It is known that the Euclidean Einstein action is not bounded from below due to the so-called "conformal modes". The use of the Euclidean formalism to study the stability of our system may therefore appear quite arbitrary. Consider, however, the following points.

(i) The phenomenon under investigation is essentially
macroscopic. Its typical distance and momentum scale is such that
terms in the action of the order of d*R*
can be disregarded. Equivalently, it is possible to insert in the
action a cut-off on the momenta which eliminates the conformal modes.

(ii) The Euclidean formalism is employed only in
a perturbative context, as the analytical continuation of Minkowski
space + small fluctuations. We use the Euclidean formalism in order
to ensure the positivity of the function m^{2}(*x*)
(see 2a) and in order to take advantage of some known techniques
for the computation of the static potential (see 2c).

Furthermore, we use the Euclidean formulation in order to exhibit the stabilizing effect of a negative cosmological term and the de-stabilizing effect of a positive comological term, but still in the context of a weak field approximation. The same results can be obtained in the Minkowskian theory, where a positive comological term corresponds to an imaginary mass.

After realizing that an instability arises, we
do not try to describe the behavior of the field through the Euclidean
formalism; we just suppose that the runaway process stops at some
field strength, in such a way that the strengths *h*_{i}
have certain unknown probabilities x_{i}
(see 2b).

Intuitively, one would expect the shielding region above the superconducting disk to have the shape of cone, instead of a cylinder (see also Fig. 9). The observed cylindrical shape implies that: (i) the source of the field of the earth is seen as pointlike; (ii) there isn't any kind of "diffraction" at the disk border.

**Fig. 9 -** Why is the shielding region cylinder-like
instead of cone-like?

This is not easy to explain. Intuition would suggest a cone-like shielding region above the superconducting disk (the basis of the cone being on the disk), instead of the infinite cylinder-like region observed by Podkletnov. In fact, when the elevation of the proof mass above the disk increases, the proof mass apparently "sees shielded" only a part of the Earth. This geometrical factor can be computed exactly (see G. Modanese, supr-con/9601001).

Nevertheless, it turns out that when computing
the shielding factor for the Newtonian part of the gravitational
force (that with dependence 1/*r*^{2}), the mass of
the Earth can still be regarded, as usual in gravity problems, as
concentrated in the center.

Condition (ii) is consistent with our theoretical model, as shown by the computation in GAU, under the reasonable hypothesis that the coherence domains in the superconducting disk are very small in comparison to the disk size.

Condition (i) is a general consequence of our theory,
too. In fact, in order to compute the static potential energy of
the interaction of two bodies with masses *M* and *m*
one adds to the gravitational action * S*_{g} the actions
*S*_{M} and *S*_{m} of the two bodies
and then computes the quantum average <exp(*S*_{M}+*S*_{m}>_{g}.
For pointlike bodies, the action is simply given by ò*ds*,
where the invariant interval *ds* computed along the trajectory
of the body, which is a geodesic line in any field configuration
*g*.

For an extended body, the center of mass (CM) still
follows a geodesic line, but in addition to the line integral we
must also insert a volume integration over the parts of the body:
*S*=ò*dV*r(**x**)ò*ds*(**x**).
This implies in practice that in addition to the minimal gravitational
coupling, of the form (*h*_{mn}*v*_{CM,m}*v*_{CM,n}),
there is a coupling with the *derivatives* of *h* and
quadrupole, octupole etc. terms of the form (¶_{a}¶_{b}*h*_{mn}*Q*_{abmn}),
etc.

These terms do give some contributions to the static
interaction potential, when inserted into the quantum average <exp(*S*_{M}+*S*_{m}>_{g},
but these contributions depend on the distance *r* as *r*^{-2},
*r*^{-3}, etc. So if we keep the multipole terms into
account when computing the "shielding correction", we
find that the correction to the Newtonian term *U*»*r*^{-1}
depends only on the coordinates of the center of mass. In other
words we can regard the two bodies as pointlike, as long as we are
interested only in the Newtonian potential, which is usually by
far the dominant part of the interaction.

For a scalar field *f* one can impose a local
constraint (see GAU) through a double well potential of the form
*U*(*f*)»[*f*^{2}(*x*)-*f*_{i}^{2}].
(More precisely, the r.h.s. is multiplied by a function *f*_{W}(*x*)
which has support in the critical region W.)
In the gravitational case we could write instead *U*(*h*)»[*h*_{00}^{2}(*x*)-*h*_{00,i}^{2}],
but in principle *h*_{00,i} cannot be a constant, because
the zero-mode, which is the new minimum of the action in the presence
of instability, depends on *x*.

Since there are several zero-modes close to the
minimum of the action, we could take an average of these modes with
respect to *x* and obtain a constant field. It is better, however,
to consider the field at one single point: here the field takes
values *h*_{i}, with probabilities x_{i},
and the same holds for any point, with the same *h*_{i}'s
and x_{i}'s; the only difference
is that for points close to the border of the critical region W_{i},
*f*_{W}(*x*) is not
equal to 1, but goes gradually to zero. Thus we have a constraint
of the form

å_{i}
x_{i} ò_{i}*dx**f*_{W}(*x*) [*h*_{00}^{2}(*x*)-*h*_{00,i}^{2}]^{2}

Actually, it would be impossible to assign different probabilities to different points, except for those at the border of the critical region.

One easily checks that the effect of the change above on the final formula for the shielding factor is just to replace the parameter g with another unknown factor.

Table II - Unproved assumptions of our model. |

(i) The Euclidean theory near equilibrium can be applied to quantum gravity. |

(ii) The instability leads
to a constraint, with probability x |

(iii) There exists a small intrinsic negative L and thus a threshold. |

In our earlier work we hypothesized that a local increase in the superconducting carriers' density could be obtained in the composite disk in non-static conditions, due to the very fast flow of the "superfluid" in the upper part of the disk near the border of the lower (non superconducting) part. (Fig. 10)

**Fig. 10 - **Fluid dynamical model
for the composite disk.

Schematic diagram representing the velocity distribution of the superconducting charge carriers in the upper layer of the toroidal disk (B) during rotation.

The lower layer (A) is not superconducting at the temperature of operation (slightly below 70 K). The relative velocity of the charge carriers with respect to the interface between the two layers is represented by thick arrows. It grows from zero to some velocity Vmax according to a power law.

This is a side view. In stationary conditions, the relative velocity has no component along the vertical direction z. We suppose that the disk is rotating clockwise, so the short arrow means in reality that near the interface the superconducting charge carriers are almost co-rotating with the disk. The red region is a critical region, with over-critical density. Its thickness is exaggerated, for clarity.

The carriers farther away from the interface "follow slower", thus with larger relative velocity, up to Vmax.

In fact, admitting that the relative drift velocity
goes to zero at the border, one concludes from simple fluid dynamical
arguments that the fluid density must *increase* at the border
- the opposite of what usually happens in static conditions at any
normal/superconducting interface. More precisely, the carriers density
would reach a peak near the interface and then go rapidly to zero
at the interface.

A velocity gradient near a flow boundary is the typical signature of a viscous fluid however, and this cannot be the case of the supercurrents. How could it then be possible to justify a velocity gradient?

We could regard the velocity as a mean velocity, and keep into account the "strong Type II" nature of the ceramic superconductor as well as the role of defects and impurities. These are much more frequent near the interface, and the latter is not clear-cut, being obtained through a thermal process which melts the upper part of the disk.

The mean free path of the superconducting carriers could be very long in the upper part of the disk and become gradually shorter towards the bottom of the disk. Therefore the superfluid would slow down at the interface - in the sense of the average motion - due to the interaction with obstacles and impurities in the lattice.

This could also explain why the disk tends to heat up just when the largest values of the shielding factor are observed: the reason would be that the almost-resistive behavior depicted above is just what is needed to achieve a local increase in the density of superconducting carriers.

Or perhaps the causes of the heat production are different (for instance, related to the pumping process), and a "mean path" of the superconducting carriers is meaningless in this context? All these questions are still unsolved; as we pointed out above, a proper theoretical treatment of composite Type II superconductors (not mentioning the peculiar properties of HTCs...) subjected to fast rotation and high-frequency e.m. fields is not available yet.

Another useful observation could be the following. We can say that the composite disk exhibits, in a certain sense, a "Tc gradient", from the top to the bottom, from 92 K to approx. 60 K. The shielding effect was observed by Podkletnov slightly below 70 K.

Remember that the distance between the vortex cores in Type II material tends to zero when the temperature approaches Tc (Fig. 11). It is then clear that if a supercurrent flowing in the "good" part of the disk in deviated to the lower part, where there is less space available for the superconducting carriers, the density of the latter can increase even if their velocity keeps constant. The deviation can be caused by the external variable magnetic field, or by fluctuations and defects in the lattice. The density increase would in any case also be connected to the relative drift velocity of the carriers with respect to the rotating lattice.

This possibility is currently under investigation.

**Fig. 11 - **Size of the flux cores in dependence
on the temperature.

As we saw in the first part, it is possible to
keep into account the non conservative character of the gravitational
field in the context of a phenomenological theory, in which the
external energy source has been included and therefore we do not
expect that the static force field is still equal to the gradient
of the potential *h*_{00}.

We still expect, however, that the minimum of the
action outside the superconductor corresponds to the Einstein equations;
it follows, for weak fields, the equation ¶_{iG 00}^{i}=0, plus
suitable boundary conditions. This admits as a solution the observed
cylinder-like shielding region.

It is natural for a general-relativist to ask, at this point: what happens to a light ray when it crosses the shielding region? Is it deviated? And will a precision clock show any gravitational slow-down when it is first placed into the shielding region and then pulled out?

These phenomena are typically studied using the
metric tensor. We know that in the presence of shielding the relation
between the static force G _{i}
and the gradient of *h*_{00} is spoiled. The question
is: is it still possible to use the metric tensor to predict the
trajectory of a light ray or the relation between the proper time
of a frequency standard and the coordinate time?

These effects could also be tested experimentally,
at least in principle. Light propagation across the shielding region
could be studied, for instance, through interference measurements
on a laser beam after several reflections at both sides of the region.
It can also be shown (details will appear elsewhere) that the modification
of the metric tensor inside the shielding region may lead to a shift
in the frequency of a clock of about 1 part in 10^{11} -
approx. 2 orders of magnitude larger than the precision of atomic
clocks.

But apart from the several practical difficulties connected to these measurements, it is not clear yet on the theoretical side whether the gravity anomaly affects in the same way all the components of the metric, and whether it affects in the same way the static behavior of the field and its variations on a short time scale. Much work is on the way in this direction.

Another important issue is the compatibility between the shielding phenomenon and the equivalence principle (see Fig. 12).

**Fig. 12 -** A "flying machine" based upon
the gravity shielding effect would violate the equivalence principle.

Imagine a box divided in two sections 1 and 2.
Suppose that the lower part of the box, with mass *m*_{2},
contains a shielding apparatus, complete with power supply generator
and everything. Now let the box be in free fall. If "the shielding
is OFF", the acceleration of the box is equal to *g*.

Then you "turn ON" the shielding, say
with efficiency a ; this means that the
gravitational force felt by the mass *m*_{1} over the
apparatus is multiplied by a factor a
<1 (for instance, a =0.98). Let us
admit that the weight of *m*_{2} itself is not affected.

It is easy to see that in this case the acceleration
of the box becomes less than *g*. This is actually what desired,
if we aim at building a flying machine. It means, however, that
the gravitational mass and the inertial mass of the box are not
equal any more. And this represents a violation of the equivalence
principle.

Note that the box is supposed to be isolated from
the enviroment: it does not expel any jet of air or gas, nor does
it interact with any external electric field, etc. In these conditions
of free fall, an observer inside the box should experience total
absence of gravity. He doesn't, however, if the shielding is ON.
He feels some gravity, because its acceleration is lower than *g*.
This, again, shows that the equivalence principle is violated.

If we do not accept the possibility of such a violation, we must admit that the shielding effect does not work like this. We must admit that if the shielding apparatus is rigidly connected to the Earth, then there is effective weight reduction of the samples suspended over the apparatus; but if the whole shielding apparatus is in free fall, then a reaction force from the samples on the apparatus arises, which makes the total weight variation vanish.

This means of course that it is impossible to build a flying machine using the gravity shielding effect. It is still possible however, in principle, to build a "lift".

It has been reported that the amplitude of the transient effect first described by J. Schnurer appears to depend much on the method employed for its detection. This led us to the following conclusions.

(1) Probably the transient weight variation at the superconducting transition has its own characteristic frequency spectrum. This is because it is short-lasting - which brings in a frequency spectrum - and also because there could be some kind of temporal spikes in the effect.

(2) The frequency spectrum of the effect should match well the proper frequency (or frequencies) of the detector, in order to cause a good response in the detector - much like in a resonance effect.

So, roughly speaking, one should not picture the measurement as "the effect comes, the detector feels a transient weight diminution and displays this, then the effect goes and the detector ceases to display the weight diminution". It could be really so, only if the duration of the effect was much longer than the response time of the balance, say 30 seconds or more, and without any spikes.

The correct picture would be: the effect comes and acts like a force which perturbates the proof mass with a certain frequency spectrum. If the proper frequency of the mechanical system holding the mass is close to the typical frequencies of the effect, then an oscillation starts; if not, almost nothing happens, because the mechanical system holding the mass is either to too rigid or too loose (compare also Fig. 6).

Probably the wooden arm used by Schnurer in his measurements or some other element in the arrangement has a certain vibration frequency which resonates with the effect. It would be very important to clarify this point.

While doing this hypothesis, we are figuring that in fact no pure weight diminutions are seen in the transient effect, but weight oscillations in both directions, due to this kind of resonance. Several balances take a few seconds to give an accurate response, and are therefore not suitable for measuring a weight which is supposed to vary quickly - or they start oscillating if the weight oscillates, but in some strange and unpredictable way. As a matter of fact, people who build balances never consider the possibility that weight can vary with high frequency, due to some little "kicks" coming from the bottom...