**March 18, 2006: E-Mail-discussion appended**

**March 29, 2006: Reply to Hauser's "rebuttal" appended**

J. Hauser and W. Droescher and their Theory

1. The IGW (Institute of Border Sciences)

Since 1999 there existed a web site of the
**I**nstitut für **G**renzgebiete der **W**issenschaft (IGW) on the server of the
Leopold-Franzens-University Innsbruck (Austria).

http://info.uibk.ac.at/c/cb/cb26/

The indicated owner of that web site was Prof. Dr. Andreas Resch. Under scientific personel were nominated Prof. Dr. Jochem Hauser and Dipl.-Ing. Walter Droescher (originally in German: Häuser and Dröscher).

According to his own biographic data A. Resch was Professor for clinical psychology and paranormology at the Accademia Alfonsiana, Pontifical Lateran University Rom from 1969 to 2000. And J. Hauser is professor at the Braunschweig-Wolfenbüttel Fachhochschule at Salzgitter, Germany. Walter Droescher, probably retired now, was a theorist at the Vienna Patent Office in the 80s, when he began to work with B. Heim.

in order to obtain reputation by devious means.

However, on Jan 27 2006 the **Leopold-Franzens-University** became aware of the abuse
of universitary facilities by the IGW, and the IGW web site was instantly removed.

Typically enough for the ambitions of the IGW, informations on the IGW and the Resch-Verlag
can still be found on the *pseudo-science* web sites

http://www.datadiwan.de/igw/

http://www.datadiwan.de/netzwerk/index.htm?/igw/rv_019d_.htm

2. The Hauser-Droescher Extension of the Heim Theory

The Heim theory did never pass a rigorous peer reviewing. In 2002 the authors J. Hauser
and W. Droescher started to extend the former Heim theory which a reviewer (below) considered to be
*almost (!?!)* error-free. For a version of that new theory they got "*the*"
AIAA research award 2004.
That formulation pretends as if there is only *one* AIAA award a year. However, there are
more than 50 kinds of such awards a year.
And the award under consideration here is the *Best-Paper Award* of just that
Technical Committee, Nuclear and Future Flight TC, a member of which is J. Hauser *himself*.
In 2005 the TCs of AIAA gave away
21 Best-Paper Awards.

The "best paper of 2004" [2] was replaced by the authors with a revised version [1]
which we are referring here to. We consider a basic section where the concept of a so-called
*poly-metric* is introduced:

Quote from [1]:

## PHYSICAL PRINCIPLES OF GRAVITOPHOTON FIELD PROPULSION

In

GRthe metric has the meaning as physical potential for gravitation. As was mentioned before this view is extended to Heim space H^{8}. We now present the most general transformation that is responsible for all physical interactions. Most important is the double transformation as described in Eq. (1). A curve inR^{4}can be specified by either Cartesian coordinates x^{m}or by curvilinear coordinates η^{i}. However, sinceR^{4}is a subspace of Heim space H^{8}with so called internal coordinates η^{i}(Dröscher and Hauser, 2004), there exists a general coordinate transformation x^{m}(ξ^{α}(η^{i})) fromR^{4}→ H^{8}→R^{4}resulting in the metric tensor (this is a major difference toGR)g

_{ik}=^{∂xm}/_{∂ξα}^{∂ξα}/_{∂ηi}^{∂xm}/_{∂ξβ}^{∂ξβ}/_{∂ηk}, g_{ik}: =Σ_{μ,ν=1 ... 8}g_{ik}^{(μν)}, g_{ik}^{(μν)}=^{∂xm}/_{∂ξ(μ)}^{∂ξ(μ)}/_{∂ηi}^{∂xm}/_{∂ξ(ν)}^{∂ξ(ν)}/_{∂ηk}, (1)where indices α,β = 1,...,8 and i,m,k = 1,...,4. The Einstein summation convention is used. The above transformation

is instrumental for the construction of the poly-metricutilized to describing all possible physical interactions. The metric tensor can be written in the form as expressed in the second term of Eq. (1). Parentheses indicate that there is no index summation. In Dröscher and Hauser (2004) it was shown that 12 hermetry forms can be generated having direct physical meaning, by constructing specific combinations from the four subspaces. The following denotation for the metric describing hermetry form H_{l}withl= 1,...,12 is used:g

_{ik}(H_{l}): =Σ_{μ,ν Î Hl}g_{ik}^{(μν)}(2)where summation indices are obtained from the definition of the hermetry forms. The expressions g

_{ik}(H_{l}) are interpreted as different physical interaction potentials caused by hermetry form H_{l}, extending the interpretation of metric employed inGRto the poly-metric of H^{8}.

(End of quote)

We pointed out by email to the authors that already the first equation in (1) cannot be true: We have

^{∂xm}/_{∂ξμ}
^{∂ξμ}/_{∂ηj} dη^{j}
=
^{∂xm}/_{∂ξμ} dξ^{μ}
=
dx^{m}

and thus

g_{jk} dη^{j} dη^{k}
= Σ_{m} (dx^{m})^{2} > 0 for each d**x** ≠ **0**,

since each dx^{m} is *real* due to the assumption **x** =
(x^{1}, x^{2}, x^{3}, x^{4})
Î **R**^{4}
= **R**×**R**×**R**×**R**,
where **R** denotes the well-known set of all real numbers.
This shows that the matrix (g_{jk}) is *positive definite* while in *GR*
the matrix (g_{jk})
must be *indefinite* like the Minkowski-matrix diag(1, 1, 1, −1).

The detected error is already contained in Hauser's and Dröscher's first paper [3; Section 3.2].

Jochem Hauser replied:

With regard to your Objections 1, 2 it must be said thatR^{4}only stands for an abbreviation that has the following meaning inGR:a manifold M with the following characteristcs:

4-dimensional curvilinear space which is locally Minkowskian,

i.e. g_{ik}= nonconstant in general

We answered with:

**R**^{4} cannot denote a manifold which is used in *GR*:

Your "manifold M" (= **R**^{4}) is such that you can
identify its points with the quadrupels (x^{1},x^{2},x^{3},x^{4})
where the coordinates are *numbers* such that you can define the metric (g_{ik})
by

g_{jk}dη^{j}dη^{k}
= (dx^{1})^{2}
+ (dx^{2})^{2}
+ (dx^{3})^{2}
+ (dx^{4})^{2}.

N.B. The coefficients g_{jk} being *nonconstant* is a necessary but *no sufficient*
condition for the *nonflatness* of the manifold.

(1) One can introduce polar coordinates in the (flat = Euclidean) plane which -
as is well-known - have *nonconstant* metric coefficients g_{jk}
though its curvature is zero; cf also the 3D-example
in S.M. Carroll [5; p.48, (2.30-32) and [6; p.41, (2.41-43)].

(2) S.M. Carroll considers the surface of the two-sphere in **R**^{3}
as a simple example of a *non-Euclidean* manifold

ds^{2}
= (dη^{1})^{2}
+ sin^{2}η^{1}
(dη^{2})^{2}
(= dθ^{2}
+ sin^{2}θ
dΦ^{2}
in Carroll's notation),

the curvature of which is 1 [5; p.48, (2.33) and [6; p.71, (2.45)].

However, with respect to the x-coordinates the "manifold M" has zero curvature, i.e. your "manifold M"
is flat, so also with respect to the curvilinear coordinates η^{j} (j=1,2,3,4).

The calculation of the curvature with respect to the x-coordinates is trivial and yields zero, while the calculation w.r. to the curvilinear coordinates would take a lot of trouble to yield the same zero result at the end. This is the reason why tensor calculus is used in multidimensional differential geometry.

If you like to construct a manifold (locally) Minkowski, then you must *give up*
the assumption
**x** = (x^{1},x^{2},x^{3},x^{4})
Î**R**^{4} and modify it to
x^{4}Î i**R**.
Setting x^{4} = i ct we start with the Minkowskian metric

g_{jk}dη^{j}dη^{k}
= (dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} − (cdt)^{2}

This defines a metric locally Minkowski - however,
this metric is even *globally* Minkowski,
i.e. the defined manifold M is *flat* again.

[1]
Walter Droescher, Jochem Hauser:
*Heim Quantum Theory for Space Propulsion Physics*, 2005,

URL presently unknown,

(available also at www.cle.de/hpcc or http://www.hpcc-space.de/publications/index.html ???)

[2]
Walter Droescher, Jochem Hauser:
*Guidelines for a Space Propulsion Device Based on Heim's Quantum Theory*,

AIAA 2004-3700, AIAA/ASME/SAE/ASE, Joint Propulsion Conference & Exhibit, Ft. Lauderdale, FL,

July 2004, 28 pp.,
http://www.hpcc-space.com/publications/documents/aiaa2004-3700-a41.pdf

(available also at www.cle.de/hpcc or http://www.hpcc-space.de/publications/index.html ???)

[3]
Walter Droescher, Jochem Hauser: *Physical Principles of Advanced
Space Transportation based on Heim's Field Theory*,

AIAA/ASME/SAE/ASE, 38th Joint Propulsion
Conference & Exhibit, Indianapolis, Indiana,

7-10 July, 2002, AIAA 2002-2094, 21 pp.,

URL presently unknown,

(available also at www.cle.de/hpcc or http://www.hpcc-space.de/publications/index.html ???)

[4]
Walter Droescher, Jochem Hauser: *MAGNET EXPERIMENT TO MEASURING
SPACE PROPULSION HEIM-LORENTZ FORCE
*,

IGW Innsbruck 2003

http://www.hpcc-space.de/publications/documents/AIAA2005-4321-a4.pdf

[5]
Sean M. Carroll: *Lecture Notes on General Relativity*,

Addison Wesley 2004, ISBN 0-8053-8732-3

http://arxiv.org/pdf/gr-qc/9712019

[6]
Sean M. Carroll: *Spacetime and Geometry*,

Addison Wesley 2004, ISBN 0-8053-8732-3

Quote from http://encyclopedia.worldvillage.com/s/b/Burkhard_Heim

1980

Up to this point, Heim had not yet confided in other theoretical physicists on the details of the mass formula derivation. Hence, the DESY results were not widely published and disseminated for academic scrutiny. Fortuitously in the same year,Walter Droescher, a theorist at the Vienna Patent Office, began to work with Heim. The first result of their collaboration cumulated into the second volume of Heim's major work, appearing in 1984. It isalmost(!?!) error-free, in contrast to the first volume which was not reviewed to this extent.

3. Further Discussion

Though the authors J. Hauser and W. Droescher were repeatedly informed by email about the above objections, there was no further reply till February 04, 2006. The discussion will be continued when we receive further contributions.

I received two longer emails from J. Hauser with no contributions to the question under consideration. Especially J. Hauser gave no equations or proofs to correct his criticized article or my objections above.

Hugh Deasy wrote on March 18, 2006:

You say:

"**R**^{4} cannot denote a manifold which is used in GR:

Your "manifold M" (=**R**^{4}) is such that you can identify its points with the
quadrupels (x^{1},x^{2},x^{3},x^{4})
where the coordinates are numbers such that you can define the metric (g_{ik}) by

g_{jk} dη^{j}dη^{k}
= (dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}
+ (dx^{4})^{2}.

This is nothing but the space **R**^{4}, on the one hand with Euclidian
rectilinear coordinates x^{m} (m = 1,2,3,4),"

But in

http://www.engon.de/protosimplex/summary/summary.htm

Heim says

"Extending the ideas of Einstein, Kaluza, Klein and Jordan the theory
described in this report shows how to geometrize in principle not only the
gravitational field but the other force fields as well. They appear as geometrical
structures of spacetime, **R**^{4} (a Minkowski space with
x^{4} = ict) subject to the
usual conservation laws, and lead to a general non-Hermitian geometry in **R**^{4}"

So this is not a flat real space.

Regards,

Hugh Deasy

G.W. Bruhn replied:

First of all: When physicists are using the language of mathematics then they have to notice the precise mathematical definitions of the symbols in use. If someone thinks that he can give own meanings to the symbols then he should not complain the confusions and misunderstandings that arise by that way.

**Example**:
**R** denotes the set of all *real* numbers. **R**^{4} is the set of
all *quadrupels* (x^{1},x^{2},x^{3},x^{4}) of REALS
x^{m}. Thus, the Minkowski space M, where x^{4} = ict is imaginary, is
not **R**^{4}.

It is less complicated to introduce the Minkowski space M as **R**^{4}
in mathematical sense but equipped with the (pseudo-)metric (since indefinite)

ds^{2} = (dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}
− (dx^{4})^{2} = γ_{mn}dx^{m}dx^{n}

where (γ_{mn}) denotes the Minkowski-matrix diag(1,1,1,−1).

It is well-known that M is flat. The reason is that all derivatives of the Minkowski
metric coefficients vanish. This implies that the Ricci tensor R^{mn} vanishes.
Hence we have the curvature R = R^{mn} γ_{mn} = 0, which means that M is flat.

We now come to Hauser's paper. His equation (1) is equivalent to

(*) ds^{2} = (dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}
− c^{2} dt^{2} = g_{ij} dη^{i}dη^{j}

where the coordinates x^{m} and the "curvilinear" coordinates η^{j} are linked
by a smooth coordinate transform x^{m} = x^{m}(η^{j}).

NB. Both sides of (*) define the metric of the same manifold M referring to different coordinates.
**That does not change the curvature**: Let R^{ij} denote the Ricci tensor relative to the eta-coordinates
and R^{mn} the Ricci tensor relative to the x-coordinates, then we obtain the curvature from

(**) R = R^{ij} g_{ij}
= R^{mn} γ_{mn} = 0 γ_{mn} = 0.

(I) The Minkowski spacetime M is flat. See [5; p.1 ff.]

(II) The curvature R does not change under coordinate transforms. See [5; p.81]

(III) Hauser's transformed M is flat.

[5]
S. M. Carroll, *Lecture Notes in Relativity"*,
arXiv [math-ph/0411085]

The topics **1.**, **2.**, **4.** ... **9.** do not concern my objections.
So I have to reply to topic **3.** only:

3.Therefore, spacetime is assumed to be a differentiable 4-dimensional manifold, M^{4}, as long as quantum effects are not considered. This manifold comprises a collection of points where each point is specified by a set of four real numbers and has the same local topology asR^{4}, i.e., it is locally but not globally (as you wrongly assume) likeR^{4}. This is why we refer to this spacetime sometimes asR^{4}, but from the physics context its meaning is always clear, see Figs. 1 and 2 on pages 3 and 4 of AIAA 2005-4521. A different question is the embedding of 4D spacetime in an Euclidean space. In GR there exist 10 independent components of the metric tensor, and thus aR^{10}would be needed. Your example is for embedding a 2D manifold that is, a surface of a sphere, inR^{3}. But this is not relevant for the construction of the poly-metric tensor.

As I have repeatedly pointed out above my criticism refers to your equation (1)

g_{ik}
=
^{∂xm}/_{∂ξα}
^{∂ξα}/_{∂ηi}
^{∂xm}/_{∂ξβ}
^{∂ξβ}/_{∂ηk} ,
g_{ik}
**: =**
Σ_{μ,ν=1 ... 8} g_{ik}^{(μν)} ,
g_{ik}^{(μν)}
=
^{∂xm}/_{∂ξ(μ)}
^{∂ξ(μ)}/_{∂ηi}
^{∂xm}/_{∂ξ(ν)}
^{∂ξ(ν)}/_{∂ηk} ,
(1)

which using the rules of calculus can be written in the form

(*)
g_{jk} dη^{j} dη^{k}
= Σ_{m} (dx^{m})^{2}.

That would be a *positiv definite* metric and would not fulfil the requirements even
of SRT, much less that ones of GR, if one assumes
(x^{1}, x^{2}, x^{3}, x^{4}) in **R**^{4} as you did.

So something in your definitions must be changed.

We could replace x^{4} → i x^{4} = i ct where x^{4} = ct is *real*,
i.e. we could define an indefinite metric on the real space **R**^{4} by

ds^{2} = (dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2}
− (dx^{4})^{2}
= γ_{mn}dx^{m}dx^{n}

where (γ_{mn}) denotes the Minkowski-matrix diag(1,1,1,−1).
Then we have the metric equation

(**)
g_{jk} dη^{j} dη^{k}
= γ_{mn}dx^{m}dx^{n}
.

This defines a manifold M that is locally Minkowskian indeed.
As can be seen from [5; p.1 ff.] it is Minkowskian even globally.
In addition, the metric equation (**)
enables us to calculate its curvature R [5; p.81], which is coordinate independent.
And the representation of M by means of the coordinates x^{m} yields R = 0
because all derivatives of the metric coefficients γ_{mn}
vanish, hence the Ricci tensor
R^{mn} as well to obtain R = R^{mn} γ_{mn} = 0.
The same result would turn out for any other coordinates η^{j} specified by
Equ. (**).

Concerning the **question of embedding**:

The spacetime manifold of GR is usually defined *without* referring to any embedding,
see [5; p.31].
Possibly you have misunderstood my remarks: What you are doing by your equation (1) is an
embedding of M into the space **R**^{4} equipped with an Euclidean or Minkowskian metric
respectively. By that way the manifold inherits the metric of the surrounding space, and that is
no way to the situations in GR.

Possibly, the whole discussion arises from your not well formulated definitions. If so,
there are several good textbooks available where correct methods of definitions of manifolds
can be found. Your Equ. (1) e.g. is *not compatible* with the method described in [5].
So my proposal is that you think over the whole issue and try it with a better formulated theory.

[5]
S. M. Carroll, *Lecture Notes in Relativity"*,
arXiv [math-ph/0411085]

[7] Jochem Hauser and Walter Dröscher:
*Rebuttal:
Critiscm of a Flat Metric by Prof. Bruhn,
Technical University of Darmstadt, Germany*

http://www.hpcc-space.de/news/RebuttalProfBruhnTUD.pdf