**Abstract -** In this report we summarize
in an informal way the main advances made in the last 3 years and
give a unified scheme of our theoretical work. This scheme aims
at connecting in a consistent physical picture (by the introduction
of some working hypotheses when necessary) the technical work published
in several single articles. The part of our model concerning the
purely gravitational aspects of the weak shielding phenomenon is
almost complete; the part concerning the density distribution of
the superconducting carriers in the HTC disks is still qualitative,
also due to the very non-standard character of the experimental
setup. The main points of our analysis are the following: coherent
coupling between gravity and a Bose condensate; induced gravitational
instability and "runaway" of the field, with modification
of the static potential; density distribution of the superconducting
charge carriers; energetic balance; effective equations for the
field; existence of a threshold density.

Copyright 1999 Giovanni Modanese. All rights reserved. Work deposited at the Los Alamos Electronic Preprint Library. This is an informal version, for educational purposes only. The author's biographical data recorded in Marquis' "Who is Who in the World, 1999". The "critical density model" for gravity anomalies first published in Europhys. Lett. 35, 413 (1996) and in subsequent papers (see references below); first public presentation by the author at the 1997 I.A.F. World Congress.

Main points of the research project | References |

(1) Relevance and innovation of the effect. Impossibility of an explanation within General Relativity.. | SHI |

(2) Logical core of our model: | |

(a) Peculiar coupling betw. gravity and a Bose condensate ® | SHI, LCC |

(b) ® Induced instability and "runaway" of the zero modes ® | ISS; earlier in SHI, LCC |

(c) ® Constraint on the gravitational field and "tunneling". | GAU; earlier in SHI, LCC |

(3) Density pattern of the superconducting carriers in the disk. | BOS |

(4) Energetic balance and effective equations for the gravitational field | BOS |

(5) Threshold density of the condensate | BOS |

References

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The weak gravitational shielding effect by HTC superconductors discovered by E. Podkletnov and the transient gravitational anomalies observed by J. Schnurer open an incredibly wide research field, at the boundary between condensed matter physics and gravitation.

All the observed phenomenology strongly supports our proposed theoretical model, based on the idea that an instability of the gravitational field arises in the so-called "critical regions" of the superconducting charge carriers condensate.

The task of summarizing and connecting in a logical way all the conceptual and theoretical implications of these effects is hard, for two reasons at least.

(i) Up till now, we do not have any consistent theory which, starting from well known paradigms, leads in a purely technical way to an explanation of the experiments. It is impossible to "prove theoretically" the existence of the anomalies. Even though the starting point of our proposed model is entirely orthodox (quantum mechanics + general relativity + Ginzburg-Landau theory of superconductors), simplifications and unproven hypotheses are needed at several points.

This situation is not unusual in physics, particularly not for very complex systems. It happens quite often that subsequent developments justify "a posteriori" previous assumptions. The art of the theoretician consists also in making reasonable assumptions, which do not hurt any fundamental principle or prior knowledge and fit well with the experimental intuition.

(ii) There is a wide interest towards these new results, but specialists of different areas have different attitudes and questions. Some wonder how the effect is connected to the perturbative or non-perturbative dynamics of quantum gravity or its generalizations; others take for granted that there is an exotic effect at the basis of the anomalous coupling, and focus on phenomenological issues. We tried to address several "frequently asked questions" throughout the paper and the reader may notice by himself the variety of the question spectrum.

The organization of this article is a direct consequence of the two points above. On one hand, the fundamental logical structure of the theoretical model is stressed, with reference to the specific articles for more details; new assumptions and obscure points are discussed. On the other hand, the various issues are also listed as single items, like in a handbook, with some redundant repetitions or logical simplifications when necessary.

The main points of our analysis are the following.

This is a necessary premise for any serious analysis of these effects. One should realize that the gravitational "anomalies" claimed by Podkletnov et al. are at least 10 orders of magnitude larger than those predicted by the General Relativity theory. For this reason, these results are hard to accept and understand. The scientific community reacted to these claims with a mixture of interest and scepticism.

It is important to notice, however, that no basic physical principles (like for instance symmetry or conservation laws) are contradicted by the claimed effects. The latter can be just ascribed to a peculiar dynamical behaviour of the gravitational field in certain circumstances. We do not see in the results of Podkletnov et al. any "revolutionary" content, such to justify a strong preconceived opposition. Furthermore, the effects have precise energetical limitations and can be observed only under certain conditions. See also on this point Section 4 and the remarks on the Equivalence Principle.

The following Points (2a), (2b), (2c) are the "logical core" of our theoretical model.

A Bose condensate has the property to be macroscopically
coherent and is thus described by a classical field f_{0}(*x*)
- also called "order parameter" - whose squared module
|f_{0}(*x*)|^{2}
is related to the condensate density.

This coherence property implies that the condensate
contributes not only to the linear part of the gravitational lagrangian
*L*, with standard coupling d*L*^{(1)}=8p*h*_{mn}*T*_{mn}(f_{0}),
but also to the quadratic part. The contribution to the quadratic
part affects locally the "potential" of the gravitational
field, that is, that part of *L* which does not depend on the
field derivatives.

More precisely, the coherent coupling term is d*L*_{Coherent}^{(2)}=
{Ödet[*g*(*x*)]}^{(2)}m^{2}(*x*),
where m^{2}(*x*) is a positive
definite quantity defined as

(1) m^{2}(*x*)
= ¶_{m}f_{0}^{*}(*x*)
¶_{n}f_{0}^{*}(*x*)
+ * m*^{2}|f_{0}(*x*)|^{2}

and {Ödet[*g*(*x*)]}^{(2)}
denotes the quadratic part of Ödet[*g*_{mn}(*x*)=d_{mn}+*h*_{mn}(*x*)],
which in terms of the field *h*_{mn}(*x*)
is negative definite and thus will be symbolically denoted in the
following simply as "-*h*^{2}".

In the gravitational lagrangian there is already
a term of this kind, namely {Ödet[*g*(*x*)]}^{(2)}(L/8p*G*),
where L is the very small negative intrinsic
cosmological constant of spacetime. We define as "critical
regions" those where m^{2}(*x*)>|L|/8p*G*,
that is, those where the coherent coupling term is larger than the
"threshold" term. Since the absolute value of f_{0}(*x*)
is related to the density of the Bose condensate, the critical regions
are defined through eq. (1) by the condition that either the density
of superconducting charge carriers or the gradient of this density
are particularly large.

There exists a class of unstable modes of the pure
Einstein action, called "zero modes", which have the same
probability to occur as the * h*=0 configuration (flat space)
and a higher probability than all other field configurations. (The
probability is proportional to exp(*iS*[*g*]/*h*_{Planck}),
and for the zero-modes one has *S*=0, like for flat space;
see the Discussion of Point 2a, "Quantum nature of the effect".)

Thus a gravitational field would always tend to
"run away" towards these configurations and loose memory
of its initial state, if it was not regulated by a small negative
cosmological term, intrinsically present in nature. This term has
the form d*L*_{Cosm.}^{(2)}=+|L|*h*^{2},
so it favours the *h*=0 configuration.

The coherent coupling d*L*_{Coherent}^{(2)}=-m^{2}*h*^{2}
of the gravitational field to the condensate amounts to a local
positive cosmological term (note that the actual signs are unfortunately
opposite to the conventional denomination). This cancels the intrinsic
stabilizing term d*L*_{Cosm.}^{(2)}
and leads to a local instability.

Therefore, while the regular coupling of gravity
to incoherent matter produces a *response* - a gravitational
field - approximately proportional to the strength *T*_{mn
}of the source, the coherent coupling induces an *instability*
of the field.

Things go as if a potential well for the field was suddenly opened. The field runs away towards those configurations which are now preferred (compatibly with the global energetic balance - compare Section 4). The runaway stops at some finite strength of the field, where higher order terms in the lagrangian, which usually can be disregarded, come into play. This strength is not known because it depends mainly on the non perturbative dynamics of the field, and very little on the initial conditions.

It is important to stress that not all the field modes become unstable and undergo this runaway; only the so-called zero modes have a high probability x to do so (compare Fig. 1) and these modes are a minor part of all possible configurations. Also for this reason (ratio of volumes in probability space, or "phase space"), the total probability of the process is quite small.

This figure represents, for illustration purposes, the behavior across a critical region of:

*Bottom -* The density
of superconducting charge carriers (SCCC). By definition, the critical
region begins where the density of SCCC exceeds the threshold density
r_{c.
}We are not interested at this stage into the causes of this
local density increase.

*Middle -* The potential
part of the gravitational lagrangian *L*(*h*), that is,
the part which does not contain the field derivatives. This well-like
shape of the potential is due to the coherent coupling and is responsible
for the instability. The dotted line means that higher order terms
in the lagrangian come into play and stop the instability - but
we do not know exactly how and for which strength of *h*. Note
that the instability arises quite sharply at the border of the critical
region (we admit that a threshold value of L
exists - see Section 5).

*Top -* The static
field produced by a far source, supposed to be on the left of the
critical region. In the critical region the field "runs away"
and takes strengths* h*_{i}^{2} with probabilities
x_{i}.
This is a quantum process, and the strengths *h*_{i}
as well as their probabilities are non-observable parameters. Only
the parameter g=
S_{i}
x_{i}* h*_{i}^{2}
is actually observed, and must be inferred from the experimental
data.

We did not represent in the graph the strength
of *h* within the critical region, just because only probabilities
can be defined.

The computation in GAU allows us to conclude that the pre-existing static field is decreased in the critical region by a % amount proportional to g and to the thickness and width of the region (corresponding to a coherence domain of the condensate or to a part of it).

The field strength is most affected just outside the critical region (see 2c), then the original value is almost completely restored, apart from a little residual modification ("tail") which is the one actually observed.