AIAA 2003-4990

FUTURE SPACE PROPULSION

BASED ON HEIM'S FIELD THEORY Walter Dröscher1, Jochem Häuser 1,2

Institut für Grenzgebiete der Wissenschaft (IGW),

Leopold - Franzens Universität Innsbruck, Innsbruck, Austria2Department of Transportation, University of Applied Sciences

andDepartment of High Performance Computing, CLE GmbH, Salzgitter, Germany

39TH AIAA/ASME/SAE/ASEE

JOINT PROPULSION CONFERENCE & EXHIBIT, HUNTSVILLE, ALABAMA,

20-23 JULY, 2003

1 Senior scientist, 2 Senior member AIAA, member SSE, www.cle.de/HPCC, www.uibk.ac.at/c/cb/cb26/

2 The upper ring (special material) is rotating in a magnetic field possessing a radial component, giving rise to theHeim-Lorentz force as predicted in Heim's Unified Field Theory, generated by so called gravitophoton particles.

Br

B I

NOW IS THE TIME TO TAKE LONGER STRIDES

-TIME FOR A GREAT NEW AMERICAN ENTERPRISE-

TIME FOR THIS NATION TO TAKE A CLEARLY LEADING ROLE IN SPACE ACHIEVEMENT,

WHICH IN MANY WAYS MAY HOLD THE KEY TO OUR FUTURE ON EARTH

PRESIDENT KENNEDY'S MESSAGE TO THE CONGRESS ON MAY 25, 19613

2003

Institut für Grenzgebiete der Wissenschaft, Leopold - Franzens Universität Innsbruck,Innsbruck, Austria

3 It does not seem that the old Europe currently has any vision of space

ABSTRACT

Effective space propulsion for interplanetary or inter-stellar missions cannot be based on the momentumprinciple of classical physics. This paper, being a con-tinuation of [1], that discussed the physical principlesof Heim's unified field theory, focuses on the propertiesof the predicted gravitophoton particle (similar to thegraviton) as a means for a revolutionary propulsion sys-tem in the sense of NASA's Breakthrough PropulsionPhysics Program (BPP) [2]. With regard to Mass, thepropulsion method does not require a propellant, sinceusing the gravitophoton field predicted by an extensionof Heim's field theory [1, 5-7], does allow for the con-version of electromagnetic radiation into a gravitationallike field (i.e., gravitational interaction takes placethrough both the graviton and the postulated gravito-photon particles) that reduces the inertia of a materialbody. Concerning the achievable Speed, the gravito-photon field, as a new interaction (force) that followsfrom Heim's completely geometrized unified field the-ory, in principle, admits superluminal travel. The cor-responding inertial transformation (conversion of elec-tromagnetic radiation into a gravitophoton field) thatdoes not exist in Einstein's four-dimensional spacetimecontinuum, however, is Lorentz invariant. Since thelaws of energy and momentum conservation [1] arestrictly adhered to, the vacuum speed of light is not thelimiting velocity for an inertial transformation. Withregard to energy, there are two different modes of pro-pulsion. In the first mode, energy will be extracted fromthe magnetic field (see Figs. 1 and 2), and the space-craft will be accelerated over a certain period in time,for instance at 1g, up to a certain velocity. For a mis-sion to Mars the velocity would be some 1.5×106 m/s.For an interstellar mission, a velocity of some 0.1 c isneeded. Now the second propulsion mode is em-ployed, reducing the inertial mass of the space craft bya factor 104. The inertial transformation will increasethe speed of the spacecraft by a factor of 104, withoutincreasing its kinetic energy4. Energy is needed onlyfor the generation of a very strong magnetic field. Oth-erwise, it would be impossible to fly at speeds compa-rable or larger than the vacuum speed of light. Thecost of the necessary energy to fly, for instance, a 100ton spacecraft close to the speed of light would be pro-hibitive [19, 20].

The paper comprises five sections. The first sectiongives a qualitative discussion of the role of the gravito-photon field and its double effect on matter, namely ac-celeration and reduction of inertia of a material body.Section 2 contains material about the role of the twonovel particles, the gravitophoton and probability parti-cles. In Section 3 a derivation of the metric tensor formoving electrical charges in 8-dimensional Heimspace, ℝ

8 , is given. In the next section, an experimentis presented to measuring the double nature of thegravitophoton field, and the strength of the interactionis calculated. The most important result is that an equa-

4 Kinetic energy is calculated using the speed of lightc, c', respectively.

tion was found, termed the Heim-Lorentz equation, thathas a form similar to the electromagnetic Lorentz force,except, that it is a gravitational like force, while the Lo-rentz force acts upon moving charged particles only. Inother words, there seems to exist a direct coupling be-tween matter and electromagnetism. Section 5 showsthe performance of the gravitophoton field as a propul-sion device. The performance of a gravitophoton pro-pulsion device is calculated, and a discussion is pre-sented of the interaction of a spacecraft traveling insome type of hyperspace where the speed of light isgreater than the vacuum speed of light. In addition, thetwo-stage nature of a gravitophoton propulsion systemis elucidated (two stage device: first acceleration, theninertia reduction). Moreover, the performance of gravi-tophoton propulsion for an interplanetary (Mars) mis-sion and interstellar (flight to an earthlike planet some100 light years away) is discussed. In the summary, thepaper is concluded with an assessment of the physicalcredibility of Heim's theory and an outlook on the ac-tual construction of a gravitophoton propulsion device.

It should be emphasized that the introduction of newphysics, i.e., the complete geometrization of existingphysical forces, will require that some concepts of to-day's physics need to be changed. It could be that insome circumstances the introduction of the transcoordi-nates, Section 3.1, may invalidate the second law ofthermodynamics. It should be clear, however, if thenew physics allows for breakthrough propulsion, sub-stantial changes to currently established physical prin-ciples are mandatory.

Nomenclature and physical constants

Compton wave length of the electron

C=

ℏ

mec

=2.43×10−12

m .

c speed of light in vacuum 299,742,458 m/s ,

(1/c2 = ε0

µ0).

D diameter of the primeval universe, some 10125 m,that contains our optical universe.

DO diameter of our optical universe, some 1026 m.

d diameter of the rotating torus, see caption Table .

dT vertical distance between magnetic coil and rotat-ing torus (see Fig. 2).

e electron charge -1.602 × 10-19 C.

ez

unit vector in z-direction.

Fe electrostatic force between 2 electrons.

Fg gravitational force between 2 electrons.

Fgp gravitophoton force, also termed Heim-Lorentz

force, Fgp=

pe

0vT×H , see Eq. (31).

3 of 25

G gravitational constant 6.67259 × 10-11 m3 kg-1 s-2.

Ggp gravitophoton constant, Ggp≈1 /672G

gi k

gpmetric subtensor for the gravitophoton in sub-

space I2∪S2 (see glossary for subspace description).

gi k

phmetric subtensor for the photon in subspace

I2∪S2∪T1 (see glossary for subspace description) .

h Planck constant 6.626076 × 10-34 Js, ℏ=h /2

hik metric components for an almost flat spacetime.

me electron mass 9.109390 × 10-31 kg.

mM Maximon mass, mM=

4

2ch

G=6.5×10

−8kg ,

without the factor 4

2 the Planck mass is obtained.

m0 mass of proton or neutron 1.672623 × 10-27 kg

and 1.674929 × 10-27 kg.

Nn number of protons or neutrons in the universe.

re classical electron radius

re=

1

40

e2

mec2=3 × 10

−15m

rge ratio of gravitational and electrostatic forces be-tween two electrons.

v velocity vector of charges flowing in the magneticcoil (see Secs. 3 and 4), some 103 m/s in circumferen-tial direction.

vT bulk velocity vector for rigid rotating ring (torus)(see Sections. 3 and 4), some 103 m/s in circumferentialdirection.

wgp probability amplitude (the square is the couplingcoefficient) for the gravitophoton force

wgp

2 =Ggp

me

2

ℏc=3.87×10

−49 Probability amplitudes

(or coupling amplitudes) can be distance dependent (in-dicated by a prime in [7]).

wgq probability amplitude for the transformation ofgravitophotons and gravitons into a particle corre-sponding to dark energy (rest mass of some 10-33 eV).

wph probability amplitude (the square is the cou-pling coefficient for the electromagnetic force, that isthe fine structure constant α)

wph

2 =1

40

e2

ℏc=

1

137

Z atomic number (number of protons in a nucleus andnumber of electrons in an atom)

Z0 impedance of free space,

Z0=

0

0

≈376.7

α coupling constant for the electromagnetic force orfine structure constant 1/137.

αgp coupling constant for the gravitophoton force .

γ ratio of probabilities for the electromagnetic and thegravitophoton force

=wph

wgp

2

=1.87×1046

µ0 permeability of vacuum 4π × 10-7 N/m2 .

τ Metron area (minimal surface 3Gh/8c3), currentvalue is 6.15×10-70 m2.ω rotation vector (see Figs. 1 and 2).

ω propagation speed of gravitational waves, accord-

ing to Heim ω =4/3 c.

AbbreviationsGRT General Theory of Relativityrhs right hand sidelhs left hand sidely light yearls light secondQED Quantum Electro-DynamicsSRT Special Theory of RelativityVSL Varying Speed of Light

Subscriptse electrongp gravitophotongq from gravitons and gravitophotons into quintessenceph denoting the photon or electrodynamicsM MaximonR to indicate the mass of the rotating ring (torus)sp space

Superscriptsem electromagneticgp gravitophotonph photon T indicates the rotating ring (torus) of mass mR

Note: Since the discussion in this paper is on engineer-ing problems, SI units (Volt, Ampere, Tesla orWeber/m2 ) are used. 1 T = 1 Wb/m2 = 104 G = 104 Oe,where Gauss (applied to B, the magnetic induction vec-tor) and Oersted (applied to H, magnetic field strengthor magnetic intensity vector) are identical. Gauss andOersted are used in the Gaussian system of units. Inthe MKS system, B is measured in Tesla, and H ismeasured in A/m (1A/m = 4π × 10

-3 G). Exact valuesof the physical constants are given in [25].Note: For a conversion from CGS to SI units, the elec-

4 of 25

tric charge and magnetic field are replaced as follows:

ee / 40 and H 4

0H

1 Introduction to Space Propulsion

using the Gravitophoton Field

For effective interplanetary and interstellartravel a revolution in space propulsion technol-ogy is needed. Such breakthrough propulsiontechniques can only emerge from novel phys-ics, i.e., physical theories that deliver a unifica-tion of physics that are consistent and foundedan basic, generally accepted principles. Thetheory by the late B. Heim, developed in thefifties and sixties, and partly published in thefollowing three decades of the last century,seems to be compliant with these requirements.

It also makes a series of predictions with re-gard to cosmology and high energy physicsthat eventually can be checked by experiment.Most important, however, Heim's theory5 pre-dicts two additional interactions (forces) [1,5-7]. These new interactions allow the trans-formation of electromagnetic radiation into agravitational like field, the so called gravito-photon field. This gravitophoton field can beused to both accelerate a material body and toreduce its inertial mass. These effects can serveas the basis for advanced space propulsion

technology, and are dealt with quantitatively inthis paper.

2 Cosmological Consequences from

Heim's Theory

2.1 Origin of the Universe

In this section we will address several funda-mental questions concerning the origin of theuniverse and the creation of matter. SinceHeim's theory answers these questions in a to-tally different way as does currently acceptedbig bang theory, it is of great importance tofind out how to experimentally test those pre-

dictions. Heim's theory also gives an interpre-tation of the physical nature of both mass andinertia (crucial to any advanced space propul-sion system) that should be experimentally veri-

5 To be more precise, Heim's theory was extended to8-dimensions by the first author, [7], to obtain theunification of the four known interactions (forces).In this process, it was found that two additional in-teractions occur, termed the gravitophoton field andthe probability field [1, 7].

fiable. His theory provides a model for a gen-eral ab initio quantized (spacetime) cosmogonyand determines the age of the universe as wellas its size and future dynamic evolution.

2.1.1 A Quantized Cosmogony

Today, the hot big bang model of the universe

has broad acceptance [9]. According to thistheory, the universe originated from an infi-nitely dense and infinitely small space. In orderto describe the very early history of the uni-verse, i.e., its first 10-30 seconds, the inflation-ary model has been developed by Guth (e.g.,[8]), causing an expansion of the diameter ofthe universe by an extraordinary factor of some1050 during this era. This picture, however, isnot consistent with the physical laws that gov-ern the expanded universe. Moreover, the as-sumption of infinities or arbitrarily large num-

bers seems to lack the proper physical basisand is physically inconsistent with the quantumprinciple. This principle needs to be employedwhenever wave packets cannot be isolated.Moreover, according to [4], the probabilitythat the universe had an initial singularity lead-

ing to its current shape is 1 over 1010

123

, mean-

ing the universe should not have developedfrom the big bang singularity. In other words,in order to understand the beginning of thephysical universe, nonphysical assumptionshave to be made and physical events have to beconceived that are in stark contradiction of es-tablished physical principles.

According to Heim, such a hot big bang didnot took place. Instead, a quantized bang didtake place. The quantization principle, that is

Nature counts, i.e., only integers and not float-

ing point numbers are used, was present fromthe very beginning. Thus any kind of singularitywas impossible. In particular, spacetime, as aphysical entity, was quantized from the verybeginning. The quantization principle, being atthe very foundation of physics, is used by Heimin a rigorous way to obtain a unified geo-metrized field theory. At the foundation ofHeim's theory is the derivation of an elemental,

discrete surface quantum, τ , denoted as Met-ron. The value of the Metron size is given by

=G h /2 c2 . It should be noted that in

Heim's theory, gravitational waves propagate

5 of 25

at a speed of ω=4 /3 c , where c denotes the

vacuum speed of light [18].

Therefore, any surface, A∈ ℝ2, cannot be a

point-continuum, but comprises a finite num-

ber, n ∈ ℕ, of these Metrons. The current sur-

face area of a Metron is 6.15×10-70 m2 . The cur-rent granularity of spacetime is extremely fineand therefore has eluded experimental detec-tion.

The most radical concept in Einstein's GeneralTheory of Relativity (GRT) is the removal ofthe idea of gravity as a force. Instead, it is con-

sidered to be a feature of spacetime curvature.

Heim extends this idea to all physical forcesand also employs the equations of GRT, i.e.,their structure, to the microcosm by quantizingthese equations in a higher-dimensional space.This approach leads to a set of eigenvalueequations, whose eigenvalues are the massspectrum of all existing material particles. Aswas outlined in6 [1], the phenomenon of massthus is a purely geometrical feature. In thiscontext, space and time are not the container

for things, but are, due to their dynamic (cy-clic) nature, the things themselves. For a quan-tized unification of gravitation and electromag-netism a 6 dimensional space is needed. If allknown interactions are to be incorporated,

space becomes 8-dimensional, ℝ8. It was de-

scribed in [1] that the metric tensor for the 8-space can be written as a composition of sub-tensors that are functions of the coordinatesfrom these subspaces. There are, however, se-lection rules for the combination of subspacesas described in [1]. For instance, constructing ametric tensor from the three real coordinates of

physical space only, would not result in a met-ric tensor associated with a physical interac-tion. Any metric tensor that can be associated

with a physical interaction is termed Hermetryform by Heim. Heim extends Einstein's idea,namely that the geometry (metric tensor) offour-dimensional spacetime causes gravity, tothe 8-space. Associated with each of these met-ric subtensors is a specific physical interaction,and thus a correspondence principle betweenthe metric and the actual physical interaction is

6 In order to understand the present paper, reference[1], AIAA-2002-2094, needs to be studied. An up-dated version can be freely downloaded, see Refer-ences.

established. This set of metric subtensors, re-sponsible for all known as well as two new

physical interactions, is denoted as poly-metricby Heim [1, 3, 5].

However, before matter (i.e., form and inertia)could come into existence (being representedby the proper metric subtensor), the corre-sponding length scale of the quantized space-

time of the universe had to reach a certainthreshold (minimal) length. In other words, themetric scale had to be fine enough to allow for

the proper curvature in physical space ℝ3.

Since the universe starts out from a quantizedspace, when a single Metron covers the sur-face of the whole universe, there are no prob-lems with the initial conditions in this picture.Evidently the Metron size is time dependent

and has decreased since the quantized bang.

For most of the time of its existence this pri-meval universe 7 possessed only structure, untilits associated elementary length scale satisfieda certain condition. While the universe was ex-

panding, its associated length scale was de-creasing. When this length scale came close tothe Planck length, a phase transition occurred,triggered by fluctuations in the length scale.This phase transition led to the generation of aparticle having the mass of the Planck mass.Heim's formula for the mass spectrum for ele-mentary particles (which contains all particles,i.e. including those with zero rest mass as wellas ponderable particles) has the form

mn=2n

1 /4

2n−11 /2qwith n∈ℕ (1)

and = c h /G . Since we are interested in

the upper bound of the possible energy quanta,i.e., in the maximum mass a particle can carry,

we choose n=1 and let q=1. The exact mass

formula is given in [3, 5 see Section 3]. There-fore, we obtain for the mass of the heaviest

(neutral) particle, denoted as Maximon (it is

interesting to note that its Schwarzschild ra-dius is equal to the range of its attractive gravi-tational field)

7 From now on the word universe will be reserved toour optical universe, which is embedded in this pri-meval universe.

6 of 25

mM=

4

2 =4

2c h

G=6.5×10

−8kg

(2)

It should be noted that in the literature the

Planck mass is defined using ℏ and without the

factor 4

2 .

This kind of phase transition occurred at manylocations in the primeval universe, in a statisti-cally (random) distributed manner, and led tothe creation of many universes, separated from

each other, i.e., no optical signal can reach ouruniverse from such a parallel universe. Theseuniverses should be similar with regard to theirphysical laws, since they are all created by thedecay of Maximon particles. At a time of about10100 s, when the time dependent Metron size,

(t), became sufficiently small, a break of a

global symmetry group must have occurred.Each of these Maximon particles was the cen-ter of a process for the generation of a uni-verse, in which ponderable particles are exist-ing. In other words, our own universe is the re-sult of the decay of one of these Maximons.

This is in contrast to the original or primevaluniverse that only holds geometrical structure.The Maximon particle decayed, cascading intomesons and baryons (this process might be in-terpreted as inflationary universe), with finalproducts as neutrons, protons, and electrons.This avalanche process was accompanied bythe emission of gamma quanta that might per-haps explain the existence of the cosmic back-ground radiation. In addition, particles thatcould be interpreted as vacuum energy mighthave been created (these particles could be in-

terpreted as dark energy because of their verysmall rest mass). Their masses corresponds tothe greatest possible wavelength possible,namely the diameter of our optical universe,

i.e., DO≈4 × 1026 m.

It should be stressed that Heim's cosmogonycomprises a primeval universe that originatedfrom a quantized bang. Our optical universe,that is of much smaller diameter, is embeddedin this primeval universe. This optical universeis one of many other universes, created simul-taneously throughout the primeval universe,caused by the phase transition mentionedabove. This phase transition triggered the pro-

duction of the heaviest elementary particle, theMaximon, whose subsequent decay eventuallylead to our universe. This rapid decay, how-ever, must have taken place by some kind of in-flationary process or through a varying speedof light (VSL), which is allowed in Heim's the-

ory, if connected to an inertial transformation.

It is important to note that during this phase

transition, mass was not conserved. Heim's uni-verse is a purely geometrical universe, and thusthere is a fundamental conservation law basedon length scale considerations. All other physi-cal conservation laws are a consequence of thisfundamental principle, see [5, Chap. 4]. Thecontents of this conservation law can be statedas

mM

4

=Nnm

0

4

(3)

where m0 is the mass of the proton or the neu-tron, i.e., these particles are the final products,

and Nnℕ is the number of protons or neu-

trons in the universe.

It should be noted that mass is a feature of ge-ometry, i.e., mass is connected with a length

scale through the general equation m=ℏ c / l ,

where l is a length and m denotes a mass.

In other words, since mass is inversely propor-tional to a length (for instance, Compton wavelength), a decrease of the length scale leads toan increase in mass. In that sense, the conser-vation of mass is not satisfied, since it is causedby and also depends on the geometry of thespacetime. The generation of a Maximon parti-

cle acts as catalyzer that triggers geometricalchange in neighboring spatial cells (denoted asPlanck), bounded by metronic (Metron) sur-faces. Corresponding to these geometricalchanges is the appearance of material particles.Knowing the end products of the decay chain,we can compute the number of neutrons, thatare finally produced (see Eq. (3)). We thus ob-tain the total mass of our universe, which isembedded in the primeval universe of muchgreater diameter of some 10125 m. This prime-val universe is without mass, but contains a

large number of universes that all have theirorigin in the decay of a Maximon particle. Thecalculation leads to a mass of 3.71×1051 kg ofthe ponderable (possessing a nonzero rest

7 of 25

mass) ordinary, visible matter in our universe.With the generation of the neutron, the corre-sponding interactions that result from the met-

ric subtensors of space ℝ8, give rise to two ad-

ditional, heretofore unknown particles, namelygravitophotons, gph, (rest mass zero) and a very

light (unnamed) particle as well as two newphysical interactions that are the basis for thenovel space propulsion.

2.2 Cosmic Numbers

An interesting fact is that there is a relation be-tween the diameter of the primeval universe, D,

and the Metron size, τ. In addition, all other

physical constants can be expressed by D. Inother words, all cosmic numbers exclusivelydepend upon the current diameter of the pri-meval universe.

The following relations hold:

~D−6/11

ℏ~D−8/11

and G~D−13/11

0~D

13/11 and

0~D

−3/11

(4)

Eqs. (4) show that all relevant empirical physi-cal quantities can be derived from the macro-

scopic structure ℝ3 of the primeval universe.However, the primeval universe has existed foran extremely long time, so that in our own uni-verse all physical constants are practically in-variant, since D, at present (this presence in-cludes the last 15 billion years), is almost con-stant. Eqs. (4) are a direct consequence of the

metrization of the Heim space ℝ8.

The ratio, rge , of the gravitational and electro-static forces between two electrons separatedat a distance r is given by

rg e=∣F g

/Fe∣=4

0Gm

e

e2

(5)

with the mass and charge of the electron givenby (see p. 33 [6])

me=a

3

Ghh

cG and e=−b

h

Z0

(6)

where a and b are real numbers depending on

π, and Z0=

0

0

≈376.7 denotes the imped-

ance of free space. Inserting Eq. (6) into (5)delivers the surprising result that also the ratioof the two forces only depends on the size ofthe Metron

rg e=16

3

3

3 a

b2

2/3 (7)

Since, according to Heim, matter was gener-ated quite recently, compared to the cosmictime scale, τ remained practically unchangedduring the last 15 billion years. Therefore, theintensity of the intra-stellar thermonuclear pro-cesses must have remained unchanged in our

universe, and thus the abundances of 3He and4He cannot be attributed to the change of theratio of gravitational and electromagneticforces.

Cold dark matter provides for some 25% of themass of the universe, but is invisible. Accordingto supersymmetric theories, dark matter parti-cles have a mass of about 1011 eV, heavier thanany known particles. The name of this stableparticle is neutralino, and it would be a leftoverfrom the hot big bang. The existence of darkmatter is inferred from galaxy dynamics [13]that must be present in the vicinity of each gal-axy in order to explain the observed fact thatthe angular frequency of orbiting stars remains

constant. Star systems that are circling the cen-ter of a galaxy are moving too fast with respectto the visible mass of the galaxy. The standardexplanation is that there exists dark matter thatcompensates for the missing mass. However,so far no sign of the necessary amount of darkmatter has ever been observed, neither directlynor indirectly.

As mentioned before, in Heim's theory gravita-tional waves propagate at a speed of

ω=4 /3 c 8, which causes a major difference in

the explanation of the redshift.

Since the attractive range of gravitation is finite

(the gravitational limit depends on the compo-sition of the atomic mass), a photon outsidethis gravitational limit, on its way toward a dis-

8 There is doubt about this relation. ω = c could also

be possible.

8 of 25

tant galaxy, would be going against a repulsiveforce, caused by the total mass ahead of it, andthus experiences an increase of its wavelength.Thus most of the observed redshift must be at-

tributed to this phenomenon, and is not causedby the Doppler effect.

3 Force Equations for Gravitophoton

Fields from poly-metric Tensor in dis-

crete 8-Dimensional Heim Space

3.1 8D Heim Space and Subspaces

In the following, a brief roadmap for the deri-vation of the force equations for the gravito-photon field is presented. Before the gravito-photon field can exert any force on a materialbody, it needs to be generated. This is achievedby creating a strong stationary magnetic field

by a current, for instance, in a superconduct-

ing coil, above which a material ring (torus, fly-wheel) is rotated at a high circumferentialspeed of some 103 m/s.

The derivation for the gravitophoton force pro-ceeds in three stages. First, the metric tensorfor moving electric charges is derived usingmodified Einsteinian field equations, demon-

strating that from this tensor the electromag-netic Lorentz force can be deduced. Second,the gravitophoton potential generated in the ro-tating torus is presented, and third, the physi-cal model for the generation gravitophotonfield and its interaction are shown. These equa-tions are the physical guidelines for the experi-mental setup, as depicted in Fig. 1. In Section4, the experiment to measure the effect of thegravitophoton field on the mass of the rotatingtorus is outlined in detail.

9 of 25

Figure 1: The picture shows the physical principle for generating a gravitophoton field. A ring or a torus(flywheel) of a given mass is rapidly rotating in a magnetic field created by a current loop as depicted.The charges in the current loop (lower part), which is below the rotating mass, are moving in the opposite

direction, generating a magnetic field H (or magnetic induction B=μ0H, in practice the velocity of the

charges in the current loop are of importance), as indicated in the upper right. Using cylindrical coordi-

nates r, θ, and z, there should be a gravitophoton force in the negative z-direction (downward), according

to Eq. (31). This force component is the result of the circumferential velocity of the rotating ring and theradial component of the H field. There also exists a gravitophoton force component in radial direction, r,that is, however, balanced by the forces within the material of the rotating mass, and therefore is of no in-terest to propulsion purposes.

Br

B I

We showed in [1] that the metric tensor in 8-space comprises several subtensors, such thateach subtensor is responsible for a differentphysical interaction. In the same way the met-ric tensor of Einstein's GRT acts as a tensorpotential for gravitation, the additional subten-

sors constructed from the quantized Heim

space, ℝ8, are responsible for all physical in-

teractions in our universe. In other words, the

subspaces in ℝ8 , in which the individual metric

tensors are specified, are the cause of physicalforces. In that respect, we can speak of a com-

pletely geometrized theory. In Heim space ℝ8

four groups of coordinates are discerned:

1. ℝ 3 , spatial coordinates (real) (1,

2, 3),

2. T1, time coordinate (imaginary) (4),

3. S2, entelechial and aeonic coordinates(imaginary) (5,6),

4. I2, information coordinates (imaginary)(7,

8).

In [1] the rules of combining the various coor-dinates to form the respective metric subten-

sors were described. Coordinates 5,..,

8are

termed trans-coordinates. There are, however,no extra space dimensions. All trans-coordi-nates are imaginary. Any metric subtensor, inorder to describe a physical interaction, mustcontain coordinates from subspaces S2 or I2.

Although a space ℝ8 is considered, all measur-

able events take place in spacetime ℝ4. We

consider therefore three types of coordinates,namely Euclidean coordinates x, and non-

Euclidean (curvilinear) coordinates η in physi-

cal spacetime ℝ4 , while denote coordinates

in Heim space ℝ8.

The physical nature of the coordinates is such

that their mapping into our spacetime gives rise

to all known physical fields in ℝ4. The non-

Euclidean structure of these coordinates is theunderlying cause of all observed physical fields.

The following coordinate transformation, Eq.(8), therefore represents the physical fact that

Heim space ℝ8 directly influences the events in

four-dimensional spacetime. In Heim's termi-nology, all known physical fields are repre-

sented by their respective hermetry forms (met-ric tensor in admissible subspaces) (see glos-sary). Since, according to Heim, the structureof Einstein's equations, supplemented by aquantization condition, is the fundamental setof equations governing physical interactions in

ℝ8 , the various metric subtensors can be used

to determine these physical interactions. Thegravitophoton field (hermetry form H

11, see [1]

and glossary), under the action of conversion

operator S1

(glossary), is transformed into the

so called probability field, described by herme-try form H

10[1]. This can formally be written

as S1

H11

= H10

. The physical interpretation of

this conversion could be as follows: one darkenergy particle (quintessence) is producedfrom one gravitophoton and one graviton. Thismeans that gravitophotons and gravitons usingthe transformation field wgq, are transformedinto so called dark energy particles, q, that hasa mass of some 10-33 eV, which is in goodagreement with recent findings from [13].

3.2 The metric Tensor in 8D Heim Space

In GRT a coordinate transformation betweenEuclidean and curvilinear coordinates in con-

tinuous spacetime ℝ4 is considered. In Heim's

theory a similar transformation is used, ac-counting, however, for the influence of the 8D

space on events in our spacetime ℝ4. There-

fore, a third set of coordinates, , is involved in

the transformation.

Making use of the general coordinate transfor-

mation xm

i , one obtains for the metric

tensor

gi k=∂ x

m

∂

∂

∂ i

∂ xm

∂

∂

∂ k

(8)

where indices α, β = 1,...,8 and i, m, k = 1,...,4.The Einstein summation convention is used,that is, indices occurring twice are summedover. The quantum aspect of the theory is onlyneeded for the derivation of the spectrum ofelementary particles. For the purpose of thispaper, a continuous transformation can beused.

10 of 25

This metric tensor can be represented, definingso called fundamental kernels, Eq. (9). Thevarious subtensors are characterized by their

fundamental kernels, i m

. It can be shown

that the respective Christoffel symbols (termedcondensors by Heim), derived from their met-ric subtensors, have tensor character, exceptfor gravitation. According to Heim there are socalled sieve (conversion) operators that can beapplied to a hermetry form with the effect thatone or more of the fundamental kernels be-come Euclidean, i.e., the resulting metric nowdescribes a different physical field, marked bythe new hermetry form. For instance, this is thecase for the electromagnetic interaction, Eq.(12), that contains the metric for the gravito-

photon interaction. If the part termed gi k

em,

could, by some experimental means, be madeEuclidean, the gravitophoton force would oc-cur.

In other words, in Heim's theory operators ex-ist that convert one hermetry form into anotherone. Our main interest in this paper is the inter-action between electrodynamics and the gravi-tational like field, the gravitophoton field thatcan be used to both accelerate a material bodyas well as to reduce its inertial mass. Such anacceleration does not exist in GRT, neither is aLorentz transformation based on a reduced in-ertial mass conceivable in the framework ofGRT. Thus, within GRT, there is no way for amaterial body flying at superluminal speed.

Contrary to GRT, in Heim's theory the exis-tence of the gravitophoton interaction, reduc-ing the inertia of a material body, does allowfor superluminal speeds without violating GRT.

3.3 The Metric Tensor Describing Photons

Next, the metric tensor is separated into severalsubtensors. Using fundamental kernels, themetric tensor can be written in the form

gi k= ∑

,=1

8

i m

m k

=: ∑

, =1

8

gi k

, (9)

where we defined the gi k

different from

[1], this being a matter of convenience only. Aswe showed in [1, Eq. 12], the hermetry form

H5=H

5I2 , S 2

,T1 9 is responsible for pho-

tons and depends on the subspaces I2, S2, and

T1 with coordinates 4, 5,...,8. The respective

metric tensor for photons is

gi k

ph= ∑

,=4

8

gi k

(10)

where the superscript ph 10, denotes the metricsubtensor for the photon. The new gravitopho-ton field that originates from the poly-metric ofHeim's extended theory [1, 7] is described by

hermetry form H11, H11=H

11I

2

, S2

depend-

ing on the subspaces I2 and S2 with coordinates

5,...,8. Its metric tensor is given by

gi k

gp = ∑

, =5

8

gi k

(11)

where the superscript gp denotes the metricsubtensor for the gravitophoton. Comparisonof Eqs. (10) and (11) leads to

gi k

ph =g

i k

gpg

i k

em (12)

where gi k

emis defined by

gi k

em := ∑

, =5

8

gi k

4 g

i k

4 g

i k

4 4 (13)

and is part of the electromagnetic interactionand thus the index em was used. However, itshould be noted that this part of the metric ten-sor has no physical meaning. It is only the met-ric tensors of Eqs. (11 and 12) that correspondto physical interactions. This interpretation be-comes clear, since Eq. (12) shows that herme-try form H5, denoted as the photon field, actu-

ally contains the metric of the gravitophotonfield. We interpret this part of the metric in Eq.(10) as coupling potential between the electro-magnetic and the gravitophoton field. It is ex-actly the metric of the gravitophoton particle.However, there is no gravitophoton interaction

9 Contrary to [1], subspaces are denoted by super-scripts instead of subscripts.

10 We adopt the convention of using an abbreviation

as a superscript to denote the physical meaning ofvarious metric subtensors, instead of using symbolslike ' , '' or ~ etc., for the sake of clarity.

11 of 25

coming from Eq. (12). Only by employing theproper sieve (conversion) operator, the metricof Eq. (12) can be converted into the metric ofthe gravitophoton field. Only then, a gravito-photon field would occur. In that respect, sepa-rating the metric for the photon, Eq. (12), into

two terms is somewhat misleading. It does notmean that the electromagnetic field comprises agravitophoton field and a second part. It doesshow, however, that conversion between thefields is mathematically possible. According toHeim's extended theory (from 6 to 8 dimen-sions, see [7]) there exists a transition operatorS2 (not to be confused with subspace S2) thatcauses this photon-gravitophoton interaction,that is, a transformation of a photon into agravitophoton, and is symbolically written asS2 H5 = H11.

Since only metric tensors (geometry) were con-sidered so far, no guidelines are available how

this conversion can be realized by experiment.How this purely mathematical transformationcan be brought into concrete physical existenceis a most important question, and is addressedin sections 3.4 to 3.7.

3.4 The Metric Tensor for Moving ChargedParticles

Making use of the coordinate transformation

xm

i one obtains from Eq. (10) the

representation for gi k

in the following form

gi k

=

∂ xm

∂

∂

∂ i

∂ xm

∂

∂

∂ k

(14)

For weak gravitational fields, spacetime is al-

most flat, so the contribution of∂

4

∂ 4

is large

in comparison to ∂

4

∂ l

, l=1,2,3 . Introducing

the abbreviations

hm 4:=∑

=5

8

gm 4

4

h4 l:=∑

=5

8

g4 l

4

h4 4:=g

4 4

4 4

hm l:= ∑

, =5

8

gm l

(15)

where m=1,...,4 and l=1,2,3.

Now the first stage of the derivation can beperformed. It is investigated whether there ex-ist modified Einsteinian field equations that

provide a metric for describing the motion ofelectric charges. The important difference toEinstein's GRT is that a transformation

ℝ4 ℝ8 ℝ4 takes place, resulting in a metric

tensor not available in GRT. Einstein's fieldequations are therefore used as structural equa-tions only in a discrete 8D Heim space. In thisspace a metric exists that is rich enough to ac-count for the four fundamental forces and theirmediator particles, but, as was stated before,gives rise to two additional interactions.

Ri k= T

i k−1

2gi kT (16)

And T=Tk

k

. In the gravitational case, κ is of

the form =8G

c4

, while for our considera-

tions κ needs to be adjusted to the electromag-netic hermetry form.

For weak electrodynamic and also for weakgravitational fields, spacetime must be almostflat, so one obtains

gi k=g

i k

0 hi k

where g0 0

0 =−1 and

gi i

0 =1 , and all other components are 0.

The hi k are small quantities whose products

are negligible. It is well known that the linear-ized Ricci tensor can be written as (see, for in-stance, [17], p. 298)

Ri k=½□2 hi k (17)

12 of 25

where □2 = g 0 i k ∂

2

∂ xi∂ x

k=∇

2−

1

c2

∂2

∂ t2

is

the D'Alembertian operator. Einstein's equa-tions can thus be written in the form

□2 hi k = ∂

2hi k

∂ xn

2=2 T

i k−

1

2gi kT (18)

with summation over index n. Eq. (18) is an in-homogeneous wave equation, whose solutionis given by the so called retarded potentials.

The components of the stress-energy-momen-tum tensor are of the following form [17]

Ti k= v

ivk

and T= c2

since vi, v

k≪c .

For a point-like mass M, the following relationholds

∫Ti kdV=M v

iv

k. The partial differential

equation (18) can be solved by means of Fou-

rier transformation (see, for instance [17], page217). This eventually leads to the result

hi k=

1

40

eQ

mec

2r

vi

c

vk

c(19)

where the charge Q is a multiple of the electroncharge and i,k =1, 2, 3. It should be noted thathik are dimensionless quantities.

It is well known from QED that the electriccharge is proportional to the coupling ampli-tude (or probability amplitude), see [26] Chap

13, e= 40ℏ c and thus Eq. (19) can

be written as

hi k

=Q

e

ℏ

mec

1

r

vi

c

vk

c(20)

The meaning of the above equations is that themovement of charges causes a metric hik, as de-scribed in Eq. (19). From Fig. 1 it can be seenthat a charge Q (here the simplification is

made describing charges in the magnetic coil bya single charge Q) is moving with velocitycomponents vi in a magnetic coil (current).

3.5 The Metric Tensor for Gravitophotons

In the second stage, the gravitophoton metrictensor can be directly found from Eq. (20), ob-

serving that the metric for the moving chargedparticle is proportional to the fine structureconstant α. In the same way, it is concludedthat the metric for the gravitophoton tensorcomponents is proportional to αgp, the couplingconstant for the gravitophoton force. There-

fore, the ratio of the metric for a movingcharged particle and a gravitophoton particleequals α /αgp.

Thus the metric for the gravitophoton particleis given by

hi k

=gp

Q

e

ℏ

mec

1

r

vi

c

vk

c(21)

where :=wph

2

=1

40

e2

ℏ cand

gp:=w

gp

2=G

gp

me

2

ℏ cInserting αgp, Eq.(21)

takes the form

hi k=G

gp

Q

e

me

c2

1

r

vi

c

vk

c (22)

The potential derived from the metric of Eq.

(22) is denoted as gravitophoton potential.The constant for the gravitophoton field, Ggp, isanalogous to the gravitational constant G.

The following relation holds

Ggp=w

gp

wg

2

G≈

1

672G .

It is important to note that from the metric ten-sor components hik,, Eqs. (20, 22), respectivepotentials for a moving charged particle and agravitophoton particle can be obtained. How-

ever, the ratio between these two potentials isequal to the ratio of their corresponding prob-ability amplitudes, and not of their couplingconstants (probabilities). The reason for this isstraightforward to observe. For instance, in Eq.(20), the product of two charges e and Q oc-

curs. Their product is proportional to α. The

respective electrostatic potential contains a

single charge only, and thus the factor is .

13 of 25

It is important to observe that the metric ofEq.(20) corresponds to a Lorentz force, where

the first part of the metric, eQ / r vi/ c , is as-

sociated with the magnetic field and the second

part, vk

T/ c with the velocity of a material

body (in our experiment this is the rotatingtorus). Since the gravitophoton metric differsfrom the metric of Eq. (20) only by the factorαgp / α the force that a single gravitophoton ex-

erts is given by

Fgp=

gp

e

0vT×H (23)

If the force acts on an electron, me has to beused, for a proton the mass mp has to be in-serted into αgp. Gravitophotons can either beabsorbed by electrons or by protons.

In order to get an appreciable gravitophotonforce, a large number of gravitophotons pervolume and time unit needs to be generated,see Figs. 1 and 2 for the experimental setup.The interesting fact is that Eq. (23) allows bothto control the magnitude and the sign of the

gravitophoton force. The gravitational actionof a gravitophoton is similar to a graviton, ex-cept that gravitophotons can be generatedthrough electromagnetic interaction and theforce can reverse sign.

The question arises under which conditionsgravitophoton fields exist that can propagateinto the surrounding space. Following Heim[7], a brief description how to compute theprobability amplitudes for the photon conver-sion into gravitophotons was presented in [1].Similar to a dielectric, quantum theory andrelativity require that for short times so calledvirtual pairs of electrons and positrons may be

present in the vacuum. If there is an electricfield present, suppose in form of a pointcharge, the positive charges of the virtual parti-cles are displaced relative to the negative ones,known as polarization of the vacuum. For largedistances, r, this leads to a shielding of thepoint charge. In a macroscopic experiment onesees this shielded charge, whereas only for verysmall r, the bare, unshielded charge becomesvisible. The scale of the spatial displacement is

set by the length ℏ/mec, associated with the

mass of the electron. The time of existence for

a virtual electron-positron pair is about ℏ/mec2,

which is some 10-21 s.

3.6 The Physical Picture: From Photons toGravitophotons

Before the force equations for the gravitopho-ton field are derived, i.e., their interaction with

a material body, it is appropriate to present aqualitative picture of the transmutation of pho-tons into gravitophotons. The gravitophotonfield is a gravitational like field, except that itcan be repulsive or attractive. It can both accel-erate a material body and reduce its inertia.Any propulsion device therefore would be atwo-stage system, first accelerating the bodyand then reducing the inertial mass of the body.

The mathematical description for the conver-sion process is given by the first equation inEq. (24). In the following, the physical mecha-nism is presented, responsible for the conver-sion of photons into gravitophotons.

The physical mechanism for the generation ofthe postulated gravitophoton particles is basedon the concept of vacuum polarization from

QED. In QED the vacuum behaves like a di-electric absorbing and producing virtual parti-cles.

The high current in the possibly superconduct-

ing coil produces a magnetic field H where vk isthe speed of the charge in the current loop or

coil (Figs. 1 or 2, respectively). Together withthe velocity vk

T of the rotating torus, this mag-netic field generates the conversion potentialaccording to Eq. (22). This potential interactswith the virtual photons generated in the vicin-ity of each of the atomic nuclei (field point),comprising the material of the rotating torus.

The nuclei (positive charge) are continuallyemitting and absorbing virtual photons. Someof the photons create electron-positron pairs

that subsequently annihilate. Virtual electronsare attracted, positrons are repelled by thepositive charge of a nucleus, resulting in vac-uum polarization. Thus the real positive chargeis partially shielded and cannot be measured,until one is inside the shielding region. Thepositive charge is higher closer to the nucleus,since the shielding effect is reduced. Theshielding distance is given by the Comptonwavelength of the electron, which is

14 of 25

C=

ℏ

mec=2.43×10

−12

m . Therefore the prob-

ability amplitude, wph, which is proportional tothe electron charge, will increase at distances

smaller than λC. Virtual photons are the inter-

action particles between the electric field of anucleus and the virtual electron. In addition, atthe location of each nucleus, the coupling po-tential of Eq. (22) is effective. The higher thedifference between the unshielded and theshielded probability amplitudes, wph, the higherthe number of virtual photons, as shown in thesecond equation in Eq. (24). The number of

virtual electrons is proportional to the numberof virtual photons. Gravitophotons are emittedby the virtual electrons. Thus the number ofgravitophotons should be proportional to thedifference in the probability amplitudes. Hence,the conversion from photons to gravitophotonsmust take place close to the nucleus within the

the shielding distance λC. In order to achieve

this goal the conversion potential must bestrong enough to generate a distance r, meas-ured from the nucleus to the location of the vir-

tual electron, that is much smaller than λC. This

condition sets a strict requirement for the pa-

rameters determining the magnitude of the con-version potential, namely the velocities vk

T

and vk.

The gravitophotons are subsequently absorbedby the protons in the torus which have a largeabsorption cross section compared to the elec-trons. In the non-relativistic case, the scatteringcross section for photon-electron interaction is

given by =8

3re

2 , see [27], where re is the

classical electron radius, given by

re=

1

40

e2

mec2=w

ph

2 ℏ

mec

. For gravitophotons

wph has to be replaced by Nwgp, since in theconversion process from photon to gravitopho-ton, N gravitophotons are generated accordingto Eq. (24). It should be noted that the factorN does not occur in Eq. (23), since it dependson the conversion process. Thus the absorptioncross section for a gravitophoton particle by amaterial particle (here the electron is used) is

given as gp=8

3Nw gp

4 ℏ

mec

2

. However,

if the absorption is by a proton, the electronmass me is to be replaced by the proton massmp. Therefore the absorption cross section of aproton is larger by the factor mp/me. Hence, theabsorption of gravitophotons by electrons canbe neglected.

In the next section, this formula will be used tocalculate the strength of the gravitophoton

field. To increase the strength of the interac-tion, a material containing hydrogen atomsshould be used, because of the small value of r.

3.7 Force Equation on a Material Body ex-erted by Gravitophotons

In the third and final stage of our derivation,the bulk equations of motion for a materialbody exerted by gravitophoton particles needto be determined. To this end, Eqs. (24) areused

wphr −w

=Nw

gp

wphr −w

ph=Aw

ph

(24)

The first equation in (24) describes the produc-tion of N2 gravitophoton particles11 from pho-tons with respect to so called conversion po-

tential wκ. This equation is obtained from

Heim's theory in 8D space, in combination witha set algorithm, and predicts the conversion ofphotons into gravitophoton particles. In par-ticular, it is emphasized that the metric for thephoton, Eq. (12), already contains the metricfor the gravitophoton particle, Eq. (11)12.

The second equation in (24), obtained fromQED, see [27], describes the screening of thecharge of a nucleus by vacuum polarization

through virtual electron-positron pair produc-tion. It should be noted that probability ampli-tudes correspond to physical potentials [26].

The coupling amplitude wphr is the prob-

ability amplitude depending on the distancefrom the nucleus, and describes the partially

11 The factor N2 results from the fact that in Eq. (24)

probability amplitudes are considered, but the gen-eration of particles depends on actual probabilities.

12 This essential equation is stated without proof. Thetheory of the coupling constants is too comprehen-sive to be treated in this paper. However, since themetric of the photon contains the metric of thegravitophoton, this could be considered as somekind of evidence for the possibility of such a conver-sion.

15 of 25

shielded potential of the nucleus [26], Chap.13. At distances larger than the Compton

wavelength, w phr =

1

137which is the square

root of the fine structure constant α.

Next, the potentials corresponding to theprobability amplitudes need to be identified :

1. wphr is interpreted as the potential of

the nucleus seen by a virtual electron,1

40

Ze

r .

2. Since the potential in 1. is a scalar potential,the potential representing wκ should also bea scalar potential. From the discussion inSection 3.5 the following potential is as-

sumed:

1

40

1

mec2

eQ

R

vi

c

vi

T

c(25)

This leads to the following equation betweenthe associated potentials

1

40

1

mec2e Ze

r−eQ

R

vi

c

vi

T

c =

Nwgp

wph

gp

Ggp

Q

e

me

c2

1

R

vi

c

vi

T

c

(26)

where the factor α/αgp is the ratio of the cou-

pling constants of the electromagnetic and thegravitophoton force. Since the rhs of Eq. (26)

is very small, it is treated as 0. This leads to theequation for r

r≈Z e

QR

c

vi

c

vi

T (27)

From Eq. (27) it is obvious that in order to

have a small value of r, that is r < λC, the total

charge Q, the velocity of the charges vi, and therotation speed of vi

T of the torus should be cho-sen as large as possible. The torus should alsohave material that contains hydrogen atoms.

In the next step, the unknown gravitophotonproduction factor Nwgp has to be calculated.This factor comprises two terms, namely theabsorption of the gravitophoton by an electronor by a proton. The absorption of gravitopho-tons by electrons is not taken into account, be-

cause of the much smaller absorption cross sec-tion.

From the physical model outlined in Section3.7, it is concluded that the number of gravito-photons emitted is proportional to the numberof virtual electrons, which depend direcly onthe difference of the coupling amplitudes, sec-ond equation in (24). Therefore, it is assumedthat the relation

Nwgp

= Awph (28)

holds. Here the proton mass is used. The func-tion A is obtained from radiation correction ,

A=2

3 ∫

1

∞

e−2m

er

11

22

2−1

1/2/

2d ,

i.e., from the virtual electron-positron pairshielding of the charge of the nucleus, for aderivation see [27].

For the emission of a gravitophoton by a vir-tual electron, the coupling constant is given by

wgpe

2=G

gp

me

2

ℏc. For the absorption process the

coupling constant has the form wgpa

2

=Ggpm

p

me

ℏc

. Using the absorption cross section for protons

from Section 3.7, the probability for this proc-ess is obtained as

w=32

3Nwgp

4 ℏ

mpc

2

d

d0

3Z (29)

With Eq. (29) the total force on the rotatingtorus can be determined. The first equation in(24) describes the conversion of photons into

N gravitophotons. Therefore, αgp needs to be

replaced by N2 αgp .

Fgp=w N

2

gp

e

0vT×H (30)

16 of 25

Multiplying Eq. (30) by probability w from Eq.(29) results in the total force of the gravitopho-tons on the rotating body

Fgp=

pe

0vT×H (31)

where p

indicates that only proton absorp-

tion processes were considered. From Eqs.

(29) and (30) p

is determined as

p=

32

3Nw

gpe

wph

2

Nwgpa4 ℏ

mpc

2

d

d0

3Z ¿

(32)

Λp (dimensionless) is a highly nonlinear func-

tion of the probability amplitude of the gravito-photon particle. d is the diameter of the torus,d0 the diameter of the atom in its ground state,and Z denotes the atomic number of the atom.For the experiment outlined in Section 4 it isassumed that a material ring of a given mass ofsome 100 kg (torus) is rotating in the x-y (orx1-x2) plane. Below that ring, a current I carry-ing a total charge Q is flowing through a mag-netic coil, located in the x-y plane (see Fig. 12).According to Eq. (31) there should be an inter-

action between the moving charge Q (currentthrough the magnetic coil) and the movingcharges (electrons) of the rotation ring.

We therefore have shown, starting from Eq.(12), which describes the underlying metric forthe electromagnetic interaction, that this metriccan actually be used to produce the so called

Heim-Lorentz force. Hence, the correspon-dence between metric and physical interactionhas been demonstrated.

4 Experiment to determining the dou-

ble Nature of the Gravitophoton field

The experiment comprises a rotating ring(torus) of a certain mass and a super-conduct-ing magnetic coil, whose inner radius is small in

comparison to its outer radius. Let the mag-netic coil be located at a distance, dT, below thetorus. The circumferential speed, vT, of the

torus is supposed to be 103 m/s. The coordinatesystem is chosen such that the midpoint of themagnetic coil is at the origin. The torus rotatesin the x-y plane, with the z-axis pointing up-wards. It is assumed that the volume of the coilis sufficiently small, so that the retardation ef-

fect for all points within its volume may be ne-glected.

The gravitophoton force on the rotating torusof mass mR is given by Eq.(32), which is sur-prisingly similar to the electromagnetic Lorentz

force. It was termed the Heim-Lorentz forceby these authors. This equation describes thecomplete conversion of magnetic field energyinto kinetic energy. This equation is the basisfor the gedanken-experiment depicted in Fig. 1.

It should be noted that the sign of the forcedepends on the direction of the velocity of therotating body. As a rule, the velocity of thecharges in the current loop and the circumfer-ential velocity of the rotating ring must be inopposite directions, see Fig. 1.

As numerical examples, four cases are investi-gated. It turns out that the Heim-Lorentz forceis strongly nonlinear, and without proper ad-justment of current I, rotation speed, and theproper number of turns for the magnetic coil, it

cannot normally be observed.

Three different magnetic coils are considered,with 104, 105, and 106 turns. A wire-thicknessof 1 mm (10-3 m) is chosen. The resultingmagnetic induction is 1.2 T, 6.3 T, 20 T, and50 T. The thickness of the torus (ring) is 0.05m with a mass of 100 kg.

It is known that the current density decreasesexponentially within a conductor. The skin

depth, δ, measures, for a particular material

and frequency, the depth at which the current

density in the material has decreased to 1/e,compared to the value at the surface. The value10-7 m was chosen for our example calculation.Together with an electron velocity of 103 m/s,

this results in a total charge Q of 4×105 As.

With a value of 0.5 m for dR, Table shows thegravitophoton force acting on the rotatingtorus. Using a magnetic field strength of some20 T, a force of some 44 N is obtained for aring rotating at a speed of 103 m/s and a mass

17 of 25

18 of 25

Figure 2: Instead of a simple current loop, a coil with many turns can be used. The field of this coil gives riseto an inhomogeneous magnetic field that has a radial field component. The radial component and the gradi-

ent in z-direction are related through Hr=−

r

2

∂ Hz

∂ z. It should be noted, however, that if the ring possesses a

magnetic moment, M, there is a magnetic force in the z-direction of magnitude F=M∂ H

z

∂ z. This force does

not depend on the rotation of the ring. For a diamagnetic material the force acts in the positive z-direction(up), while para- and ferromagnetic materials are drawn toward the region of increasing magnetic fieldstrength (down). The gravitophoton force comes into play as soon as the ring starts rotating, and superim-poses these effects. Perhaps equipment used to measuring magnetic moments can be employed to determinethe gravitophoton force. For instance, if a diamagnetic substance is used, the gravitophoton force (down)could be used to balance the magnetic force, so that the resulting force is 0. From Refs. [23 and 24] it is found

that a quartz sample (SiO2, diamagnetic) of a mass of 10-3 kg experiences a force of 1.6 × 10-4 N in a field of

Bz=1.8 T and a gradient of dBz/dz=17 T/m. A calcium sample (paramagnetic) of the same mass would be

subject to a force of -7.2 × 10-4 N. It is important that the material of the rotating ring is an insulator to avoid

eddy currents.

r

Br

B I

N

n Nwgp

0H

(T)

Fgp

(N)

104

2.6× 10-14

2.0 1.4×10-58

105

1.1 ×10-5

6.3 2.8 ×10-6

106

1.5×10-4

20.0 4.4 ×101

106

2.5×10-4

50.0 2.7 ×103

Table1: The right most column shows the total gravitophotonforce in Newton that would act on the rotating ring. The forceresults from both processes, namely the absorption of thegravitophoton by an electron and a proton. The absorptionthrough a proton results in a much larger force, so that inprinciple the interaction of a gravitophoton with an electron,regardless whether real or virtual, can be neglected. Thenumber of turns of the magnetic coil is denoted by n, themagnetic field is given in Tesla, and the current through thecoil is 100 A, except for the last row where 250 A were used.

The mass of the rotating torus is 100 kg, its thickness, d (di-ameter) 0.05m, and its circumferential speed is 103 m/s. Thewire cross section is 1 mm2. The meaning of the probabilityamplitude is given in the text. Because of the highly nonlin-

ear character of Λp with respect to Nwgp, the resulting force

varies from actually 0 to 2.7g. It should be mentioned thatthere are type II superconductors that can sustain a magneticfield of up to 34 T13.

The experiment described is based on two wellknown ingredients, namely a magnetic coil and

a rotating body of mass or flywheel. The inter-action is between the charges (electrons) flow-ing in the magnetic coil and the electrons of theatoms, rotating with the flywheel.

There is perhaps another way to measure thegravitophoton field, namely directly on theatomic scale. Eq. (23) describes the gravito-photon force between a single electron and acharge Q. Let us consider a single atom in a socalled magnetic micro trap [21]. This trap com-prises micro-electromagnets with micro-fabri-

cated Cu-wires of a width of several μm

through which a current is flowing. Special po-tentials can be produced to manipulate theatom that can move in the axial direction. Ac-

cording to Eq. (31) a gravitophoton force be-tween the charges in the Cu-wire and the atomshould occur. However, there is also a forceacting on the magnetic moment of the atom be-cause of the inhomogeneous magnetic field,which would superimpose the gravitophotonfield. This experiment needs to be considered in

13 The most recent analysis, too late to be included inthis paper, shows that substantially larger forcesmay occur if the recoil virtual electrons are subjectto due to emission of gravitophotons is included inthe momentum budget.

more detail in order to find out, whether thegravitophoton force could be detected.

Another possible source for the gravitophotonfield is on the cosmological scale. It is reportedthat neutron stars that are pulsars have a mag-netic induction of some 108 T. Atoms or mole-cules moving in this very strong field should besubject to a gravitophoton force, resulting from

Eq. (31). The question of course is, how to ac-tually observe this effect, separated from allother forces. A neutron star of some 10 km di-ameter and a mass of about three times thesun's mass, may rotate rapidly at hundreds ofrevolutions per second [22].

5 Performance of the Gravitophoton

Field as a Propulsion Device

In the following, we will do two gedanken-ex-periments for a gravitophoton propulsion de-vice. First, we consider an interplanetary mis-sion to Mars. Second, an interstellar mission toa planet 100 ly away from earth is discussed.

5.1 Interplanetary Mission

In order to use a gravitophoton device as apropulsion system that can launch a spacecraftfrom the surface of the earth into outer space,the gravitophoton field that acts normal to theplane of rotation should be able to lift thespacecraft, i.e., the acceleration of the space-craft must be larger than 1 g (9.8 m/s2). Usingthese values, the magnetic induction of 50 Tshould be able to launch a spacecraft with a

mass of 3× 104 kg, accounting for losses, see

Table 1. For a mission to Mars, whose average

distance from earth is some 900 ls (light sec-onds), which amounts to a distance s of

2.7×1011 m. A non relativistic calculation leads

to a flight time t=2s

g=1.6×10

5s for half

of the distance, and a total flight time of 3.7

days. The peak velocity of the spacecraft

would be some 1.5×106 m/s, which is, com-

pared to chemical propulsion a very high, butstill non-relativistic speed. For this interplane-tary mission, only the accelerative nature of thegravitophoton field has been used. In order todo an interstellar mission, superluminal speedsare necessary, which can only be achieved bythe so called inertial transformation, where the

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gravitophoton field is used to reduce the iner-tial mass of the spacecraft by converting elec-tromagnetic radiation into gravitophotons.

5.2 Interstellar Mission

The interstellar mission to a planet some 100 lyaway from earth would take place in twostages. In stage one, lasting 30 days, the space-craft reaches a speed of some 0.1c, using gravi-tophoton acceleration. In stage two, the inertialmass of the spacecraft is reduced by a factor of10-4. To this end a magnetic field is needed thatis of the same magnitude as during the accel-eration phase. In addition, fine tuning is neededto reduce the gravitational field of the space-craft. Because the ratio of the initial and the re-duced inertial masses is proportional to the ra-tio of the final and initial velocities of thespacecraft (see [1], which follows directly fromthe conservation of momentum and energy),the final speed of the spacecraft is 103 c. Thespacecraft would travel in some kind of hyper-space in which the speed of light c' = 104 c.The total travel time would be 0.1 y + 2×30 d,which is approximately 3 months. A return tripwould be feasible in 6 months time. A majoradvantage would be that during 4 months, theastronauts would be subjected to an accelera-tion of 1 g.

The question arises of what will happen to theastronaut flying at a cruising speed close orhigher to the vacuum speed of light and even-tually flying back to earth. The so called twinparadox should not play a role, since the de-nominator in the Lorentz transformation does

not change, because v'/ c

'=v / c where

primed quantities denoted values in hyper-space. This relation follows directly from mo-mentum conservation. Thus the question whichtwin aged more is not a relevant one.

SRT introduces the vacuum speed of light, c,as the upper speed limit. One might argue thatexceeding the vacuum speed of light during aninterstellar flight, might cause a change in the

uncertainty relation x px≥ ℏ . Inserting

the value mc for the uncertainty of the momen-

tum leads to x≥ℏ/mc , it is not the value

of c, but the total momentum that restricts theuncertainty in the location. However, as was

said before, an inertial transformation leavesthe momentum of the vehicle unchanged.

Conclusions and Future Work

In the present paper an outline of some of thefeatures of Heim's fully geometrized, unifiedfield theory was given. The most important as-

pect is his discrete spacetime in 8 dimensions,with a minimal (quantized) surface element, theso called Metron. As a physical consequence,the universe started in a quantized bang, withwell determined initial conditions. During theexpansion of this primeval universe, the associ-ated length scale became smaller. When thelength scale reached the value of the so calledPlanck length, matter could be created, and aphase transition took place. According toHeim's formula for the mass spectrum for allexisting particles, the heaviest particle, the

Maximon, associated with this length scale,was generated. This effect took place at thesame time at many locations in the primevaluniverse. The Maximon rapidly decayed, withthe stable particles, namely electrons and pro-tons along with high energy photons as endproducts. This decay process took place assome kind of inflationary process. Each ofthese Maximons was the cause of a new uni-verse. Matter was not conserved, instead, theinflationary process was governed by Eq. (3)that allows to calculate the mass of our present

universe, which is embedded in the primevaluniverse.

Heim's theory is an extension of Einstein's the-ory in that each physical interaction and its as-sociated interaction particle is described in aquantized higher dimensional space. In otherwords, all forces and all material particles areof geometric origin. Elementary particles pos-sess a complex dynamic structure that also ex-hibits zones within such a structure. In the 8-dimensional space, termed Heim space by theauthors, several metric subtensors can beformed. Each of these subtensors, called aHermetry form, is responsible of a physical in-

teraction or interaction particle [1]. When thesemetric subtensors are formed, two new addi-tional interactions along with their interactingparticles occur. One of these particles, termedthe gravitophoton, is responsible for the reduc-tion of the inertial mass of a material body(spacecraft). This physical effect would lead to

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an inertial transformation in the Lorentz matrix,that, in principle, allows for superluminaltravel, because of the conservation of momen-tum and energy. The kinetic energy of thespacecraft, flying at a velocity greater than thevacuum speed of light, has not increased, since

its inertial mass decreased. Otherwise, anyspacecraft, flying at velocities close to c, wouldneed an amount of kinetic energy that is impos-sible to supply and to pay for.

In that respect, the goals of NASA's Break-through Physics Propulsion Program, namely,no fuel, superluminal speed, and no excessiveamounts of energy needed for a revolutionaryspace propulsion system can be met, provided,of course, that Heim's theory represents physi-cal reality.

Again, as was said in [1], the authors are awareof several shortcomings in this paper. Not all ofthe physical features of Heim's theory were de-rived properly. Some of the conclusions arebased on a somewhat speculative physicalmodel concerning the generation of gravito-

photon particles.

It should be mentioned that Heim's legacy is

very large, several thousand pages, and hispresentation style is not the one of contempo-rary physics. Heim uses his own terminologythat needs to be translated into the language ofmodern physics. In addition, since his theory iscompletely geometric, there are many conceptsthat have no counterpart in modern physics.Whether his theory is actually true, can only bedetermined by experiment. One of the most im-portant predictions is that of Section 4, exploit-ing the nature of the gravitophoton field.

In conclusion, it can be said that the gain, if thistheory were true, will be close to infinity, whilethe probability of success may be close to zero,the product of these two numbers remains un-

determined.

The risk, however, to investigate in the sug-

gested experiment seems to be relatively low. Iffound to be true, a genuine revolution of spaceflight could be the outcome. Such a propulsionsystem might even be simpler than existingrockets, based on highly complex chemical pro-pulsion.

Needless to say, if the proposed reduction ininertial mass could be confirmed by experi-

ment, not only a revolution in space transporta-tion, but also in ground transportation wouldtake place.

Heim's theory currently is not mainstreamphysics, but it contains several highly interest-ing ideas, and its geometric origin of the physi-cal world, is appealing, at least to the authors.As far as the authors understand Heim's theory

(many of his calculations remained uncheckedso far, simply because of the amount and thedifficulty of his work), Heim seems to haveachieved a consistent mathematical formulationthat describes all physical interactions in geo-metrical terms. In that respect, he has realizedEinstein's original idea, but ascribing space,namely the 8 dimensional Heim space (3 realcoordinates comprising physical space and 5imaginary coordinates) many additional, un-usual features.

Future work will focus on a more precise pre-diction of the gravitophoton field with empha-sis on the experiment suggested in order tomeasure the reduction of inertial mass. Compu-

tations will be refined to give a better predic-tion of the performance of the proposed pro-pulsion device. Furthermore, the physicalmodel underlying this propulsion system willbe given a more extensive description.

Acknowledgment

The authors are grateful to Prof. Dr. Dr. A.Resch, director of IGW at Innsbruck Univer-sity, for providing a stimulating working at-mosphere and for numerous discussions con-cerning Heim's theory during the last twoyears.

The authors are very much indebted to Dipl.-Phys. I. von Ludwiger, former manager and

physicist at DASA, for making available rele-vant literature and for many helpful discussions

as well as explanations concerning the implica-

tions of Heim's theory.

The second author was partly funded by Ar-beitsgruppe Innovative Projekte (AGIP), Min-istry of Science and Education, Hanover, Ger-many.

We are grateful to Prof. Dr. T. Waldeer of theUniversity of Applied Sciences at SalzgitterCampus for helpful discussions. The help of T.Gollnick and O. Rybatzki of the University of

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Applied Sciences at Salzgitter Campus in thepreparation of the manuscript is appreciated.

Glossary

aeon Denoting an indefinitely long period oftime. The aeonic dimension can be inter-preted as steering structures governed bythe entelechial dimension toward a dy-namically stable state.

anti-hermetry Coordinates are called anti-her-metric if they do not deviate from Carte-sian coordinates, i.e., in a space with anti-hermetric coordinates no physical eventscan take place.

condensation For matter to exist, as we areused to conceive it, a distortion fromEuclidean metric or condensation, a termused by Heim, is a necessary but not a suf-ficient condition.

condensor The Christoffel symbols k mi be-

come the so called condensor functions,i

km , that are normalizable. This denota-tion is derived from the fact that thesefunctions represent condensations ofspacetime metric. A condensor corre-sponds to a physical force.

coupling constant Value for creation and de-

struction of messenger (virtual) particles,relative to the strong force (whose value isset to 1 in relation to the other couplingconstants).

coupling potential (Kopplungspotential) Thecoupling potential is the first term of the

metric in Eq. (12), denoted as gi k

gp . The

reason for using the superscript gp is thatthis part of the photon metric equals themetric for the gravitophoton particle and

that a sieve (conversion) operator exists,which can transform a photon into a gravi-tophoton by making the second term in themetric anti-hermetric. In other words, theelectromagnetic force can be transformedinto a gravitational like force, and thus canbe used to reduce the inertial mass of a ma-terial body.

cosmogony (Kosmogonie) The creation ororigin of the world or universe, a theory ofthe origin of the universe (derived from thetwo Greek words kosmos (harmoniousuniverse) and gonos (offspring)).

entelechy (Greek entelécheia, objective, com-pletion) used by Aristotle in his work ThePhysics. Aristotle assumed that each phe-nomenon in nature contained an intrinsicobjective, governing the actualization of aform-giving cause. The entelechial dimen-sion can be interpreted as a measure of thequality of time varying organizational

structures (inverse to entropy, e.g., plantgrowth) while the aeonic dimension issteering these structures toward a dynami-cally stable state. Any coordinates outsidespacetime can be considered as steeringcoordinates.

eschatology Concerned with the final events

in the history of the universe.

fundamental kernel (Fundamentalkern)

Since the function i m

occurs in

xm

=∫

i m

d

ias the kernel in the inte-

gral, it is denoted as fundamental kernel ofthe poly-metric.

geodesic zero-line process This is a processwhere the square of the length element in a6- or 8-dimensional Heim space is zero.

gravitational limit(s) There are three dis-tances at which the gravitational force iszero. First, at any distance smaller than R_,the gravitational force is 0. Second,

gravitophoton field Denotes a gravitationallike field, represented by the metric sub-

tensor, gi k

gp , generated by a neutral mass

with a smaller coupling constant than theone for gravitons, but allowing for the pos-sibility that photons are transformed intogravitophotons. This field can be used toreduce the gravitational potential around aspacecraft.

graviton (Graviton) The virtual particle re-sponsible for gravitational interaction.

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Heim-Lorentz force Resulting from the newlypredicted gravitophoton particle that is a

consequence of the Heim space ℝ8. Ametric subtensor is constructed in the sub-space of coordinates I2, S2 and T1, denotedas hermetry form H5, see [1, 5, 6]. Theequation describing the Heim-Lorentzforce has a form similar to the electromag-netic Lorentz force, except, that it exer-cises a force on a moving body of mass m,while the Lorentz force acts upon movingcharged particles only. In other words,there seems to exist a direct coupling be-tween matter and electromagnetism. Inthat respect, matter can be consideredplaying the role of charge in the Heim-Lo-rentz equation. The force is given by w

Fgp=

pe

0vT×H . Here Λp is a coeffi-

cient, v the velocity of a rotating body (in-sulator) of mass m, and H is the magneticfield strength. It should be noted that thegravitophoton force is 0, if velocity andmagnetic field strength are perpendicular.Thus, any experiment that places a rotatingdisk in a uniform magnetic field that is ori-ented parallel or anti-parallel to the axis ofrotation of this disk, will measure no ef-fect.

hermetry form (Hermetrieform) The wordhermetry is an abbreviation of hermeneu-tics, in our case the semantic interpretationof the metric. To explain the concept of ahermetry form, the space ℝ6 is considered.There are 3 coordinate groups in this

space, namely s 3=1 ,2 ,3 forming

the physical space ℝ3, s 2=4 for

space T1, and s1=5 ,6 for space S2.

The set of all possible coordinate groups isdenoted by S={s1, s2, s3}. These 3 groupsmay be combined, but, as a general rule(stated here without proof, derived, how-ever, by Heim from conservation laws inℝ6

, (see p. 193 in [2])), coordinates 5 and

6 must always be curvilinear, and must be

present in all metric combinations. An al-lowable combination of coordinate groupsis termed hermetry form, responsible for aphysical field or interaction particle, and

denoted by H. H is sometimes annotatedwith an index, or sometimes written in theform H=(1, 2 ,...) where 1, 2 ,... ∈ S.This is a symbolic notation only, andshould not be confused with the notationof an n-tuple. From the above it is clearthat only 4 hermetry forms are possible in

ℝ6 . A 6 space only contains gravitationand electrodynamics. It needs a Heimspace ℝ8 to incorporate all known physicalinteractions. Hermetry means that onlythose coordinates occurring in the herme-try form are curvilinear, all other coordi-nates remain Cartesian. In other words, Hdenotes the subspace in which physicalevents can take place, since these coordi-nates are non-euclidean. This concept is atthe heart of Heim's geometrization of allphysical interactions, and serves as thecorrespondence principle between geome-try and physics.

hermeneutics (Hermeneutik) The study ofthe methodological principles of interpret-ing the metric tensor and the eigenvaluevector of the subspaces. This semantic in-terpretation of geometrical structure iscalled hermeneutics (from the Greek wordto interpret).

hermitian matrix (self adjoint, selbstad-jungiert) A square matrix having the prop-erty that each pair of elements in the i-throw and j-th column and in the j-th rowand i-th column are conjugate complex

numbers (i - i). Let A denote a square

matrix and A* denoting the complex conju-

gate matrix. A† := (A*)T = A for a hermitian

matrix. A hermitian matrix has real eigen-values. If A is real, the hermitian require-ment is replaced by a requirement of sym-

metry, i.e., the transposed matrix AT = A .

homogeneous The universe is everywhere uni-form and isotropic or, in other words, is ofuniform structure or composition through-out.

inertial transformation (Trägheitstransfor-mation) Such a transformation, fundamen-tally an interaction between electromagnet-ism and the gravitational like gravitopho-

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ton field, reduces the inertial mass of a ma-terial object using electromagnetic radia-tion at specific frequencies. As a result ofmomentum and energy conservation in 4-dimensional spacetime, v/c = v'/c', the Lo-rentz matrix remains unchanged. It follows

that c < c' and v < v' where v and v' denotethe velocities of the test body before andafter the inertial transformation, and c andc' denote the speeds of light, respectively.In other words, since c is the vacuumspeed of light, an inertial transformation al-lows for superluminal speeds. An inertialtransformation is possible only in a 8-di-mensional Heim space, and is in accor-dance with the laws of SRT. In an Ein-steinian universe that is 4-dimensional andcontains only gravitation, this transforma-

tion does not exist.

isotropic The universe is the same in all direc-tions, for instance, as velocity of lighttransmission is concerned measuring thesame values along axes in all directions.

partial structure (Partialstruktur) For in-

stance, in ℝ6, the metric tensor that is her-mitian comprises three non-hermitian met-rics from subspaces of ℝ6. These metricsfrom subspaces are termed partial struc-ture.

poly-metric The term poly-metric is used withrespect to the composite nature of the met-ric tensor in 8D Heim space. In addition,there is the twofold mapping ℝ4

→ ℝ8→ ℝ4.

quantized bang According to Heim, the uni-verse did not evolve from a hot big bang,but instead, spacetime was discretized fromthe very beginning, and such no infinitelysmall or infinitely dense space existed. In-stead, when the size of a single Metroncovered the whole (spherical volume) uni-verse, this was considered the beginningof this physical universe. That conditioncan be considered as the mathematical ini-tial condition and, when inserted intoHeim's equation for the evolution of theuniverse, does result in the initial diameterof the original universe [1]. Much later,when the Metron size had decreased far

enough, did matter come into existence asa purely geometrical phenomenon.

transformation operator or sieve operator(Sieboperator) The direct translation ofHeim's terminology would be sieve-selec-tor. A transformation operator, however,converts a photon into a gravitophoton by

making the coordinate 4 Euclidean.

unitary matrix (unitär) Let A denote a square

matrix, and A* denoting the complex con-

jugate matrix. If A† := (A*)T = A-1, then A is

a unitary matrix, representing the generali-zation of the concept of orthogonal matrix.If A is real, the unitary requirement is re-placed by a requirement of orthogonality,i.e., A-1 = AT. The product of two unitarymatrices is unitary.

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