What is the effect of the instability induced by the peculiar coherent coupling on a pre-existing static gravitational field, generated by a far source? Why does a weak shielding result?

This crucial point of our model was already studied qualitatively in Ref.s SHI, LCC, and has been clarified through an explicit calculation in GAU.

Intuitively, it is clear that the static field is affected by the presence of the potential well. As we mentioned earlier (compare also Fig.1) the field is forced to assume within the critical region, with certain probabilities, strengths independent of its original strength. Which is the residual effect of this constraint outside the critical region?

One could try some analogy with a tunneling effect in ordinary quantum mechanics. After doing a Fourier decomposition of the static field produced by the far source, one can identify those modes which have a large probability to be stopped by the potential well. It turns out, however, that it is impossible to carry on this qualitative analysis in a satisfying way. It is necessary to implement a rigorous "ab initio" calculation. This has been done in GAU, and the positive results give us further confidence in our model.

In the special case of a massless field (like the
gravitational field), we found that if the field, denoted by *f,*
is forced to assume the values *f*_{i} within a given
region, with probabilities x_{i},
then the static potential of the interaction mediated by *f*
between two sources placed at the opposite sides of that region
(Fig. 2) is decreased by the factor

(2) [1 - g*abF*(e,r,*n*)],

where g=S_{i}x_{i}
|*f*_{i}|^{2}, *a* and *b* represent
width and thickness of the region, e
is the inverse of the distance of the first source in units of *a*,
r is the ratio *b/a* and *n*
is the distance of the second source from the region, measured in
units of *a*. Evaluating the function *F*(*n*) for
e and r constant
one obtains a decreasing exponential, with a non-zero asymptotic
value (Fig. 3).

**Fig. 3 **- Plot of the function *F*(*n*)
for different values of r .

This predicted behavior corresponds to the experimental findings, according to which there is: (1) a cylindrical shielding region, (2) without diffraction at its border.

In fact, the coherence domains inside the disk
are very small on a macroscopic scale and a single critical region
cannot be larger than one of these domains. Thus the total shielding
factor is given by the contribution of several domains. (See Fig.
4. Also note that the effect of different domains can superpose
over the thickness of the disk: Fig. 5.) The distance of the proof
masses from the superconducting disk corresponds to a large value
of *n*, such that *F*(*n*) has already reached its
constant asymptotic value. This explains the observed features (1)
and (2) above.

This graph shows, for purely illustrative purposes,
the behaviorof the shielding factor produced by a single critical
region (coherence domain) along its vertical axis. The shielding
factor is proportional to the adimensional function *F*(*n*),
where *n* is the distance from the critical region in units
of the region's radius. The present graph is in practice the same
one as in Fig. 3, transformed into a bars diagram, rotated by 90
degrees and placed in space.

The coherence domain corresponds in our model to a region with over-critical density, and in our calculations we assumed it for simplicity to have the form a squeezed ellipsoid (ra>1). It lies within the superconducting disk and is surrounded by similar domains.

The total macroscopic shielding factor is given
by the sum of the effects of the single domains. Note that the size
of the domain depicted in the figure is exaggerated as compared
to the size of the disk. Since the parameter *n* represents
the distance *in units of the disk radius*, it is clear that
the exponential decay of the shielding factor predicted by our model
actually takes place very close to the disk.

What remains at macroscopic distances, and gives
the observed effect, is the constant "tail" of the exponential,
which amounts to approx. the 10% of its value for *n*=1 and
does not vary with increasing distance.

The effects of more coherence domains, each of
thickness *b*, are summed in such a way that the total shielding
factor is proportional to the thickness of the layer of the disk
where critical density can be reached. We recall (arrow) that this
layer is quite thin compared to the whole disk (see Point 3: "Density
pattern of the superconducting carriers in the composite disk").

The shielding produced by a single domain is proportional
also to its width *a*; thus the total shielding factor is proportional
to the average value of *a* but not to the radius of the disk.

(Actually, a larger disk radius enhances the effect in an indirect way, because it means a larger tangential velocity during rotation andthis leads in turn to a thicker "critical layer" - see Point 3.)

This is a more "phenomenological" issue. While the Points 1-3 above were mainly concerned with the dynamics of the gravitational field, this point is concerned with the properties of the superconducting material.

It is important to know the density distribution of the condensate of the superconducting charge carriers in the ceramic disk, because this distribution defines the critical regions, where the density or its gradient exceed the threshold value.

In our earliest work (SHI) we limited ourselves
to state that the fast rotation of the disk and the applied magnetic
fields were probably responsible for strong variations in the condensate
density. Here we push our analysis a bit further. It is not necessary,
to this end, to make any special hypothesis about the microscopic
mechanism of high-*T*_{c} superconductivity. It is
sufficient to consider an effective model, based upon an order parameter,
like the Ginzburg-Landau (GL) theory.

In the GL theory the variations of the order parameter,
and thus the variations of the condensate density, are always such
that this density vanishes at the boundaries between superconducting
and non superconducting regions. In our case, on the contrary, we
are interested into a local * increase* of the density. How
is it possible to achieve this?

All the phenomenology of the experiment shows that: (i) the two-phases structure of the superconducting disk is essential in order to obtain the shielding phenomenon; (ii) the effect takes place in conditions - fast rotation, high frequency fields applied - which are very far from the static limit, and thus cannot be adequately described by the GL theory.

Therefore it seems that the presence of an interface between two phases of the disk having different superconducting properties, combined with rapid rotation, is able to produce a local increase in the density which is not predicted by the GL theory and as such quite rare.

We do not have any proper formalism to describe this situation yet, namely a composite superconductor in rapid rotation. We can only resort at present to a qualitative model.

In our earlier work BOS we depicted the superconducting carriers in the upper part of the disk as a perfect fluid which flows, during rotation, close to a "border" represented by the lower part of the disk, which is non superconducting at the operation temperature. The relative sliding velocity, due to the fast rotation of the disk, is very large in comparison to the average velocities typical of the superconducting charge carriers in stationary conditions. This - we hypothesized - would produce a density increase near the border.

This phenomenological model is probably incorrect at the microscopic level, because the motion of the superconducting carriers cannot really be a "viscous" motion exibiting velocity gradients; moreover, the microscopic structure typical of any Type II superconductor must be taken into account. Nevertheless, the model has the merit to stress the importance of the interface between the two parts of the disk, and is also supported by several experimental observations, which give a clear feeling of a fluid-dynamical behavior of the system.

In particular, one observes that the shielding effect is considerably increased during the disk braking phase, when the relative sliding velocity of the superfluid in the upper part of the disk is much increased by virtue of the inertia of the fluid.

Furthermore, it has been observed since the very
first experiments in 1992 that the effect remains even when, after
reaching a rotation speed of 4000-5000 *rpm*, all lateral magnets
are turned off and the disk rotates by inertia. As long as the disk
rotates, the shielding effect persists. This confirms the purely
kinematical role of rotation.

See the "Discussion" below for a closer analysis; further work is in progress.

This point, like Point 3, also represents a wide phenomenological issue, which can be studied independently from the microscopic gravitational mechanism.

Several questions arise spontaneously here. First of all, it is clear that any physical mechanism which expels from a region even a small part of the gravitational field must have some consequences on the global energetical balance. If a mechanical process inside the shielded region is affected, the total energy must be conserved.

The most frequent consequence of the shielding
phenomenon seems to be an *increase* in the mechanical energy
of the shielded objects (see BOS). This energy must come from an
external "pumping", and if the latter is absent or inefficient,
the shielding will be inhibited.

The whole phenomenology of Podkletnov's experiment indicates that the pumping is done by the high-frequency components of the applied magnetic field. This kind of pumping is familiar in atomic physics, for instance in lasers or in systems with magnetic resonance.

We believe that the high-frequency field allows the forming of regions with overcritical density of superconducting carriers, or it "activates" these regions and triggers the runaway of the field as described in Section 2b. It is a common experience that the runaway of an unstable system cannot take place unless a channel for the energetic exchange is available, in such a way that the total energy balance is respected (compare for instance phase transition phenomena).

**Fig. 6** - Some energy pumping process
is needed, for gravitational shielding to arise.

When in a region of the superconducting disk (red square in the figure) the density of the condensate exceeds the critical value, the gravitational field is virtually affected in that region. This produces a shielding cylinder: a proof mass placed above that region will become slightly lighter. However, this (still virtual) process can in turn produce mechanical energy! For instance, if there is a counterweight, we can compress a spring in this way; when the shielding ceases, the two masses will start to oscillate.

All this means that the state "region in the disk with overcritical density, shielding activated" has higher energy than the "normal" state. A transition to this state is possible only if energy is supplied from the outside. In Podkletnov's apparatus, this "activation" energy comes from the external high-frequency e.m. field.

This representation of the role of the high-frequency field is obviously approximated, and a complete theory should eliminate the distinction, useful but artful, between the shielding process, the pumping which activates it and its energetic consequences. From the causal point of view, the three phases are in the following order:

® ® |

We could also put an arrow that goes from the last
point up to the first one, with a sort of *feedback*; this
is because any variation in the mechanical energy of the objects
inside the shielded region requires an energy supply, and in the
absence of this the runaway of the field in the critical regions
with subsequent shielding is inhibited.

In tandem with a direct interaction between the high frequency e.m. field and the superconducting carriers, indirect interactions of several kinds could be present. The e.m. field might first excite some mode peculiar of the crystal lattice of the ceramic material, or a mode belonging to the spectrum of the superconducting state; then the energy would be released for the shielding process. This could explain the resonant behavior of the system at certain frequencies.

Given our poor knowledge of the details of the pumping process, Podkletnov's strategy to send on the disk a strong and wide spectrum AC field appears to be drastic but somehow effective - at least as long as the superconductor is not heated too much.

It was early realized that the observed clear-cut cylindrical form of the shielding region and its vertical extension (at least a few meters) are very unusual and represent a puzzle from the conceptual point of view. For comparison, a metal shield placed in an electrostatic field produces in general a cone-like shielding region, with relevant border effects. On the other hand, if the SC disk would emit an hypothetical secondary field, or some radiation, this would probably appear like a divergent beam. (Also note that in this case one should expect some emission downwards, which does not appear to be the case.)

**Fig. 7** - Non-conservative character of
the modified field.

The observed modified field pattern is clearly non conservative (Fig. 7). If a test mass makes a round trip going up inside the shielding cylinder and coming down outside, the gravitational field exerts a net work on the mass. Formally this amounts to saying that the observed field is not the gradient of a gravitational potential. The Einstein equation for a weak static field still holds, namely

div**G**_{00}
= 0, with *a* = **G**_{00}/*m*

(*a* is the acceleration of the test mass
*m*, in the limit when the velocity of *m* is much smaller
than the light velocity); however, the associated equation

**G**_{00}
= grad *h*_{00}

is not true.

The non conservative character of the observed field configuration originates from the peculiar coupling of the gravitational field to the pumping e.m. field - coupling mediated by the condensate in the superconductor. The mechanical energy gained by a test mass along a round trip is furnished by the e.m. field and of course there is no global gain.

An interesting experimental observation in support of our interpretation is the following. If the energetic pumping is inefficient (that is, the applied e.m. field is too weak, or its frequency too low, or the test mass too heavy) one observes a weak "expulsive" force acting on the test mass at the border of the shielding cylinder. This force tends to restore the zero-work condition along a round trip.