergamon Press, Braunschweig: Friedrich
Vieweg \& Sohn, 1969.\\
\\
{}[2] \textsc{Eisenhart}, L. P.: Riemannian Geometry.\\
Princeton: University Press, 1949.\\
\\
{}[3] \textsc{Bruchholz}, U.: Zur Berechnung stabiler elektromagnetischer Felder
(On Calculation of Stable Electromagnetic Fields).\\
IET \textbf{10} (1980) 481, Leipzig.\\
Possible archives: Technische Universit\"at Dresden (www.tu-dresden.de),
Deutsche B\"ucherei.\\
\\
{}[4] \textsc{Bruchholz}, U.: Berechnung elementarer Felder mit Kontrolle durch die
bekannten Teilchengr\"o\ss{}en (Calculation of Elementary Fields with Control
by Means of the Known Quantities of Particles).\\
Experimentelle Technik der Physik \textbf{32} (1984) 377, Jena.\\
Possible archives: Friedrich-Schiller-Universit\"at Jena (www.uni-jena.de),
Deutsche B\"ucherei.\\
\\
{}[5] Article in sci.physics.research from 2001/7/19.\\
From: Ulrich \textsc{Bruchholz},\\
Subject: Ricci Main Directions in an EM Vacuum,\\
Message-ID: \textless9j71vm\$lp8mr\$1@ID-28113.news.dfncis.de\textgreater,\\
archived at http://www.lns.cornell.edu/spr/~.\\
\\
{}[6] \textsc{Bruchholz}, U.: Quantities of Particles as Integration Constants from
Tensor Equations.\\
http://bruchholz.psf.net/feld\_ber.tgz (printing only)\\
http://bruchholz.psf.net/feld\_ber.zip\\
Paper archived at the German patent office, Munich.\\
\\
{}[7] \textsc{Bruchholz}, U.: Relativit\"atstheorie und Geometrische Theorie von
Gravitation und Elektromagnetismus, ganz leicht verst\"andlich\\
(Theory of Relativity and Geometric Theory of Gravitation and Electro-
magnetism, quite easily understandable).\\
http://bruchholz.psf.net/lehrbuch.tgz (printing only)\\
\\
See generally\\
http://bruchholz.psf.net/index.html~.\\
\\
{}[8] \textsc{Gerthsen}, Ch.: Physik.\\
Berlin/Heidelberg/New York: Springer-Verlag, 1966.\\
\\
{}[9] http://pdg.lbl.gov/ (actual list of the particle numbers)\\
\\
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\\
{\footnotesize
The author is available on\\
info@bruchholz-acoustics.de\\
WEB site: http://www.bruchholz-acoustics.de\\}
\newpage
\section*{Appendix: The used field equations}
{\large
\begin{displaymath}
R_{ik} = \kappa \cdot ( \frac{1}{4}~g_{ik} F_{ab} F^{ab} - F_{ia} F_k{}^a )
\end{displaymath}
\begin{displaymath}
F^{ia}{}_{;a} = 0
\end{displaymath}
\begin{displaymath}
F_{ik} = A_{i,k} - A_{k,i}
\end{displaymath}\\}
\normalsize
$\mathcal{A}$ electromagnetic vector potential,
$\mathcal{F}$ electromagnetic field tensor,
$g_{ik}$~metrics, $\mathcal{R}$~Ricci tensor,
$\kappa$ Einstein's gravitation constant,
all indices from~1~to~4, or from~0~to~3~. Solved is to $g_{ik}$~, $A_i$~.\\
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{\footnotesize This document has been later composed with \LaTeX{}
from a plain text.\\}
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