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\date{31 October 2005}
\author{Ulrich~\textsc{Bruchholz}\footnote{Dipl.-Ing.
\textsl{Ulrich Bruchholz},~http://www.bruchholz-acoustics.de}}
\title{What~is~Geometric~Theory~of~Fields~?}

\begin{document}
\sloppypar
\maketitle

The present physical thinking is dominated by the imagination of
``matter'' in space and time that determine all events around us.
As well, nobody can say what matter be, and physicists make it their
business to fathom this.

The effect to a body known for long time is a force at accelerated
motion on the one hand, and caused by gravitation on the other hand.
With the mass term, Isaac \textsc{Newton} has found a measure
depending only on the body itself. As well, it occurs as though
there be two qualitatively different masses, namely an inert
and a heavy mass.

\textsc{Newton} could clear up that each mass builds up a
gravitation field around itself, that decreases in its effect
for the factor $ 1/r^2 $ with the distance. \textsc{Newton}
has experimentally determined the proportion factor, the
gravitation constant, for it too. He took as working hypothesis
the assumption that space and time be given forever, and
independent on the matter being situated in them. That means
a distant action, because the one body works with its mass
immediately to the other body with its mass via the gravitation.
In the daily life, the dominant body is the earth with its
gravitation field. - However, \textsc{Newton} himself was
never happy with the working hypothesis and the distant action
resulting from it. The inertia keeps mysterious as well.
\textsc{Newton} could only take notice that the second
derivative of the way, done by the body, to the time
is the deciding factor.

The exploration of the electromagnetism involved a great step
of discovery. \textsc{Faraday} described the electric and
magnetic fields with force lines so called by him. These work
only from one point up to the adjacent point, from this up to
the next again, and so on as well in the space as in the time
too. This action does not jump-like happen, but there are only
appropriately small changes from point to point. One can
answer the legitimate question for the point distances, that
these are chosen \emph{any} small. That is called a continuum, and
the action is a \emph{near action}~.

An essential consequence from the near action is the prediction
of electromagnetic waves by \textsc{Maxwell}, which Heinrich
\textsc{Hertz} has experimentally detected first. The light is
under them, so that the electromagnetic waves propagate
with light speed.

Now, the search for the assumpted medium begun, in which the
electromagnetic waves are supposed to propagate. As well,
preferably the total independence of the light speed on the
observer's relative motion to the light source irritated.
But there is no dependence on an absolute motion (e.g. of
the earth at the rotation around the sun) too, as the known
\textsc{Michelson} experiment demonstrated. With it, the
existence of such medium, also known as ether, was questioned
at all.

These \emph{seeming} discrepancies made Albert \textsc{Einstein}
deal with relative motions. - In mechanics are so called inertial
systems moving in straight lines and unacceleratedly. The behaviour
of bodies is the same on all inertial systems. That does not
work in electrodynamics. Moved charge induces a magnetic field,
and a moved magnet an electric field to a resting system. 
\textsc{Lorentz} designed transformation formulae for the
inertial systems, which are supposed to consider the constant
light speed. As well, time and length in the moved system are
changed for the resting observer. The moved body appears shorter
in direction of the motion, and the clocks go slower on the body.

\textsc{Einstein} saw that the \textsc{Lorentz} transformations
return the \emph{real} change of the scales and clocks at
relative motion. With it, everybody has his own time. Time and
space are nothing absolute but connected with each other via
\textsc{Lorentz} transformations. With this interpretation,
the principle of relativity is valid not only in mechanics
but also in electrodynamics. The induction laws become a part
of transformation relations of the electric and magnetic fields,
which are unified this way.

The \textsc{Lorentz} transformations appear pretty arbitrary
with the postulate of the constant light speed alone. We owe
Hermann \textsc{Minkowski} the deciding comprehension jump
with his geometric interpretation of the time. \textsc{Minkowski}
took the time as fourth co\"ordinate (together with the three
spatial) by setting $ x_4 = \mathsf{j} c t $ with
$ \mathsf{j}^2 = -1 $~. Therefore, the time is imaginary length,
or a length is imaginary time. Since we live in the time,
a line element in the space-time performed this way becomes
time-like, i.e. with the real time
$ \mathsf{d} s^2~=~(c \mathsf{d} t)^2~-~\mathsf{d} x^2~-~\mathsf{d} y^2
~-~\mathsf{d} z^2~>~0 $~. However, for the light is
$ \mathsf{d} s~=~0 $~.\\
The right of this geometric interpretation of the time is given by
the fact that, with the \textsc{Lorentz} transformations, the time
itself becomes a co\"ordinate. The electromagnetic wave equations
are a visible certification of it. In the space-time, the
\textsc{Lorentz} transformations themselves become a simple rotation
of the time co\"ordinate and the x co\"ordinate (at motion in x
direction) for an \emph{imaginary} angle $ \psi $. The relative
velocity is the tangent of this angle
$ v~=~\mathsf{j} c \mathsf{tan} \psi $ then. Mathematics results
from it in the addition theorem of the velocities inclusive the fact
that a body can never reach the light speed.\\
{\footnotesize
\textsc{Minkowski} has tried to trace these facts on the paper,
i.e. the time like a real length. For this, it is referred to the
relevant literature under the headword ``\textsc{Minkowski} cone''.
This analogy has its limits, however, the so-called twin paradoxon
and similar things can be well understood with it. See also [1].\\}
\normalsize \\

What are fields~? - \textsc{Einstein} could answer this question
for the gravitation via the equivalence principle:

\textsc{Einstein} has identified the \emph{special relativity},
i.e. the relativity of motion of inertial systems (that is valid
also for electromagnetism), via the \textsc{Lorentz} transformations.
Consequently, \textsc{Einstein} has been led to the search for
a \emph{general} relativity of motion. That means the same as
the question for the relativity of accelerated reference systems.

We notice in accelerated reference systems that a force is effective
against a massive body. The same force action is in the gravitation
field too. If the observer does not \emph{know}, where the force comes
from, he cannot notice it. For that reason, the question for the
general relativity leads to a new question for the origin of the
two force actions. \textsc{Einstein} raised the equivalence of
inert mass and of heavy mass to a principle, the \emph{equivalence
principle}.

Now, the geometry can help again. -\\
Each body describes a curve in \textsc{Minkowski}'s four-dimensional
space-time. For the unmoved body, this curve is identical with the
time axis, at unaccelerated motion a straight line inclined to the
time axis, as explained in the special relativity. An acceleration
leads to a bent curve. The most important parameter of a curve
is its curvature vector. (It is the total derivative of the
tangent vector.) A comparison of the physical parameters
with the curvature vector results in remarkable:
\textsc{Newton}'s force equation $ \mathcal{F}~=~m~(\frac{\partial^2 \mathcal{X}}
{\partial t^2}~+~\mathcal{G}) $ consists of the two equivalent parts.
The second derivative of the way to the time is the accelerated
motion, during $ \mathcal{G} $ summarizingly expresses the field strength
of the gravitation. The curvature vector contains the second
derivative of the local vector to the distance on the curve,
which means the local time of the accelerated body and, with it,
is identical with the observer's time in first approximation.
It follows cogently from the physical fact that the curvature
vector \emph{must} have a second part. That is the case exactly
then, if the space-time itself is curved~! Any curvatures of
the space-time go into the parameters of the curve.\\
The curve in the space-time for \emph{force-free} motion is a
geodesic, because the curvature vector vanishes for it. The
equation of the geodesic $ \mathcal{K} = 0 $ is physically
an equation of motion with it.

One may understand the curvature of the four-dimensional
space-time (not the three-dimensional space alone~!) analogously
to the curvature of a surface, as known from the daily life.
This generalization of the geometry goes back upon Bernhard
\textsc{Riemann}.\\
The line element on a curved surface results with any co\"ordinates
on the surface from $ \mathsf{d} s^2~=~
g_{11} (\mathsf{d} x_1)^2~+~2g_{12} \mathsf{d} x_1 \mathsf{d} x_2
~+~g_{22} (\mathsf{d} x_2)^2 $~. The general relation for many
dimensions is then $ \mathsf{d} s^2~=~\sum_{\mu,\nu} g_{\mu\nu}
\mathsf{d} x_{\mu} \mathsf{d} x_{\nu} $ with $ g_{\nu\mu} = g_{\mu\nu} $~.

The co\"efficients $ g_{\mu\nu} $ perform a symmetric tensor, which
completely returns metrics of the manifold. With metrics, each
distance in the manifold can be determined. However, metrics essentially
depends on the chosen co\"ordinates, and, with it, is no measure
for the curvature of the surface respectively the space-time.
(However, the curvature goes into metrics~!)\\
\textsc{Gauss} found out that the properties of a surface at each
point of the surface are described with a single quantity~!
This \textsc{Gauss}ian curvature is the product from maximum
and minimum \emph{vertical} curvature.\\
{\footnotesize
The further above described curvatures are horizontal curvatures, i.e.
\emph{in} the surface resp. space-time. A geodesic can be vertically
curved. So one can understand, why a sheet of paper can be arbitrarily
rolled. The \textsc{Gauss}ian curvature of the paper keeps always
zero~!\\}
\normalsize
\textsc{Riemann} had the brilliant realization that the properties
of an \textit{n}-dimensional manifold can be expressed from the
\textsc{Gauss}ian curvatures of $ n (n - 1)/2 $ mutually orthogonal
surfaces being situated in the manifold. That are 6 surfaces in
the space-time, namely the 3 known spatial surfaces, and the 3
surfaces being stretched by the time and one spatial co\"ordinate
each.\\
{\footnotesize
Above mentioned \textsc{Gauss}ian curvatures are now replaced by
parallel shift of vectors, that makes possible the use of the
tensor calculus. Tensors are invariant quantities in their entity.
The tensor components follow generally valid transformation laws.\\
This mathematics has been founded by a school of mathematicians
under \textsc{Ricci}, \textsc{Levi-Civit\'a}, \& al. at \textsc{Einstein}'s
times.  \textsc{Einstein} was the first user.\\}
\normalsize
These curvature measures are \emph{not} identical with the gravitation~!
However, there is the close context of these curvatures \emph{and,
with it,} metrics with the curve parameters.

From the context of gravitation with curvature of the space-time
follows that general relativity can be defined only \emph{locally},
i.e. in immediate surroundings of a point in the space-time.
We can define \textsc{Euklid}ian conditions there. But since
metrics at gravitation is different from metrics without
gravitation, the local scales and clocks behave differently
from scales and clocks out of the gravitation field.
The clocks tick slower for the outer observer, and the scales
become longer. That means for the outer observer a smaller
light speed in the gravitation field, what led to \textsc{Einstein}'s
famous prediction of the bending of rays of light in gravitation
fields. However, the \emph{local} light speed is constantly $ c $~!

It should be mentioned, that the gravitation field itself contains
no energy. However, observers on distant positions can notice
different energy states of the same system due to differences in
metrics. With it, gravitation does not transfer energy but
arranges this.\\

With the general theory of relativity, we have the paradoxical
situation that the gravitation is derived from the geometry
of the space-time, during the masses are seen as ``generating''
gravitation, and the structure of them keeps unsolved. This
paradoxon can be only seen in the context with another till then
unsolved questions:\\
~- What is electromagnetism~?\\
~- What about the quantization~?\\
The quantization of physical quantities is a fact of experience,
which is manifested also from it, that statistical methods
have been successfully used. We have to suppose that these three
unsolved questions are to solve only together.\\

One may hope for reasonable answers to questions to the nature,
only if these questions are unbiased. That means, no answers may
be expected or even given. In the context with the three unsolved
questions, following concrete questions appear useful:\\
~1) What quantities are conserved~?\\
~2) What quantities have discrete values~?\\
Conserved are the ``material'' quantities mass, spin, electrical
charge, and magnetical momentum. For the equivalence of mass and
energy, following from special relativity, the conservation of
energy follows from the conservation of mass. Mathematics gives
the surprising answer, when these quantities take on discrete
values: \emph{as integration constants of source-free partial
differential equations~!}\\
{\footnotesize
That is the case for example with a present boundary, in which
one may not expect the classical margin of the potential theory.
The concrete circumstances can be found out by means of e.g.
numerical simulation.\\}
\normalsize
Mass and spin are the first integration constants of \textsc{Einstein}'s
gravitation equations, and charge and magnetical momentum are those
in \textsc{Maxwell}'s equations. Distributed masses and momenta
as well as distributed charges and currents do not exist~!\\
{\footnotesize
The non-existence of distributed charges and currents becomes
clear also from the mesh and knot laws known in electrotechnique,
when the meshes and knots become very small. That is no contradiction
to the fact, that one can measure a current in a mesh and a voltage
between two knots.\\}
\normalsize
On these conditions, \textsc{Einstein}'s and \textsc{Maxwell}'s
equations can be unified via the energy tensor of the electromagnetic
field. The resulting source-free \textsc{Einstein}-\textsc{Maxwell}
equations\footnote{quoted in [1]} obtain purely geometric meaning
then~!\\
{\footnotesize
The sources are an equivalent representation of accumulated
singularities from the integration constants. However, the
energy law is violated with the sources. - Numerical simulations
do not lead to singularities, if one accepts the existence
of \emph{geometric boundaries}.\\}

\normalsize
The possibility, describing the material quantities as integration
constants, is known, however, it is not accepted up to now. Why~?
It contradicts the imagination established by Ernst \textsc{Mach},
that ``matter'' be suspended in the space and (secondarily)
generate the fields and, with it, determine the structure of the
universe. Actually, this kind of matter is not needed. Matter
is manifested in the integration constants. All physical (energy
law~!) \emph{and} mathematical difficulties are cancelled for
the price of the \emph{traditional} matter.\\
The particles are discrete (elementary) solutions of the source-free
\textsc{Einstein}-\textsc{Maxwell} equations then. There are already
significant indications from numerical simulations~!~[1]~-\\
{\footnotesize
Within tolerances of $ \pm 5 $\%~, the equivalent integration
constants lead to most stable solutions, when the values of them
are identical with the measured values of spin, charge, magnetic
momentum, and geometric boundaries appear at the presumed
particle radius. These four values are mutually conditional~!\\}
\normalsize
Though solutions exist only from discrete values of the integration
constants, such solution is ambiguous in the near field. The
near field has an extension about $ 10^{-15} $m for nuclei,
for more complex solutions (atoms, molecules, \&c.)
essentially more.

With the question, how photons result from the
\textsc{Einstein}-\textsc{Maxwell} equations, it be referred to~[1].
There are to find also detailed reports and results from numerical
simulations.
\\

We have noticed that in the geometric analogy the gravitation
together with the accelerated motion is a curve parameter,
namely the curvature vector of the curve in the space-time.
The very similar properties of the electromagnetism, mainly
the propagation in the vacuum, force to the conclusion that
electromagnetism means curve parameters too. That must so be,
because each measuring tool describes such curve in the space-time.
The entire set of these curves is sufficient to describe the
curvature circumstances in the space-time.

During gravitation and accelerated motion perform a vector
accompanying the curve, electromagnetism can be described from
two accompanying surfaces, which are stretched by two vectors each.~-
The curvature vector is space-like and directly to feel. The term
``below'' means nothing else than the \emph{direction} of the
curvature vector from the gravitation field of the earth.
Unfortunately, the surfaces from electromagnetic fields are to
feel not as directly. At this place, it can be said only, that it
concerns two dual surfaces with quite special curvature properties
(more see~[1]). The source-free \textsc{Einstein}-\textsc{Maxwell}
equations involve a special geometry, that is possible exclusively
in the four-dimensional space-time. It \emph{is the} geometry of
the space-time, and this is unique~! The space-time does not
allow another geometry~!\\
{\footnotesize
The accompanying surfaces manifest the difference between
electric and magnetic field, because the one surface is
time-like in one dimension. There is no symmetry of electricity
and magnetism for it~!\\}
\normalsize
The unique geometry of the space-time expresses itself also
with the incomplete causality. The source-free
\textsc{Einstein}-\textsc{Maxwell} equations perform only 10
independent equations for 14 variables, what means very many
degrees of freedom. In the physical reality, however, the special
role of the time becomes essential. Since we live in the time,
the world is causal in first approximation. (One can give
mathematical reasons for this.) That is not more true in the
micro range~!
\\

Metrics is not the field itself, however, one can easily understand
that metrics is influenced by curvatures of the space-time. Quite
practical calculations result in following tendencies:\\
Gravitation expresses itself \emph{for the distant observer} in it,
that the clocks tick slower and the scales become longer. However,
electromagnetism goes quadratically into metrics. The clocks tick
faster and the scales become shorter, at the electric field in
direction of the field strength and at the magnetic field vertically
to the field strength.

The shortening of the scales at the electric field has quite
practical consequences: With the shorter distance, the field
strength becomes greater, with it the distance shorter and
shorter and so on. That means a feedback, that leads to a
zero-distance between two points with finite distance in the
co\"ordinate system. The unbiased reader may wonder, how
lightning and tunnel effects with super light speeds come about.
\\

Numerical simulations regularly lead to a geometric boundary,
where is no time. Within this boundary is nothing, neither space
nor time~! For this reason, it is useless to speculate what may
be within a particle.\\
\\
\\
{\footnotesize \emph{The author owes interesting suggestions
for this work to a discussion circle with
the gentlemen Prof. Manfred Geilhaupt, Dr. Gerhard Herres and
Werner Mikus . Here are
especially to emphasize valuable hints by Werner Mikus, who
saw to it that the article is intelligible.}\\}
%\newpage
\normalsize
\textsl{Reference~:}{}\\
\\
{}[1]~\textsc{Bruchholz}, U.~: http://bruchholz.psf.net/~.\\
\\
More references in ./article2.txt\\
On the special role of the geometry (intelligible)
see \mbox{./Geometry.pdf}\\
The geometric theory as textbook (only in German)
see \mbox{./Textbook.pdf}\\
On numerical simulations ./feldber-.htm or \mbox{./feldber.zip}\\
See also ./selfdoing.html\\
Derivation of photons in ./h-article.pdf\\
Derivation of the geometry of the electromagnetism in the textbook
and in \mbox{./Ricci\_Main\_Dir.txt}\\
\\
\\
\\
\\
{\footnotesize This document has been composed with \LaTeX{}.\\}
\end{document}