AIAA 2002-4094

PHYSICAL PRINCIPLES OF

ADVANCED SPACE PROPULSION

BASED ON HEIMS'S FIELD THEORY

Walter Dröscher1, Jochem Häuser2

1Institut für Grenzgebiete der Wissenschaft,

Leopold - Franzens Universität Innsbruck, Innsbruck, Austria

2Department of Transportation, University of Applied Sciencesand

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

THE SUGGESTION THAT SPACESHIPS MAY EVENTUALLY TRAVEL FASTER THAN LIGHT

WILL CAUSE MOST ASTRONOMERS TO THROW UP THEIR HANDS IN HORROR, AND FOR EX-CELLENT MATHEMATICAL AND PHYSICAL REASONS OUTLINED IN EINSTEIN'S THEORY OF

RELATIVITY. BUT SCIENCE IS STILL IN ITS EXTREME INFANCY. IT IS A BIT FATUOUS TO

THINK THAT WE HAVE APPROACHED ANY ULTIMATE REALITIES, OR ARE VERY LIKELY TO

UNTIL SCIENCE IS SEVERAL THOUSAND YEARS OLDER THAN IT IS NOW.

FROM SPACE FLIGHT BY C. C. ADAMS, MCGRAW HILL, 1956, P. 236.

38TH AIAA/ASME/SAE/ASEE

JOINT PROPULSION CONFERENCE & EXHIBIT,

INDIANAPOLIS, INDIANA, 7-10 JULY, 2002

1Senior scientist, 2 Senior member AIAA, member SSE, www.cle.de/cfd, @IGW Innsbruck Univ, Austria

2002 Institut für Grenzgebiete der Wissenschaft,Leopold - Franzens Universität Innsbruck,Innsbruck, Austria

ABSTRACT

In this paper an overview is given of the results of acompletely geometrized unified field theory that givesrise to a novel concept for an advanced space transporta-tion technology, permitting, in principle, superluminaltravel. This theory predicts the existence of a quasi anti-gravitational force, and allows the design of an experi-ment for the verification of this theory. This theory ofquantum gravity, based on publications by B. Heim etal. [1-6], introduces new physics at the quantum scale,predicting that a transformation of electromagnetic waveenergy at specific frequencies into gravitational like en-ergy, is possible. This transformation thus is reducingthe inertial mass of a material (ponderable) body. Thetheory was recently extended to 8 dimensions by thefirst author [4]. The predicted reduction of inertia (mass)can be used as the design principle for an innovativespace transportation system. In this paper, Heim’s fieldequations along with the extensions mentioned above,will be presented and discussed. In addition, the magni-tude of the coupling constant between the conversion ofelectromagnetic energy and gravitational like energy (socalled gravito-photons) will be given.

The authors have investigated the fundamental physicalassumptions as well as some of the predictions ofHeim’s theory, and carefully checked its logical consis-tency [7]. Although several attempts of a formulation fora unified quantum field theory (including gravity) havebeen made, their success has been very limited. There-fore, because Heim’s theory satisfies several moderntheoretical requirements, unknown at the time of its de-velopment, there is evidence for the physical correct-ness of this work. In fact, cosmological data stronglysuggest the possible existence of an anti-gravitationalforce (i.e. repulsive), and Heim’s concepts, developeddecades ago, may have striking solutions to this newlyfound experimental evidence. We believe that further in-vestigation in this theory is justified in view of its com-pliance with recently established criteria for a generalunified quantum field theory. Furthermore, the theoryseems to be logically and physically consistent, and, iffound to be correct, would offer an extremely high pay-off. The particular coupling between gravitation andelectromagnetism cannot be obtained from a manipula-tion of Einstein's equations of general relativity, e.g., bylinearizing the equations of general relativity, or by ex-tending Newton's gravitation law to a time dependentformulation, assuming that the gravitational equationsare of the same form as the Maxwell equations [8]. Thecoupling obtained from Heim’s theory is derived fromfundamental principles, and is very different from theones obtained by other ad-hoc approaches. Heim's the-ory is therefore much more interesting, since it may al-low gravity manipulation at lower energy densities, andis based on new physics, thereby leading to new predic-tions. There may be several new and surprising physicalphenomena with far reaching consequences that are pre-dicted by Heim's theory. Some of these can be checkedagainst presently available experimental data, both fromcosmology and quantum physics. The physical principleis presented of how to construct a space propulsion de-vice that does not use any propellant, instead is based onan energy transformation process. In a Gedanken ex-

periment an order of magnitude estimate of the neces-sary energies will be given.

This interaction is rooted in the unification of quantumtheory, gravitation and electromagnetism in an 8-dimen-sional discrete and spin-oriented space. In Einstein's 4-dimensional spacetime continuum that only containsgravitation, this effect cannot take place. Since the inter-action between gravitation and electromagnetism re-duces the inertial mass of a material object, it is calledinertial transformation. Since conservation laws for mo-mentum and energy are strictly adhered to, the theory re-quires superluminal velocities, without contradictingEinstein's theory of relativity. Heim's physical theory,provided it reflects physical reality, has the potential tolead to a completely new concept of space transporta-tion.

Nomenclature and physical constants

c speed of light in vacuum 299,742,458 m s-1

(c2 = ε0 µ0. )

D diameter of the physical universe.

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

h Planck constant 6.6260755 × 10-34 J s.

i, j, k indices in �4 , ranging from 1 to 4.

lp Planck length (ħG ⁄c3)

1 ⁄2=1.61605×10� 35 m .

gab components of the Hermitian metric tensor in �6,or �8.

mp proton mass, 1.672623 × 10-27 kg.

R_, R+ smallest and largest radii between which gravita-tional law, Eq. (24) is valid.

�3 3-dimensional physical space (3 real coordinates).

�4 4-dimensional physical space-time (1 imaginary co-ordinate).

�6 6-dimensional physical space or 6D-Heim space, �4 ⊂ �6, (discrete space, a 6D volume element isbounded by oriented metrons, in general an n - di-

mensional cube has (nk)2(n�k ) k - dimensional

surfaces) (3 imaginary coordinates).

�8 8-dimensional space or 8D-Heim space (5 imagi-nary coordinates).

Tαβ components of the Hermitian energy-stress-mo-mentum tensor in �6 or�8.

x1,...,x6 Cartesian coordinates in 6D-Heim space.

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x1, x2, x3 spatial coordinates in physical space �3.

x4 time coordinate (imaginary).

x5, x6 entelechial and aeonic coordinates (imaginary)in Heim space �6.

x7, x8 information coordinates (imaginary) in Heimspace �8 .

α, β indices in �6, or �8 ranging from 1 to 6 or 1 to 8,respectively. In a Heim space, Greek indices (ifpossible) are used for tensor components.

ε0 permittivity constant 8.854187817 × 10-12 F m-1.

µ0 permeability constant 12.566370614 × 10-7 N A-2.

ξ1,...,ξ8 curvilinear coordinates in Heim space.

λp Compton wave length of the proton mass,(λ p=ħ⁄m p c), 2.10308937 10-16 m.

�ikm normalizable tensor fields for the microcosm that

correspond to the Christoffel symbols in the macrorealm.

ρ radius at which gravitational force is 0, see Eq. (24).

τ Metron area (minimal surface 3Gh/8c3)

6.15 × 10-70 m2.

Source for physical constants: Cohen, E.R., TaylorB.N., The fundamental Physical Constants, Physics To-day, August 1996, pp. BG9-BG13. Measurement uncer-tainties of constants not listed.

1. The Need for Advanced SpacePropulsionSpace transportation, as we know it today, isbased on the century-old rocket equation for-mulating momentum conservation in theframework of classical mechanics. Althoughthere still is room for improvement in the cur-rent designs of space transportation systems,there are strict engineering limits associatedwith the use of the rocket equation. With thiskind of propulsion system interplanetary mis-sions are at best cumbersome, interstellarflights are impossible.

NASA2 has the difficult task of evaluating andselecting candidate technologies for novel

2 There is no activity at ESA or anywhere in Europeconcerning a breakthrough physics space propulsionprogram.

revolutionary space transportation systems,based on breakthrough physics. Very often, thepotentially most valuable technologies arebased on the most innovative theoretical con-cepts. The return on investment is more diffi-cult to estimate, since it is basically the prod-uct of two quantities, one being very large (theusefulness for propulsion, e.g. anti-gravity, un-limited energy or superluminal travel) and theother being small (the likelihood of success).The product of infinity and zero being undeter-mined, the problem of selecting which con-cepts are worthy of further investigation isduly appreciated.

If we wish to have a revolutionary space trans-portation system, incremental improvements inthe technology are not sufficient. We need touse different physical laws that are not subjectto these barriers. Hence, the quest for break-through physics propulsion. If there were aunified quantum field theory, it would bestraightforward to examine this theory fornovel physical principles of advanced spaceflight. At present, such a theory is not avail-able. Current, incomplete unified field theoriesallow for the remote physical possibility oftransforming a so called wormhole into a timemachine as described in [12-14]. However, de-riving a proper space flight technology fromthese principles does not seem to be feasible inthe near future.

The alternative is to check for observed un-usual physical effects that might give a hint tohitherto unknown physical laws, or to searchfor novel physical theories not widely known,but offering physical principles for advancedspace flight. At present, such a theory cannotbe found in mainstream physics. According toW. Pauli a theory must be crazy enough, buton the other hand must be logical. The authors,having studied Heim's theory for several years,feel that his theory satisfies Pauli's as well asDirac's requirements for a unified field the-ory [7]. They also believe to have identifiednovel physical concepts that might have thepotential to evolve into an advanced spaceflight technology. Heim's theory makes severalpredictions regarding cosmology, and also pro-vides a formula for calculating the mass spec-trum of elementary particles along with theirlifetimes. In the section entitled Speculative

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Cosmology, some of these predictions will bepresented to provide data for the experimentalvalidation of the theory. In the space propul-sion approach presented below, the vacuumspeed of light is not the limiting velocity, al-though conservation principles are strictly ad-hered to.

2. Introduction to Heim's FieldTheoryHeim's theory is a generalization and an exten-sion of the GRT (General Theory of Relativity)to the microcosm in that it computes the physi-cal properties such as lifetime, mass etc. ofeach microparticle and also unifies all physicalforces. To this end, it geometrizes all physicalinteractions, but does account for, however,the principle of quantization of the physicalquantity called action (energy × time). Heimextends the nonlinear equations of GRT to thequantum world. This is important, since theequations of GRT must be obtained in thetransfer from to the micro- to the macrocosm.Hence his theory extends the principle of geo-metrization of physics to all physical interac-tions. In that respect, his approach is similar toGauge theory that extends the classical theoryof electromagnetism to a unified theory of theweak and the electromagnetic forces. How-ever, in formulating his theory to unify quan-tum theory, gravitation, and electromagnetism,Heim uses a higher dimensional space that isquantized in a particular way (see below).

There is, however, an important, radically dif-ferent view from Einstein's GRT. In GRT thephysical picture of matter curves spacetime isused, and the coefficients of the metric tensorform a tensor potential for gravitation. In thisregard, matter and spacetime curvature areequal. In Heim's view, the physical picture istotally different. The energy-stress-momentumtensor contains both the source (particle) andits gravitational field. This means, the compo-nents of the metric tensor cannot be interpretedas gravitational potentials, since they are al-ready contained in the energy-stress-momen-tum tensor. Therefore, there is an equivalencebetween the metric and matter (including allfields). In other words, matter is caused by themetric and does not exist independently. In thisrespect, it would be correct to say that matter,as we are used to conceive it, is an illusion. All

interactions or fields, namely gravitation, elec-tromagnetics, weak and strong forces are a dis-tortion of their proper Euclidean metrics in ahigher-dimensional space. This idea was firstpresented by Heim in 1952 at the InternationalCongress on Aeronautics in Stuttgart, Ger-many, and later on published in a series ofthree articles in 1959, in an obscure Germanjournal on spaceflight. Heim's view is similarto Wheeler's geometrodynamics [16]. How-ever, it seems that the first one who publishedthis idea was Rainich in 1925 [17]. Heimclaims to have developed a truly universal uni-fied field theory along the lines suggested byEinstein, but including quantum gravity and allother physical interactions. The difficulties inderiving a theory of quantum gravitation aredescribed by Isham in [10]. Isham suggests aprogram (only the outline is presented withoutany mathematical derivations) how these diffi-culties may be overcome, at least in principle.It is very interesting to learn that in Heim's the-ory, developed much earlier than 1991 (whenIsham's article was published), the most impor-tant requirements, as identified by Isham, arealready included. A state of the art discussionon multi-dimensional theories can be found in[22].

In the following the physical concepts that areunderlying Heim's field theory are presented,along with the major cosmological predictionsthat follow from this geometrized theory. Itshould be noted that the essential parts of thetheory were developed in the fifties and six-ties3, some 50 years ago.

The basic idea of Heim's theory is the repre-sentation of a quantum of matter, Mq, (elemen-tary particles or resonances) as a geometricalentity. Space itself is assigned significantphysical features. Space itself is quantized andis 6 or 8 dimensional, depending on the num-ber of physical interactions to be considered. A6-dimensional space is needed for a unificationof gravitation and electromagnetic theory,while 8 dimensions are needed to represent allknown interactions. Thus the space �6 needsto be extended to a space for �8 a unified the-ory that describes all known 4 interactions, but

3 Heim's work was supported for a about a decade bythe CEO of MBB. MBB later became known asDASA, and is now called Astrium and part ofEADS.

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also gives rise to 2 additional interactions.However, there are dimensional laws that ex-clude, for instance, spaces with 7 or 9 dimen-sions. Furthermore, there are only 3 real coor-dinates (the usual spatial coordinates) that areequivalent and thus interchangeable, all othercoordinates are imaginary and are not inter-changeable. This is important for the buildingof the poly-metric, see below, from the varioussubspaces. A more detailed discussion will begiven in Chapter 3.

In contrast to current string theory, Heim is us-ing so called Metrons, quantized minimal sur-faces with orientation (spin) whose size hasvaried in time. From the beginning of the uni-verse up to today, the Metron size has de-creased, and is now approximately the size ofthe Planck length squared, i.e., its physical di-mension is m2. At the same time, the numberof Metrons has increased. Thus, the beginningof the universe is identified with the eventwhen there was only one Metron, whose sur-face covered the whole universe. According tothis quantum picture, the universe started at afinite size without developing a singularity,naturally avoiding the problem of infinite self-energies. In other words, there are no space-time points, a concept actually in conflict withHeisenberg's uncertainty principle.

According to Heim, the whole universe com-prises a grid of Metrons or metronic lattice.Space that does not contain any informationconsists of a discrete uniform Euclidean grid,bounded by Metrons (e.g., a 6 dimensionalvolume element is bounded by 240 orientedMetrons). However, empty space must be iso-tropic with regard to spin orientation. If allmetronic spins of a 6D volume pointed out-ward or inward, such a world would not have aspin potentiality. Therefore, cells with all spinsoutward have to have neighboring cells withall spins inward and vice versa. This alternat-ing spin structure satisfies the isotropy require-ment, but provides empty space with spin po-tentiality. If two neighboring volumes are in-terchanged, a spin structure has been realized.In other words, empty space is both isotropicwith regard to its metric as well as its spinstructures, but is is capable to develop discretestructures. Thus, empty space is void of physi-cal events, but has inherent potentiality forphysical events to happen. In order for an

event to happen, a distortion of the Euclideanmetric is necessary. In this sense, the theory isa geometric dynamic theory, a term coined byA. Wheeler. The whole theory as it is formu-lated by Heim, is based on geometrical lan-guage. It should be mentioned for matter to beexisting, as we are used to conceive it, a distor-tion from Euclidean metric or condensation, aterm used by Heim, is a necessary but not asufficient condition.

Before we enter into some of the mathematicalformulations, the main physical features of thetheory are summarized below:

1. For a quantized unification of gravitationand electromagnetism a 6 dimensionalspace �6 is needed. If all known interac-tions are to be incorporated space becomes8-dimensional. In �6 the two transcoordi-nates x5 and x6 are imaginary coordinates.Only the 3 spatial coordinates can be real.Any higher-dimensional space with real co-ordinates will not permit stable ellipticalplanetary orbits. Spacetime �4 is a subset of�6 Transcoordinates (in this context it doesnot matter that we use the Euclidean coordi-nates instead of the curvilinear coordinates)

x5=0 or x6=0 denote virtual (latent)events and are outside manifest events

( x5≠0 and x6≠0 ) in 4 dimensionalspacetime �4. The transcoordinate x5 is de-noted as entelechial coordinate, and x6 iscalled the aeonic coordinate. The semanticmeaning of these coordinates stems fromthe Greek word entelechy, governing the ac-tualization of a form-giving cause, andaeon, denoting an indefinitely long periodof time. The entelechial dimension can beinterpreted as a measure of the quality oftime varying organizational structures (in-verse to entropy, e.g., plant growth) whilethe aeonic dimension is steering these struc-tures toward a dynamically stable state. Anycoordinates outside spacetime can be con-sidered as steering coordinates. The 2 addi-tional coordinates in �8 are denoted as in-formation coordinates.

2. Spacetime itself is quantized. The currentarea of a Metron, τ, is 3Gh/8c3 where G isthe gravitational constant, h denotes thePlanck constant, and c is the speed of light

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in vacuum. The Metron size is a derivedquantity and is not postulated.

3. Novel Cosmology and poly-metrics. In� 6 the metric tensor, gαβ (α, β= 1,...,6) is

Hermitian (symmetric in complex space),and is the union of a set of non-Hermitianmetric tensors of subspaces �3 formed byspatial coordinates (x1, x2, x3), space T1 gen-erated by the time coordinate (x4), and thestructural space S2, built from the two trans-coordinates (x5, x6). Formally, �6 is the un-ion �3 ∪ T1 ∪ S2. This is an importantpoint, since the various metrics resultingfrom a combination of these subspaces arethe generators of the physical interactions(or forces). In �8 there are 2 additionalimaginary coordinates. The respective sub-space is denoted as I2. Hence, �8 is the un-ion �3 ∪ T1 ∪ S2 ∪ Ι2.. Therefore, there are4 different coordinate groups in �8. It willbe shown below, that the metric tensor forthe 8 space can be written as a compositionof subtensors that are functions of the coor-dinates from these subspaces. Associatedwith each metric subtensor is a physical in-teraction, and thus a correspondence princi-ple between the metric and the actual phys-ics is established.

In this context, space and time are not thecontainer for things, but are, due to their dy-namic (cyclic) nature, the things them-selves. This is an entirely different physicalpicture from the approach of simply addingthe stress-energy-momentum tensor of theelectromagnetic field to the right-hand sideof Einstein's field equations, for instance,see [15] in order to obtain a geometrizationof gravity and electromagnetism. Other at-tempts by H. Weyl (modifying the Christof-fel symbols by adding a 4-vector to be inter-preted as the electromagnetic 4-vector po-tential), Kaluza and Klein (5 dimensionalspace), or Einstein-Schrödinger (unsymmet-ric metric tensor) have not been successfuleither. Instead of regarding the field equa-tions, as established by Heim, as a set ofdifferential equations for the gαβ, they are tobe regarded as a set of equations for thecomponents of the stress-energy-momen-tum equations, Tαβ, see also Wheeler [16].

However, Heim eventually derives a set ofeigenvalue equations.

4. Geometrization of elementary particles andthe physical interpretation of geometricstructure. Einstein's field equations are ex-tended to the micro area. The energy-stress-momentum tensor is proportional to geo-metrical entities termed tensor fields,, that�l

km are normalizable and correspond to theChristoffel symbols in the macro realm. Ei-genvalue equations of purely geometricalcharacter are set up, using the quantizationprinciple.

5. The only empirical constants (non-derivedquantities) are G , h , ε0 , µ0 . All otherconstants are derived quantities. This in-cludes the coupling constants, too.

6. Interpretation of elementary particles asgeometrical entities that possess an internaldynamic structure which is changing cycli-cally in time. Elementary particles do pos-sess an internal spatial structure (zones), butare elementary in a sense that they are notcomposed of subparticles. Elementary parti-cles are not point entities, but do consist ofMetrons.

7. Derivation of strictly enforced symmetrylaws for elementary particles. The massspectrum and the life times of elementaryparticles are computed.

As an empirical and logical basis for Heim'sfield theory the following assumptions aremade:

1. There exist general conservation principles,for example, for mass, momentum, energy,or electrical charge.

2. There are extremum principles, for instance,the entropy law in the macroscopic worldthat can be described by variational theo-rems.

3. The principle of quantization of action, i.e.,there is a smallest unit of action, h, and allall other actions are multiples of h. Thusmatter is not continuous, but is quantized(i.e., discrete). The quantization of charge,light, and energy is a consequence of thisquantization principle.

4. As a logical basis it is assumed that both inthe macroscopic and the microscopic realms

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material structures interact through actionfields : in the micro- and macroscopic areathere exist electromagnetic and gravita-tional fields, while on nuclear distancesshort range fields are present.

3. The Field Equations Accordingto HeimEinstein's 1915 theory explained gravity asnothing more than a property of spacetime,namely its intrinsic curvature. The distributionof energy (i.e., the 4-vector of energy and mo-mentum) causes spacetime curvature. Thisgeometrical style of physics, is extended byHeim to all physical interactions but, as wasstated before, with a radically different physi-cal interpretation, namely that spacetime cur-vature is equivalent to the combined (sourceand field) energy-stress-momentum tensor. Inother words, there are geometrical explana-tions for electromagnetism, as well as for theweak and strong forces. As already mentioned,however, the approach is different in that themetric gives rise to an energy-momentum dis-tribution in a higher dimensional, so calledHeim space. A purely discrete and uniformCartesian grid will not provide any informationthat could be interpreted as energy-momentumdistribution, and hence is a sign of an emptyspace. The physical laws that cause a distor-tion from this uniform grid, are the generatorsof all physical phenomena.

From Einstein's theory of relativity it is knownthat spacetime curvature gives rise to 10 gravi-tational potentials. Furthermore, Einstein'sequations have been verified for several dec-ades [20], and were found to be correct. Inother words, it is an established empirical factthat the deviation from a Cartesian metric inspacetime gives rise to gravitational interac-tion. Heim's idea was to extend this principleof geometrization to all other physical interac-tions. This means that the structure of theequations of general relativity should be re-tained, being valid also in the quantum range.Since the curvature of spacetime is responsiblefor gravitation, a higher dimensional space isneeded whose curvature accounts for all fur-ther physical interactions. In addition, the prin-ciple of quantization has to be taken into ac-count, giving rise to a set of eigenvalue equa-tions.

According to Heim, since a particle has a non-linear, completely geometrized structure, anonlinear operator is necessary to produce a setof geometrical eigenwert equations. In GRT,the relation between the curvature tensor andChristoffel symbols is Rkml

i=K l Γk m

i , whereKl denotes the well known differential operatorfor the curvature tensor. In analogy to GRT,we write Cl �

ikm , when going from the macro-

to the microcosm, describing the field of a mi-croparticle. Because of the correspondenceprinciple from the macro- to the microcosm,Cl=Kl . The Christoffel symbols Γ k m

i becomethe so called normalizable condensor func-tions, �i

km . This denotation is derived from thefact that these functions represent condensa-tions of the spacetime metric. The fundamentalidea in Heim's theory is that matter can be ex-plained as a geometrical phenomenon andthus, Einstein's field equations should be validin the microcosm, too. On the other hand, it iswell known that Schrödinger's equation de-scribes the probability amplitude for the loca-tion of a microparticle. The stationary Schrö-dinger equation is an eigenvalue equation forthe probability amplitude, ψ , with discrete en-ergy values as eigenvalues. This equation,however, does not make any statements aboutthe physical properties of the microparticle.

Therefore, if, in contrast, one wants to have aneigenvalue equation for the particle itself, todescribe its geometrical structure as well as itsphysical properties that are independent of anexternal field, only depending on the underly-ing spacetime metric, a different set of eigen-value equations needs to be conceived.

These equations can be obtained by observingthat the Christoffel symbols can be interpretedas physical fields, and consequently Heim as-sociates the condensor functions �i

km withphysical fields in the microcosm. Einstein'stheory is in excellent agreement with observa-tion in the limiting case of a spacetime contin-uum on a macroscopic level. Any unified the-ory ought to agree with his theory in this limit-ing case. Furthermore, since particle and waveare a unity, it is concluded that the eigenvalueequations for the condensor functions shouldhave a form similar to Schrödinger's equationand are written in the form

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C (l )� k m(l )=λ(l )� k m

(l )=εk m

l (1)

The LHS is a tensor of 4th rank (the operatorand the condensor functions considered sepa-rately), λl is a vector of eigenvalues, and�l

km is a tensor of 3rd rank. The Christoffelsymbols and thus the condensor functions are3rd rank tensors only with respect to affine (lin-ear) coordinate transformations. Any index inparentheses is not summed over. The RHS rep-resents quantized energy densities, denoted byεl

km . The 3 indices independently run from 1 to4, representing a set of 64 eigenvalue equa-tions. As Heim shows by symmetry considera-tions, 28 of the eigenvalue spectra are empty.Rearranging the remaining 36 eigenvalue spec-tra in a 6×6 tensor, Heim eventually constructsa 6-dimensional space. Since Eqs. (1) are ei-genvalue equations, not subject to any externalfield, there will be no curvilinear transforma-tions, leaving the tensor character of thesefunctions intact.

3.1 Eigenvalue Equations in Heim'sTheory

In order to give physical meaning to the vari-ous metrics obtained from the metrics of sub-spaces of �8 , some kind of hermeneutics isneeded. The hermeneutics of the �8 geometry(abbreviated in the following as hermetry)means the study of the methodological princi-ples of interpreting the metric tensor and theeigenvalue vector of the subspaces. This se-mantic interpretation of geometrical structureis called hermeneutics (from the Greek word tointerpret).

Interactions in Heim's theory are causing amodification of the underlying space time met-rics. Although there is an 8-dimensional Heimspace, we need to consider that the actualphysics takes place in 4-dimensional space-time. In order to see physical events happen, adeviation from empty space, i.e., from the uni-form grid must occur. The uniform grid isgiven by an Euclidean space using coordinatesx1,...,x4. Originally, Heim arrived at a 6-dimen-sional space with coordinates x1,...,x6. For aphysical event to happen, a 4-dimensional cur-vilinear coordinate system needs to be intro-duced, ξ1,...,ξ4 that denotes the deviation fromEuclidean metric and, according to our inter-

pretation, gives rise to physical interactions.The four additional curvilinear coordinates in�8, ξ5,...,ξ8 are termed transcoordinates and areimaginary. It should be noted that the Heimspace �8 can also have a Euclidean structurewith coordinates x1,...,x8. A physical interac-tion taking place in �4, not only changes coor-dinates η1,...,η4 in physical spacetime, but alsoinfluences the transcoordinates in �8. In otherwords, there exists a mapping from �4 to �8.This means, coordinates ξi depend uponspacetime coordinates ηk or ξi = ξi(ηk). Themodified coordinates ξi in turn influence thespacetime metric. Hence, there is a mappingfrom �4 to �8 and back to �4. This double co-ordinate transformation can be written as xl =xl(ξi(ηk))

From the structure of the metric tensor it fol-lows that Heim's theory postulates the exis-tence of two additional interactions, denoted asfield Ψ1 (transformation of photons intogravito-photons, hermetry form H11, see Sec.3.4) and Ψ2 (transformation of gravito-photonsinto the probability field, hermetry form H10,see Sec. 3.4) so that there are now 6 differentfundamental interactions. In addition, there ex-ist two conversion fields allowing for thetransformation of photons into gravito-photons(Ψ1 field), provided proper external conditionsare generated, and subsequently these gravito-photons are converted via the conversion field,Ψ2 , into the probability field, see below.

In Eq. (2) the structure of the energy-stress-momentum tensor for Heim space �6 is shown.The tensor contains 12 vanishing components,as shown by Heim in [1]. Since Tα5=Tα6= 0 andT5α=T6α= 0, the transspace (with regard to �4),

namely the components T55, T56 and T65, T66, in-teract only via time components Tα4.and T4α

with spacetime �4. This means that in the mi-crocosm where Heim's equations are valid, thefuture is not predetermined, i.e., only prob-abilities for future possibilities are possible.Causality is appearing only in the case of a su-perposition of many microstates, forming acollective macrostate.

Zero entries in Tαβ may become non-zero ac-cording to Heisenberg's uncertainty principle.

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(2)

3.2 Poly-metrics and Physical Forces

We mentioned that the metric tensor is com-prised of several components, such that eachcomponent is responsible for a different physi-cal interaction. There are several subspaces inin �8 in which individual metric tensors arespecified, that in turn are the cause of differentphysical forces. The association of each coor-dinate group (or subspace) follows certain se-lection rules. Its corresponding physical inter-action is listed below. The physical meaning ofa coordinate or a group of coordinates is re-sponsible for the physical interaction. First,four groups of coordinates are discerned:

spatial coordinates (real) (ξ1, ξ2, ξ3 ),

time coordinate (imaginary) (ξ4),

entelechial and aeonic coordinates (imagi-nary) (ξ5, ξ6),

information coordinates (imaginary) (ξ7, ξ8).

As was outlined before, empty space is a dis-crete uniform Euclidean space. We now askthe following question: let us suppose that onlycoordinates (x5, x6� undergo a deformation δξ5

and δξ6, rendering these coordinates curvilin-

ear. The change in these coordinates has animpact on the 4-dimensional spacetime geome-try, too and appears as some kind of physicalphenomenon. In order to calculate this changeof spacetime metric, we write

gi k=∂ xm

∂ξα

∂ξα

∂ηi

∂ xm

∂ξβ

∂ξβ

∂ηk

(3)

where indices α, β = 1,...,6 and i, m, k = 1,...,4.This follows directly from the transformationrule given in Sec. 3.1. The same structure

holds for the metric tensor of �8 where indicesα, β = 1,...,8. The terms on the RHS can bewritten in the form

(4)

where indices in parentheses are not summedover, and the definition of the factors κi,m

(α)

and κm,k

(β) follows directly from the above for-mula. Since the function occurs as the kernelin the integral

(5)

it is denoted as fundamental kernel of thepoly-metric. The term poly-metric is usedwith respect to the composite nature of themetric tensor as well as the twofold mapping�4→ �8→ �4. Using the fundamental kernels,we can write the metric tensor in �8 in theform

gi k=∑

α=1

8

κi m

(α)∑β=1

8

κmk

(β)=

(∑α=1

3

κi m

(α)+κ

i m

4+∑

α=5

6

κii m

(α)+∑

α=7

8

κi m

(α))(∑

β=1

3

κmk

(β)+κ

mk

4+∑

β=5

6

κmk

(β)+∑

β=7

8

κmk

(β))(6)

In the next step, we write the components ofthe metric tensor in such a way that there is acorrespondence with the four subspaces I2, S2,T1, and�3.

gi k=∑α=0

3

χi m(α)∑β=0

3

χm k(β) (7)

where the relationship between the new func-tions χ and the fundamental kernels κ isobtained from the comparison with the previ-ous formula of gi k . That is,

χi m(1 )=∑

5

6

κi m(α) (8)

It should be noted that χi m(α)≠χm i

(α).

10 of 21

T αβ=(T 11 T 12 T 13 T 14 0 0T 21 T 22 T 23 T 24 0 0T 31 T 32 T 33 T 34 0 0T 41 T 42 T 43 T 44 T 45 T 46

0 0 0 T 54 T 55 T 56

0 0 0 T 64 T 65 T 66

) κi m

(α)κmk

(β)=

∂ xm

∂ξ(α)

∂ξ(α)

∂ηi

∂ xm

∂ξ(β)

∂ξ(β)

∂ηk

.

xm

(α)=∫κ

i m

(α)dη

i

χi,m(0 )=∑

α=7

8

κi,m(α),

χi m(3 )=∑

α=1

3

κi m(α).χi m

(2 )=κi m

(4 ) and

From Eq. (7) it can be seen that the metric ten-sor in �8 is composed of 16 different termswhere each term belongs to one of the 4 sub-spaces of�8. The metric tensor can also be ex-pressed as

gi k= ∑α , β=0

3

gi k(α β) (9)

where

gi k(α β)

=χi m(α)χm k

(β) . (10)

As was mentioned before, values of α, β areassociated with the subspaces of �8. The fol-lowing relationship holds:

(11)

If the eigenvalue vector, λp, of Eqs. (1) is dif-ferent from 0, not all of its components need tobe different from 0 in �8.The spectrum of λp

may refer to a k-dimensional subspace Vk , de-noted as λp(Vk), such that its k coordinates arehermetric (curvilinear), while the remaining8-k coordinates are Euclidean. If the k coordi-nates of the subspace are hermetric, i.e., giverise to a hermetric form as described inEq. (13), the remaining 8-k Euclidean coordi-nates outside Vk are termed anti-hermetric. Asubspace Vk of �8 could be built by combiningcoordinates from any of the four spaces I2, S2,T1, and �3. The number of physically relevantsubspaces is, however, restricted, because thereal coordinates are interchangeable and aretaken as a semantic unit. All other coordinatesare not interchangeable and thus are separatesemantic coordinate entities. In addition, trans-coordinates can only occur in pairs, that is bothcoordinates from S2 or I2 must be present simul-taneously. Otherwise λp = 0, as Heim shows in[2, pp. 192-195]. If both the transcoordinatesof S2 and I2 are anti-hermetric, then the coordi -nates of �4 must be anti-hermetric as well. Inother words, transcoordinates must always bepresent in a subspace in order that a physicalevent can take place. Stated somewhat differ -ently, at least one of the two transcoordinate

groups S2 or I2 must be present in order tosteer physical processes in �4.

Next, using these rules it can be determinedwhich of the subspaces correspond to physi-cal structures in the microcosm. In addition,these subspaces need to be assigned theirproper physical interaction. This semantic in-terpretation or hermeneutics has to be per-formed in a way to reflect physical processes.We have seen that Heim space �8 comprisesfour semantic entities, namely the subspaces I2,S2, T1, and �3. Employing the above rule thereare 12 physically meaningful combinations ofthe four subspaces in �8, describing eitherphysical particles or interactions. Ten of thesesubspaces can be identified with known virtualparticles or physical interactions. They can beassociated with the four known physical inter-actions (strong, electromagnetic, weak, gravi-tation) and the four types of known virtual par-ticles (gluons, photons, bosons, gravitons). Thehermetry forms H2, H6, H8, and H9 correspondto the four known interactions, namely thestrong, weak, gravitational, and electromag-netic forces, respectively.

There are, however, two additional interac-tions (fields) that have not been known before.In the light of recent cosmological observa-tions, they could possibly be associated withdark matter and dark energy, described by her-metry forms H10 and H11. Since H10 containsonly the space, I2, this field is termed probabil-ity field. In H11 both types of transcoordinatesare present, and the particle associated withthis hermetry form is termed gravito-photon.Perhaps these two fields are the explanationfor dark matter and dark energy or quintes-sence. Quintessence [23], [24] has the strikingphysical characteristic that it causes the expan-sion of the universe to speed up. Energy, eitherin form of matter or radiation, causes the ex-pansion to slow down due to the attractiveforce of gravity. For quintessence, however,the gravitational force is repulsive, and thiscauses the expansion of the universe to accel-erate. In addition, it is interesting to compare,see Sec. 5, with Heim's modified Newtonianlaw, derived in the fifties of the last century.

Below are listed the 12 hermetry forms that re-sult from the 8-dimensional Heim space �8.

Arguments in parentheses specify the subspace

11 of 21

α ,β=0 : I 2=(ξ7,ξ8) ,

α ,β=1 : S 2=(ξ5,ξ6) ,

α ,β=2 : T 1=(ξ4) ,

α ,β=3 :� 3=(ξ1,ξ2,ξ3) .

Vk in which the physical interaction takesplace.

(12)

The hermetry forms can also be represented bythe components of the metric tensor of the cor-responding subspace Vk. The superscripts,ranging from 0 to 3, in the χ quantities referto the respective coordinate groups.

H 5=(χi m(0 ) ,χi m

(1 ) ,χi m(2 )) photons (13)

It is reasoned that hermetry forms H10 and H11

are similar to the graviton field H12, since theyare both caused by transcoordinates, and thuswill have a small coupling constant. The im-portant point is that in Heim's theory there aretransformation operators, S1 or S2 (not to beconfused with space S2), that, when applied toone hermetry form can transform it into an-other one. Mathematically, these operatorstransform the respective coordinate from a nonEuclidean to a Euclidean one. For instance, S2

applied to hermetry form H11 will transformelectromagnetic radiation into gravito-photons.

4. Physical Principles of SpaceFlightHeim's field theory predicts - provided one isin the low energy range since in the energyrange of some 1016 GeV all interactions are ofthe same strength - the four known couplingconstants (gravitation, weak, electromagnetic,strong), but predicts the existence of two addi-tional, hitherto unknown interactions, namelythe hermetry forms H10 and H11 (see Eqs. (13)).His analysis, like most discussions of gravita-tional radiation, proceeds by analogy withelectromagnetic radiation. Just as changes inan electric or magnetic field trigger electro-magnetic waves, changes in a gravitationalfield trigger gravitational waves.

The theory of the coupling constants is mostimportant for the physical interactions. If onemeasures the product of two charges in units ofħ c and uses the proton mass, mp, as a refer-

ence mass, the following relations hold for thefour known interactions:

12 of 21

H 1=H 1 (I 2 ,T 1) gluons

H 2=H 2 (I 2 ,T 1 , R 3) color charges

H 3=H 3 (I 2 , S 2 ,T 1 , R3) W +_ bosons

H 4=H 4 (I 2 , S 2 , R3) Z 0 boson

H 5=H 5 (I 2 , S 2 ,T 1) photons

H 6=H 6 (I 2 ,T 1)∗H 7=H 7 (S 2 ,T 1)

weak charge

H 8=H 8 (S 2 ,R3)

neutral field (particle) with mass

H 9=H 9 (S 2 ,T 1 , R3)

field (particle) with electric charge and mass

H 10=H 10 (I 2)probability field

H 11=H 11 (I 2 , S 2)gravito-photon

H 12=H 12 (S 2) graviton.

H 1=(χi m(0 ) ,χi m

(3 )) gluons

H 2=(χi m(0 ) ,χi m

(2 ),χi m(3 )) color charges

H 3=(χi m(0 ) ,χi m

(1 ) ,χi m(2 ) ,χi m

(3 )) W +_ bosons

H 8=(χi m(1 ) ,χi m

(3 ))

neutral field (particle) with mass

H 10=(χi m(0 )) probability field

H 11=(χi m(0 ) ,χi m

(1 )) gravito-photon

H 12=(χ i m(1 )) graviton.

H 4=(χi m(0 ) ,χi m

(1 ) ,χi m(3 )) Z 0 boson

H 6=H 6 (χ i m(0 ) ,χi m

(2 ))∗H 7=H 7 (χi m

(1 ) ,χi m(2 ))

weak charge

H 9=(χi m(1 ) ,χi m

(2 ),χi m(3 ))

field (particle) with electric charge and mass

Gmp

2

ħ c=5.9×10�38

λ p

cβ

ħ c=2×10�8

e2

ħ c=

1137

=7.3×10�3

g2

ħ c=15

(14)

where Eq. (14) denotes the relative strength ofthe gravitational, weak, electromagnetic, andstrong interactions, respectively, and cβ is thecoupling constant of the beta decay andλ p=ħ ⁄mp c is the Compton wavelength of

the proton. The relative strength of the forcesis approximately 1: 10-3: 10-9: 10-40, the strongforce assigned the value 1. According to Feyn-man coupling constants can be interpreted as aprobability for the exchange of virtual parti-cles. Heim's eigenvalue equations allow tocompute the spectrum of the ponderable parti-cles. The physical constants that determine thecoupling constants are depending on the eigen-values. The set of eigenvalues itself is deter-mined by geometrical symmetries. These sym-metries are related to the coordinates of theHeim space. The sets of eigenvalues can becharacterized by their cardinal numbers. Thecardinal numbers, therefore, point out a way tothe mathematical description of all couplingconstants, and hence a set algorithm was con-structed that is behind the derivation of themagnitude of the coupling constants4.

It turns out that for Heim space �8 not only thevalues for the 4 known coupling constants areobtained, but four additional probability am-plitudes occur. In other words, there exists aset of 8 probability amplitudes, wi, where w1 tow4 describe the 4 known interactions, whosecarrier particles are gravitons, vector bosons,photons, and gluons. Probability amplitudesw5, w6 are interpreted as transmutation fields(mathematically represented as transformationoperators S1 and S2). The other two couplingconstants, w7, w8 are interaction fields, and are

4 A complete theory was derived by the first authorcalculating the exact values of the coupling con-stants, based on the theory of cardinal numbers. Apaper on the derivation of the coupling constants isin preparation. The theory comprises some 50 pagesand cannot be presented in this paper.

interpreted as gravito-photon and probabilityfields. They are gravitational like fields, char-acterized by hermetry forms H10 and H11, seeEqs.(13).

The physical meaning of the transmutationfields is that photons, characterized in �8 bythe hermetry form H5 , are transformed by theaction of the transmutation operator S2 (not tobe confused with space S2) into gravito-pho-tons (hermetry form H11). In a more formalway, one could write S2 H5 = H11. The relationbetween the corresponding probability ampli-tudes is

3w3 -w5 = 3w7 (15)

where operator S2 is associated with w5, herme-try form H5 with w3, and hermetry form H11

with probability amplitude w7. It should benoted that the equations were slightly simpli-fied.

In a second step, the gravito-photon field (her-metry form H11), under the action of transmuta-tion operator S1, is transformed into the prob-ability field described by hermetry form H10.This can be written as S1 H11 = H10 or

w7 -w1w6 = w8. (16)

The value of w7 is 1.14754864×10-21 and w8 iscalculated as 1.603810891×10-28. Again, theequations were slightly simplified. Becausethe value of the probability amplitude w7 issimilar to the value of w1, (graviton, value7.6839×10-20), the denotation of the gravito-photon field as a gravitational like field seemsto be appropriate. It should be rememberedthat w-values denote probability amplitudes,i.e., their square gives a probability, which isthe respective coupling constant for the corre-sponding interaction.

4.1 Lorentz Matrix and Inertial Trans-formations

Under the physical conditions specified above,an electromagnetic field can be transformedinto a gravitational like field, such that thegravitational field around a space vehicle is re-duced, according to Eqs. (15) and (16). Withthe reduction of the gravitational potential, Φ,in each area of space where this transformationtakes place, gravitational mass density �g and

13 of 21

thus gravitational mass mg must be reduced,too. This follows directly from

(17)

where the integration is over the volume inphysical space in which the transformationtakes place. According to the equivalence ofinertial mass (Newton's second law) and gravi-tational mass (Newton's gravitational law), areduction in gravitational mass is equivalent toa reduction in inertial mass. Therefore, m > m'where m and m' denote the inertial masses be-fore and after the transformation. Let us nowconsider the 4-momentum vector in spacetime

(18)

where p=m v is the classical momentumand i is the imaginary unit. Since the magni-tude of P is an invariant, both momentumand energy conservation hold. For a space ve-hicle with initial inertial mass m and reducedmass m', the following relations are thereforevalid,

m v=m' v' and m c=m' c' (19)

that is c' > c and v' > v, since m > m'. Quanti-ties with a prime indicate the transformed sys-tem. We denote this kind of transformation asinertial transformation. Dividing the firstequation by the second one, it immediately fol-lows that

(20)

Therefore, the corresponding Lorentz transfor-mations, namely for the first system, in whichc is the speed of light and the spacecraft ismoving with velocity v, and the second system,in which the speed of light is c' and the vehiclespeed is v', are described by the same Lorentzmatrix , that is

(21)

where β = v/c , x4 = ict or b = v'/c' , x'4 = ic't.The movement is in the x (that is x1) directiononly. Since a transformation of inertial massonly changes c to c' ( with c' > c), the otherspatial coordinates remain unchanged, andonly coordinates x1 and x4 are changing,respectively. There is no contradiction to spe-cial relativity, since an inertial transformationis not considered in SRT. The argument inSRT is, that if v > c, then β becomes imagi-nary. Thus, it is concluded that no observer canpossess a velocity greater than that of lightrelative to any other observer. In an inertialtransformation, however, β remains positive.Such a transformation is not possible in SRTor GRT, since it is a consequence of the unifi-cation of physical interactions and the poly-metric in �8.

If it were technically possible to generate a suf-ficiently strong field for the reduction of iner-tial mass in the vicinity of a moving space ve-hicle, the velocity of the space vehicle will in-crease from v to v' = α ν, where α := isthe velocity gain factor, and 1/α gives the fac-tor at which the graviton field of the space ve-hicle is reduced. In addition, the transforma-tion field for the inertial mass may by itselfhave a repulsive effect and thus further accel-erate the space vehicle. The action of thistransformation field is such that the space ve-hicle disappears from the usual spacetime withc = constant, and enters a spacetime in whichc' = constant is valid. If the transformationfield disappears, the space vehicle returns to itsoriginal spacetime. It is interesting to note thatduring the transition phase from α = 1 to α> 1, the acceleration can be arbitrarily largewithout the occurrence of a force, caused bythis acceleration. This simply follows from theconservation of momentum, namely the fact

14 of 21

Φ(x)=∫ρg (x')

|xx'|d3 x'

P=m0(1�v2⁄c2

)�1⁄2

(v , ic)= (m v , imc)=( p ,imc)

v' ⁄c'=v⁄c .

A=(1

(1β2)

0 0iβ

(1β2)

0 1 0 00 0 1 0

iβ

(1β2)

0 01

(1β2))

that m v = m' v' = constant (m = α m'), whichmeans that the force, responsible for the accel-eration, is 0. Owing to the invariance of theLorentz matrix with respect to an inertial trans-formation, which is rooted in the fact that v'/c'= v/c, superluminal velocities should be pos-sible. Most interesting, this fact is not in con-tradiction with GRT, allowing, in principlespace flight at superluminal velocities. Al-though superluminal velocities have been con-jectured for some time, the difference now isthat there is a physical theory, according towhich this phenomenon can be computed.

It goes without saying, that there remain twoimportant questions to be settled. First, an ex-perimental proof of the validity of Heim's the-ory, and second, this validity assumed, whatare the technological challenges to construct aviable propulsion system from this inertialtransformation principle. In the subsequentsection, an order of magnitude estimate for theenergies to be supplied, is presented.

4.2 Estimating the Order of Magnitudeof Transformation Effect andGedanken experiment

In the previous section it was shown that elec-tromagnetic radiation can be converted into agravitational like field, thus reducing the gravi-tational mass of a body that is under the influ-ence of this radiation. First, it was observedthat the corresponding coupling constant isweak, i.e., the interaction and thus the actualforce will be small. Second, the frequency ofthe radiation needs to be determined at whichthis transformation takes place.

From the theory of the coupling constants avalue of 28.66 keV is computed for the photonenergy at which, according to Eq. (15), pho-tons are completely converted into gravito-photons (w7 field) by the transmutation field,w5, that is present in vacuum. The photon den-sity has to satisfy an additional constraint. Forexample, this could be achieved by oscillatingelectrons in a free electron laser. InterpretingEq. (16), one sees that the generated w7 field istransformed by a graviton field (w1) into aprobability field (w8). In this process the neces-sary gravitons are converted at the same loca-tion at which they were generated, i.e., thegraviton field, associated with a spacecraft, isreduced in its strength. This means a decrease

of the inertial mass of the spacecraft. As longas the gravito-photon field exists, the spacecraft is accelerated from velocity v to v', as cal-culated in Eq. (20). A speculative interpreta-tion would be that the spacecraft disappearsfrom our spacetime, characterized by speed oflight c, and enters the spacetime of speed oflight c', and returns to the original spacetime ifthe gravito-photon field disappears. We nowgive an estimate for the two most interestingquestions, concerning the technical usefulnessas well as the technical feasibility of a spacetransportation system based on inertial trans-formation:

Question 1: If we want a velocity n-times theinitial velocity as well as a limiting velocity n-times the vacuum speed of light, what is therequired mass reduction of the spacecraft?

Question 2: What is the total photon energy re-quired that delivers this inertial reduction?

The transmutation equations, Eqs. (15), (16),are valid for single photons and gravito-pho-tons. Both, the strength of the electromagneticfield and its energy density are much largerthan the respective values of the gravito-pho-ton field, which can be seen directly by thecoupling constants. Therefore, energy will betaken from the vacuum, in accordance withHeisenberg's uncertainty principle, that needsto be returned to the vacuum, so that energyconservation is satisfied. To reduce a space-craft with mass M, gravito-photon (w7) parti-cles have to be generated. Rewriting Eq. (16)using the value for w6 from the theory of cou-pling constants

w7 - 0.014 w1 = w8 (22)

shows that some 67 w7 -particles convert 1 w1-particle. In addition, only 1 out of 4 w7 -parti-cles can be used to converting w1- particles,hence the factor 67. That is, the photon energymust be increased by a factor of 67 to compen-sate a w1 field. Let the spacecraft have aspherical body of radius R = 1 m and massM = 104 kg. Let us furthermore consider thatthe spacecraft is not subject to any externalgravitational field. Its gravitational self-energyis

15 of 21

E=12∫ρ( x)Φ ( x)d3 x =

GM 2

2R=

6.67×10�11×108

2=

3.365×10�3 J

(23)

where the factor of œ is introduced to avoiddouble counting of pairs of mass particles inthe distribution, and Φ(x) is the potential pro-duced by the mass distribution. The resultingenergy is fairly small. The photon energyneeded would therefore be a mere 0.9 J, nolosses assumed. This result holds for the con-version using a free electron laser. Using a sec-ond solution, requiring a lower photon energy,employing an electrically charged rotatingtorus, an energy of some 105 J would be neces-sary. According to our present calculations, thesituation changes drastically if we were tolaunch from the surface of a planet. Let us as-sume that both planet and spacecraft arespherical bodies, and a launch from the surfaceof the earth is intended. The mass of the earthis 5.98×1024 kg and the radius is 6.378×106 m.Eq. (23) (without factor œ) results in an energyof some 6.25×1011 J and a photon energy of67×6.25×1011 J. It was assumed that all of thepotential energy resides in the field. We wouldlike to emphasize that this extremely largevalue might not be the last word, since we can-not claim at present that all consequences ofthe theory are fully understood.

To eventually reach a velocity n-times the ini-tial veleocity, v, we need to reduce the inertialmass such that m = n m' where m' is the re-duced inertial mass. That means, in order totravel at a limiting speed that is 10-times thespeed of light, c' = 10 c, the inertial mass mmust be reduced to 10% of its original value.

The theory allows superluminal flight in prin-ciple, however, the technical realization of theinertial transformation to achieve this goal is aproblem to be solved in the future. Further re-search will be needed to establish criteria forthe effort needed.

5. Speculative CosmologySince the higher dimensional space used inHeim's theory comprises a discrete metroniclattice, there are no singularities. Hence, thebeginning of the universe is clearly defined.

The actual starting point for the universe was,when the size of a single Metron,τ, which is a function of time, dτ/dt < 0, cov-ered the surface of the universe, assumed to bespherical. During the expansion phase of theuniverse, the number of Metrons increased.

Eq. (16) has an interesting consequence for theinflationary phase of the universe. An existingw7 field transforms a gravitational field w1 intoa w8 field. A vanishing w1 field causes, becauseof the equivalence of inertial and gravitationalmasses, a reduction of the inertial mass of theuniverse. Owing to the conservation of mo-mentum and energy, the speed of light c duringthis phase will have increased to a much higherspeed c', according to Eq. (18). The same prin-ciple that might be used for superluminal spacetransportation, could have caused the inflation-ary phase of the universe.

According to Heim [1, 2] (the derivation of theformula below was not calculated independ-ently by the authors), Newton's law needs to bemodified for large distances by a negative termand thus becomes repulsive:

a=Gm (r )

r2(1�

r2

ρ2) , ρ=

h2

G m03 (24)

m0 being the mass of a single nucleon com-prising the mass of the field source. Mass m(r)is the total mass and comprises the ponderableand the field mass. According to Heim, thevalue of ρ is some 10 to 20 million light-years.This modified law has severe consequences,since the observed redshift would, at least par-tially, be a gravitational redshift.

Heim also calculates a lower bound for therange of validity of the gravitational law,which is in the microscopic range, and whosevalue is approximately given as

R -=3e16

G M 0

c2 (25)

M0 denoting the (ponderable) mass without thefield mass. This threshold is practicallyequivalent to the Schwarzschild radius. Thegravitational force is attractive for distancesR_ < r < ρ. For r = ρ, the gravitational force

is 0.

16 of 21

There is a third distance, R+, depending on themass in the universe, so that for ρ < r < R+

the gravitational force is repulsive and goes to0 for r = R+. R+ is some type of Hubbleradius,but is not the radius of the universe, instead itis the radius of the optically observable uni-verse. All optical signals are subject to a red-shift due to the anti-gravitational effect fromEq. (24). For distances larger than R+, the red-shift becomes infinite, and thus signals cannotbe received. R+ would be largest if the universecontained only a single particle of minimalmass which could be a neutral particle of amass 1% less than the electron mass that mightexist. Then one would obtain the largest radiuspossible, Rmax or D = 2 Rmax as maximal di-ameter of the physical universe, computedfrom the following formulas, see also [6]

32

f(14

32

Df 3

τ31)

2

=Dτ

(e D τ

πE1) f 2

=e D τ

πE

(26)

e and E being the basis of the natural logarithmand a unit surface, respectively. The readermay wish to calculate the present diameter ofthe universe himself. To calculate the diameterD0 at the beginning of time, the relation πD0

2 = τ0 has to be inserted into the secondequation of Eq. (26). τ0 denotes the metronicsize at the origin of the universe. The resultingequation of 7th order for f0 has 3 real roots. Thisresults in 3 different positive values of D0 andthree different negative values for D0. Heim in-terprets this as a trinity of spheres, separated intime by a chronon, the quantum time interval.At the end of its life cycle, the universe col-lapses into a trinity of spheres, determined bythe negative values for the diameter.

Since spacetime is quantized, black holes inform of a singularity should not exist. It shouldbe noted that for dimensions for which the deBroglie wave length is small quantum theoryshould be incorporated and the extension ofGRT to microscopic dimensions may be incor-rect.

According to Heim the age of the universe issome 10127 years. Matter as we know it wasgenerated only some 15 billion years ago,when τ, the Metron size, became smallenough. The phenomenon of gamma raybursts may be an indication of the creation ofmatter. At that time the universe was alreadyalmost flat, i.e., �τ⁄τ≈0and �D ⁄D≈0. Inother words, the universe is expanding, but at aslower rate as presently believed and is at pres-ent almost flat.

It should be noted, that Heim's theory [1,2]also provides formulas for the mass spectrumof the elementary particles as well as their life-times.

Conclusions and Future WorkWe are aware of the fact that the present articlehas several shortcomings. First, we stated sev-eral important physical assertions withoutproper mathematical proof. However, the deri-vations of the coupling constants and the her-metry forms are a subject of their own, whichis beyond the scope of this paper. Second,Heim's legacy contains a large body of unpub-lished work. The authors were not able tocheck all of his calculations. In particular, thederivation of the nonlinear potential equation,see Chap. 5, has not been derived independ-ently 5. Third, Heim, being visually impaired,used his own physical terminology, necessitat-ing a translation into the language of contem-porary physics. In addition, several completelynovel concepts needed to be introduced thatmay require additional physical interpretation.Last but not least, since this is work in prog-ress on a challenging topic, both, errors inHeim's theory as well as those introduced bythe authors, cannot be excluded. Thus, none ofthe new physics presented here should be takenfor granted.

The so called Standard Model, see for instance[19], is an amalgam of experimental observa-tions and theoretical derivations, but needssome 30 adjustable parameters to be deter-mined from experiments. Spacetime is con-

5 Heim left some 4,000 pages, now at the university li-brary in Salzgitter, of unpublished material, provid-ing, in many cases, the detailed calculations notfound in his published work. His work also includestreatises on cosmology, elementary particle physics,mathematical logic as well as on bioscience.

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tinuous and therefore leads to singularities andinfinite self energies. The lifetimes and thespectrum of elementary particles cannot bepredicted. It also is, according to these authors,logically inconsistent that 6 quarks and 6 lep-tons are the basic ingredients of matter, butfree quarks are per definition unobservable. Nophysical explanation is provided for this fact.Furthermore, there is no quantum gravity inthe context of the Standard Model. There isalso no explanation for the existence of thesebasic constituents, and it could well be thatquarks need to be comprised of even more fun-damental particles, which then might lead tothe conclusion that it is turtles all the waydown.

In this respect, Heim's theory seems to be logi-cally more consistent, based on verified physi-cal principles, and the idea of a higher dimen-sional, discrete space, composed by oriented,minimal surface elements. It leads, however, toa radically different view of matter and space,and predicts a cosmology that substantially dif-fers from the current model of the big bang.

Heim derived the theory without the fitting ofany experimental parameters, making decisivepredictions about cosmological phenomena,and delivers a formula for the life times andthe mass spectrum of elementary particles aswell as specifying appropriate selection rules.It also contains a formulation for a quantumgravity, based on the aesthetic and elegant ideathat the metric is the generator for all physicalinteractions, leading to a poly-metric. The the-ory makes several remarkable predictions,namely the existence of two additional interac-tions, hitherto unknown, that are the basis foradvanced space travel.

Whether Heim's geometrized field theory re-flects physical reality is, at present, undecided.If the theory were true, an entirely new conceptof the universe would emerge. Moreover,Heim's theory is a definite non-mechanisticconcept of nature, in contrast to our presentview of the world, initiated in the 15th centuryby Leonardo da Vinci.

Heim's theory unifies all known interactions ina 8-dimensional space and allows for a specialLorentz transformation, named inertial trans-formation, permitting, at least in principle, forsuperluminal travel.

AcknowledgmentThis paper is dedicated to Dr. William Berry(ret.), head of the Propulsion and Aerothermo-dynamics Division, European Space researchand Technology Center (ESTEC) of the Euro-pean Space Agency (ESA) in Noordwijk, TheNetherlands. The second author, while work-ing in his division, learned that taking a highrisk is acceptable as long as engineering andscientific judgment remain intact. He grate-fully acknowledges the many discussions onpropulsion principles.

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

The authors are very much indebted to Dipl.-Phys. I. von Ludwiger, former manager andphysicist at DASA, for making available rele-vant literature and for many helpful discus-sions as well as explanations concerning theimplications of Heim's theory.

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

Glossaryaeon Denoting an indefinitely long period of

time. The aeonic dimension can be inter-pretedis as steering structures governed bythe entelechial dimension toward a dy-namically stable state.

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

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

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condensor The Christoffel symbols Γ k mi be-

come the so called condensor functions, i

km , that are normalizable. This denota-tion is derived from the fact that thesefunctions represent condensations ofspacetime metric.

coupling constant Value for creation and de-struction of messenger (virtual) particles,relative to the strong force (whose value isset to 1 in relation to the other couplingconstants).

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

fundamental kernel (Fundamentalkern)Since the function κ

i m

(α) occurs in

xm

(α)=∫κ

i m

(α)dη

i

as the kernel in the in-

tegral, it is denoted as fundamental kernelof the poly-metric.

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

gravito-photon field Denotes a gravitationallike field generated by a neutral mass witha smaller coupling constant than for gravi-tons, but allowing for the possibility thatphotons are transformed into gravito-pho-tons. This field can be used to reduce thegravitational potential around a spacecraft.

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

hermetry form (Hermetrieform) The wordhermetry is an abbreviation of hermeneu-tics, in our case the semantic interpretationof the metrics. To explain the concept of a

hermetry form, the space �6 is considered.There are 3 coordinate groups in thisspace, namely s 3=(ξ1 ,ξ2 ,ξ3) formingthe physical space �3, s 2=(ξ4) forspace T1, and s1=(ξ5 ,ξ6) 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, but derived byHeim from conservation laws in �6, see p.193 in [2]), coordinates ξ5 and ξ6 must al-ways be curvilinear, and must be presentin all combinations. An allowable combi-nation of coordinate groups is termed her-metry form, and denoted by H, sometimesannotated with an index, or sometimeswritten in the form H=(ξ1 ,ξ2 ,...)where ξ1 ,ξ2 ,...∈S . This is a symbolicnotation only, and should not be confusedwith the notation of an n-tuple. From theabove it is clear that only 4 hermetry formsare possible in �6. Thus, a 6 space onlycontains gravitation and electrodynamics.It needs a Heim space �8 to incorporate allknown physical interactions. Hermetrymeans that only those coordinates denotedin the hermetry form are curvilinear, allother coordinates remain Cartesian. Inother words, H denotes the subspace inwhich physical events can take place, sincethese coordinates are non Euclidean. Thisconcept is at the heart of the geometriza-tion of all physical interactions, and servesas the correspondence principle betweengeometry and physics.

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

hermitian matrix (self adjoint, selbstad-jungiert) A square matrix having theproperty that each pair of elements in the i-th row and j-th column and in the j-th rowand i-th column are conjugate complexnumbers (i → - i). Let A denote a squarematrix and A* denoting the complex con-jugate matrix. A† := (A*)T = A for a hermi-

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tian matrix. A hermitian matrix has real ei-genvalues. If A is real, the hermitian re-quirement is replaced by a requirement ofsymmetry, i.e., the transposed matrix AT =A .

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

inertial transformation (Trägheitstransfor-mation) Such a transformation, fundamen-tally an interaction between electromag-netism and gravitational like field, reducesthe inertial mass of a material object usingelectromagnetic radiation at specific fre-quencies. As a result of momentum andenergy conservation in 4-dimensionalspacetime, v/c = v'/c', and thus the Lorentzmatrix remains unchanged. It follows thatc < c' and v < v' where v and v' denote thevelocities of the test body before and afterthe inertial transformation, and c and c' de-note the speeds of light, respectively. Inother words, since c is the vacuum speedof light, an inertial transforemation allowsfor superluminal speeds. An inertial trans-formation is possible only in a 8-dimen-sional Heim space, and is in accordancewith the laws of SRT. In an Einsteinianuniverse that is 4-dimensional and con-tains only gravitation, this transformationdoes not exist.

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

partial structure (Partialstruktur) For in-stance, in �6, the metric tensor that is Her-mitian comprises three non-Hermitianmetrics from subspaces of �6. These met-rics from subspaces are termed partialstructure.

poly-metric The term poly-metric is used withrespect to the composite nature of the met-ric tensor. In addition, there is the twofoldmapping �4 → �8→ �4.

transformation operator (Sieboperator) Thedirect translation of Heim's terminologywould be sieve-selector. A transformationoperator, however, converts a photon intoa gravito-photon by making the coordinateξ4 Euclidean.

unitary matrix (unitär) Let A denote a squarematrix, and A* denoting the complex con-jugate matrix. If A† := (A*)T = A-1, then A isa unitary matrix, representing the generali-zation of the concept of orthogonal matrix.If A is real, the unitary requirement is re-placed by a requirement of orthogonality,i.e., A-1 = AT. The product of two unitarymatrices is unitary.

REFERENCES

1. Heim, B., Vorschlag eines Weges einer einheitlichenBeschreibung der Elementarteilchen, Z. für Natur-forschung, 32a, 1977, pp. 233-243.

2. Heim, B., Elementarstrukturen der Materie, Band 1,Resch Verlag, 2nd ed., Innsbruck, 1989

3. Heim, B., Elementarstrukturen der Materie, Band 2,Resch Verlag, Innsbruck, 1984.

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

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