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\huge Theses on the Geometric Theory of~Fields
\end{center}
\bigskip
\begin{center}
\large from\\ \Large Ulrich \textsc{Bruchholz}
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\end{titlepage}
\large
\begin{center}
{\LARGE \textsc{Galileo}'s Golden Rule\\}
\end{center}
\bigskip
\begin{center}
{\Large \emph{Take nature as is, not as you'd like that it had to be~!}}
\end{center}
\newpage
\begin{center}
{\LARGE \textsc{Newton}'s Grand Idea\\}
\end{center}
\bigskip
\textsc{Newton} defined a property of each body, independent on location
and time, from force-actions to the body. He called this property \emph{mass}.
\newpage
\begin{center}
{\LARGE \textsc{Faraday}'s Grand Idea\\}
\end{center}
\bigskip
\textsc{Faraday} saw that electromagnetism does not act across a
finite distance.
It acts always from one point to the next, as well in time as spatially.
The interval respectively distance from one point to the next is
\emph{very} small. The change from one point to the next is very
small too. We call that \emph{near action}.
\newpage
\begin{center}
{\LARGE \textsc{Gauss}' Grand Idea\\}
\end{center}
\bigskip
\textsc{Gauss} has decided the properties of a bent surface, independent
on the location. As well, he saw that the surface is completely described
by exactly one quantity at each point of the surface. It is the
\textsc{Gauss}ian curvature, and results from the \emph{product}
of minimal and maximal vertical curvature.
\newpage
\begin{center}
{\LARGE \textsc{Riemann}'s Grand Idea\\}
\end{center}
\bigskip
Bernhard \textsc{Riemann} defined a general geometry with any number of
dimensions. In which, he saw that the properties of the entire
\mbox{$n$-manifold} (with $n$ dimensions) are fixed from
\quad $ n~(n~-~1)~/~2 $ \quad
mutually orthogonal surfaces in this manifold. With it, the entire
manifold can be bent.\\
By the way, \textsc{Riemann} saw the physical meaning of this step !
\newpage
\begin{center}
{\LARGE \textsc{Minkowski}'s Grand Idea\\}
\end{center}
\bigskip
Hermann \textsc{Minkowski} saw from the law of the light propagation
(with the unchanging light-speed), that time is a geometric category
like length. One can define a fourth co\"ordinate
\quad $ x_{4}~=~\mathsf{j}~c~t $ \quad with \quad
$ \mathsf{j}^{2}~=~-1 $~, and treat it like a spatial co\"ordinate.\\
People call ~j~ the imaginary unit, but this name is definitively
false. This number is \emph{very} real~!
\newpage
\normalsize
\begin{center}
{\LARGE \textsc{Einstein}'s Grand Idea\\}
\end{center}
\smallskip
{\large
Albert \textsc{Einstein} saw the dependence of the scales and clocks
on the observer.\\}
That means first the relative velocity (special \mbox{relativity).}\\
\\
With the try to define a general relativity (under acceleration),
he found the dependence of the scales and clocks on the gravitation
via the \emph{equivalence principle}. It says: Acceleration and
gravitation act same way to a body with its mass.
\newpage
\large
\begin{center}
{\LARGE What is Force ?\\}
\end{center}
\bigskip
It is surely a kind of action. One can say:\\
% \bigskip
\begin{center}
{\LARGE Force \quad = \quad Property times Action}
\end{center}
\bigskip
What properties and actions are that ? There are two of each
$ \Longrightarrow $
\newpage
Remember \textsc{Minkowski}'s Grand idea. It unifies space and time
to the four-dimensional space-time. With \textsc{Riemann}, the
entire space-time is bent, not the space alone.\\ \\
So, each body describes a time-like curve in the space-time,
a so-called world-line. This world-line can be bent within
the space-time.
\newpage
\emph{Accelerated motion as well as gravitation act to the body
with its mass.}
{\Large
\begin{displaymath}
\mathcal{F}~=~m~\cdot~(\frac{\partial^2 \mathcal{X}}
{\partial t^2}~+~\mathcal{G})
\end{displaymath}}
That is nothing else than the first approximation of the \emph{curvature
vector} of the body's world-line, multiplied with the constant rest
mass ! The curvature vector is the curvature parameter of any curve !
It ``accompanies'' the curve.
\newpage
{\Large
\begin{displaymath}
\mathcal{F}~=~Q~\cdot~(\mathcal{E}~+~\mathbf{v}~\times~\mathcal{B})
\end{displaymath}}
Yet a force ! The charge $Q$ is another property of the body. Thus,
the electromagnetic field were another parameter of the world-line~?\\
Let us see $ \Longrightarrow $
\newpage
Each curve in an $n$-manifold is accompanied by a set of $n$ mutually
orthogonal unit vectors, an orthogonal ennuple or $n$-Bein. With it,
each world-line has its orthogonal quadruple or Vierbein. The first
vector is the tangent vector (time-like), and the second the main
normal (space-like). The direction of the main normal is that of
the curvature vector. The remaining vectors are physically
irrelevant.\footnote{This statement is supported by the fact that
the normals of geodesics are indefinite.}
\newpage
Just the world-lines can be accompanied also by congruences of
two dual surfaces~! These are mutually orthogonal, and meet at
the current point of the world-line. There is a special pair of
surfaces, which performs the electromagnetic field tensor, like
the main normal performs the curvature vector. With it, the
electromagnetic field tensor is a geometric quantity~!\\
\emph{These surfaces reflect unique geometric properties of the
four-manifold~!}
\newpage
\begin{center}
{\LARGE What are Mass and Charge ?\\}
\end{center}
\bigskip
{\Large These are integration constants of the source-free
\textsc{Einstein-Maxwell} equations. $ \Longrightarrow $\\
Spin and magnetical momentum are such integration constants too.\\}
(Also higher momenta are.)
\newpage
\normalsize
{\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 \quad \textnormal{(even~NOT}~~S^{i}~
\textnormal{!)}
\end{displaymath}
\begin{displaymath}
\textnormal{with} \quad F_{ik} = A_{i,k} - A_{k,i}
\end{displaymath}\\}
These tensor equations have been derived from \textsc{Maxwell}'s equations
and \mbox{\textsc{Einstein}'s} gravitation equations, together with
\textsc{Lorentz}' energy and momentum components. They involve mentioned
unique geometry.
\newpage
Are you surprised at the lack of sources, i.e. distributed charges
and currents~? The reason is quite simple~:
\begin{enumerate}
\item
The material quantities, i.e. mass, spin, charge, magnetical momentum,
are the first integration constants, as already mentioned.
\item
The sources are only a mathematical trick to avoid singularities.
But this trick
\begin{enumerate}
\item
is unnecessary, and
\item
makes huge problems.
\end{enumerate}
\end{enumerate}
\newpage
\emph{There are good reasons for the general validity of the source-free
\textsc{Einstein-Maxwell} equations~:}
\begin{enumerate}
\item
The math runs seamlessly. (Headword \textsc{Bianchi} identities)\\
Nobody has to invent any additional terms making additional problems.
\item
We have 10 independent equations for 14 components of metrics and
electromagnetic vector potential. However, in first approximation
(\textsc{Maxwell}'s equations respectively four components of
gravitation equations alone), that become 8~wave equations for
8~components. This mathematical fact says all about causality.
\item
Plausible interpretation of electrical conductivity and tunnel effects
as well as quantum ``coupling''.
\item
Last but not least, particles are discrete solutions of the source-free
\textsc{Einstein-Maxwell} equations. It should be known that integration
constants take on discrete values as soon as limits are present. These
limits are special geometrical structures of the space-time.
\end{enumerate}
\newpage
\large
Are you aware that algorithms based on finite differences lead to
solutions that \emph{fundamentally} differ from so-called ``exact''
solutions, even also if the differences become very small ?
It has to do with chaos.\\
Nature says, that the first way is the right. We find chaos everywhere.\\
Numerical simulations have already provided eleven particle numbers
with great confidence. Everybody can repeat it.
\newpage
\normalsize
{\large \emph{In view of overwhelming evidence, are there obstacles ?}\\}
Philosophers take the geometry as a mathematical construction.
Is it not more, even reality ?\\
That could be an interesting philosophical subject.\\
However, physicists say about the fact that the material properties
of a body are integration constants:
\begin{center}
\emph{That may not be !}
\end{center}
But it is so. Sorry.
\begin{center}
{\large \emph{What is so hard to grasp ?}}
\end{center}
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