AIAA 2002-4094 PHYSICAL PRINCIPLES OF ADVANCED SPACE PROPULSION BASED ONHEIMS'S FIELD THEORY
20/05/2014 20:44
AIAA 2002-4094
PHYSICAL PRINCIPLES OF
ADVANCED SPACE PROPULSION
BASED ONHEIMS'S FIELD THEORY
Walter Dröscher1 , Jochem Häuser2
1Institut für Grenzgebiete derWissenschaft,
Leopold - Franzens Universität Innsbruck, Innsbruck, Austria
2Department of Transportation, University of Applied Sciences
and
Department of High Performance Computing, CLE GmbH, Salzgitter, Germany
THE SUGGESTION THAT SPACESHIPS MAY EVENTUALLY TRAVEL FASTER THAN LIGHT
WILL CAUSE MOST ASTRONOMERS TO THROW UP THEIR HANDS IN HORROR, AND FOR EXCELLENT
MATHEMATICAL AND PHYSICAL REASONS OUTLINED IN EINSTEIN'S THEORY OF
RELATIVITY. BUT SCIENCE IS STILL IN ITS EXTREME INFANCY. IT IS A BIT FATUOUS TO
THINK THAT WE HAVE APPROACHED ANY ULTIMATE REALITIES, OR ARE VERY LIKELY TO
UNTIL SCIENCE IS SEVERAL THOUSAND YEARS OLDER THAN IT IS NOW.
FROM SPACE FLIGHT BY C. C. ADAMS,MCGRAWHILL, 1956, P. 236.
38TH AIAA/ASME/SAE/ASEE
JOINT PROPULSIONCONFERENCE&EXHIBIT,
INDIANAPOLIS, INDIANA, 7-10 JULY, 2002
1Senior scientist, 2 Senior member AIAA, member SSE, www.cle.de/cfd, @IGW Innsbruck Univ, Austria
2002 Institut für Grenzgebiete der Wissenschaft,
Leopold - Franzens Universität Innsbruck,
Innsbruck, Austria
ABSTRACT
In this paper an overview is given of the results of a
completely geometrized unified field theory that gives
rise to a novel concept for an advanced space transportation
technology, permitting, in principle, superluminal
travel. This theory predicts the existence of a quasi antigravitational
force, and allows the design of an experiment
for the verification of this theory. This theory of
quantum gravity, based on publications by B. Heim et
al. [1-6], introduces new physics at the quantum scale,
predicting that a transformation of electromagnetic wave
energy at specific frequencies into gravitational like energy,
is possible. This transformation thus is reducing
the inertial mass of a material (ponderable) body. The
theory was recently extended to 8 dimensions by the
first author [4]. The predicted reduction of inertia (mass)
can be used as the design principle for an innovative
space transportation system. In this paper, Heim’s field
equations along with the extensions mentioned above,
will be presented and discussed. In addition, the magnitude
of the coupling constant between the conversion of
electromagnetic energy and gravitational like energy (so
called gravito-photons) will be given.
The authors have investigated the fundamental physical
assumptions as well as some of the predictions of
Heim’s theory, and carefully checked its logical consistency
[7]. Although several attempts of a formulation for
a unified quantum field theory (including gravity) have
been made, their success has been very limited. Therefore,
because Heim’s theory satisfies several modern
theoretical requirements, unknown at the time of its development,
there is evidence for the physical correctness
of this work. In fact, cosmological data strongly
suggest the possible existence of an anti-gravitational
force (i.e. repulsive), and Heim’s concepts, developed
decades ago, may have striking solutions to this newly
found experimental evidence. We believe that further investigation
in this theory is justified in view of its compliance
with recently established criteria for a general
unified quantum field theory. Furthermore, the theory
seems to be logically and physically consistent, and, if
found to be correct, would offer an extremely high payoff.
The particular coupling between gravitation and
electromagnetism cannot be obtained from a manipulation
of Einstein's equations of general relativity, e.g., by
linearizing the equations of general relativity, or by extending
Newton's gravitation law to a time dependent
formulation, assuming that the gravitational equations
are of the same form as the Maxwell equations [8]. The
coupling obtained from Heim’s theory is derived from
fundamental principles, and is very different from the
ones obtained by other ad-hoc approaches. Heim's theory
is therefore much more interesting, since it may allow
gravity manipulation at lower energy densities, and
is based on new physics, thereby leading to new predictions.
There may be several new and surprising physical
phenomena with far reaching consequences that are predicted
by Heim's theory. Some of these can be checked
against presently available experimental data, both from
cosmology and quantum physics. The physical principle
is presented of how to construct a space propulsion device
that does not use any propellant, instead is based on
an energy transformation process. In a Gedanken experiment
an order of magnitude estimate of the necessary
energies will be given.
This interaction is rooted in the unification of quantum
theory, gravitation and electromagnetism in an 8-dimensional
discrete and spin-oriented space. In Einstein's 4-
dimensional spacetime continuum that only contains
gravitation, this effect cannot take place. Since the interaction
between gravitation and electromagnetism reduces
the inertial mass of a material object, it is called
inertial transformation. Since conservation laws for momentum
and energy are strictly adhered to, the theory requires
superluminal velocities, without contradicting
Einstein's theory of relativity. Heim's physical theory,
provided it reflects physical reality, has the potential to
lead to a completely new concept of space transportation.
Nomenclature and physical constants
c speed of light in vacuum 299,742,458 m s-1
(c2 = ε0 μ0. )
D diameter of the physical universe.
G gravitational constant 6.67259 × 10-11 m3 kg-1 s-2.
h Planck constant 6.6260755 × 10-34 J s.
i, j, k indices in 4 , ranging from 1 to 4.
lp Planck length (ħG⁄ c3)1⁄2=1.61605×1035m .
gab components of the Hermitian metric tensor in 6,
or 8.
mp proton mass, 1.672623 × 10-27 kg.
R_, R+ smallest and largest radii between which gravitational
law, Eq. (24) is valid.
3 3-dimensional physical space (3 real coordinates).
4 4-dimensional physical space-time (1 imaginary coordinate).
6 6-dimensional physical space or 6D-Heim space,
4 ⊂ 6, (discrete space, a 6D volume element is
bounded by oriented metrons, in general an n - dimensional
cube has (n
k)2(nk ) k - dimensional
surfaces) (3 imaginary coordinates).
8 8-dimensional space or 8D-Heim space (5 imaginary
coordinates).
Tαβ components of the Hermitian energy-stress-momentum
tensor in 6 or8.
x1,...,x6 Cartesian coordinates in 6D-Heim space.
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x1, x2, x3 spatial coordinates in physical space 3.
x4 time coordinate (imaginary).
x5, x6 entelechial and aeonic coordinates (imaginary)
in Heim space 6.
x7, x8 information coordinates (imaginary) in Heim
space 8 .
α, β indices in 6, or 8 ranging from 1 to 6 or 1 to 8,
respectively. In a Heim space, Greek indices (if
possible) are used for tensor components.
ε0 permittivity constant 8.854187817 × 10-12 F m-1.
μ0 permeability constant 12.566370614 × 10-7 N A-2.
ξ1,...,ξ8 curvilinear coordinates in Heim space.
λp Compton wave length of the proton mass,
(λp=ħ⁄mp c) , 2.10308937 10-16 m.
i
km normalizable tensor fields for the microcosm that
correspond to the Christoffel symbols in the macro
realm.
ρ radius at which gravitational force is 0, see Eq. (24).
τ Metron area (minimal surface 3Gh/8c3)
6.15 × 10-70m2.
Source for physical constants: Cohen, E.R., Taylor
B.N., The fundamental Physical Constants, Physics Today,
August 1996, pp. BG9-BG13. Measurement uncertainties
of constants not listed.
1. The Need for Advanced Space
Propulsion
Space transportation, as we know it today, is
based on the century-old rocket equation formulating
momentum conservation in the
framework of classical mechanics. Although
there still is room for improvement in the current
designs of space transportation systems,
there are strict engineering limits associated
with the use of the rocket equation. With this
kind of propulsion system interplanetary missions
are at best cumbersome, interstellar
flights are impossible.
NASA2 has the difficult task of evaluating and
selecting candidate technologies for novel
2 There is no activity at ESA or anywhere in Europe
concerning a breakthrough physics space propulsion
program.
revolutionary space transportation systems,
based on breakthrough physics. Very often, the
potentially most valuable technologies are
based on the most innovative theoretical concepts.
The return on investment is more difficult
to estimate, since it is basically the product
of two quantities, one being very large (the
usefulness for propulsion, e.g. anti-gravity, unlimited
energy or superluminal travel) and the
other being small (the likelihood of success).
The product of infinity and zero being undetermined,
the problem of selecting which concepts
are worthy of further investigation is
duly appreciated.
If we wish to have a revolutionary space transportation
system, incremental improvements in
the technology are not sufficient. We need to
use different physical laws that are not subject
to these barriers. Hence, the quest for breakthrough
physics propulsion. If there were a
unified quantum field theory, it would be
straightforward to examine this theory for
novel physical principles of advanced space
flight. At present, such a theory is not available.
Current, incomplete unified field theories
allow for the remote physical possibility of
transforming a so called wormhole into a time
machine as described in [12-14]. However, deriving
a proper space flight technology from
these principles does not seem to be feasible in
the near future.
The alternative is to check for observed unusual
physical effects that might give a hint to
hitherto unknown physical laws, or to search
for novel physical theories not widely known,
but offering physical principles for advanced
space flight. At present, such a theory cannot
be found in mainstream physics. According to
W. Pauli a theory must be crazy enough, but
on the other hand must be logical. The authors,
having studied Heim's theory for several years,
feel that his theory satisfies Pauli's as well as
Dirac's requirements for a unified field theory
[7]. They also believe to have identified
novel physical concepts that might have the
potential to evolve into an advanced space
flight technology. Heim's theory makes several
predictions regarding cosmology, and also provides
a formula for calculating the mass spectrum
of elementary particles along with their
lifetimes. In the section entitled Speculative
4 of 21
Cosmology, some of these predictions will be
presented to provide data for the experimental
validation of the theory. In the space propulsion
approach presented below, the vacuum
speed of light is not the limiting velocity, although
conservation principles are strictly adhered
to.
2. Introduction to Heim's Field
Theory
Heim's theory is a generalization and an extension
of the GRT (General Theory of Relativity)
to the microcosm in that it computes the physical
properties such as lifetime, mass etc. of
each microparticle and also unifies all physical
forces. To this end, it geometrizes all physical
interactions, but does account for, however,
the principle of quantization of the physical
quantity called action (energy × time). Heim
extends the nonlinear equations of GRT to the
quantum world. This is important, since the
equations of GRT must be obtained in the
transfer from to the micro- to the macrocosm.
Hence his theory extends the principle of geometrization
of physics to all physical interactions.
In that respect, his approach is similar to
Gauge theory that extends the classical theory
of electromagnetism to a unified theory of the
weak and the electromagnetic forces. However,
in formulating his theory to unify quantum
theory, gravitation, and electromagnetism,
Heim uses a higher dimensional space that is
quantized in a particular way (see below).
There is, however, an important, radically different
view from Einstein's GRT. In GRT the
physical picture of matter curves spacetime is
used, and the coefficients of the metric tensor
form a tensor potential for gravitation. In this
regard, matter and spacetime curvature are
equal. In Heim's view, the physical picture is
totally different. The energy-stress-momentum
tensor contains both the source (particle) and
its gravitational field. This means, the components
of the metric tensor cannot be interpreted
as gravitational potentials, since they are already
contained in the energy-stress-momentum
tensor. Therefore, there is an equivalence
between the metric and matter (including all
fields). In other words, matter is caused by the
metric and does not exist independently. In this
respect, it would be correct to say that matter,
as we are used to conceive it, is an illusion. All
interactions or fields, namely gravitation, electromagnetics,
weak and strong forces are a distortion
of their proper Euclidean metrics in a
higher-dimensional space. This idea was first
presented by Heim in 1952 at the International
Congress on Aeronautics in Stuttgart, Germany,
and later on published in a series of
three articles in 1959, in an obscure German
journal on spaceflight. Heim's view is similar
to Wheeler's geometrodynamics [16]. However,
it seems that the first one who published
this idea was Rainich in 1925 [17]. Heim
claims to have developed a truly universal unified
field theory along the lines suggested by
Einstein, but including quantum gravity and all
other physical interactions. The difficulties in
deriving a theory of quantum gravitation are
described by Isham in [10]. Isham suggests a
program (only the outline is presented without
any mathematical derivations) how these difficulties
may be overcome, at least in principle.
It is very interesting to learn that in Heim's theory,
developed much earlier than 1991 (when
Isham's article was published), the most important
requirements, as identified by Isham, are
already included. A state of the art discussion
on multi-dimensional theories can be found in
[22].
In the following the physical concepts that are
underlying Heim's field theory are presented,
along with the major cosmological predictions
that follow from this geometrized theory. It
should be noted that the essential parts of the
theory were developed in the fifties and sixties3,
some 50 years ago.
The basic idea of Heim's theory is the representation
of a quantum of matter, Mq, (elementary
particles or resonances) as a geometrical
entity. Space itself is assigned significant
physical features. Space itself is quantized and
is 6 or 8 dimensional, depending on the number
of physical interactions to be considered. A
6-dimensional space is needed for a unification
of gravitation and electromagnetic theory,
while 8 dimensions are needed to represent all
known interactions. Thus the space 6 needs
to be extended to a space for 8 a unified theory
that describes all known 4 interactions, but
3 Heim's work was supported for a about a decade by
the CEO of MBB. MBB later became known as
DASA, and is now called Astrium and part of
EADS.
5 of 21
also gives rise to 2 additional interactions.
However, there are dimensional laws that exclude,
for instance, spaces with 7 or 9 dimensions.
Furthermore, there are only 3 real coordinates
(the usual spatial coordinates) that are
equivalent and thus interchangeable, all other
coordinates are imaginary and are not interchangeable.
This is important for the building
of the poly-metric, see below, from the various
subspaces. A more detailed discussion will be
given in Chapter 3.
In contrast to current string theory, Heim is using
so called Metrons, quantized minimal surfaces
with orientation (spin) whose size has
varied in time. From the beginning of the universe
up to today, the Metron size has decreased,
and is now approximately the size of
the Planck length squared, i.e., its physical dimension
is m2. At the same time, the number
of Metrons has increased. Thus, the beginning
of the universe is identified with the event
when there was only one Metron, whose surface
covered the whole universe. According to
this quantum picture, the universe started at a
finite size without developing a singularity,
naturally avoiding the problem of infinite selfenergies.
In other words, there are no spacetime
points, a concept actually in conflict with
Heisenberg's uncertainty principle.
According to Heim, the whole universe comprises
a grid of Metrons or metronic lattice.
Space that does not contain any information
consists of a discrete uniform Euclidean grid,
bounded by Metrons (e.g., a 6 dimensional
volume element is bounded by 240 oriented
Metrons). However, empty space must be isotropic
with regard to spin orientation. If all
metronic spins of a 6D volume pointed outward
or inward, such a world would not have a
spin potentiality. Therefore, cells with all spins
outward have to have neighboring cells with
all spins inward and vice versa. This alternating
spin structure satisfies the isotropy requirement,
but provides empty space with spin potentiality.
If two neighboring volumes are interchanged,
a spin structure has been realized.
In other words, empty space is both isotropic
with regard to its metric as well as its spin
structures, but is is capable to develop discrete
structures. Thus, empty space is void of physical
events, but has inherent potentiality for
physical events to happen. In order for an
event to happen, a distortion of the Euclidean
metric is necessary. In this sense, the theory is
a geometric dynamic theory, a term coined by
A. Wheeler. The whole theory as it is formulated
by Heim, is based on geometrical language.
It should be mentioned for matter to be
existing, as we are used to conceive it, a distortion
from Euclidean metric or condensation, a
term used by Heim, is a necessary but not a
sufficient condition.
Before we enter into some of the mathematical
formulations, the main physical features of the
theory are summarized below:
1. For a quantized unification of gravitation
and electromagnetism a 6 dimensional
space 6 is needed. If all known interactions
are to be incorporated space becomes
8-dimensional. In 6 the two transcoordinates
x5 and x6 are imaginary coordinates.
Only the 3 spatial coordinates can be real.
Any higher-dimensional space with real coordinates
will not permit stable elliptical
planetary orbits. Spacetime 4 is a subset of
6 Transcoordinates (in this context it does
not matter that we use the Euclidean coordinates
instead of the curvilinear coordinates)
x5=0 or x6=0 denote virtual (latent)
events and are outside manifest events
( x5≠0 and x6≠0 ) in 4 dimensional
spacetime 4. The transcoordinate x5 is denoted
as entelechial coordinate, and x6 is
called the aeonic coordinate. The semantic
meaning of these coordinates stems from
the Greek word entelechy, governing the actualization
of a form-giving cause, and
aeon, denoting an indefinitely long period
of time. The entelechial dimension can be
interpreted as a measure of the quality of
time varying organizational structures (inverse
to entropy, e.g., plant growth) while
the aeonic dimension is steering these structures
toward a dynamically stable state. Any
coordinates outside spacetime can be considered
as steering coordinates. The 2 additional
coordinates in 8 are denoted as information
coordinates.
2. Spacetime itself is quantized. The current
area of a Metron, τ, is 3Gh/8c3 where G is
the gravitational constant, h denotes the
Planck constant, and c is the speed of light
6 of 21
in vacuum. The Metron size is a derived
quantity and is not postulated.
3. Novel Cosmology and poly-metrics. In
6 the metric tensor, gαβ (α, β= 1,...,6) is
Hermitian (symmetric in complex space),
and is the union of a set of non-Hermitian
metric tensors of subspaces 3 formed by
spatial coordinates (x1, x2, x3), space T1 generated
by the time coordinate (x4), and the
structural space S2, built from the two transcoordinates
(x5, x6). Formally, 6 is the union
3 ∪ T1 ∪ S2. This is an important
point, since the various metrics resulting
from a combination of these subspaces are
the generators of the physical interactions
(or forces). In 8 there are 2 additional
imaginary coordinates. The respective subspace
is denoted as I2. Hence, 8 is the union
3 ∪ T1 ∪ S2 ∪ Ι2.. Therefore, there are
4 different coordinate groups in 8. It will
be shown below, that the metric tensor for
the 8 space can be written as a composition
of subtensors that are functions of the coordinates
from these subspaces. Associated
with each metric subtensor is a physical interaction,
and thus a correspondence principle
between the metric and the actual physics
is established.
In this context, space and time are not the
container for things, but are, due to their dynamic
(cyclic) nature, the things themselves.
This is an entirely different physical
picture from the approach of simply adding
the stress-energy-momentum tensor of the
electromagnetic field to the right-hand side
of Einstein's field equations, for instance,
see [15] in order to obtain a geometrization
of gravity and electromagnetism. Other attempts
by H. Weyl (modifying the Christoffel
symbols by adding a 4-vector to be interpreted
as the electromagnetic 4-vector potential),
Kaluza and Klein (5 dimensional
space), or Einstein-Schrödinger (unsymmetric
metric tensor) have not been successful
either. Instead of regarding the field equations,
as established by Heim, as a set of
differential equations for the gαβ, they are to
be regarded as a set of equations for the
components of the stress-energy-momentum
equations, Tαβ, see also Wheeler [16].
However, Heim eventually derives a set of
eigenvalue equations.
4. Geometrization of elementary particles and
the physical interpretation of geometric
structure. Einstein's field equations are extended
to the micro area. The energy-stressmomentum
tensor is proportional to geometrical
entities termed tensor fields,, that
l
km are normalizable and correspond to the
Christoffel symbols in the macro realm. Eigenvalue
equations of purely geometrical
character are set up, using the quantization
principle.
5. The only empirical constants (non-derived
quantities) are G, h, ε0 , μ0 . All other
constants are derived quantities. This includes
the coupling constants, too.
6. Interpretation of elementary particles as
geometrical entities that possess an internal
dynamic structure which is changing cyclically
in time. Elementary particles do possess
an internal spatial structure (zones), but
are elementary in a sense that they are not
composed of subparticles. Elementary particles
are not point entities, but do consist of
Metrons.
7. Derivation of strictly enforced symmetry
laws for elementary particles. The mass
spectrum and the life times of elementary
particles are computed.
As an empirical and logical basis for Heim's
field theory the following assumptions are
made:
1. There exist general conservation principles,
for example, for mass, momentum, energy,
or electrical charge.
2. There are extremum principles, for instance,
the entropy law in the macroscopic world
that can be described by variational theorems.
3. The principle of quantization of action, i.e.,
there is a smallest unit of action, h, and all
all other actions are multiples of h. Thus
matter is not continuous, but is quantized
(i.e., discrete). The quantization of charge,
light, and energy is a consequence of this
quantization principle.
4. As a logical basis it is assumed that both in
the macroscopic and the microscopic realms
7 of 21
material structures interact through action
fields : in the micro- and macroscopic area
there exist electromagnetic and gravitational
fields, while on nuclear distances
short range fields are present.
3. The Field Equations According
to Heim
Einstein's 1915 theory explained gravity as
nothing more than a property of spacetime,
namely its intrinsic curvature. The distribution
of energy (i.e., the 4-vector of energy and momentum)
causes spacetime curvature. This
geometrical style of physics, is extended by
Heim to all physical interactions but, as was
stated before, with a radically different physical
interpretation, namely that spacetime curvature
is equivalent to the combined (source
and field) energy-stress-momentum tensor. In
other words, there are geometrical explanations
for electromagnetism, as well as for the
weak and strong forces. As already mentioned,
however, the approach is different in that the
metric gives rise to an energy-momentum distribution
in a higher dimensional, so called
Heim space. A purely discrete and uniform
Cartesian grid will not provide any information
that could be interpreted as energy-momentum
distribution, and hence is a sign of an empty
space. The physical laws that cause a distortion
from this uniform grid, are the generators
of all physical phenomena.
From Einstein's theory of relativity it is known
that spacetime curvature gives rise to 10 gravitational
potentials. Furthermore, Einstein's
equations have been verified for several decades
[20], and were found to be correct. In
other words, it is an established empirical fact
that the deviation from a Cartesian metric in
spacetime gives rise to gravitational interaction.
Heim's idea was to extend this principle
of geometrization to all other physical interactions.
This means that the structure of the
equations of general relativity should be retained,
being valid also in the quantum range.
Since the curvature of spacetime is responsible
for gravitation, a higher dimensional space is
needed whose curvature accounts for all further
physical interactions. In addition, the principle
of quantization has to be taken into account,
giving rise to a set of eigenvalue equations.
According to Heim, since a particle has a nonlinear,
completely geometrized structure, a
nonlinear operator is necessary to produce a set
of geometrical eigenwert equations. In GRT,
the relation between the curvature tensor and
Christoffel symbols is Rkml
i
=Kl
Γ
k m
i , where
Kl denotes the well known differential operator
for the curvature tensor. In analogy to GRT,
we write Cl i
km , when going from the macroto
the microcosm, describing the field of a microparticle.
Because of the correspondence
principle from the macro- to the microcosm,
Cl=Kl . The Christoffel symbols Γk m
i become
the so called normalizable condensor functions,
i
km . This denotation is derived from the
fact that these functions represent condensations
of the spacetime metric. The fundamental
idea in Heim's theory is that matter can be explained
as a geometrical phenomenon and
thus, Einstein's field equations should be valid
in the microcosm, too. On the other hand, it is
well known that Schrödinger's equation describes
the probability amplitude for the location
of a microparticle. The stationary Schrödinger
equation is an eigenvalue equation for
the probability amplitude, ψ , with discrete energy
values as eigenvalues. This equation,
however, does not make any statements about
the physical properties of the microparticle.
Therefore, if, in contrast, one wants to have an
eigenvalue equation for the particle itself, to
describe its geometrical structure as well as its
physical properties that are independent of an
external field, only depending on the underlying
spacetime metric, a different set of eigenvalue
equations needs to be conceived.
These equations can be obtained by observing
that the Christoffel symbols can be interpreted
as physical fields, and consequently Heim associates
the condensor functions i
km with
physical fields in the microcosm. Einstein's
theory is in excellent agreement with observation
in the limiting case of a spacetime continuum
on a macroscopic level. Any unified theory
ought to agree with his theory in this limiting
case. Furthermore, since particle and wave
are a unity, it is concluded that the eigenvalue
equations for the condensor functions should
have a form similar to Schrödinger's equation
and are written in the form
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C(l )k m
(l ) =λ(l )k m
(l ) =εk m
l (1)
The LHS is a tensor of 4th rank (the operator
and the condensor functions considered separately),
λ
l is a vector of eigenvalues, and
l
km is a tensor of 3rd rank. The Christoffel
symbols and thus the condensor functions are
3rd rank tensors only with respect to affine (linear)
coordinate transformations. Any index in
parentheses is not summed over. The RHS represents
quantized energy densities, denoted by
l
km . The 3 indices independently run from 1 to
4, representing a set of 64 eigenvalue equations.
As Heim shows by symmetry considerations,
28 of the eigenvalue spectra are empty.
Rearranging the remaining 36 eigenvalue spectra
in a 6×6 tensor, Heim eventually constructs
a 6-dimensional space. Since Eqs. (1) are eigenvalue
equations, not subject to any external
field, there will be no curvilinear transformations,
leaving the tensor character of these
functions intact.
3.1 Eigenvalue Equations in Heim's
Theory
In order to give physical meaning to the various
metrics obtained from the metrics of subspaces
of 8 , some kind of hermeneutics is
needed. The hermeneutics of the 8 geometry
(abbreviated in the following as hermetry)
means the study of the methodological principles
of interpreting the metric tensor and the
eigenvalue vector of the subspaces. This semantic
interpretation of geometrical structure
is called hermeneutics (from the Greek word to
interpret).
Interactions in Heim's theory are causing a
modification of the underlying space time metrics.
Although there is an 8-dimensional Heim
space, we need to consider that the actual
physics takes place in 4-dimensional spacetime.
In order to see physical events happen, a
deviation from empty space, i.e., from the uniform
grid must occur. The uniform grid is
given by an Euclidean space using coordinates
x1,...,x4. Originally, Heim arrived at a 6-dimensional
space with coordinates x1,...,x6. For a
physical event to happen, a 4-dimensional curvilinear
coordinate system needs to be introduced,
ξ1,...,ξ4 that denotes the deviation from
Euclidean metric and, according to our interpretation,
gives rise to physical interactions.
The four additional curvilinear coordinates in
8, ξ5,...,ξ8 are termed transcoordinates and are
imaginary. It should be noted that the Heim
space 8 can also have a Euclidean structure
with coordinates x1,...,x8. A physical interaction
taking place in 4, not only changes coordinates
η1,...,η4 in physical spacetime, but also
influences the transcoordinates in 8. In other
words, there exists a mapping from 4 to 8.
This means, coordinates ξi depend upon
spacetime coordinates ηk or ξi = ξi(ηk). The
modified coordinates ξi in turn influence the
spacetime metric. Hence, there is a mapping
from 4 to 8 and back to 4. This double coordinate
transformation can be written as xl =
xl(ξi(ηk))
From the structure of the metric tensor it follows
that Heim's theory postulates the existence
of two additional interactions, denoted as
field Ψ1 (transformation of photons into
gravito-photons, hermetry form H11, see Sec.
3.4) and Ψ2 (transformation of gravito-photons
into the probability field, hermetry form H10,
see Sec. 3.4) so that there are now 6 different
fundamental interactions. In addition, there exist
two conversion fields allowing for the
transformation of photons into gravito-photons
(Ψ1 field), provided proper external conditions
are generated, and subsequently these gravitophotons
are converted via the conversion field,
Ψ2 , into the probability field, see below.
In Eq. (2) the structure of the energy-stressmomentum
tensor for Heim space 6 is shown.
The tensor contains 12 vanishing components,
as shown by Heim in [1]. Since Tα5=Tα6= 0 and
T5α=T6α= 0, the transspace (with regard to 4),
namely the components T55, T56 and T65, T66, interact
only via time components Tα4.and T4α
with spacetime 4. This means that in the microcosm
where Heim's equations are valid, the
future is not predetermined, i.e., only probabilities
for future possibilities are possible.
Causality is appearing only in the case of a superposition
of many microstates, forming a
collective macrostate.
Zero entries in Tαβ may become non-zero according
to Heisenberg's uncertainty principle.
9 of 21
(2)
3.2 Poly-metrics and Physical Forces
We mentioned that the metric tensor is comprised
of several components, such that each
component is responsible for a different physical
interaction. There are several subspaces in
in 8 in which individual metric tensors are
specified, that in turn are the cause of different
physical forces. The association of each coordinate
group (or subspace) follows certain selection
rules. Its corresponding physical interaction
is listed below. The physical meaning of
a coordinate or a group of coordinates is responsible
for the physical interaction. First,
four groups of coordinates are discerned:
spatial coordinates (real) (ξ1, ξ2, ξ3 ),
time coordinate (imaginary) (ξ4),
entelechial and aeonic coordinates (imaginary)
(ξ5, ξ6),
information coordinates (imaginary) (ξ7, ξ8).
As was outlined before, empty space is a discrete
uniform Euclidean space. We now ask
the following question: let us suppose that only
coordinates (x5, x6 undergo a deformation δξ5
and δξ6, rendering these coordinates curvilinear.
The change in these coordinates has an
impact on the 4-dimensional spacetime geometry,
too and appears as some kind of physical
phenomenon. In order to calculate this change
of spacetime metric, we write
gi k=
∂ xm
∂ξα
∂ξα
∂ηi
∂ xm
∂ξβ
∂ξβ
∂ηk
(3)
where indices α, β = 1,...,6 and i, m, k = 1,...,4.
This follows directly from the transformation
rule given in Sec. 3.1. The same structure
holds for the metric tensor of 8 where indices
α, β = 1,...,8. The terms on the RHS can be
written in the form
(4)
where indices in parentheses are not summed
over, and the definition of the factors κi,m
(α)
and κm,k
(β) follows directly from the above formula.
Since the function occurs as the kernel
in the integral
(5)
it is denoted as fundamental kernel of the
poly-metric. The term poly-metric is used
with respect to the composite nature of the
metric tensor as well as the twofold mapping
4→ 8→ 4. Using the fundamental kernels,
we can write the metric tensor in 8 in the
form
gi k=Σ
α=1
8
κi m
(α)Σ
β=1
8
κmk
(β)
=
(Σ
α=1
3
κi m
(α)κi m
4
Σ
α=5
6
κii m
(α)Σ
α=7
8
κi m
(α))
(Σ
β=1
3
κmk
(β)κmk
4
Σ
β=5
6
κmk
(β)Σ
β=7
8
κmk
(β))
(6)
In the next step, we write the components of
the metric tensor in such a way that there is a
correspondence with the four subspaces I2, S2,
T1, and3.
gi k=Σ
α=0
3
χi m
(α)Σ
β=0
3
χmk
(β) (7)
where the relationship between the new functions
χ and the fundamental kernels κ is
obtained from the comparison with the previous
formula of gi k . That is,
χi m
(1 )=Σ
5
6
κi m
(α) (8)
It should be noted that χi m
(α)≠χm i
(α) .
10 of 21
T αβ
=(T 11 T 12 T 13 T 14 0 0
T 21 T 22 T 23 T 24 0 0
T 31 T 32 T 33 T 34 0 0
T 41 T 42 T 43 T 44 T 45 T 46
0 0 0 T 54 T 55 T 56
0 0 0 T 64 T 65 T 66) κi m
(α)κmk
(β)=
∂ x
m
∂ξ(α)
∂ξ(α)
∂ηi
∂ x
m
∂ξ(β)
∂ξ(β)
∂ηk
.
x
m
(α)=∫κi m
(α)d ηi
χi,m
(0 )=Σ
α=7
8
κi,m
(α) ,
χi m
(3 )=Σ
α=1
3
κi m
χ (α) . i m
(2 )=κi m
(4) and
From Eq. (7) it can be seen that the metric tensor
in 8 is composed of 16 different terms
where each term belongs to one of the 4 subspaces
of8. The metric tensor can also be expressed
as
gi k= Σ
α, β=0
3
gi k
(α β) (9)
where
gi k
(α β)=χi m
(α)χm k
(β) . (10)
As was mentioned before, values of α, β are
associated with the subspaces of 8. The following
relationship holds:
(11)
If the eigenvalue vector, λp, of Eqs. (1) is different
from 0, not all of its components need to
be different from 0 in 8.The spectrum of λp
may refer to a k-dimensional subspace Vk , denoted
as λp(Vk), such that its k coordinates are
hermetric (curvilinear), while the remaining
8-k coordinates are Euclidean. If the k coordinates
of the subspace are hermetric, i.e., give
rise to a hermetric form as described in
Eq. (13), the remaining 8-k Euclidean coordinates
outside Vk are termed anti-hermetric. A
subspace Vk of 8 could be built by combining
coordinates from any of the four spaces I2, S2,
T1, and 3. The number of physically relevant
subspaces is, however, restricted, because the
real coordinates are interchangeable and are
taken as a semantic unit. All other coordinates
are not interchangeable and thus are separate
semantic coordinate entities. In addition, transcoordinates
can only occur in pairs, that is both
coordinates from S2 or I2must be present simultaneously.
Otherwise λp = 0, as Heimshows in
[2, pp. 192-195]. If both the transcoordinates
of S2 and I2 are anti-hermetric, then the coordi -
nates of 4 must be anti-hermetric as well. In
other words, transcoordinates must always be
present in a subspace in order that a physical
event can take place. Stated somewhat differ -
ently, at least one of the two transcoordinate
groups S2 or I2 must be present in order to
steer physical processes in 4.
Next, using these rules it can be determined
which of the subspaces correspond to physical
structures in the microcosm. In addition,
these subspaces need to be assigned their
proper physical interaction. This semantic interpretation
or hermeneutics has to be performed
in a way to reflect physical processes.
We have seen that Heim space 8 comprises
four semantic entities, namely the subspaces I2,
S2, T1, and 3. Employing the above rule there
are 12 physically meaningful combinations of
the four subspaces in 8, describing either
physical particles or interactions. Ten of these
subspaces can be identified with known virtual
particles or physical interactions. They can be
associated with the four known physical interactions
(strong, electromagnetic, weak, gravitation)
and the four types of known virtual particles
(gluons, photons, bosons, gravitons). The
hermetry forms H2, H6, H8, and H9 correspond
to the four known interactions, namely the
strong, weak, gravitational, and electromagnetic
forces, respectively.
There are, however, two additional interactions
(fields) that have not been known before.
In the light of recent cosmological observations,
they could possibly be associated with
dark matter and dark energy, described by hermetry
forms H10 and H11. Since H10 contains
only the space, I2, this field is termed probability
field. In H11 both types of transcoordinates
are present, and the particle associated with
this hermetry form is termed gravito-photon.
Perhaps these two fields are the explanation
for dark matter and dark energy or quintessence.
Quintessence [23], [24] has the striking
physical characteristic that it causes the expansion
of the universe to speed up. Energy, either
in form of matter or radiation, causes the expansion
to slow down due to the attractive
force of gravity. For quintessence, however,
the gravitational force is repulsive, and this
causes the expansion of the universe to accelerate.
In addition, it is interesting to compare,
see Sec. 5, with Heim's modified Newtonian
law, derived in the fifties of the last century.
Below are listed the 12 hermetry forms that result
from the 8-dimensional Heim space 8.
Arguments in parentheses specify the subspace
11 of 21
α,β=0: I 2=(ξ7,ξ8) ,
α,β=1: S2=(ξ5,ξ6) ,
α,β=2: T 1=(ξ4) ,
α,β=3:3=(ξ1, ξ2,ξ3).
Vk in which the physical interaction takes
place.
(12)
The hermetry forms can also be represented by
the components of the metric tensor of the corresponding
subspace Vk. The superscripts,
ranging from 0 to 3, in the χ quantities refer
to the respective coordinate groups.
H5=(χi m
(0 ) ,χi m
(1 ) ,χi m
(2 )) photons (13)
It is reasoned that hermetry forms H10 and H11
are similar to the graviton field H12, since they
are both caused by transcoordinates, and thus
will have a small coupling constant. The important
point is that in Heim's theory there are
transformation operators, S1 or S2 (not to be
confused with space S2), that, when applied to
one hermetry form can transform it into another
one. Mathematically, these operators
transform the respective coordinate from a non
Euclidean to a Euclidean one. For instance, S2
applied to hermetry form H11 will transform
electromagnetic radiation into gravito-photons.
4. Physical Principles of Space
Flight
Heim's field theory predicts - provided one is
in the low energy range since in the energy
range of some 1016 GeV all interactions are of
the same strength - the four known coupling
constants (gravitation, weak, electromagnetic,
strong), but predicts the existence of two additional,
hitherto unknown interactions, namely
the hermetry forms H10 and H11 (see Eqs. (13)).
His analysis, like most discussions of gravitational
radiation, proceeds by analogy with
electromagnetic radiation. Just as changes in
an electric or magnetic field trigger electromagnetic
waves, changes in a gravitational
field trigger gravitational waves.
The theory of the coupling constants is most
important for the physical interactions. If one
measures the product of two charges in units of
ħ c and uses the proton mass, mp, as a reference
mass, the following relations hold for the
four known interactions:
12 of 21
H1=H1 ( I 2 ,T 1) gluons
H2=H2 ( I 2 ,T 1 , R3) color charges
H3=H3 ( I 2 , S 2 ,T 1 , R3) W+_ bosons
H4=H4 ( I 2 , S 2 , R3) Z0 boson
H5=H5 ( I 2 , S 2 ,T 1) photons
H6=H6 ( I 2 ,T 1)∗H7=H7 (S 2 ,T 1)
weak charge
H8=H8 (S 2 ,R3)
neutral field (particle) with mass
H9=H9 (S 2 ,T 1 , R3)
field (particle) with electric charge and mass
H10=H10 ( I 2) probability field
H11=H11 ( I 2 , S 2) gravito-photon
H12=H12 (S 2) graviton.
H1=(χi m
(0 ) ,χi m
(3 )) gluons
H2=(χi m
(0 ) ,χi m
(2 ) ,χi m
(3 )) color charges
H3=(χi m
(0 ) ,χi m
(1 ) ,χi m
(2 ) ,χi m
(3 )) W+_ bosons
H8=(χi m
(1 ) ,χi m
(3 ))
neutral field (particle) with mass
H10=(χi m
(0 )) probability field
H11=(χi m
(0 ) ,χi m
(1 )) gravito-photon
H12=(χi m
(1 )) graviton.
H4=(χi m
(0 ) ,χi m
(1 ) ,χi m
(3 )) Z0 boson
H6=H6(χi m
(0 ) ,χi m
(2 ))∗H7=H7(χi m
(1 ) ,χi m
(2 ))
weak charge
H9=(χi m
(1 ) ,χi m
(2 ) ,χi m
(3 ))
field (particle) with electric charge and mass
G
mp
2
ħ c
=5.9×1038
λp
cβ
ħ c
=2×108
e2
ħ c
=
1
137
=7.3×103
g2
ħ c
=15
(14)
where Eq. (14) denotes the relative strength of
the gravitational, weak, electromagnetic, and
strong interactions, respectively, and cβ is the
coupling constant of the beta decay and
λ
p
=ħ⁄mp c is the Compton wavelength of
the proton. The relative strength of the forces
is approximately 1: 10-3: 10-9: 10-40, the strong
force assigned the value 1. According to Feynman
coupling constants can be interpreted as a
probability for the exchange of virtual particles.
Heim's eigenvalue equations allow to
compute the spectrum of the ponderable particles.
The physical constants that determine the
coupling constants are depending on the eigenvalues.
The set of eigenvalues itself is determined
by geometrical symmetries. These symmetries
are related to the coordinates of the
Heim space. The sets of eigenvalues can be
characterized by their cardinal numbers. The
cardinal numbers, therefore, point out a way to
the mathematical description of all coupling
constants, and hence a set algorithm was constructed
that is behind the derivation of the
magnitude of the coupling constants4.
It turns out that for Heim space 8 not only the
values for the 4 known coupling constants are
obtained, but four additional probability amplitudes
occur. In other words, there exists a
set of 8 probability amplitudes, wi, where w1 to
w4 describe the 4 known interactions, whose
carrier particles are gravitons, vector bosons,
photons, and gluons. Probability amplitudes
w5, w6 are interpreted as transmutation fields
(mathematically represented as transformation
operators S1 and S2). The other two coupling
constants, w7, w8 are interaction fields, and are
4 A complete theory was derived by the first author
calculating the exact values of the coupling constants,
based on the theory of cardinal numbers. A
paper on the derivation of the coupling constants is
in preparation. The theory comprises some 50 pages
and cannot be presented in this paper.
interpreted as gravito-photon and probability
fields. They are gravitational like fields, characterized
by hermetry forms H10 and H11, see
Eqs.(13).
The physical meaning of the transmutation
fields is that photons, characterized in 8 by
the hermetry form H5 , are transformed by the
action of the transmutation operator S2 (not to
be confused with space S2) into gravito-photons
(hermetry form H11). In a more formal
way, one could write S2 H5= H11. The relation
between the corresponding probability amplitudes
is
3w3 -w5 = 3w7 (15)
where operator S2 is associated with w5, hermetry
form H5 with w3, and hermetry form H11
with probability amplitude w7. It should be
noted that the equations were slightly simplified.
In a second step, the gravito-photon field (hermetry
form H11), under the action of transmutation
operator S1, is transformed into the probability
field described by hermetry form H10.
This can be written as S1H11= H10 or
w7 -w1w6 = w8. (16)
The value of w7 is 1.14754864×10-21 and w8 is
calculated as 1.603810891×10-28. Again, the
equations were slightly simplified. Because
the value of the probability amplitude w7 is
similar to the value of w1, (graviton, value
7.6839×10-20), the denotation of the gravitophoton
field as a gravitational like field seems
to be appropriate. It should be remembered
that w-values denote probability amplitudes,
i.e., their square gives a probability, which is
the respective coupling constant for the corresponding
interaction.
4.1 Lorentz Matrix and Inertial Transformations
Under the physical conditions specified above,
an electromagnetic field can be transformed
into a gravitational like field, such that the
gravitational field around a space vehicle is reduced,
according to Eqs. (15) and (16). With
the reduction of the gravitational potential, Φ,
in each area of space where this transformation
takes place, gravitational mass density g and
13 of 21
thus gravitational mass mg must be reduced,
too. This follows directly from
(17)
where the integration is over the volume in
physical space in which the transformation
takes place. According to the equivalence of
inertial mass (Newton's second law) and gravitational
mass (Newton's gravitational law), a
reduction in gravitational mass is equivalent to
a reduction in inertial mass. Therefore, m > m'
where m and m' denote the inertial masses before
and after the transformation. Let us now
consider the 4-momentum vector in spacetime
(18)
where p=mv is the classical momentum
and i is the imaginary unit. Since the magnitude
of P is an invariant, both momentum
and energy conservation hold. For a space vehicle
with initial inertial mass m and reduced
mass m', the following relations are therefore
valid,
mv=m' v' and mc=m' c' (19)
that is c' > c and v' > v, since m> m'. Quantities
with a prime indicate the transformed system.
We denote this kind of transformation as
inertial transformation. Dividing the first
equation by the second one, it immediately follows
that
(20)
Therefore, the corresponding Lorentz transformations,
namely for the first system, in which
c is the speed of light and the spacecraft is
moving with velocity v, and the second system,
in which the speed of light is c' and the vehicle
speed is v', are described by the same Lorentz
matrix , that is
(21)
where β = v/c , x4 = ict or b = v'/c' , x'4 = ic't.
The movement is in the x (that is x1) direction
only. Since a transformation of inertial mass
only changes c to c' ( with c' > c), the other
spatial coordinates remain unchanged, and
only coordinates x1 and x4 are changing,
respectively. There is no contradiction to special
relativity, since an inertial transformation
is not considered in SRT. The argument in
SRT is, that if v > c, then β becomes imaginary.
Thus, it is concluded that no observer can
possess a velocity greater than that of light
relative to any other observer. In an inertial
transformation, however, β remains positive.
Such a transformation is not possible in SRT
or GRT, since it is a consequence of the unification
of physical interactions and the polymetric
in 8.
If it were technically possible to generate a sufficiently
strong field for the reduction of inertial
mass in the vicinity of a moving space vehicle,
the velocity of the space vehicle will increase
from v to v' = α ν, where α := is
the velocity gain factor, and 1/α gives the factor
at which the graviton field of the space vehicle
is reduced. In addition, the transformation
field for the inertial mass may by itself
have a repulsive effect and thus further accelerate
the space vehicle. The action of this
transformation field is such that the space vehicle
disappears from the usual spacetime with
c = constant, and enters a spacetime in which
c' = constant is valid. If the transformation
field disappears, the space vehicle returns to its
original spacetime. It is interesting to note that
during the transition phase from α = 1 to α
> 1, the acceleration can be arbitrarily large
without the occurrence of a force, caused by
this acceleration. This simply follows from the
conservation of momentum, namely the fact
14 of 21
Φ(x)=∫
ρg ( x')
|x x'|
d3 x'
P=m0 (1v2 ⁄ c2)1⁄2 (v, ic)
= (mv, imc)=( p ,imc)
v' ⁄ c'=v⁄ c.
A=( 1
(1 β2)
0 0
i β
(1 β2)
0 1 0 0
0 0 1 0
i β
(1 β2)
0 0
1
(1 β2))
that m v = m' v' = constant (m = α m'), which
means that the force, responsible for the acceleration,
is 0. Owing to the invariance of the
Lorentz matrix with respect to an inertial transformation,
which is rooted in the fact that v'/c'
= v/c, superluminal velocities should be possible.
Most interesting, this fact is not in contradiction
with GRT, allowing, in principle
space flight at superluminal velocities. Although
superluminal velocities have been conjectured
for some time, the difference now is
that there is a physical theory, according to
which this phenomenon can be computed.
It goes without saying, that there remain two
important questions to be settled. First, an experimental
proof of the validity of Heim's theory,
and second, this validity assumed, what
are the technological challenges to construct a
viable propulsion system from this inertial
transformation principle. In the subsequent
section, an order of magnitude estimate for the
energies to be supplied, is presented.
4.2 Estimating the Order of Magnitude
of Transformation Effect and
Gedanken experiment
In the previous section it was shown that electromagnetic
radiation can be converted into a
gravitational like field, thus reducing the gravitational
mass of a body that is under the influence
of this radiation. First, it was observed
that the corresponding coupling constant is
weak, i.e., the interaction and thus the actual
force will be small. Second, the frequency of
the radiation needs to be determined at which
this transformation takes place.
From the theory of the coupling constants a
value of 28.66 keV is computed for the photon
energy at which, according to Eq. (15), photons
are completely converted into gravitophotons
(w7 field) by the transmutation field,
w5, that is present in vacuum. The photon density
has to satisfy an additional constraint. For
example, this could be achieved by oscillating
electrons in a free electron laser. Interpreting
Eq. (16), one sees that the generated w7 field is
transformed by a graviton field (w1) into a
probability field (w8). In this process the necessary
gravitons are converted at the same location
at which they were generated, i.e., the
graviton field, associated with a spacecraft, is
reduced in its strength. This means a decrease
of the inertial mass of the spacecraft. As long
as the gravito-photon field exists, the space
craft is accelerated from velocity v to v', as calculated
in Eq. (20). A speculative interpretation
would be that the spacecraft disappears
from our spacetime, characterized by speed of
light c, and enters the spacetime of speed of
light c', and returns to the original spacetime if
the gravito-photon field disappears. We now
give an estimate for the two most interesting
questions, concerning the technical usefulness
as well as the technical feasibility of a space
transportation system based on inertial transformation:
Question 1: If we want a velocity n-times the
initial velocity as well as a limiting velocity ntimes
the vacuum speed of light, what is the
required mass reduction of the spacecraft?
Question 2: What is the total photon energy required
that delivers this inertial reduction?
The transmutation equations, Eqs. (15), (16),
are valid for single photons and gravito-photons.
Both, the strength of the electromagnetic
field and its energy density are much larger
than the respective values of the gravito-photon
field, which can be seen directly by the
coupling constants. Therefore, energy will be
taken from the vacuum, in accordance with
Heisenberg's uncertainty principle, that needs
to be returned to the vacuum, so that energy
conservation is satisfied. To reduce a spacecraft
with mass M, gravito-photon (w7) particles
have to be generated. Rewriting Eq. (16)
using the value for w6 from the theory of coupling
constants
w7 - 0.014 w1 = w8 (22)
shows that some 67 w7 -particles convert 1 w1-
particle. In addition, only 1 out of 4 w7 -particles
can be used to converting w1- particles,
hence the factor 67. That is, the photon energy
must be increased by a factor of 67 to compensate
a w1 field. Let the spacecraft have a
spherical body of radius R = 1 m and mass
M = 104 kg. Let us furthermore consider that
the spacecraft is not subject to any external
gravitational field. Its gravitational self-energy
is
15 of 21
E=
1
2
∫ρ( x)Φ ( x) d3 x =
GM2
2R
=
6.67×1011×108
2
=
3.365×103 J
(23)
where the factor of oe is introduced to avoid
double counting of pairs of mass particles in
the distribution, and Φ(x) is the potential produced
by the mass distribution. The resulting
energy is fairly small. The photon energy
needed would therefore be a mere 0.9 J, no
losses assumed. This result holds for the conversion
using a free electron laser. Using a second
solution, requiring a lower photon energy,
employing an electrically charged rotating
torus, an energy of some 105 J would be necessary.
According to our present calculations, the
situation changes drastically if we were to
launch from the surface of a planet. Let us assume
that both planet and spacecraft are
spherical bodies, and a launch from the surface
of the earth is intended. The mass of the earth
is 5.98×1024 kg and the radius is 6.378×106 m.
Eq. (23) (without factor oe) results in an energy
of some 6.25×1011 J and a photon energy of
67×6.25×1011 J. It was assumed that all of the
potential energy resides in the field. We would
like to emphasize that this extremely large
value might not be the last word, since we cannot
claim at present that all consequences of
the theory are fully understood.
To eventually reach a velocity n-times the initial
veleocity, v, we need to reduce the inertial
mass such that m = n m' where m' is the reduced
inertial mass. That means, in order to
travel at a limiting speed that is 10-times the
speed of light, c' = 10 c, the inertial mass m
must be reduced to 10% of its original value.
The theory allows superluminal flight in principle,
however, the technical realization of the
inertial transformation to achieve this goal is a
problem to be solved in the future. Further research
will be needed to establish criteria for
the effort needed.
5. Speculative Cosmology
Since the higher dimensional space used in
Heim's theory comprises a discrete metronic
lattice, there are no singularities. Hence, the
beginning of the universe is clearly defined.
The actual starting point for the universe was,
when the size of a single Metron,
τ, which is a function of time, dτ/dt < 0, covered
the surface of the universe, assumed to be
spherical. During the expansion phase of the
universe, the number of Metrons increased.
Eq. (16) has an interesting consequence for the
inflationary phase of the universe. An existing
w7 field transforms a gravitational field w1 into
a w8 field. A vanishing w1 field causes, because
of the equivalence of inertial and gravitational
masses, a reduction of the inertial mass of the
universe. Owing to the conservation of momentum
and energy, the speed of light c during
this phase will have increased to a much higher
speed c', according to Eq. (18). The same principle
that might be used for superluminal space
transportation, could have caused the inflationary
phase of the universe.
According to Heim [1, 2] (the derivation of the
formula below was not calculated independently
by the authors), Newton's law needs to be
modified for large distances by a negative term
and thus becomes repulsive:
a=G
m(r)
r2 (1
r2
ρ2 ) , ρ=
h2
Gm0
3 (24)
m0 being the mass of a single nucleon comprising
the mass of the field source. Mass m(r)
is the total mass and comprises the ponderable
and the field mass. According to Heim, the
value of ρ is some 10 to 20 million light-years.
This modified law has severe consequences,
since the observed redshift would, at least partially,
be a gravitational redshift.
Heim also calculates a lower bound for the
range of validity of the gravitational law,
which is in the microscopic range, and whose
value is approximately given as
R-=
3e
16
GM0
c2 (25)
M0 denoting the (ponderable) mass without the
field mass. This threshold is practically
equivalent to the Schwarzschild radius. The
gravitational force is attractive for distances
R_ < r < ρ. For r = ρ, the gravitational force
is 0.
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There is a third distance, R+, depending on the
mass in the universe, so that for ρ < r < R+
the gravitational force is repulsive and goes to
0 for r = R+. R+ is some type of Hubbleradius,
but is not the radius of the universe, instead it
is the radius of the optically observable universe.
All optical signals are subject to a redshift
due to the anti-gravitational effect from
Eq. (24). For distances larger than R+, the redshift
becomes infinite, and thus signals cannot
be received. R+ would be largest if the universe
contained only a single particle of minimal
mass which could be a neutral particle of a
mass 1% less than the electron mass that might
exist. Then one would obtain the largest radius
possible, Rmax or D = 2 Rmax as maximal diameter
of the physical universe, computed
from the following formulas, see also [6]
3
2
f(1
4
3
2
Df 3
τ
3 1)2
=
D
τ
(eD τ
πE
1)f 2=
eD τ
πE
(26)
e and E being the basis of the natural logarithm
and a unit surface, respectively. The reader
may wish to calculate the present diameter of
the universe himself. To calculate the diameter
D0 at the beginning of time, the relation π
D0
2 = τ0 has to be inserted into the second
equation of Eq. (26). τ0 denotes the metronic
size at the origin of the universe. The resulting
equation of 7th order for f0 has 3 real roots. This
results in 3 different positive values of D0 and
three different negative values for D0. Heim interprets
this as a trinity of spheres, separated in
time by a chronon, the quantum time interval.
At the end of its life cycle, the universe collapses
into a trinity of spheres, determined by
the negative values for the diameter.
Since spacetime is quantized, black holes in
form of a singularity should not exist. It should
be noted that for dimensions for which the de
Broglie wave length is small quantum theory
should be incorporated and the extension of
GRT to microscopic dimensions may be incorrect.
According to Heim the age of the universe is
some 10127 years. Matter as we know it was
generated only some 15 billion years ago,
when τ, the Metron size, became small
enough. The phenomenon of gamma ray
bursts may be an indication of the creation of
matter. At that time the universe was already
almost flat, i.e., τ ⁄τ≈0and D ⁄D≈0. In
other words, the universe is expanding, but at a
slower rate as presently believed and is at present
almost flat.
It should be noted, that Heim's theory [1,2]
also provides formulas for the mass spectrum
of the elementary particles as well as their lifetimes.
Conclusions and Future Work
We are aware of the fact that the present article
has several shortcomings. First, we stated several
important physical assertions without
proper mathematical proof. However, the derivations
of the coupling constants and the hermetry
forms are a subject of their own, which
is beyond the scope of this paper. Second,
Heim's legacy contains a large body of unpublished
work. The authors were not able to
check all of his calculations. In particular, the
derivation of the nonlinear potential equation,
see Chap. 5, has not been derived independently
5. Third, Heim, being visually impaired,
used his own physical terminology, necessitating
a translation into the language of contemporary
physics. In addition, several completely
novel concepts needed to be introduced that
may require additional physical interpretation.
Last but not least, since this is work in progress
on a challenging topic, both, errors in
Heim's theory as well as those introduced by
the authors, cannot be excluded. Thus, none of
the new physics presented here should be taken
for granted.
The so called Standard Model, see for instance
[19], is an amalgam of experimental observations
and theoretical derivations, but needs
some 30 adjustable parameters to be determined
from experiments. Spacetime is con-
5 Heim left some 4,000 pages, now at the university library
in Salzgitter, of unpublished material, providing,
in many cases, the detailed calculations not
found in his published work. His work also includes
treatises on cosmology, elementary particle physics,
mathematical logic as well as on bioscience.
17 of 21
tinuous and therefore leads to singularities and
infinite self energies. The lifetimes and the
spectrum of elementary particles cannot be
predicted. It also is, according to these authors,
logically inconsistent that 6 quarks and 6 leptons
are the basic ingredients of matter, but
free quarks are per definition unobservable. No
physical explanation is provided for this fact.
Furthermore, there is no quantum gravity in
the context of the Standard Model. There is
also no explanation for the existence of these
basic constituents, and it could well be that
quarks need to be comprised of even more fundamental
particles, which then might lead to
the conclusion that it is turtles all the way
down.
In this respect, Heim's theory seems to be logically
more consistent, based on verified physical
principles, and the idea of a higher dimensional,
discrete space, composed by oriented,
minimal surface elements. It leads, however, to
a radically different view of matter and space,
and predicts a cosmology that substantially differs
from the current model of the big bang.
Heim derived the theory without the fitting of
any experimental parameters, making decisive
predictions about cosmological phenomena,
and delivers a formula for the life times and
the mass spectrum of elementary particles as
well as specifying appropriate selection rules.
It also contains a formulation for a quantum
gravity, based on the aesthetic and elegant idea
that the metric is the generator for all physical
interactions, leading to a poly-metric. The theory
makes several remarkable predictions,
namely the existence of two additional interactions,
hitherto unknown, that are the basis for
advanced space travel.
Whether Heim's geometrized field theory reflects
physical reality is, at present, undecided.
If the theory were true, an entirely new concept
of the universe would emerge. Moreover,
Heim's theory is a definite non-mechanistic
concept of nature, in contrast to our present
view of the world, initiated in the 15th century
by Leonardo da Vinci.
Heim's theory unifies all known interactions in
a 8-dimensional space and allows for a special
Lorentz transformation, named inertial transformation,
permitting, at least in principle, for
superluminal travel.
Acknowledgment
This paper is dedicated to Dr. William Berry
(ret.), head of the Propulsion and Aerothermodynamics
Division, European Space research
and Technology Center (ESTEC) of the European
Space Agency (ESA) in Noordwijk, The
Netherlands. The second author, while working
in his division, learned that taking a high
risk is acceptable as long as engineering and
scientific judgment remain intact. He gratefully
acknowledges the many discussions on
propulsion principles.
The authors are grateful to Prof. Dr. Dr. A.
Resch, director of IGW at Innsbruck University,
for providing a stimulating working atmosphere
and for numerous discussions concerning
Heim's theory.
The authors are very much indebted to Dipl.-
Phys. I. von Ludwiger, former manager and
physicist at DASA, for making available relevant
literature and for many helpful discussions
as well as explanations concerning the
implications of Heim's theory.
The second author was partly funded by Arbeitsgruppe
Innovative Projekte (AGIP), Ministry
of Science and Education, Hanover, Germany.
Glossary
aeon Denoting an indefinitely long period of
time. The aeonic dimension can be interpretedis
as steering structures governed by
the entelechial dimension toward a dynamically
stable state.
anti-hermetry Coordinates are called anti-hermetric
if they do not deviate from Cartesian
coordinates, i.e., in a space with antihermetric
coordinates no physical events
can take place.
condensation For matter to exist, as we are
used to conceive it, a distortion from
Euclidean metric or condensation, a term
used by Heim, is a necessary but not a sufficient
condition.
18 of 21
condensor The Christoffel symbols Γk m
i become
the so called condensor functions,
i
km , that are normalizable. This denotation
is derived from the fact that these
functions represent condensations of
spacetime metric.
coupling constant Value for creation and destruction
of messenger (virtual) particles,
relative to the strong force (whose value is
set to 1 in relation to the other coupling
constants).
entelechy (Greek entelécheia, objective, completion)
used by Aristotle in his work The
Physics. Aristotle assumed that each phenomenon
in nature contained an intrinsic
objective, governing the actualization of a
form-giving cause. The entelechial dimension
can be interpreted as a measure of the
quality of time varying organizational
structures (inverse to entropy, e.g., plant
growth) while the aeonic dimension is
steering these structures toward a dynamically
stable state. Any coordinates outside
spacetime can be considered as steering
coordinates.
fundamental kernel (Fundamentalkern)
Since the function κi m
(α) occurs in
x
m
(α)=∫κi m
(α)d ηi
as the kernel in the integral,
it is denoted as fundamental kernel
of the poly-metric.
geodesic zero-line process This is a process
where the square of the length element in a
6- or 8-dimensional Heim space is zero.
gravito-photon field Denotes a gravitational
like field generated by a neutral mass with
a smaller coupling constant than for gravitons,
but allowing for the possibility that
photons are transformed into gravito-photons.
This field can be used to reduce the
gravitational potential around a spacecraft.
graviton (Graviton) The virtual particle responsible
for gravitational interaction.
hermetry form (Hermetrieform) The word
hermetry is an abbreviation of hermeneutics,
in our case the semantic interpretation
of the metrics. To explain the concept of a
hermetry form, the space 6 is considered.
There are 3 coordinate groups in this
space, namely s3=(ξ1 ,ξ2 ,ξ3) forming
the physical space 3, s2=(ξ4) for
space T1, and s1=(ξ5 ,ξ6) for space S2.
The set of all possible coordinate groups is
denoted by S={s1, s2, s3}. These 3 groups
may be combined, but, as a general rule
(stated here without proof, but derived by
Heim from conservation laws in 6, see p.
193 in [2]), coordinates ξ5 and ξ6 must always
be curvilinear, and must be present
in all combinations. An allowable combination
of coordinate groups is termed hermetry
form, and denoted by H, sometimes
annotated with an index, or sometimes
written in the form H=(ξ1 ,ξ2 ,...)
where ξ1 ,ξ2 ,...∈S . This is a symbolic
notation only, and should not be confused
with the notation of an n-tuple. From the
above it is clear that only 4 hermetry forms
are possible in 6. Thus, a 6 space only
contains gravitation and electrodynamics.
It needs a Heim space 8 to incorporate all
known physical interactions. Hermetry
means that only those coordinates denoted
in the hermetry form are curvilinear, all
other coordinates remain Cartesian. In
other words, H denotes the subspace in
which physical events can take place, since
these coordinates are non Euclidean. This
concept is at the heart of the geometrization
of all physical interactions, and serves
as the correspondence principle between
geometry and physics.
hermeneutics (Hermeneutik) The study of
the methodological principles of interpreting
the metric tensor and the eigenvalue
vector of the subspaces. This semantic interpretation
of geometrical structure is
called hermeneutics (from the Greek word
to interpret).
hermitian matrix (self adjoint, selbstadjungiert)
A square matrix having the
property that each pair of elements in the ith
row and j-th column and in the j-th row
and i-th column are conjugate complex
numbers (i → - i). Let A denote a square
matrix and A* denoting the complex conjugate
matrix. A† := (A*)T = A for a hermi-
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tian matrix. A hermitian matrix has real eigenvalues.
If A is real, the hermitian requirement
is replaced by a requirement of
symmetry, i.e., the transposed matrix AT =
A .
homogeneous The universe is everywhere uniform
and isotropic or, in other words, is of
uniform structure or composition throughout.
inertial transformation (Trägheitstransformation)
Such a transformation, fundamentally
an interaction between electromagnetism
and gravitational like field, reduces
the inertial mass of a material object using
electromagnetic radiation at specific frequencies.
As a result of momentum and
energy conservation in 4-dimensional
spacetime, v/c = v'/c', and thus the Lorentz
matrix remains unchanged. It follows that
c < c' and v < v' where v and v' denote the
velocities of the test body before and after
the inertial transformation, and c and c' denote
the speeds of light, respectively. In
other words, since c is the vacuum speed
of light, an inertial transforemation allows
for superluminal speeds. An inertial transformation
is possible only in a 8-dimensional
Heim space, and is in accordance
with the laws of SRT. In an Einsteinian
universe that is 4-dimensional and contains
only gravitation, this transformation
does not exist.
isotropic The universe is the same in all directions,
for instance, as velocity of light
transmission is concerned measuring the
same values along axes in all directions.
partial structure (Partialstruktur) For instance,
in 6, the metric tensor that is Hermitian
comprises three non-Hermitian
metrics from subspaces of 6. These metrics
from subspaces are termed partial
structure.
poly-metric The term poly-metric is used with
respect to the composite nature of the metric
tensor. In addition, there is the twofold
mapping 4 → 8→ 4.
transformation operator (Sieboperator) The
direct translation of Heim's terminology
would be sieve-selector. A transformation
operator, however, converts a photon into
a gravito-photon by making the coordinate
ξ4 Euclidean.
unitary matrix (unitär) Let A denote a square
matrix, and A* denoting the complex conjugate
matrix. If A† := (A*)T = A-1, then A is
a unitary matrix, representing the generalization
of the concept of orthogonal matrix.
If A is real, the unitary requirement is replaced
by a requirement of orthogonality,
i.e., A-1 = AT. The product of two unitary
matrices is unitary.
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