AIAA 2004 - 3700

GUIDELINES FOR A SPACE PROPULSION DEVICE BASED ON HEIM'S QUANTUM THEORY

Walter Dröscher1, Jochem Häuser 1,2

1Institut für Grenzgebiete der Wissenschaft (IGW),

Leopold - Franzens Universität Innsbruck, Innsbruck, Austria

2Department of Transportation, University of Applied Sciencesand

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

40TH AIAA/ASME/SAE/ASEE

JOINT PROPULSION CONFERENCE & EXHIBIT, FORT LAUDERDALE, FLORIDA,

11-14 JULY, 2004

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

The text of the calligraphy on the front page means Cosmos, comprising the two chinese sym-bols for space and time. This calligraphy was done by Hozumi Gensho Roshi, Professor of Ap-plied Sciences at Hanazono University, Kyoto, Japan in September 2003. The two red squaresdepict the seal of Hozumi Gensho Roshi.

2004 Institut für Grenzgebiete der Wissenschaft, Leopold - Franzens Universität Innsbruck,Innsbruck, Austria

Table of Contents1 Space Propulsion and Higher-Dimension Quantized Spacetime Physics....................................4

1.1 Basic Concepts of HQT ....................................................................................................51.2 LQT and HQT......................................................................................................................61.3 Fundamental Physical Interactions in 8-D Quantized Space...........................................7

2 The Physical Principles for Field Propulsion .............................................................................72.1 The Physics of Hermetry Forms..........................................................................................82.2 The Metric for Electromagnetic Interactions.....................................................................102.3 The Metric for Coupling Electromagnetism and Gravitation............................................112.4 Physical Model for Gravitophoton Generation...................................................................132.5 Conversion of Photons into Gravitophotons......................................................................13

3 Space Flight Dynamics of Gravitophoton Field Propulsion......................................................143.1 Gravitophoton Interaction Equations for Space Propulsion .............................................143.2 Technical Data for Acceleration Gravitophoton Field Propulsion ...................................163.3 Space Flight in Parallel Space...........................................................................................173.4 Lunar, Interplanetary, and Interstellar Missions................................................................19

4 Cosmology from HQT and LQT ..............................................................................................204.1 Deficiencies in Current Fundamental Physical Theories...................................................204.2 Common Concepts in HQT and LQT................................................................................214.3 Cosmological Consequences..............................................................................................224.4 Dark Matter .......................................................................................................................224.5 Dark Energy.......................................................................................................................22

5 Conclusions and Future Work...................................................................................................23Acknowledgment...........................................................................................................................23Appendix A: Mass Spectrum of Elementary Particles.................................................................24Appendix B: Gravitational Coupling Constants............................................................................24Glossary.........................................................................................................................................24References......................................................................................................................................27

Abstract

This paper is the third one in a series of publi-cations, describing a novel and revolutionaryspace propulsion technique, based on a unifiedfield theory in a quantized, higher-dimensionalspace, developed by the late B. Heim and thefirst author, termed Heim quantum theory(HQT) in the following. It is interesting to notethat this theory shares a similar physical pic-ture, namely a quantized spacetime, with therecently published loop quantum theory (LQT)by L. Smolin, A. Ashtektar, C. Rovelli, M. Bo-jowald et al. [11, 24-28]. LQT, if proved cor-rect, would stand for a major revision of cur-rent physics, while HQT would cause a revolu-tion in the technology of propulsion.

For effective and efficient interplanetary aswell as interstellar travel, NASA's Break-through Propulsion Physics Program (BPP)specified three basic features, namely little orno fuel mass, limited amount of energy con-sumption (a spacecraft approaching the speedof light would not satisfy this requirement,since its mass becomes infinite), and (prefera-bly) superluminal speed. To satisfy these re-quirements a revolution in space propulsiontechnology is needed. Such breakthrough pro-pulsion techniques can only emerge fromnovel physics. If we believed that current phys-ics held the answer to all questions, a BPP de-vice would not be possible. Recently, how-ever, more and more evidence has been pilingup that current physics is far from final an-swers and, in addition, exhibits fundamentalinconsistencies, even on the classical level.Furthermore, quantum theory (QT) in its cur-rent form does not lead to an explanation ofthe elementary structures of matter, and doesnot lead to a consistent cosmology either.

For a revolutionary space transportation sys-tem, however, the physical concepts of matterand inertia as well as the nature of space andtime have to be understood. In QT the exis-tence of matter is taken for granted, definingan elementary particle as a point-like structure[17]. In classical physics, including the Gen-eral Theory of Relativity (GR), science startsfrom the belief that space and time are infi-nitely divisible, in other words, that spacetimeis continuous (a differentiable manifold in themathematical sense). Both ideas contradictNature's all pervading quantization principle

and immediately lead to contradictions in theform of infinite self-energies etc. or self-accel-erations [18]. HQT is an extension of Ein-stein's GR, using his field equations as a tem-plate in a quantized higher-dimensional space,but also extending these equations into thesubatomic range. This eventually leads to apoly-metric, whose partial metric structuresare interpreted as fundamental physical inter-actions. This theory, seems to complementboth QT and GR, in explaining the nature ofelementary particles as well as their discretemass spectrum and life times, based on the ba-sis of a quantized geometrodynamics (quan-tized elemental surfaces of some 10-70 m2,termed metron by Heim) in a 12 dimensionalspace. Heim derives a dimensional law that de-termines the maximum number of possible di-mensions that can exist, along with admissiblesubspaces, and also gives their physical inter-pretation. HQT seems to be able to explain thenature of matter (physicists deemed this ques-tion to be of importance in the early fifties, see[21]). The physical features of the postulated12 dimensional, quantized hyperspace, de-noted as Heim space by the authors, is de-scribed in detail. The 12D Heim space com-prises five semantic units, namely, the sub-spaces ℝ3 (space), T1 (time), S2 (organization),I2 (information), and G4 (steering of I2) wheresuperscripts denote dimension. Except for the3 spatial dimensions, all other coordinates areimaginary. Several metric tensors can be con-structed from these subspaces. Each metrictensor is associated with a specific physical in-teraction, similar to Einstein's GR, wherespacetime curvature is interpreted as gravita-tion (graviton). Analyzing the metric tensorsacting in ℝ4, the theory predicts six fundamen-tal interactions, instead of the four experimen-tally known ones. These interactions representphysical fields that are carrying energy. Ac-cording to HQT, a transformation of electro-magnetic energy into gravitational energyshould be possible. It is this interaction that isused as the physical basis for the novel spacepropulsion concept, termed field propulsion [1,2], which is not conceivable within the frame-work of current physics.

The paper comprises four technical chapters.In the first chapter, a qualitative discussion ofthe six fundamental interactions, derived fromthe concept of Heim space and its conse-

1

quences for a novel propulsion system, arepresented. In addition, a qualitative discussionof the physical principle that serves as the ba-sis for advanced propulsion is given. In chap-ter two, the physical principles of the so calledfield propulsion system are quantitatively ad-dressed, explaining their application in futurespaceflight2. In particular, it is shown how thepoly-metric from Heim space leads to a metricdescribing electromagnetic phenomena and itsconversion into a gravitational3 like metric,postulating a novel particle, the gravitophoton.In chapter three, the equations of the gravito-photon interaction are derived, and a physicalmodel is presented to calculating the magni-tude of the gravitophoton interaction. This isalso the main chapter, presenting the quantita-tive physical model of the gravitophoton inter-action and the concept of parallel space fromwhich the physical guidelines for the field pro-pulsion device can be derived. In particular,the technical requirements for a gravitophotonpropulsion device will be discussed. The ex-perimental set up of such a device will also bepresented. In addition, a lunar mission, an in-terplanetary, and an interstellar mission will beinvestigated. In chapter four, similarities be-tween HQT and LQT are discussed. Cosmo-logical consequences dealt with concern theamount of dark matter (concept of parallelspace) and the cause of dark energy. The ac-celeration of the cosmic expansion is ex-plained qualitatively, since it turns out that thepostulated sixth fundamental force (interpretedas quintessence) and represented by the postu-lated vacuum particle, is of repulsive nature.

Nomenclature and physical constants

à value for the onset of conversion of pho-tons into gravitophotons, see Eq. (39).

A denotes the strength of the shielding poten-tial caused by virtual electrons, see Eq. (37).

Compton wave length of the electron

C= hme c

=2.43×10−12 m , ƛC=C /2 .

2 This chapter contains a certain amount of mathemat-ics. The reader may wish to skip the derivations andcontinue with the gravitational Heim-Lorentz force,Eq. (47).

3 The term gravitational is reserved to the two addi-tional interactions represented by the gravitophotonand the vacuum particle acting on material particles.

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

( 1 /c2=0 0 ).

D diameter of the primeval universe, some10125 m, that contains our optical universe.

DO diameter of our optical universe, some1026 m.

d diameter of the rotating torus, see captionTable .

dT vertical distance between magnetic coiland rotating torus (see Fig.1).

-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, F gp=p e0 vT×H , see Eq. (47).

G = Gg + Ggp + Gq = 6.67259 × 10-11 m3 kg-1 s-2,

gravitational constant .

Gg graviton constant, Gg≈G that is Gg de-

cribes the gravitational interaction without thepostulated gravitophoton and quintessence in-teractions.

Ggp gravitophoton constant, Ggp≈1/672 Gg .

Gq quintessence constant, Gq≈4×10−18 Gg .

gi kgp metric subtensor for the gravitophoton

in subspace I2 S∪ 2 (see glossary for subspacedescription).

gi k ph metric subtensor for the photon in sub-

space I2 S∪ 2 T∪ 1 (see glossary for subspace de-scription).

h Planck constant 6.626076 × 10-34 J s,ℏ=h /2 .

hik metric components for an almost flatspacetime.

ℓ p= G ℏ3

c3 =1.6×10−35 m Planck length.

me electron mass 9.109390 × 10-31 kg.

2

m0 mass of proton or neutron1.672623 × 10- 27 kg and 1.674929 × 10-27 kg.

Nn number of protons or neutrons in the uni-verse.

q electric charge.

R distance from center of coil to location ofvirtual electron in torus, see Fig.(1).

rN distance from nucleus to virtual electronin torus, see Fig.(1).

R_ is a lower bound for gravitational struc-tures, comparable to the Schwarzschild ra-dius. The distance at which gravitationchanges sign, , is some 46 Mparsec. ρ

R+ denotes an upper bound for gravitationand is some type of Hubble radius, but is notthe radius of the universe, instead it is the ra-dius of the optically observable universe.Gravitation is zero beyond the two bounds,that is, particles smaller than R- cannot gener-ate gravitational interactions.

re classical electron radius

r e=1

40

e2

me c2 =3 × 10−15 m .

rge ratio of gravitational and electrostaticforces between two electrons.

v velocity vector of charges flowing in themagnetic coil, see Eq. (27), some 103 m/s incircumferential direction.

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

wgp probability amplitude (the square is thecoupling coefficient) for the gravitophotonforce (fifth fundamental interaction)

wgp2 =Ggp

me2

ℏ c=3.87×10−49 probability ampli-

tudes (or coupling amplitudes) can be distancedependent (indicated by a prime in [9]).

wgpe probability amplitude for emitting agravitophoton by an electron

wgpe=wgp .

wgpa probability amplitude for absorption of agravitophoton by a proton or neutron

wgpa2 =Ggp mp

me

ℏ c.

wg_q conversion amplitude for the transforma-tion of gravitophotons and gravitons into thequintessence particle, corresponding to thedark energy (rest mass of some 10-33 eV).

wph probability amplitude (the square is thecoupling coefficient for the electromagneticforce, that is the fine structure constant )α

w ph2 = 1

40

e2

ℏ c= 1

137.

wph_qp conversion amplitude for the transfor-mation of photons into gravitophotons (see Eq.(35)).

wq probability amplitude for the quintessenceparticle,(sixth fundamental interaction), corre-sponding to dark energy (rest mass of some10- 33 eV).

Z atomic number (number of protons in a nu-cleus and number of electrons in an atom)

Z0 impedance of free space,

Z 0= 0

0

≈376.7 .

α coupling constant for the electromagneticforce or fine structure constant 1/137.

αgp coupling constant for the gravitophotonforce .

γ ratio of probabilities for the electromagneticand the gravitophoton force

= w ph

wgp

2

=1.87×1046 .

0 permeability of vacuum 4 × 10-7 N/mπ 2 .

metron area (minimal surface 3Gh/8c3),current value is 6.15 x 10-70 m2.

Φ gravitational potential, =GM/R.

ω rotation vector (see Fig. 1 ).

3

Abbreviations

BPP breakthrough propulsion physics

GR General Relativity

HQT Heim Quantum Theory

LQT Loop Quantum Theory

LHS left hand side

ls light second

ly light year

QED Quantum Electro-Dynamics

RHS right hand side

SR Special Relativity

VSL Varying Speed of Light

Subscripts

e electron

gp gravitophoton

gq from gravitons and gravitophotons intoquintessence

ph denoting the photon or electrodynamics

sp space

Superscripts

em electromagnetic

gp gravitophoton

ph photon

T indicates the rotating ring (torus)

Note: Since the discussion in this paper is onengineering problems, SI units (Volt, Ampere,Tesla or Weber/m2 ) are used. 1 T = 1 Wb/m2

= 104 G = 104 Oe, where Gauss (applied to B,the magnetic induction vector) and Oersted(applied to H, magnetic field strength or mag-netic intensity vector) are identical. Gauss andOersted are used in the Gaussian system ofunits. In the MKS system, B is measured inTesla, and H is measured in A/m (1A/m = 4π× 10-3 G). Exact values of the physical con-stants are given in [22].

Note: For a conversion from CGS to SI units,the electric charge and magnetic field are re-placed as follows:

1 Space Propulsion and Higher-Di-mension Quantized Spacetime Phys-icsFor effective and efficient lunar space trans-portation as well as interplanetary or even in-terstellar space flight a revolution in space pro-pulsion technology is needed.

Regarding the requirements of NASA's Break-through Physics Propulsion Program (BPP) arevolutionary space propulsion system should

• use no or a very limited amount of fuel,

• possibility for superluminal speed, and

• requirement for a low energy budget. Thisimmediately rules out any device flyingclose to the speed of light, since its mass isgoing to infinity, according to SR. A space-craft having a mass of 105 kg, flying at aspeed of 1% of the speed of light, carries anenergy content of 4.5x1017 J. Even if thespacecraft can be provided with a 100 MWnuclear reactor, it would take some 143years to produce this amount of energy.

It is understood that the laws of current phys-ics do not allow for such a revolutionary spacepropulsion system. Propulsion techniques ofthis type can only emerge from novel physics,i.e., physical theories that deliver a unificationof physics that are consistent and founded anbasic, generally accepted principles, either re-moving some of the limits, or giving rise to ad-ditional fundamental forces, and thus provid-ing alternatives to current propulsion princi-ples. Theories like HQT and LQT are thereforeof great interest, since they might offer the po-tential for these advanced technologies, see,for instance, the remark on p. 9 in [29].

Hopes for such a unified theory are, indeed,not futile. In classical physics, science startsfrom the belief that space, time, and matter areinfinitely divisible, in other words, that space-time is continuous (a differentiable manifold inthe mathematical sense) and not subjected tothe quantization principle. Regarding the mi-crocosm, there exists a large number of ele-mentary particles that cannot be subdivided

4

ee /40 and H 40 H .

any further. In quantum physics arbitrary di-visibility of matter has proved to be an illu-sion. On the other hand, the existence of mat-ter is taken for granted, i.e., the occurrence ofelementary particles is accepted as such, andthe cause for the existence of matter cannot berevealed. There is substantial evidence that thecurrently favored Standard Model is far frombeing the final theory.

In the twentieth century there has been enor-mous progress in physics, based on both Ein-stein's theory of general relativity and quantumtheory. Both theories are very successful intheir own range, but could not be unified sofar. The reason for the unification is ... De-spite the successes of the two theories, the cur-rent status of physical theory lacks the under-standing of the most fundamental physicalfacts. First, it has not been possible, despitenumerous attempts over the last eight decades,to extent Einstein's idea beyond the range ofgravitation. Second, QT has not been able todeliver the mass spectrum of elementary parti-cles, nor is there a theoretical explanation fortheir lifetimes, neither can quantum numbersbe derived. None of these theories is able toexplain the nature of matter and inertia, topicsthat are essential for the physics of a com-pletely novel propulsion system.

1.1 Basic Concepts of HQT

Einstein's view was eventually deemed unten-able, because next to gravitation other forcesbecame known. The recent article by L. Smo-lin [11] on Atoms of Space and Time, however,seems to be a sign that physics may be return-ing to the Einsteinian picture, namely the geo-metrization of the physical world, meaningthat all forces (interactions) are ultimately de-termined by the structure of spacetime. Thetwo important ingredients that Einstein did notuse are a discrete spacetime and a higher-di-mensional space, provided with special, addi-tional features.

It is known that the general theory of relativityin a 4-dimensional spacetime delivers only onepossible physical interaction, namely gravita-tion. Since Nature shows us that there exist ad-ditional interactions, and because both GR andthe quantum principle are experimentally veri-fied, it seems logical to extend the geometricalprinciple to a discrete, higher-dimensional

space. Consequently, Heim’s quantum theory,HQT, of gravity and elementary structures ofmatter is based on the geometric view of Ein-stein, namely that geometry itself is the causeof all physical interactions, but it uses thestructure of Einstein's field equations only as atemplate for physical interactions in a higher-dimensional discrete space, and extends themalso to the microcosm.

Eventually developed by Heim and the firstauthor, the theory utilizes an 8-dimensionaldiscrete space4 in which a smallest elementalsurface, the so-called metron, exists. HQT, de-veloped first by Heim in the fifties and sixties,and partly published in the following threedecades of the last century, seems to be com-pliant with these modern requirements. It alsomakes a series of predictions with regard tocosmology and high energy physics [12] thateventually can be checked by experiment.

Most important, however, Heim's extendedtheory predicts two additional interactions[1, 6-9] identified as quintessence, a weak re-pulsive gravitational like interaction (dark en-ergy) and gravitophoton interaction, that en-ables the conversion of electromagnetic radia-tion into a gravitational like field, representedby the two hypothetical gravitophoton (nega-tive and positive energies) particles. The gravi-tophoton interaction is discussed in Chaps. [2,3.1]. Quintessence (dark energy) is briefly dis-cussed in Chap. [4].

The interpretation of the physical equations forthe gravitophoton field leads to the conclusionthat this field could be used to both acceleratea material body and to cause a transition of amaterial body into some kind of parallelspace, possibly allowing superluminal speed.These effects could serve as the basis for ad-vanced space propulsion technology, and aredealt with quantitatively in the following chap-ters.

4 To be more precise, Heim's theory was extendedfrom 6 to 8-dimensions by the first author andHeim, [7], to obtain the unification of the fourknown interactions (forces). In this process, it wasfound that two additional gravitational like interac-tions should occur, termed the gravitophoton field(attractive and repulsive) and the vacuum field (re-pulsive, interpreted later on as quintessence) [1, 7].The dimensional law derived by Heim requires a12-dimensional space, but the additional four coor-dinates are needed only in the explanation of thesteering of probability amplitudes (information).

5

According to Heim's theory, gravitation, as weknow it, is comprised of three interactions,namely by gravitons, the postulated gravito-photons, and by the quintessence particle. Thismeans that the gravitational constant G con-tains contributions from all three fields. Thequintessence interaction, however, is muchsmaller than the first two contributions.

It is interesting to note, that the mass spectrumfor elementary particles, calculated fromHeim's mass formula, and partly shown in Ap-pendix A as taken from [12], is very sensitiveto G. A corrected value of G obtained by thefirst author, accounting for the contribution ofthe gravitophoton field, led to substantially im-proved results of the mass values when com-pared to experimental data. In Heim's theorythe existence of matter as an independent en-tity is replaced by the features of a dynamic 8-dimensional discrete space, and as such is cre-ated by space itself. In other words, matter iscaused by a non-Euclidean metric in space ℝ8,termed 8D Heim space, comprised by a largenumber of elemental space atoms (called met-rons by Heim), interacting in a dynamic andhighly complex way.

A few words about the history of HQT seem tobe in place. Heim first published his theory ofa higher-dimensional discrete spacetime in anobscure German journal [10] in a series of fourarticles in 1959. In 1977, following the adviceof Heisenberg’s successor, H.-P. Dürr, Heimpublished an article entitled Vorschlag einesWeges zur einheitlichen Beschreibung der Ele-mentarteilchen (Recommendation of a Way toa Unified Description of Elementary Particles)[4], which in today's terminology was a sum-mary of his theory for a unified field theory in-cluding quantum gravity. Later on, he wrotetwo text books Elementarstrukturen der Ma-terie [6, 7] that were eventually published byA. Resch (see Acknowledgment). However,to be fair, it should be mentioned that Heim'spublications are difficult to read, and needed tobe modified and extended by the first author inseveral ways, for instance [9].

1.2 LQT and HQT

In order to understand how to categorizeHeim's quantum theory, it seems worthwhileto determining the similarities between recentloop quantum theory by L. Smolin, A.

Ashtekar, and C. Rovelli [11, 24, 25] andHQT, and also to learn how HQT compareswith GR and QT. A major difference betweenGR and Heim is that in GR the material fieldsource does not appear in geometrized form,but occurs as a phenomenological quantity (inthe form of matter that is an entity of its own,whose existence is taken for granted).

It should be noted that HQT complements bothQT and GR, in explaining the nature of ele-mentary particles as well as their discrete massspectrum and life times, based on the basis of aquantized geometrodynamics (quantized ele-mental areas of some 10-70 m2, termed met-ron by Heim) in a 12 dimensional space (3 realand 9 imaginary coordinates). As shown byHeim the fact that all additional coordinatesare imaginary leads to real eigenvalues in themass spectrum for elementary particles [4, 6].The idea of geometrization is extended to thesub-atomic range. However, in that case, theChristoffel symbols need to be replaced byreal tensor components5. Heim derives a di-mensional law that restricts the maximumnumber of dimensions to 12 and requires theexistence of subspaces. From the metric ofsubspace ℝ 6, originally conceived by Heim,the premises of QT cannot be derived, and thequantization principle had to be introduced adhoc. For instance, Dirac's equations cannot bederived within ℝ6. However, when the metricin space ℝ8 is considered, all possible physicalinteractions are reproduced. The completespace ℝ12 is needed to explain how probabilityamplitudes (immaterial) are steering events inspacetime ℝ4.

In the following, a Heim space is a quantizedspace comprising elemental surfaces with ori-entation (spin), the metron, whose size is thePlanck length (apart from a factor) squared,comprising 6, 8, or 12 dimensions. A Heimspace may comprise several subspaces, eachequipped with its individual Riemannian met-ric. The union of these individual metricspaces is termed a poly-metric.

Furthermore, in GR the gravitational potentialis associated with the metric tensor, and thushas a direct physical meaning. Extending this

5 This can be shown by employing the double trans-formation described in Eq. (2) to Heim's eigenvalueequations for the mass spectrum of elementary parti-cles. Otherwise masses of particles could be trans-formed away which is unphysical.

6

concept to the poly-metric in Heim space ℝ8,and forming special combinations of these par-tial metrics, all possible fundamental physicalinteractions are obtained. Since GR has beenextremely well verified experimentally, this in-terpretation seems to be justified.

1.3 Fundamental Physical Interactions in8-D Quantized Space

In GR the gravitational force is nothing but aneffect of the geometric curvature of spacetime.The predictions of GR have been tested exten-sively, and today GR arguably is the experi-mentally best verified theory. Therefore, thereis some confidence that this concept can beextended to all physical forces, and that thestructure of the equations of GR is valid for allphysical interactions in a higher-dimensionalspace.

A Heim space ℝ12, where the superscript de-notes dimension, comprises five subspaces orpartial structures that form semantic units.Combining these semantic units by employingcertain selection rules a set of so called herme-try forms or partial metric tensors is obtained,forming the poly-metric, that represents allphysical interactions [2]. Each of the semanticunits (or subspaces) has its own metric. Thereare the subspaces ℝ3 with real coordinates (x1,x2, x3), T1 with imaginary time coordinate (x4),S2 with imaginary coordinates for organizationof structures (x5, x6,), I2 with imaginary coordi-nates for information (x7, x8), and G4 withimaginary coordinates for steering of probalityamplitudes and thus events in ℝ4 (x9, x10, x11,x12). The space ℝ12 is comprised of the twospaces ℝ6 = ℝ3 T∪ 1 S∪ 2 and V6 =I2 G∪ 4. Theconcept of energy exists in6 ℝ6, while V6 is de-noted as immaterial. Considering the spaceℝ8 = ℝ3 T∪ 1 S∪ 2 I∪ 2, that is omitting the spaceG4, the theory predicts six fundamental inter-actions, instead of the four experimentallyknown ones. These interactions emerge in ourspacetime and represent physical fields carry-ing energy. According to the theory, a trans-formation of electromagnetic energy intogravitational energy (gravitophoton) should bepossible (see Chap. 2.5). It is this conversionthat is used as the physical basis for the novel

6 This is not completely correct, since the vacuum orquintessence particles of hermetry form H10(I2) (seeglossary and [2]) with I2⊂ℝ8 possess (small) ener-gies.

space propulsion concept [1, 2], which is notconceivable within the framework of currentphysics. This is a direct consequence of the di-mension of Heim space, and the interpretationof a partial metric (hermetry form, see glos-sary) as a physical interaction or particle. Inother words, if Einstein's view, namely of ge-ometry being the cause of gravitation is ex-tended to the poly-metric in Heim space, theinterpretation of all physical interactions is anatural consequence. Moreover, the two addi-tional interactions should also be consideredreal.

It seems that space G4 does not have a directphysical meaning in a sense that it is responsi-ble for physical interactions. Its role seems tobe acting as a symmetry breaking principle, re-sponsible for a quantized bang, and much lateron (some 1010 years ago), for the creation ofmatter.

Starting from Einstein's equations, Heim de-rives a set of nonlinear eigenvalue equationsfor microscopic particles (mass spectrum ofelementary particles), first in ℝ 6. In Heim”stheory, quantum mechanics is not contained inℝ6, but in space ℝ8. In this regard, Heim's the-ory can be understood as being complementaryto the wave picture, taking care of the particlenature of physical objects (see [7] pp. 360 forthe linear Dirac equations.

2 The Physical Principles for FieldPropulsion7 In the following a roadmap for the derivationof the gravitophoton interaction is presented.

The proposed propulsion concept works in twostages. First, a gravitophoton field is generatedthrough interaction with an electromagneticfield that exerts a force on the space vehicle.Second, there exist parallel spaces in whichcovariant laws of physics are valid that allowspeeds larger than the vacuum speed of light inℝ4. Under certain conditions, a spacecraft canenter a parallel space, see Sec. (3.3).

For the gravitophoton interaction to exist, Ein-stein's principle of geometrization of physics

7 The term breakthrough propulsion is not used, sinceit does not relate to the propulsion principle that isbased on the conversion of photons (electromag-netic field) into gravitophotons (gravitational likefield).

7

needs to be valid for all physical interactions.According to HQT this is the case in 8D space.

For instance, in classical physics, there is nodifficulty to include electromagnetism in gen-eral relativity by adding the stress-momentum-

tensor of the electromagnetic field T i kem to the

RHS of Einstein's field equations in 4D space-time

Ri k=T i kg T i k

em−12

gi k T (1)

and T =T kk contains both the gravitational and

electromagnetic contribution. The parameter κ

is of the form =8G

c4 . To complete the set

of equations, Maxwell's equations have to beadded. This is, however, not the approach of aunified field theory, because the electrody-namic field is added from the outside withoutany geometrical interpretation. In the next sec-tion, the concept underlying the unification ofall physical interactions is derived, extendingEinstein's principle of geometrization.

2.1 The Physics of Hermetry Forms

As described in [1] there is a general coordi-

nate transformation xm i from

ℝ 4ℝ8ℝ4 resulting in the metric tensor

gi k=∂ xm

∂

∂

∂i

∂ xm

∂

∂

∂k

(2)

where indices , = 1,...,8 and α β i, m, k = 1,...,4.The Einstein summation convention is used,that is, indices occurring twice are summedover. The above transformation is instrumentalfor the construction of the poly-metric used todescribing physical interactions. The Euclid-ean coordinates xm and curvilinear coordinatesi are in ℝ4, while curvilinear coordinates

are in ℝ8. The metric tensor can be written inthe form

gi k=: ∑ ,=1

8

gi k (3)

and the individual components are given by

gi k=

∂ xm

∂

∂

∂i

∂ xm

∂

∂

∂k

. (4)

Parentheses indicate that there is no indexsummation. In [2] it was shown that 12 herme-try forms (see glossary) can be establishedhaving direct physical meaning, involving spe-cific combinations from the four subspaces.The following denotation for the metric de-scribing hermetry form Hℓ with ℓ=1,...,12 isused:

gi k H ℓ=: ∑ ,∈H ℓ

gi k

(5)

where summation indices are obtained fromthe definition of the hermetry forms. The ex-

pressions gi k H ℓ are interpreted as differ-

ent physical interaction potentials caused byhermetry form Hℓ, following the interpretationemployed in GR. The combination of coordi-nates and are characteristic for the inter-action, and also characterize the subspace. Ap-plying the sieve operator formed fromKronecker symbols, namely

s 0 ,0:=0 0 (6)

to Eq. (5) selects the term gi k0 0. A sieve

operator (or sieve transformation) can be ap-plied repeatedly, and thus serves to convertone hermetry form into another one. At themoment a sieve operator is a mathematicalconstruction only, but it is the aim of this dis-cussion to show how such a conversion can beobtained in physical reality. For the sake ofsimplicity, the following short form, omitting

subscripts ik, is introduced :=gi k .

Next the hermetry forms pertaining to thethree subspaces S2 , I2, S2 × I2 are investi-gated. Cosmological data clearly show that theuniverse is expanding, which indicates a re-pulsive interaction. Gravitational attraction iswell known since Newton. Both interactionsact on matter, so that there should be two her-metry forms having anti-symmetric properties.The spaces corresponding to these interactionare identified as S2 and I2. The gravitationalfield, as described by gravitons, is given byhermetry form H12

8

gi k H12=55566566 , (7)

while the vacuum field (quintessence) is givenby

gi k H10=77788788. (8)

There is a third hermetry form whose metric isin the space S2 × I2. Since this metric is a com-bination of an attractive and a repulsive inter-action, it is assumed that there are exist twofields. The first partial metric is considered tobe attractive, since its components contain thegravitational metric of Eq. (7). For the samereason the second part is considered to be re-pulsive. The particle for mediating this interac-tion is called the gravitophoton because of thepossible interaction with the electromagneticfield. The reasons will become clear in thenext section. It is postulated from the metricsof Eqs.(9, 10) that there are two types of gravi-tophotons associated with the attractive andthe repulsive gravitophoton potentials. Theirrespective coupling constants are denoted by

Ggp- and Ggp

+ that will be described below.

The attractive gravitophoton particle is de-scribed by Eq. (9), the minus sign denotingnegative energy density, because it containsthe metric of the graviton which is directlyvisible from Eq. (7). The repulsive gravitopho-ton particle is described by Eq. (10), the plussign denoting positive energy density, becauseit contains the metric of the vacuum or quin-tessence particle that describes a repulsiveforce. Their partial metric have the form

gi k H11- =55566566+

57675868 (9)

gi k H11+ =77788788+

75768586. (10)

To conclude this section, it has been shownthat in Heim space ℝ8 there are three physicalinteractions acting on material particles,namely, gravitation represented by hermetryform H12 (S2) (attractive), the quintessence orvacuum field hermetry form H10 (I2) repulsive),and the gravitophoton field, hermetry form H11

(S2, I2) (both attractive and repulsive). Nega-tive and positive gravitophotons are generated

simultaneously in pairs, and H11 is the onlyhermetry form that is identically 0, that is

gik H11=gik S2×I 2=0. (11)

It is a strange fact that a hermetry form that iszero should have any physical effect at all.This reflects the fact that the total energy beingextracted from the vacuum by pair productionof gravitophotons is zero. However, the physi-cal effect lies in the different absorption coef-ficients of negative and positive gravitopho-tons. As it turns out in Chap. 3, gravitophotonsare generated by virtual electrons, that is, theyare generated by vacuum polarization 8. In thisprocess energy is conserved, but two differenttypes of energy both negative and positive areobtained, adding up to zero. H11 is the onlyhermetry form that is comprised by spaceS2 × I2, the so called transcoordinates9. Noother of the hermetry forms is identical to 0,since this is the only hermetry form associatedwith creating particles from the vacuum.

Hence, the gravitational constant G is com-prised of the three individual couplingstrengths of these interactions

G=GgGgpGq (12)

where10

Ggp≈1/672 Gg and Gq≈4×10−18 Gg .

In the following section the metric describingphotons, given by hermetry form H5 (T1, S2,I2), and its interaction with the gravitophotonmetric is investigated.

8 A nonzero vacuum density during the early universeresulted in an exponential expansion (inflationaryphase), and also is the cause of the Casimir effect,although extremely small. GR alone cannot providethe physics for field propulsion. Moreover, vacuumenergy is also considered to be responsible for theexpansion of the universe.

9 Coordinates of S2 are associated with GR and thoseof I2 are assigned to QT.

10 In the physical interaction picture, generally the firstpartner emits and the second one absorbs a messen-ger particle. Since the quintessence is formed fromthe vacuum itself, there is no generating mass, forinstance, a proton mass emitting a photon. Thus thevalue Gq is some kind of fictitious value.

9

2.2 The Metric for Electromagnetic Interac-tions

The metric tensor for photons depends on sub-spaces I2, S2, and T1 with coordinates 4,5,...,8, see [2].

gi k ph:=gi k H 5= ∑

,=4

8

gi k (13)

The coupling constant of this hermetry form is

=wph2 = 1

4 0

e2

ℏ c.

For weak electrodynamic and also for weakgravitational fields, spacetime (4D) is almostflat, so one obtains

gi k=gi k0hi k where g4 4

0=−1 and

gi i0=1 where i=1,2,3 and k, =1,...,4. Theℓ

hik are small quantities whose products arenegligible. In the following the hik are used todescribe the metric for electromagnetic inter-actions. All other components are 0. The geo-desic equation [1] takes the form

x i=−12 ∂ hil

∂k

−∂ hkl

∂i

∂ hik

∂l xk x l .

where the dot denotes the time derivative.Evaluating the terms on the RHS gives

−2 x i=2 hi4 , 4−h44, i x4 x4 +

2 hi4 , lhil ,4−h4 l , i x4 x l +

hik , lhil , k−hkl , i xk x l .

(14)

with i,k, =1,2,3 and the comma denoting a parℓ -tial derivative. Investigating the first two terms

of Eq. (14) and using x4=ct 11 one obtains

x i= 12

h44, i−hi4 , 4c2h4 l , i−hi4 , l c x l .

(15)

Introducing the quantity

M i k= 11k 4

hk 4, i – hi 4, k ,

Eq. (15) can be written in shortform

11 It should be noted that in this section contravariantcoordinates are used.

x i=M ik c xk . (16)

This form can be directly compared with anelectron moving in an electromagnetic field

x i= e

me cF ik xk

(17)

where F ik=∂ Ai

∂ x k −∂ Ak

∂ x i and no distinction

needs to be made between covariant and theordinary derivative. The electromagnetic fieldtensor is obtained from the 4-vector electro-magnetic potential that is defined as

, Ai. From comparison of Eqs. (16) and

(17) the following expression for the metric isobtained

h4 k=e

me c2 14 k Ak . (18)

In the next step, the third term of Eq. (14) is

investigated. Tensor potentials hi k can be

written, see [1], by means of retarded poten-tials

hi k=1

4 0

eQ

me c2 r

vi

c

vk

c= e

me c2 Aixk

c (19)

with i, k=1,2,3. Combining Eqs. (17) and (18)with Eq. (19), one obtains

hi k=1

4 0

14 i 4 k eQ

me c2 r

vi

c

vk

c (20)

with i, k =1,...,4. Analyzing Eq. (20) shows that for i=4 and k=4the metric describes the electric potential ,while for k=4 and i=1,2,3 the metric represents

the vector potential A . For indices i, k=1,2,3

an additional tensor potential is obtained,which is not present in classical electrodynam-ics. Therefore, a 4×4 matrix is needed to de-scribe all electromagnetic potentials.

Tensor potentials hik with i, k =1,2,3 are be-

longing to the hermetry form for photons and

thus have coupling coefficient =wph2 , but

cannot be associated with the 4-potential of theelectromagnetic field. Introducing the cou-pling coefficient in Eq. (α 20) leads to

10

hi k=14 i 4 k Qe

ℏme c

1r

vi

c

vk

c. (21)

On the other hand, hik= ∑ ,=4

8

hi k is defined

by its sum of partial potentials, so that the sumof all of these potentials is determining thecoupling constant for the electromagneticfield, namely wph. Coupling constants for dif-ferent fields are thus determined by the corre-sponding sum of their partial potentials.

2.3 The Metric for Coupling Electromagne-tism and Gravitation

As was shown above, the metric tensor for thegravitophoton depends on subspaces S2 and I2

with coordinates 5, 6, 7, 8, and is written as

gi kgp:=gi k H11= ∑

, =5

8

gi k =0.

(22)

In comparison with Eq. (8), the metric for thephoton can be written in the form12

gi k ph=gi k

gpgi k4 4 ∑

, =5

8

gi k 4gi k

4 .

(23)

The second and third terms, as will be shownbelow, can be associated with the electric force(electric scalar potential) and the Lorentz force(vector potential). The first term represents thecombined metric for the negative and positivegravitophoton particles. If an experiment canbe conceived which causes the metric of thephoton to become 0, then the metric for thegravitophoton particles remains. The experi-ment needs to remove the time dependencefrom the photon metric, so that only the spaceS2 × I2 remains, responsible for the gravitopho-ton metric 13.

In addition it can be shown, see Eq. (35), thatin the presence of virtual electrons, responsiblefor the vacuum polarization and the shieldingof the charge of a nucleus [16], there exists a

12 The sum of the second and third terms were denoted

as electromagnetic metric tensor, gi kem , in [1].

13 We are aware of the fact that these theoretical pre-dictions sound highly speculative, but they are di-rect consequence of the geometrization principle.

nonzero probability amplitude for converting aphoton into gravitophotons. This gravitationalforce is the basis for the propulsion concept,termed gravitophoton field propulsion or fieldpropulsion.

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. Therefore,

only the scalar photon potential needs to beconsidered

g4 4 ph=g4 4

0h4 4 ph. (24)

For the linearized potential a formula similarto Eq. (23) holds,

h4 4 ph=h4 4

4 4 + ∑ , =5

8

h4 4 4hi k

4

+ ∑ , =5

8

h4 4 .

(25)

Next, the contributions of the partial potentialson the RHS of Eq. (25) are evaluated. Fromthe known form of the electric and Lorentz

forces, F=q Eq vT×B , vT denoting the

velocity of the rotating torus, there follows theexistence of a scalar electric potential and aϕvector potential A with components

Ai=0

Qvi

Rwhere Qvi denotes the total cur-

rent in the magnetic coil and i=1,2,3. The firstterm in (25) is associated with the electric po-tential, seen by a virtual electron with charge-e at distance rN from a nucleus, located inthe torus, see Fig. (1). The potential thus takesthe form

h4 44 4=− 1

40

1

me c2

eZer N

. (26)

For the first stage of the proposed propulsionmechanism as well as for the experiment ofFig. (1), it can be assumed that speed vi << c.No field propulsion system will accelerate aspacecraft to a velocity comparable to thespeed of light, since the required energy ren-ders such an approach impractical.

11

The second partial potential can be determinedfrom Eq. (19). The corresponding vector po-tential is of the form

∑ , =5

8

h4 4 4h4 4

4 =− 140

1me c2

eQR

vi

cvi

T

c (27)

with summation over i=1,2,3 and vi /c and Qthe speed and total charge of the electrons inthe current loop (see Fig. (1)), while vi

T /c de-notes a velocity component of the rotatingtorus (see Fig. (1)). The charge -e denoteselectron charge. R is the distance from the cen-ter of the coil to the location of a virtual elec-tron in the torus. This potential represents theLorentz force.

There is a third partial potential in Eq. (25)that has the form of a tensor potential, whichhas no counterpart in classical electrodynamicstheory and comes from the geodesic equation.According to the geometrization of forces, anew force should exist, derived from

∑ , =5

8

h4 4 =

− 140

1me c2

eQR

vi

cvi

T

cvk

cvk

T

c

(28)

with summation over i and k, assuming values1,2,3. The potential of Eq. (28) describes ascalar potential that exists at a location inspace carrying a specific charge e/me. Addingup all three contributions results in a potential

h4 4 ph= 1

40

1me c2 ×

eZ r er N

−eQR

vi

cvi

T

c−eQ

Rvi

cvi

T

cvk

cvk

T

c.

(29)

For distances r < rN, Z(r) is replacing Z, ac-counting for the shielding effect of the chargeof the nucleus by the virtual electrons that arebeing formed in the vicinity of a nucleuswithin the range of the Compton wavelengthof the electron. It should be noted that the elec-tron charge, -e, was used in the first term. Inthe second and third terms it should be notedthat eQ > 0, since electrons are involved.From Eq. (27) it is required that the 4-dimen-sional vector potential, ( , ϕ Ai) with i=1,2,3, ofclassical electrodynamics has to be replaced bythe 4-dimensional tensor potential ( , ϕ Ai, Aik)with i,k =1,2,3. Since velocities of charges in amaterial body are much smaller than the speed

of light, the value of the factor vk /c vkT /c be-

ing in the range of 10-11 to 10-16, it is under-standable that the tensor potential was notseparately identified so far. Expressing

e Z r e=eZeeZ e r where e(r) rep-

resents the additional positive charge of thenucleus resulting from the shielding effect ofthe virtual electrons (see below), Eq. (29)takes the form

h4 4 ph = 1

40

1me c2 ×

eZer N

eZ e

r N

−eQR

vi

cvi

T

c−eQ

Rvi

cvi

T

cvk

cvk

T

c.

(30)

Considering a nucleus of one of the atoms inthe material comprising the torus, there is a lo-cation rN for which the first and third terms ofEq. (30) cancel, namely for

rN= Z eQ

Rcvi

c

viT (31)

where the constant charge value Ze was used.With e=Ae , see Eq. (35), the followingequation holds

eZ er N

= AeQR

vi

c

viT

c. (32)

The value of A, derived from vacuum polari-zation, is specified in Eq. (36) and computedin Eqs. (37, 38). If the value rN is smaller than

C= hme c

=2.43×10−12 m , the Compton wave-

length of the electron, the second term in (30)is different from 0 and the speed vi can be cho-sen such that the first and the third terms can-cel, leading to

h4 4 ph= 1

40

1

me c2

eQR

vi

cvi

T

c A−

vk

cvk

T

c. (33)

From the nature of A, it is obvious that the firstterm in the above potential is generated fromthe vacuum, wile the second term comes fromthe tensor potential generated in the coil. Thetotal energy extracted from the vacuum is,however, always zero. According to L. Kraussin [3] the cosmological constant is 5×10-10

J/m3. This means that the conversion of pho-tons into gravitophotons begins to occur as

soon as the condition h4 4 ph≈0 is satisfied.

12

2.4 Physical Model for Gravitophoton Gen-eration

In the following, starting from Eq. (33), thephysical mechanism is presented, responsiblefor the conversion of photons into gravitopho-tons. The mechanism for the generation of thepostulated negative and positive gravitophotonparticles is based on the concept of vacuumpolarization known from Quantum Electrody-namics (QED). In QED the vacuum behaveslike a dielectric absorbing and producing vir-tual particles and the Coulomb potential is as-sociated with the transfer of a single virtualphoton. Vacuum polarization in form of theelectron-photon interaction changes the Cou-lomb potential of a point charge for distanceswithin the electron Compton wavelength withrespect to a nucleus.

The velocities vi , viT in combination with the

total charge Q in the current loop or magneticcoil need to be chosen such that

r NC , (34)

otherwise vacuum polarization does not occur.It should be noted that the experiment allowsto vary these three parameters. However, aswill be shown below, two more conditionshave to be satisfied. In addition, the material inthe torus should contain hydrogen atoms to geta value of Z as small as possible, that is closeto 1.

A conversion of photons into gravitophotons ispossible according to Eqs. (35). The first equa-tion describes the production of N2 gravitopho-ton particles14 from photons. This equation isobtained from Heim's theory in 8D space incombination with considerations from numbertheory, and predicts the conversion of photonsinto gravitophoton particles. The second equa-tion is taken from Landau [16]

w phr −w ph=Nwgp

wphr −wph=Awph . (35)

The physical meaning of Eqs. (35) is that anelectromagnetic potential containing probabil-ity amplitude Awph can be converted into a

14 The factor N2 results from the fact that in Eq. (35)probability amplitudes are considered, but the gen-eration of particles depends on actual probabilities.It should be noted that N is not needed, but the prod-uct Nwgp.

gravitophoton potential with associated prob-ability amplitude Nwgp. From Eqs. (35) the fol-lowing relation holds for gravitophoton pro-duction, requiring the existence of a shieldingpotential

Nwgp=Awph . (36)

The function A(r) can be calculated from Lan-dau's radiation correction [16] and is given by

A= 23

∫1

∞

e−2

me c

ℏr

1 1

2 2 2−11/2 /2 d (37)

with numerical values for A ranging from 10-3

to 10-4. For small r (r << λC) the integral in Eq.(37) can be evaluated

A=−2 3 ln

me c

ℏrCE5

6 (38)

where CE = 0.577 is Euler's constant. For r >>λC, the integral Eq. (37) falls off exponentially

as e−2

me ℏc

r. Vacuum polarization changes the

Coulomb potential of a point charge only fordistances r < λC. The radiation correction isnot only caused by electron-positron interac-tion, but interaction with muons and pions isalso possible. QED works for muons, but doesnot work for pions, since they are subject tothe strong interaction. Therefore, for r ~ h/mπc,QED will not suffice anymore, i.e., there is noapplicable theory. Hence, the physical modelpresented below is limited to this fact.

The third condition is, according to Eq. (33),to make the photon potential vanish, i.e., totrigger the conversion of a photon into nega-tive and positive gravitophotons, which re-quires that A takes on a value à that is

A=vk

cvk

T

c (39)

where the value of à depends on the velocitiesof the charges in the coil and the rotating torus.This conversion takes place at a larger value ofr, since the product on the RHS of Eq. (39) issome 10-11.

2.5 Conversion of Photons into Gravitopho-tons

To summarize, there are the following threeconditions to be satisfied in order to convert aphoton into a pair of negative and positive

13

gravitophotons while insuring that the total en-ergy extracted in form of gravitophoton parti-cles from the vacuum is zero.

A=vk

cvk

T

c

r N C= hme c

r N = Z eQ

Rcvi

c

viT

(40)

The crucial point in the interpretation of Eq.(40) is that the first equation provides a valueof Ã≈10-11. This value is needed to start con-verting photons into gravitophotons. However,for this value of à the conversion process isnot efficient, i.e., the number of gravitopho-tons produced is too small to result in an ap-preciable force. Equations two and three deter-mine the conditions at which, according to Eq.(42), an effective gravitophoton potential ex-ists for which the respective value rN is deter-mined. The corresponding value for A > à issome 10-3. It should be noted that Eq. (39) isnot interpreted as a resonance phenomenon,but sets a condition for the photon potential todisappear and the gravitophoton potential toappear that is, for the onset of the conversionof photons into gravitophotons. Once this hap-pened, the value of A can be increased further,giving rise to an efficient and effective gravi-tophoton potential for field propulsion15.

In the following these conditions will be em-ployed to determining the technical require-ments of a gravitophoton propulsion device.Since an almost flat space was assumed, theequation for the gravitophoton metric, Eq.(22), is reduced to a single component in directanalogy to Eq. (25), and thus the equation forthe gravitophoton potential can be written as

h4 4gp= 1

40

1

me c2

eQR

vi

c

viT

cA . (41)

From the nature of A, it is obvious that theabove potential is generated from the vacuum.

In addition, a factorN ' wgp

wph

=1 is introduced

in Eq. (42) to emphasize that this equationdoes not contain any electrical charges any-

15 It should be noted that this not a proof that the con-version process takes place as indicated. Only theexperiment can prove the correctness of this as-sumption.

more, since it describes a purely gravitationalfield. Replacing A by Eq. (36) and insuring that thepotential of Eq. (41) identically vanishes, theconverted gravitophoton field takes the form

h4 4gp∓ =∓

Nwgp

w ph

N ' wgp

wph

140

1me c2

eQR

vi

cvi

T

c.(42)

The ∓ sign in Eq. (42) represents the fact

that there are both attractive and repulsivegravitophotons as described by the two metricforms in Eqs. (9, 10). The sum of the two po-tentials adds up to 0, satisfying Eq. (22). Thegravitophoton field is a gravitational like field,acting on material particles, except that it canbe both attractive and repulsive, and is repre-sented by two different types of gravitopho-tons. However, the coupling constants of thetwo particles are different, and only the nega-tive (attractive) gravitophotons are absorbedby protons and neutrons, while absorption byelectrons can be neglected16. Under certain cir-cumstances, a material body may be able totransition into a postulated parallel space thatis not subject to the limit c, the vacuum speedof light. Any gravitophoton propulsion devicetherefore works as a two-stage system, firstaccelerating the spacecraft by gravitophotonforce and then, for certain values of the mag-netic field and torus properties, causes a tran-sition into parallel space.

3 Space Flight Dynamics of Gravito-photon Field PropulsionIn this chapter the two-stage gravitophotonpropulsion system is discussed. In Secs. (3.1,3.2) the acceleration phase is described, andSec. (3.3) discusses the transition into parallelspace, presenting the physical laws governingparallel space. In addition, the physical condi-tions for transition into and leaving parallelspace are outlined.

3.1 Gravitophoton Interaction Equations forSpace Propulsion

Negative gravitophotons are subsequently ab-sorbed by the protons in the torus which havea large absorption cross section compared topositive gravitophotons. In the non-relativisticcase, the scattering cross section for photon-

16 The amount of energy extracted from the vacuum ingravitophoton pair production is zero.

14

15

Figure 1: Instead of a simple current loop, a coil with many turns can be used. Both, the current in the coil andthe rotation are in counter-clockwise direction. The field of this coil gives rise to an inhomogeneous magneticfield that has a radial field component. The radial component and the gradient in z-direction are related

through H r=− r2

∂ H z

∂ 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 ferromagneticmaterials are drawn toward the region of increasing magnetic field strength (down). The gravitophoton foresuperimposes these effects.

The gravitophoton force comes into play as soon as the ring starts rotating and the condition according to Eq.(40) is satisfied. i.e., the velocity components vk and vk

T must have the same sign. Perhaps equipment used tomeasuring magnetic moments can be employed to determine the gravitophoton force. For instance, if a para-magnetic substance is used, the gravitophoton force (up) could be used to balance the magnetic force, so thatthe resulting force is 0. From Refs. [23 and 24] it is found that a quartz sample (SiO2, diamagnetic) of a massof 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 cal-cium sample (paramagnetic) of the same mass would be subject to a force of -7.2 × 10-4 N. It is importantthat the material of the rotating ring is an insulator to avoid eddy currents. For the acceleration phase, thetorus should contain hydrogen atoms. For transition into parallel space another material should be used.

r

Br

BI

N

proton interaction is given by =83

r p2 ,

where rp is the classical proton radius, givenby

r p=1

40

e2

mp c2 =w ph2 ℏ

m p c. (43)

For gravitophotons, wph has to be replaced by17

Nwgp, since in the conversion process fromphoton to gravitophotons , N2 gravitophotonpairs are generated according to Eq. (35). Theabsorption cross section for a attractive (nega-tive) gravitophoton particle by a material parti-cle (here a proton or neutron is assumed) isgiven as

gp=83

Nwgpa 4 ℏmp c

2

. (44)

However, if the absorption is by an electron,the proton mass mp has to be replaced by elec-tron mass me. Therefore, the absorption crosssection of a proton is larger by the factor(mp/me)2. Hence, the absorption of negativegravitophotons by electrons can be neglected.

For the generation (emission) of a gravitopho-ton pair from the vacuum by means of a virtualelectron, the coupling constant is given by18

wgpe2 =Ggp

me2

ℏ c. For the absorption of a negative

gravitophoton by a proton, the coupling con-

stant has the form wgpa2 =Ggp mp

me

ℏ c. Using the

absorption cross section for protons, the prob-ability for this process is obtained as

w=323

Nwgpa 4 ℏmp c

2

d

d03 Z . (45)

d is the diameter of the torus, d0 the diameterof the atom in its ground state, and Z denotesthe mass number of the atom. Since the firstequation in Eqs. (35) describes the conversionof photons into N2 gravitophoton pairs, α gp

needs to be replaced by N2αgp. The force re-sulting from this conversion process, termed

17 In the following, a distinction between emission(electron mass) and absorption (proton mass) ofgravitophotons is necessary.

18 The value of Ggp is given in Appendix B.

the Heim-Lorentz force [1], which is a gravi-tational force, has the form

Fgp=−w N 2 gp

e0 vT×H .

(46)

The total force of the negative gravitophotonson the rotating body can be expressed in aform analogous to the Lorentz force

Fgp=−p e0 vT×H (47)

where p indicates that only proton and

neutron absorption processes were considered.

From Eqs. (45, 46) p is determined as

323 Nwgpe

wph

2

Nwgpa 4 ℏmp c

2

d

d03 Z . (48)

p (dimensionless) is a highly nonlinearfunction of the probability amplitude of thegravitophoton particle.

It is important to note that Eq. (47) only de-scribes the acceleration stage of gravitophotonfield propulsion. It should be noted that thecurrent understanding is that the kinetic energyof the spacecraft is provided from the vac-uum19 and not from the magnetic field that isneeded only to maintain the conversion proc-ess. The role of the magnetic field seems to bethat of a catalyzer.

Conditions for entering a parallel space aregiven in Sec. (3.3). As will be seen, a com-pletely different physical scenario needs to beestablished, requiring magnetic induction ofsome 30 T and torus material different fromhydrogen.

3.2 Technical Data for Acceleration Gravito-photon Field Propulsion

Formulas (47, 48) will be used to calculate thestrength of the gravitophoton field. To increasethe strength of the interaction, a material con-taining hydrogen atoms should be used, be-cause of the small value of r. For interstellarmissions a different material should be used.

19 It is emphasized that the total energy extracted is 0.

16

n N wgpe 0 H

(T)

Fgp

(N)

104 2.6× 10-14 2.0 7.14×10-43

105 1.1 ×10-5 6.3 3×101

106 1.5×10-4 20.0 4.5×107

106 2.5×10-4 50.0 1.45×109

Table1: The right most column shows the total gravitophotonforce in Newton that would act on the rotating ring. Theforce results from the absorption of the gravitophoton by aproton. The absorption through a proton results in a muchlarger force, so that in principle the interaction of a gravito-photon with an electron, regardless whether real or virtual,can be neglected. The number of turns of the magnetic coil isdenoted by n, the magnetic induction is given in Tesla, andthe current through the coil is 100 A, except for the last rowwhere 250 A were used. The mass of the rotating torus is 100kg, its thickness, d (diameter) 0.05 m, and its circumferentialspeed is 103 m/s. The wire cross section is 1 mm2. The mean-ing of the probability amplitude is given in the text.

For instance, if a larger spacecraft of 105 kgwith a rotating ring of 103 kg needs to have aconstant acceleration of 1g, a magnetic induc-

tion 0 H of some 13 T is needed together

with a current density of 100 A/mm2 and a coilof 4×105 turns for a value N wgpe=4.4×10−5 .The resulting force would be 106 N. Thus alaunch of such a spacecraft from the surface ofthe earth seems to be technically feasible.

The high current in the superconducting coilproduces a magnetic field H, that can be de-rived from a vector potential. Velocity vk is thespeed of the charge in the current loop or coil(Fig. 1). It is assumed that a superconductor isproducing charge speeds of some 103 m/s. To-gether with the velocity vk

T of the rotatingtorus, this magnetic field generates the conver-sion potential according to Eq. (33). Photonsare converted into negative and positive gravi-tophotons. Negative gravitophotons are ab-sorbed by protons and neutrons, while positivegravitophotons do not interact, thus resultingin a measurable force.

3.3 Space Flight in Parallel Space

Gravitophoton propulsion takes place in twophases. In phase one a spacecraft is subject toacceleration in ℝ4. Covering large interplane-tary or interstellar distances, requires the tran-sition into parallel space, which is phase twoof the field propulsion system.

A complete mathematical discussion of paral-lel space cannot be given in the framework ofthis paper. Therefore, only the salient physicalfeatures and their consequences are presented.

As was shown in Chap. 2, an electromagneticfield can be transformed into a gravitationallike field, producing both negative and posi-tive gravitophotons from the vacuum. The fun-damental fact for transition into parallel spaceis that the gravitational potential of a space-craft with mass M is reduced by the interactionof the positive gravitophotons with gravitonsand their conversion into (repulsive) vacuumparticles. There is an additional conversionequation, similar to Eqs. (35), for convertingthe gravitons of the spacecraft together withthe positive gravitophotons into the postulatedvacuum particles (or quintessence particles),thus reducing the gravitational potential of thespacecraft. The gravitational potential, , isΦgiven by

which implies that obeys the Poisson equaΦ -tion

4G M=∮∇ r ' ⋅d S (49)

where integration is over the closed surface Sof a volume V in physical space that containsthe spacecraft. Following the arguments byPenrose [5, 18] (violation of causality) and byKrauss [3] (a signal is needed to tell spacetimeto warp, but its speed itself cannot exceed c),GR clearly does not allow to travel faster thanthe speed of light in spacetime ℝ4. Since ab-sorption of positive gravitophotons is reducing

, this would require either the mass of theΦspacecraft to be reduced in ℝ4, or the gravita-tional constant G to become smaller, owing toEq. (49). As a consequence of a reduced mass,conservation of momentum would require a

17

r =−∫Gg r ' ∣r−r '∣

d 3 r '

velocity c' > c 20 in ℝ4, which has to be ruledout. The decision for a reduced gravitationalconstant G' < G in ℝ4 is more difficult. To thisend, we refer to quantum gravity theory [26],according to which area is quantized, that is

=16ℏ G

c3 j j1. (50)

The minimal surface =8 3ℏ G

c3 is ob-

tained for j=1/2. Therefore any physical phe-nomenon that requires a gravitational constantG' < G or a speed of light c' > c in ℝ4 has to beruled out, violating the fact that τ is the mini-mum surface. On the other hand, because ofpositive gravitophoton action, actually is reΦ -duced, and thus the concept of parallel space(or parallel universe or multiverse) is intro-duced, denoted as ℝ4(n) with n . For ℕ n=1,v(1):=v (velocity of the spacecraft) and ℝ4(1):=ℝ4. It is postulated that a spacecraft, under cer-tain conditions, stated below by Eq.(52), willbe able to transition into such a parallel space.For G(n)=G/n, M(n)=nM, and c(n)= nc, thespacecraft would transition into nth-parallelspace ℝ 4(n). An indirect proof for the exis-tence of parallel spaces could be the observedphenomenon of dark matter, see Sec. (4.4).

A parallel space ℝ 4(n), in which covariantphysical laws with respect to ℝ4 exist, is char-acterized by the scaling transformation

x i n= 1

n2 x 1 , i=1,2,3 ; t n= 1

n3 t 1

v n=n v 1; c n=n c 1

G n=1n

G ;ℏn=ℏ ; n∈ℕ .

(51)

The fact that n must be an integer stems fromthe requirement in LQT for a smallest lengthscale. Hence only discrete and no continuoustransformations are possible. The Lorentztransformation is invariant with regard to thetransformations of Eqs. (51) 21. In other words,physical laws should be covariant under dis-

20 This is in contrast to [2] where a reduction in masswas assumed, and a velocity c' > c was postulated.The important fact, however, is the reduction of thegravitational potential.

21 It is straightforward to show that Einstein's fieldequations as well as the Friedmann equations arealso invariant under dilatations.

crete (quantized) spacetime dilatations (con-tractions).

The two important questions to be addressed,concern the value n, in particular, how it is in-fluenced by experimental parameters, and theback-transformation from ℝ4(n)ℝ4. The re-sult of the back-transformation must not de-pend on the choice of the origin of the coordi-nate system in ℝ4. For the lack of space, thedetailed discussion for the two mappings fromℝ4ℝ4(n)ℝ4 cannot be presented, but theresult is that the spacecraft has moved a dis-tance nvΔT when reentering ℝ4. This mappingfor the transformation of distance, time and ve-locity differences is not the identity matrix thatis, the second transformation is not the inverseof the first transformation22. The spacecraft isassumed to be leaving ℝ4 with velocity v, andΔT denotes the time difference between leav-ing and reentering 4, as measured by an obℝ -server in ℝ 4. It should be noted that energyconservation in ℝ4 has to be satisfied that is,the energy of the spacecraft remains un-changed upon reentering (provided no accel-eration occurred in ℝ4(n)), given by the relativ-istic formula

M vc2=

M 0 c2

1− v 1c 1 2

and v 1c 1

=v nc n

= nv 1nc 1

.

The value of n is obtained from the followingformula, Eq. (52), relating the field strength of

the gravitophoton field, ggp , with the gravi-

tational field strength, gg, produced by thespacecraft itself,

n=ggp

+

gg

Ggp

G. (52)

This formula will be used in the next section tocalculate the conditions for a transition intoparallel space. The positive gravitophoton fieldis generated together with the negative gravito-photon field, and, because of energy conserva-

22 In other words, a quantity v(n)=nv(1), obtained from

a quantity of ℝ4, is not transformed again when go-

ing back from ℝ4(n) to ℝ4. This is in contrast to aquantity like ΔT(n) that transforms into ΔT. The rea-son for this unsymmetrical behavior is that ΔT(n) is

a quantity from ℝ4(n) and thus is being trans-formed.

18

tion, has the same value. Therefore, itsstrength can be directly calculated from Eq.(47). Assuming a magnetic induction of 30 T,a current density of 230 A/mm2, and 4×105

turns for the magnetic coil, the positive gravi-tophoton field should result in an accelerationof 3×102 m/s2, in direct vicinity of the torus.Some 10 m away from the torus the accelera-tion should be some 0.1 g or 1 m/s2. This value

for ggp is being used in calculating the value

of n for the interplanetary and interstellar mis-sions of Sec. (3.4). The rotating torus gener-ates pairs of both negative and positive gravi-tophotons. Negative gravitophotons are ab-sorbed by protons and neutrons, while the re-maining positive gravitophotons interact withthe gravitons of the spacecraft, being con-verted into vacuum particles, thus reducing thegravitational potential of the spacecraft. Eq.(52) then determines the condition for transi-tion into parallel space ℝ4(n). Since n is an in-teger, the effect is quantized and requires a

threshold value for ggp .

3.4 Lunar, Interplanetary, and InterstellarMissions

In the following we discuss three missions, alunar mission, a Mars mission, and an inter-stellar mission. From the numbers provided, itis clear that gravitophoton field propulsion, iffeasible, cannot be compared with chemicalpropulsion or any other currently conceivedpropulsion system. Furthermore, an accelera-tion of 1g can be sustained during flight for lu-nar missions.

Gravitophoton field propulsion is a two-stageprocess. First, an acceleration is achieved bythe absorption of negative gravitophotonsthrough the protons and neutrons of the torusmaterial. In the second stage, a transition into aparallel space takes place that leads to a hugeincrease by a factor n, see Eq. (52), in speedwith regard to our spacetime ℝ4 (see previoussection).

For lunar missions only stage one is needed.The high values of magnetic induction for atransition to parallel space are not needed. Forthe lunar mission a launch from the surface ofthe earth is foreseen with a spacecraft of amass of some 1.5 ×105 kg (150 t). With a mag-netic induction of 20 T, compare Table (1), a

rotational speed of the torus of vT = 103 m/s,and a torus mass of 2×103 kg, an accelerationlarger than 1g is produced so that a launch ispossible. Assuming an acceleration of 1g dur-ing flight, the first half of the distance, dM, tothe moon is covered in some 2 hours, which

directly follows from t= 2 dM

g, resulting in a

total flight time of 4 hours. Since the distanceis very short, entering parallel space is notnecessary.

A Mars mission, under the same assumptionsas a flight to the moon, that is, if only stageone of the field propulsion is used, would needan acceleration phase of 414 hours. The finalvelocity would be v= gt = 1.49×106 m/s. Thiswould be a hypothetical value only, if theamount of energy needed would have to beprovided by the electromagnetic field. How-ever, this kinetic energy is extracted from thevacuum, although the total energy extractedfrom the vacuum that is in the form of nega-tive and positive gravitophotons, is zero. Thetotal flight time to Mars with acceleration anddeceleration is 34 days.

Using stage two field propulsion that is enter-ing parallel space, a transition is possible at aspeed of some 3×104 m/s that will be reachedafter approximately 1 hour at a constant accel-eration of 1g. The transition into parallel spacehas the effect that the velocity increases to0.4 c. In that case, total flight time would bereduced to some 2.5 hours23.

For an interstellar mission, the concept of par-allel space is indispensable. An accelerationphase of some 34 days with 1g would result ina final velocity of one per cent of the speed oflight, 0.01 c. Again, gravitophoton field pro-pulsion would obtain the kinetic energy fromthe vacuum. The transition into parallel spacewould need a repulsive strength of the gravito-photon field (positive gravitophotons), produc-

ing an acceleration ggp+ =1 m /s2 at some 10

m (order of magnitude) away from the space-

23 Today's propulsion systems demand long missiontimes to Mars, and adequate protection must be pro-vided against radiation hazards. Reinforced polyeth-ylene (hydrogen content) is being investigated, butmay significantly increase the mass of the space-craft.

19

craft. The gravitational field strength of thespacecraft itself with mass 105 kg is given by

gg=GM

R2 ≈6.67×10−8 m / s2 . Inserting these

values into Eq. (52), transition into parallelspace would cause a velocity gain by a factorof n = 3.3×104, resulting in an effective speedof 3.3×102 c. This means for an observer in ℝ4

that the spacecraft seems to have moved atsuch a superluminal speed. A distance of 10light-years could be covered within 11 days.The deceleration phase requires another 34days, so that a one-way trip will take about 80days to reach, for instance, the star Procyonthat is 3.5 pc24 from earth. There are about 30known stars within a radius of 13 light-yearsfrom earth.

4 Cosmology from HQT and LQT Despite the successes of modern physics, themost fundamental questions, as will be shownin the subsequent section, cannot be answered.It is clear that the current status of physics isfar from being the final theory. Therefore, toargue that something is not possible based onthe insights of current theory, is not necessar-ily true. Advanced propulsion systems do re-quire novel physics beyond present concepts.Quantum gravity could be a key theory.

4.1 Deficiencies in Current FundamentalPhysical Theories

The wave picture of QT does not describe theparticle aspect occurring in Nature. The cur-rent standard model, trying to describe particlefeatures by introducing new additional quan-tum numbers, has not been successful in ex-plaining fundamental physics such as themeasured mass spectrum of elementary parti-cles and their lifetimes [17], neither can thevery nature of matter be explained. It is alsonot known how many fundamental physical in-teractions exist, neither is the dimensionalityof space, nor can quantum numbers be de-rived. If all the quantum numbers describingan elementary particle have to be introducedad hoc, it would be difficult to imagine such aparticle as elementary. According to Heim thisis actually not the case [4].

24 Parsec is a distance and 1 pc= 3.26 ly.

Despite its success in predicting the mass ofsome new particles, quantum electrodynamicsand quantum chromodynamics are plagued bylogical inconsistencies, i.e., by infinities. Re-normalization gets rid of these infinities bysubtracting infinities from infinities in order toget something finite. This is only justified bythe meaningful results that follow from thisphysically inconsistent process [17].

A major question therefore is on the roles ofand the relationship between GR and QT withregard to the explanation of physical reality,and how these theories could be modified todescribe the material world in a consistentway. Furthermore, it has to be clarifiedwhether current theory has found all possiblephysical interactions. For instance, many sci-entists have already guessed that an interactionbetween electromagnetism and gravitationshould exist [3], not contained in the laws ofcurrent physics.

Einstein's GR of 1915 is a revolutionary theorythat could provide the framework of a geo-metrization of all physical interactions: gravityis no longer treated as a force, but is repre-sented by the curvature of spacetime. Over thelast two decades GR has been tested to be cor-rect to one part in 1014 by measuring theshrinking in the orbit of the Hulse-Taylor bi-nary pulsar (two neutron stars where one is apulsar) [23], i.e., measuring the periodic Dop-pler shift of the pulsar's radiation. The energyloss is attributed to gravitational waves.Hence, the accuracy of GR is greater than forQT or even QED. If gravity is not to be treateddifferent from all other physical interactions,and since it is confirmed to such a marvelousaccuracy, it seems to be justified to use this ap-proach for all physical forces, and also to ap-ply this concept to the subatomic range. There-fore, in Heim's approach Einstein's theoryserves as the paradigm, on which all otherphysical theories are to be modeled.

Consequently, Heim has extended four dimen-sional spacetime to higher dimensions (addi-tional imaginary coordinates), constructing apoly-metric, and assigning all physical inter-actions their proper metric. In the subatomicrange, the quantization of spacetime proved tobe necessary. In this way a unified theory wasobtained. In other words, physics is geometry,and matter is geometry, too.

20

When Heim solved the resulting eigenvalueequations, the mass spectrum of ponderableparticles was obtained as described in [6, 12],along with their quantum numbers. Elementaryparticles themselves are dynamical, cyclicstructures built from metrons, elemental sur-faces with spin, in a hierarchical way [4, 7].Thus, a complete geometrization of Nature isachieved, reducing matter to a cyclic feature ofhigher-dimensional space itself. Most impor-tant, however, if the admissible metric combi-nations are investigated, they lead to the con-clusion that six fundamental interactions mustexist.

4.2 Common Concepts in HQT and LQT

Though HQT is based on geometrical aspectsin a 6, 8, or 12-dimensional space, it is neithercontinuous nor smooth, but contains an ele-mental surface area, the metron, equipped witha spin vector. A quantized volume element,bounded by metron surfaces, may have allspins pointed outward (exogen) or all spins di-rected inward (endogen). Because of the isot-ropy of space the metronic lattice (also called -lattice) contains both types of volumes.Empty space comprises a dynamic lattice oforthogonal elemental cells. Any lattice that de-viates from this Cartesian lattice is called a hy-perstructure and is capable of describingphysical events. The curvature of space, inher-ent to a hyperstructure, is termed condensa-tion, since the number of metrons (because oftheir fixed size) on a curved surface must belarger than on its projection into Euclideanspace. Hyperstructures are described by an ei-genvalue problem, leading to quantized levelsof space that are associated with real physicalstates. Space itself is ascribed a structural po-tential, meaning that fundamental ontologicalqualities of space itself appear as geometricstructures.

When compared to recent ideas from loopquantum gravity, there is similarity on theideas of the quantized structure of spacetime.Furthermore, both theories start from Ein-stein's GR, requiring background independ-ence (no fixed coordinate system, but a dy-namical evolving geometry) and independenceon the coordinate values (the physics must notdepend on the choice of coordinate system,also termed diffeomorphism invariance). Spinnetworks in quantum loop theory [11] and

Heim's hyperstuctures are both used to repre-sent dynamical quantum states of space. Heimuses a higher-dimensional space, for instancein 6D space there are 30 metron surfacesbounding a volume, to construct a poly-metricto unify all physical forces [4]. Loop quan-tum theory currently is formulated in 4Dspacetime and does not explicitly rely on ametric. As mentioned by its authors the exten-sion to higher dimensional spaces should bepossible. However, if physical quantities needto be computed, a metric eventually is indis-pensable25.

While Heim's derivation of the metron is basedon heuristic physical arguments, the picture ofquantum spacetime in LQT is on firm mathe-matical ground, see, for instance, [28]. Itseems that Heim's approach resembles theBohr model of the atom, while LQT is morelike the Heisenberg picture. On the other hand,Heim constructed a unified theory through theintroduction of a poly-metric in a higher-di-mensional space, predicting the existence oftwo additional fundamental interactions.Moreover, he constructed a set of eigenvalueequations from the field equations of GR re-sulting in the derivation of the mass spectrumfor elementary particles [4, 6, 12]. Nothing canbe said at present whether HQT, in analogy toEinstein's GR, could be cast into the frame-work of Ashtekar's novel formulation in whichGR is in a similar form to Yang-Mills theory[24, 26]26. It should be noted that according toHeim any material particle has its associatedproper hermetry form, and, as a consequenceof this, there is a unity of field and fieldsource.

One important consequence of any theorybased on quantized surfaces and volumes isthat the picture of an elementary particle as apoint-like entity [17] becomes untenable. Ac-cording to GR and QT, point-like particleswith mass will collapse into black holes [26]and disappear, which is in stark contrast to thevery existence of elementary particles. Withthe classical radius of the electron of some

25 This includes the fact that all intermediate volumesconnecting initial and final volumes on the lattice orthe spin network could be identified and numbered.To this end, Heim developed his own mathematicsand coined the term selector, see Chap. 3 in [6].

26 The idea for the comparison of the two theories isdue to Dr. Jean-Luc Cambier, Senior Scientist, Pro-pulsion Directorate, EAFB.

21

3×10-15 m, and a metron size of some 10-70 m2,about 1041 metrons are needed to cover the sur-face of the electron. An electron thereforemust be a highly complex geometrical quan-tity. According to Heim, elementary particleshaving rest mass constitute self-couplings offree energy. They are indeed elementary as faras their property of having rest mass is con-cerned, but internally they possess a very sub-tle, dynamic structure. For this reason they areelementary only in a relative sense. The argu-ment, put forward by C. Rovelli [25] that had-rons cannot be elementary, because there aretoo many quantum numbers needed for theirdescription is not necessarily true. Heim, in his1977 paper [4], uses a set of 12 quantum num-bers to describe an elementary particle, andclaims that these quantum numbers can be re-duced to a single quantum number k=1 or k=2

and the decision =±1 for particle or anti-

particle.

4.3 Cosmological Consequences

Going back in time, the volume of the universebecomes smaller and smaller [1, 2, 28]. Be-cause of the quantization of area and volume asingularity in space cannot develop. The uni-verse would have started with the smallestpossible volume, namely a volume being pro-

portional to the Planck length cubed, ℓ p3 . Ac-

cording to Heim this was, however, not thecase, since the metron size, , is increasingτwhen going back in time while, in parallel, thenumber of metrons decreases, until there is asingle metron only, covering the primeval uni-verse.

This links the dynamical evolution with theinitial conditions, and allows for one singlechoice only. Thus, the problem of initial con-ditions for the universe is solved by quantiza-tion 27.

Moreover, in HFT, gravitation is attractive be-tween a distance R- < r < , and becomesρslightly repulsive for ρ < r < R+ and goes to 0for r > R+. R- is a lower bound for gravitational

27 No discussion is intended of pre-structures beforethe universe came into being via creation of the firstmetron, see [9]. The idea presented there, the apei-ron, is similar to Penrose's ideal mathematicalworld [5].

structures, comparable to the Schwarzschildradius. The distance at which gravitationchanges sign, , is some 46 Mparsec. ρ R+ de-notes an upper bound and is some type ofHubble radius, but is not the radius of the uni-verse, instead it is the radius of the opticallyobservable universe. Gravitation is zero be-yond the two bounds, that is, particles smallerthan R- cannot generate gravitational interac-tions.

Consequently, the cosmological redshift is ex-plained by the repulsive gravitational potential,and not by the Doppler shift. A more detaileddiscussion is given in Sec. 5 of [2]. Accordingto Heim the age of the universe is some 10127

years. Matter, as we know it, was generatedonly some 15 billion years ago, when , theτmetron size, became small enough.

A very interesting fact is that in HQT the con-stants G, , ћ ε0, μ0, and are all functions of theτdiameter, D, of the primeval universe in whichour optical universe is embedded. Since matteris very recent in comparison with the age ofthe primeval universe, these constants re-mained practically unchanged for the last 15billion years (for more details see Sec. 2.2 [1]).

Although HFT and LQT [28] provide a differ-ent cosmogony, they both solve the singularityproblem based on quantized spacetime as wellas the problem of initial conditions. Both theo-ries make proposals that can be confrontedwith cosmological observations.

4.4 Dark Matter

The existence of parallel spaces could indi-rectly support the observed amount of darkmatter. According to our computations, themass in all parallel spaces ℝ 4(n) should besome 7 times the mass in ℝ4, and should befelt via gravitational interaction in ℝ4. Darkmatter therefore should be around 28% withregard to the sum of visible and non-visiblematter (currently at about 4%) in ℝ4.

4.5 Dark Energy

Dark energy is explained by hermetry formH10, and is the sixth fundamental interactionproposed by HQT. This interaction is termedvacuum field and is repulsive.

22

5 Conclusions and Future WorkThe authors are aware of the fact that the cur-rent paper contains shortcomings with regardto mathematical rigor, and also proposes twohighly speculative concepts28. It should be keptin mind, however, that any type of field pro-pulsion29 necessarily must exceed conventionalphysical concepts. In addition, the importantconversion equations for photons into gravito-photons and gravitophotons into vacuum(quintessence) particles were not derived, sim-ply for the lack of space, see [9].

The first of these concepts is the complete geo-metrization of physics, extending the Einstein-ian picture to all physical interactions. This re-quires an 8D space, termed Heim space, com-prising four subspaces that are used to con-struct a polymetric. Each of the partial metric,termed hermetry form (see glossary), repre-sents a physical interaction or interaction parti-cle. As a consequence, there are six fundamen-tal interactions, instead of the four knownones. The two additional interactions aregravitational like, one allowing the conversionof photons into hypothetical gravitophotonparticles, generated from the vacuum thatcome in two forms, namely repulsive and at-tractive. This interaction is the basis of theproposed gravitophoton field propulsion.Whether or not the mechanism, described indetail in Chap. 2, on which this field propul-sion is based, is true can only be decided byexperiment. Consequently, an experimental setup along with calculated gravitophoton forceswas presented.

The sixth interaction is identified with thequintessence and is repulsive, thus giving anexplanation for the observed expansion of theuniverse.

The second concept concerns the transition ofa material object into a so called parallel space(i.e., there are other universes), in which thelimiting speed is nc 30, where c is vacuumspeed of light and n > 1 is an integer (accord-

28 The reader should remember the remark byA.Clarke that any future technology is indistin-guishable from magic.

29 Field propulsion means that the propulsion mecha-nism is not based on the classical principle of mo-mentum conservation as being used in the rocketequation.

30 The problem of causality is still present. Also, theqestion of navigation a spacecraft in

ing to our computations there is an upper limitof 6.6×1010 for n). The concept of parallelspaces could indirectly be justified, since italso can be used in calculating the amount ofdark matter. Additional matter exists in theseparallel spaces, and via gravitational interac-tion may cause the observed effects in our uni-verse, attributed to dark matter. The questionof navigation in a parallel space with regard toℝ4 cannot be answered at present.

Substantial work needs to be done to refine thecalculations for the gravitophoton force andthe experimental setup.

With respect to the theoretical framework ofHQT, the authors feel that HQT could benefitmuch from the mathematical structure of LQT.Recent LQT seems to have the potential torevolutionize the physical picture of space-time. Most interesting, from the concept ofminimal surface it follows directly that super-luminal speeds or the reduction of G are pro-hibited in our spacetime. HQT postulating theexistence of gravitophotons that can reduce agravitational potential, thus requires the con-cept of parallel space.

It is interesting to see that HQT, being devel-oped much earlier than LQT, is based on simi-lar assumptions. Therefore, in view of the pro-gress in quantum gravity and the similaritiesbetween the two theories, the attempt to castHeim's theory into the modern formulation ofloop theory seems to be worthwhile, because,if such a formulation can be achieved, it wouldbe a clear hint that these additional interactionsactually might exist, having an enormous im-pact on the technology of future flight andtransportation in general.

Acknowledgment

The authors dedicate this paper to Prof. P. Dr.Dr. A. Resch, director of IGW at InnsbruckUniversity, at the occasion of his 70th birthday.Prof. Resch has not only published the theoriesof Burkhard Heim, but was also instrumentalin editing the complete scientific work ofHeim. This proved to be an enormously diffi-cult and time-consuming task, since Heim asan author, because of his handicap, was notable to proofread complex manuscripts, andthus could not help with the typesetting of thecomplex formulas. The authors also wish toacknowledge the voluminous scientific work

23

of Dr. Resch, Imago Mundi, whose prime sub-ject was and is the creation of a consistentWeltbild, acceptable both in science and hu-manities to bridge the gap that currently di-vides these two disciplines.

The authors are very much indebted to Dipl.-Phys. I. von Ludwiger, currently secretary ofthe Heim Forschungskreis and former managerand physicist at DASA, for making availablerelevant literature and for publishing a recentarticle about Heim's mass formula calculatingthe mass spectrum of elementary particles,providing a comparison with latest experimen-tal results.

The second author was partly funded by Arbe-itsgruppe Innovative Projekte (AGIP), Minis-try 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 ofApplied Sciences at Salzgitter Campus in thepreparation of this manuscript is appreciated.

Appendix A: Mass Spectrum of Ele-mentary ParticlesMass spectrum of elementary particles as cal-culated from HQT together with comparisonsof recent experimental data are available from

http://www.uibk.ac.at/c/cb/cb26/heim/index.html.

Appendix B: Gravitational CouplingConstants for the three gravitationalinteractions as obtained from numbertheory (see [9]). For the table below, it should be noted that allcoupling constants are computed with respectto the proton mass. In this regard, wq is a ficti-tious value only, since there is no emitting realproton mass. Most likely, the generation en-ergy for vacuum particles is the Planck mass.Dimensional units used for table entries are kg,m, and s.

Gravitational Coupling Constants

wg 7.683943001×10-20 graviton

wgp 1.14754864 ×10-21 gravitophoton

wq 1.603810891×10-28 vacuum particle(quintessence)

Ggp 1/672 G wg

wgp2

= GGgp

Gq 4.3565×10-18 G wg

wq

2

= GGq

gα 5.904298005×10-39

G 6.6722037×10-11 calculated

Ge 6.673(10)×10-11 experimental value

Glossary31

aeon Denoting an indefinitely long period oftime. The aeonic dimension, x6, can be in-terpreted as a steering structure governedby the 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 termintroduced by Heim, is a necessary but nota sufficient condition.

conversion amplitude Allowing the transmu-tation of photons into gravitophotons,wph_gp (electromagnetic-gravitational inter-action), and the conversion of gravitopho-tons into quintessence particles, wgp_q

(gravitational-gravitational interaction).

coupling constant Value for creation and de-struction of messenger (virtual) particles,relative to the strong force (whose value is

31 A more comprehensive glossary is available underwww.cle.de/hpcc, see Publications

24

g=Gm p

2

ℏ ccalculated

set to 1 in relation to the other couplingconstants).

coupling potential between photon-gravito-photon (Kopplungspotential) As cou-pling potential the first term of the metric

in Eq. (23) is denoted, that is gi kgp . The

reason for using the superscript gp is thatthis part of the photon metric equals themetric for the gravitophoton particle andthat 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 repulsive gravitational like force,and thus can be used to accelerate a mate-rial body.

cosmogony (Kosmogonie) The creation ororigin of the world or universe, a theory ofthe origin of the universe (derived fromthe two 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, x5, can be interpreted as a measure ofthe quality of time varying organizationalstructures (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.

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 kgp , generated by a neutral mass

with a smaller coupling constant than the

one for gravitons, but allowing for the pos-sibility that photons are transformed intogravitophotons. Gravitophoton particlescan be both attractive and repulsive andare always generated in pairs from the vac-uum under the presence of virtual elec-trons. The total enery extracted from thevacuum is zero, but only attractive gravi-tophotons are absorbed by protons or neu-trons. The gravitophoton field representsthe fifth fundamental interaction. Thegravitophoton field generated by repulsivegravitophotons, together with the → vac-uum particle, can be used to reduce thegravitational potential around a spacecraft.

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

Heim-Lorentz force Resulting from thenewly predicted gravitophoton particlethat is a consequence of the Heim spaceℝ8. A metric subtensor is constructed inthe subspace of coordinates I2, S2 and T1,denoted as hermetry form H5, see [1, 6, 7].The equation 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

F gp=p q 0 vT×H . Here p is a coef-

ficient, vT the velocity of a rotating body(e.g., torus, insulator) of mass m, and H isthe magnetic field strength. It should benoted that the gravitophoton force is 0, ifvelocity and magnetic field strength areperpendicular. Thus, any experiment thatplaces a rotating disk in a uniform mag-netic field that is oriented parallel or anti-parallel to the axis of rotation of this disk,will measure a null effect.

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.

25

There are 3 coordinate groups in this

space, namely s3=1 ,2 ,3 forming

the physical space ℝ 3, s2=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 [6])), coordinates 5 and6 must always be curvilinear, and must bepresent in all metric combinations. An al-lowable combination of coordinate groupsis termed hermetry form, responsible for aphysical field or interaction particle, anddenoted by H. H is sometimes annotatedwith an index such as H10 , or sometimeswritten in the form H=(1, 2 ,...) where 1,2 ,... S. This is a symbolic notation∈only, and should not be confused with thenotation of an n-tuple. From the above it isclear that only 4 hermetry forms are possi-ble in ℝ6. It needs a Heim space ℝ8 to in-corporate all known physical interactions.Hermetry means that only those coordi-nates occurring in the hermetry form arecurvilinear, all other coordinates remainCartesian. In other words, H denotes thesubspace in which physical events cantake place, since these coordinates arenon-euclidean. This concept is at the heartof Heim's geometrization of all physicalinteractions, and serves as the correspon-dence principle between geometry andphysics.

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).

homogeneous The universe is everywhereuniform and isotropic or, in other words,is of uniform structure or compositionthroughout.

hyperstructure (Hyperstruktur) Any latticeof a Heim space that deviates from the iso-tropic Cartesian lattice, indicating an

empty world, and thus allows for physi-cal events to happen.

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.It can be shown that when this mapping isapplied to the Christoffel symbols theytake on tensor character.

probability amplitude With respect to the sixfundamental interactions predicted fromthe →poly-metric of the Heim space ℝ 8,there exist six (running) coupling con-stants. In the particle picture, the first threedescribe gravitational interactions, namelywg (graviton, attractive), wgp (gravitopho-ton, attractive and repulsive), wq (quintes-sence, repulsive). The other three describethe well known interactions, namely wph

(photons), ww (vector bosons, weak inter-action), and ws (gluons, strong interaction).In addition, there are two → conversionamplitudes predicted that allow the trans-mutation of photons into gravitophotons(electromagnetic-gravitational interaction),and the conversion of gravitophotons intoquintessence particles (gravitational-gravi-tational interaction).

quantized bang According to Heim, the uni-verse did not evolve from a hot big bang,but instead, spacetime was discretizedfrom the very beginning, and such no infi-nitely small or infinitely dense space ex-isted. Instead, when the size of a singlemetron covered the whole (spherical vol-ume) universe, this was considered thebeginning of this physical universe. Thatcondition can be considered as the mathe-matical initial condition and, when in-

26

serted into Heim's equation for the evolu-tion of the universe, does result in the ini-tial diameter of the original universe [1].Much later, when the metron size had de-creased far enough, did matter come intoexistence as a purely geometrical phe-nomenon.

sieve operator see → transformtion operator

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 bymaking the coordinate 4 Euclidean.

vacuum particle responsible for the accel-eration of the universe, also termedquintessence particle The vacuum parti-cle represents the sixth fundamental inter-action and is a repulsive gravitationalforce whose gravitational coupling con-stant is given by 4.3565×10-18 G, see Ap-pendix B.

References1. Dröscher, W., J. Häuser: Future Space Propulsion

Based on Heim's Field Theory, AIAA 2003-4990,AIAA/ASME/SAE/ASE, Joint Propulsion Confer-ence & Exhibit, Huntsville, AL, 21-24 July, 2003,25 pp., see also www.uibk.ac.at/c/cb/cb26 andwww.cle. de/hpcc.

2. Dröscher, W., J. Häuser: Physical Principles of Ad-vanced Space Transportation based on Heim's FieldTheory, AIAA/ASME/SAE/ASE, 38 th Joint Pro-pulsion Conference & Exhibit, Indianapolis, Indi-ana, 7-10 July, 2002, AIAA 2002-2094, 21 pp., seealso www.cle.de/hpcc andwww.uibk.ac.at/c/cb/cb26.

3. Millis, M.G.: (ed.), NASA Breakthrough PropulsionPhysics, Workshop Proceedings, NASA/CP-1999-208694 and NASA-TM-2004-213082, Prospectsfor Breakthrough Propulsion from Physics, May2004, www.grc.nasa.gov/WWW/bpp/TM-2004-213082.htm.

4. Heim, B.: Vorschlag eines Weges einer einheitlichenBeschreibung der Elementarteilchen, Zeitschrift fürNaturforschung, 32a, 1977, pp. 233-243.

5. Penrose, R.: The Small, the Large and the HumanMind, Cambridge University Press, 1997.

6. Heim, B.: Elementarstrukturen der Materie, Band 1,Resch Verlag, 3rd ed., Innsbruck, 1998.

7. Heim, B.: Elementarstrukturen der Materie, Band 2,Resch Verlag, 2nd ed., Innsbruck, 1984.

8. Heim, B.: Ein Bild vom Hintergrund der Welt, in A.Resch (ed.) Welt der Weltbilder, Imago Mundi,Band 14, Resch Verlag, Innsbruck, 1994.

9. Heim, B. and W. Dröscher: Strukturen der physi-kalischen Welt und ihrer nichtmateriellen Seite,Innsbruck, Resch Verlag, 1996.

10. Heim, B., Flugkörper, Heft 6-8, (in German only)1959.

11. Smolin, L., Atoms of Space and Time, ScientificAmerican, January 2004.

12. Ludwiger, von I., Grünert, K., Zur Herleitung derHeim'schen Massenformel, IGW, Innsbruck Univer-sity, 2003, 81 pp., paper (in German only) availablein PDF format at:

http://www.uibk.ac.at/c/cb/cb26/heim/index.html.

13. Cline, D.B.: The Search for Dark Matter, ScientificAmerican, February 2003.

14. Lawrie, I.D.: A Unified Grand Tour of TheoreticalPhysics, 2nd ed., IoP 2002.

15. Harris, E.G.: Modern Theoretical Physics, Vol I,Wiley&Sons, 1975.

16. Landau, L., Lifschitz, E., Lehrbuch der Theore-tischen Physik, Volume IV, §114, 1991.

17. Veltman, M., Facts and Mysteries in ElementaryParticle Physics, World Scientific, 2003.

18. Penrose, R., The Emperor's New Mind, Chap. 5,Vintage, 1990.

19. Ashford, D., Spaceflight Revolution, Imperial Col-lege Press, 2002.

20. Wentzel, G., Quantum Theory of Fields, Intersci-ence Publishers, 1949.

21. Jordan, P., Das Bild der mordernen Physik, StromVerlag Hamburg-Bergedorf, 1947.

22. Woan, G., The Cambridge Handbook of PhysicsFormulas, Cambridge University Press, 2000.

23. Hawking, S., Penrose, R., The Nature of Space andTime, Princeton University Press, Chap. 4, 1996.

24. Ashtekar, A., et al., Background Independent Quan-tum Gravity:A Status Report, 125 pp., arXiv:gr-qc/0404018 v1, 5 April 2004.

25. Rovelli, C., Quantum Gravity, Chap.1, 329 pp.,draft version 30 December 2003, to be published byCambridge Univ. Press, 2004.

26. Rovelli, C., Loop Quantum Gravity, Physics World,November 2003.

27. Rovelli, C., Quantum Spacetime, Chap. 4 in PhysicsMeets Philosophy at the Planck Scale, eds. C. Cal-lendar, N. Huggett, Cambridge Univ. Press, 2001.

28. Bojowald, M., Quantum Gravity and the Big Bang,arXiv:astro-ph/0309478, 2003.

29. Thiemann, T., Lectures on Loop Quantum Gravity,arXiv:gr-qc/0210094 v1, 28 Oct. 2002.

27