We mentioned earlier the condition defining a critical
region in the condensate of the superconducting carriers: a certain
function m^{2}(*x*) of the
condensate density and gradient must be larger than the "natural"
cosmological term L/8p*G*.
Therefore this term represents a threshold value for the density.

The value of L is not known a priori. There are some upper limits, deduced from astronomical observations, but also some indications that L scales with the distance and its effective value is larger at small distances.

Do Podkletnov's data suggest the existence of a
threshold density? Or does the shielding effect instead depend in
a *continuum* way on the density? To give a proper answer,
one must take into account the pumping process, because it could
happen in some cases that the condensate density is well above the
threshold value but the effect does not take place due to absent
or inefficient pumping.

This issue has been discussed in BOS. In order to conclude that a threshold exists, we must start with conditions of efficient pumping. If in these conditions the shielding effect is observed only above a certain local condensate density or density gradient, this is a point in favour of the treshold.

It was observed, for instance, that disks without two-layer structure, and thus without density gradients, do not produce any shielding effect. This is a point in favour of the existence of a treshold.

If a treshold exists and the pumping is efficient, then any increase or diminution in the strength of the effect can be interpreted as following an increase or diminution of the number or average size of the critical regions.

Note that with absent or inefficient pumping even a condensate whose density exceeds by far the threshold value fails to produce any gravitational anomaly. This is probably the case of superfluid helium, which is much more dense than an electronic condensate: since it is electrically neutral and its temperature is extremely low, it cannot be subjected to any pumping of reasonable efficiency.

According to General Relativity the dynamics of the gravitational field and its coupling to the mass-energy-momentum density which generates it are described by the (classical) Einstein equations. These are non-linear partial differential equations involving the components of the metric tensor and its first and second derivatives. They are similar, in several respects, to Maxwell equations, though more complicated and non-linear.

In very simplified terms, we can say that Einstein
equations allow for finding the gravitational field as a *response
to a source* - linear in a first approximation, or non-linear
in the presence of strong mass-energy densities. The proportionality
constant between field and source is of the order of the Newton
constant *G* for linear responses and even smaller, of the
order of *G*/*c*^{n}, for non-linear responses.
There exist static fields and fields propagating like waves, but
in any case their strength is related to the mass of the source
which has generated them.

Given the mass and proximity of the earth, it would
seem impossible that any laboratory experiment could produce even
a local modulation of the earth's gravity sufficient to be detected.
Any object or physical system available on a laboratory scale, irrespective
of its chemical composition or microscopic structure, generates
gravitational fields of exceedingly small strength. These fields
can be detected through very sensitive instruments, but they are
typically of the order of 10^{-9} *g* or less (*g*»9.8
*m*/*s*^{2} is the field generated by the earth
at its surface).

These observations are well known and lead to the
conclusion, in full agreement with Einstein equations, that the
gravitational field generated by a very massive field is in practice
unaffected by the presence of any other body whose mass is much
smaller. Therefore, it does not seem possible that the gravitational
acceleration *g* at the earth surface can be affected, through
any human-sized apparatus, by more than approx. 1 part in a billion.

The conclusion above rests, as mentioned, upon the hypothesis that the equations of classical General Relativity are appropriate to the situation.

It is known that quantum mechanics brings in some very small corrections to the classical equations of any field, including the gravitational field. In the quantum view, the field oscillates in an approximately harmonic "potential"; these oscillations take place around a minimum value corresponding to the classical field strength.

Usually the quantum fluctuations are irrelevant on a macroscopic scale. One can show, however, that the presence in a region of space of coherent vacuum energy ("zero point energy") modifies the potential in which the gravitational field oscillates. Zero point energy is present in macroscopic systems, well above the atomic scale, which are described as a whole by a single wave function. If the zero point energy term was present uniformly in all space, it would not bring any consequence: the gravitational field of the entire space would react exactly in such a way to reset the zero of energy. Things are different, however, if the zero point energy term is present only in a well-defined small region of space; in this case it produces a localized instability (see Point 2).

A Bose condensate is formally described by a classical
field f_{0}(*x*), like a
sort of ideal fluid. We assume that the lagrangian of this field
has the standard scalar form and that external conditions define
the field density, so that f_{0}
behaves in the functional integral (or "quantum partition function")
of the system like an external field and not like an integration
variable.

At the classical level, in order to find the gravitational
effects of f_{0} we take the
first variation of the action and obtain the Einstein equations
with a small cosmological term originating from the coupling to
the fluid. This gives an extremely small correction to the vacuum
Einstein equations.

Let us check this classical result formally. The energy-momentum tensor of the condensate is

*T*_{mn}
= ¶_{m}f_{0}^{*}¶_{n}f_{0} - *g*_{mn}*L*(f_{0})
=

= ¶_{m}f_{0}^{*}¶_{n}f_{0} - *g*_{mn}
(1/2 ¶_{a}f_{0}^{*}¶_{a}f_{0} + 1/2 *m*^{2}|f_{0}|^{2})
=

=¶_{m}f_{0}^{*}¶_{n}f_{0} - *g*_{mn}
(1/2 m^{2})

The total lagrangian, including the so-called minimal coupling of the condensate with the gravitational field, is

*L* = *L*(*g*) + 8p*G
T*_{mn} h_{mn};

*T*_{mn}*h*_{mn} = *h*_{mn}¶_{m}f_{0}^{*}¶_{n}f_{0} - (Tr *h*)(1/2
m^{2})

and the corresponding Einstein equations are

d*L*/d*h*_{mn}
= *R*_{mn} - 1/2 *R g*_{mn}
+ 8p*G T*_{mn}

= *R*_{mn}
- 1/2 *R g*_{mn} + 8p*G*
(¶_{m}f_{0}^{*}¶_{n}f_{0} - 1/2 *g*_{mn}m^{2})
= 0

The last term is of the form *g*_{mn}L,
corresponding to a "local" cosmological constant. It is
also interesting to take the trace of these equations. We obtain

*R* = 8p*G*
Tr *T* = 8p*G*(-¶_{m}f_{0}^{*}¶_{m}f_{0} - 2*m*^{2}
|f_{0}|^{2})

where we see that the scalar curvature produced
by the fluid is very small, due to the small value of *G* (we
work in units *c*=1).

In conclusion, the fluid generates a gravitational field which is approximately proportional to its mass-energy-momentum density. The main factor is the mass density distribution and the forces leading to this particular distribution do not play an important role. This clearly holds, in General Relativity, for any fluid, no matter if coherent or not. Equations like those above are employed in cosmological and astrophysical models, to study the internal structure of the stars etc.

Our attitude must be different if we consider the
quantum nature of the gravitational field. In this case the field
fluctuates and its dynamics is represented by a functional integral
weighed by the factor exp(*iS*[*g*]/*h*_{Planck}).
This means that the field can assume any configuration *g*,
with a probability proportional to this factor.

Clearly the preferred configurations are those
near the stationary points of *S*, i.e. those obeying the classical
Einstein equations. We know that the action has the form *S*=òÖ{det*g*}*R*
and a stationary point is given by *h*=0, *R*=0 (flat
space). There are some configurations however, the so-called *zero
modes*, for which *h* and *R* are not zero, but the
integral vanishes and *S* is zero. Therefore these configurations
are as likely as flat space.

Why does the field usually prefer to be in the flat space configuration and not in a zero mode? Because there is in the action a small additional spacetime independent term, called the negative intrinsic cosmological term, which suppresses them, as can be seen by expanding the action to second order.

The existence of a negative intrinsic cosmological term has been demostrated by numerical simulations of Euclidean quantum gravity near equilibrium. Also independently from these simulations, we take it as one of the fundamental assumptions of our model (see Table II), and there are several indirect evidences of it.

As we saw above, expanding to second order the action of a condensate coupled to gravity one finds a positive cosmological term which cancels the intrinsic negative term and leads to an instability.

It is important to stress that in a quantum mechanical
context the quantum coherence of the fluid described by f_{0}
is essential. An incoherent fluid is not represented by a field
like f_{0}, but by an ensemble
of pointlike particles whose energy-momentum tensor is of the form

*T*_{mn}
= S_{i} ò*ds*_{i}*p*_{i,m} *p*_{i,n}d(*x*-*x*_{i})

Thus an incoherent fluid is not able to cause any instability.

Finally we observe that also an electromagnetic
field in a coherent state might play a role comparable to that of
f_{0}. This point deserves further
investigation, even though a magnitude order estimate shows that
a coherent e.m. field cannot achieve the mass-energy density present
in a Bose condensate.

In the works where the coherent coupling is discussed
(SHI, LCC, BOS) the classical field f_{0}(*x*)
is introduced as the mean value of a quantum field: f_{0}=<0|f(*x*)|0>.
This definition raises formal problems for the definition of the
state |0>, because this should be at the same time a relativistic
invariant state, and the ground state of a condensate of massive
particles.

It is therefore more convenient to work just with
the functional integral of the system, without mentioning the states.
We assume that f(*x*)=f_{0}(*x*)+f'(*x*),
with f_{0}(*x*) a *relativistic*
(this is necessary for the gravitational coupling) classical field
describing the condensate.

While f' is a quantum
field, f_{0} is an external classical
field, and not an integration variable in the functional integral.
We assume this to be a natural description of the condensate in
the present context. We shall identify better f_{0}
a posteriori; in particular, we are interested in its relation to
the density of superconducting charge carriers.

The relativistic hamiltonian, or energy density,
of this field isequal to m^{2}(*x*)
as given in eq. (1). This hamiltonian does not take into account
the external fields (mainly the magnetic field) and the boundary
conditions (normal regions of the superconductor) which let f_{0}(*x*)
take its particular value at every point *x*. It just accounts
for the mechanical energy of the fluid: the rest energy + the term
due to the 4-gradient (which reduces in practice to the spatial
gradient, because the time derivative is divided by *c*).

Since in natural units the mass of the charge carriers
is of the order of 10^{10} *cm*^{-1} and usually
the spatial variations of f_{0}
take place on a scale much larger than 10^{-10} *cm*,
we can at first disregard the gradient term. Then, consistently,
we can estimate m^{2}»*mN*»*m*^{2}*V*|f_{0}|^{2}
(compare BOS), from which we get the relation between f_{0}
and the energy density. If the coherence length of the superconductor
is very small, it may be necessary to introduce a small correction
to this relation, but the magnitude order of the total energy will
be basically unaffected, and the same holds for |f_{0}|.

In Ref. BOS we mentioned a "pinning"
of the gravitational field within the critical regions. Here we
speak instead of a "runaway" of the field. The two terms
are basically equivalent, since both refer to the behavior of the
field in a potential where the *h*=0 value is unstable and
therefore the field runs away from this value and gets pinned -
almost independently from the initial conditions - at a strength
*h*' different from zero. *h*' is determined by terms
in the potential higher than second order.

**Fig. 8 - **Double-well potential for the
field, localized to the critical regions (red).

The most typical example of this kind of potential
is the double well potential (Fig. 8). Actually, in the Ref. GAU
we used a potential of this kind in order to compute the effects
of a local pinning on the field propagation in all space. That computation
describes *exactly* the case of a scalar massless field with
local pinning. We recall that the correction to the propagator is
proportional in that case to the parameter g=x*f*^{2},
in turn proportional to m^{2}
- the local imaginary mass term.

But in the gravitational case, even though we have
a term m^{2} (the "positive
cosmological term") in the lagrangian comparable to the imaginary
mass term, it is not the value of m^{2}
which enters directly in the correction to the propagator. In other
words, g is not proportional to m^{2}.
If it was so, than the shielding effect would be irrelevant, because
m^{2} is extremely small.

We are not able, in the gravitational case, to
relate directly m^{2} (which
depends on the condensate density in the critical regions) to g
(which is evaluated experimentally, through eq. (2), from the observed
shielding strength). This is because the instability induced by
the m^{2} term in the gravitational
lagrangian is much worse than even that due to a double well potential
in the scalar case. This is because there exist gravitational modes
for which the kinetic term in the action - usually stabilizing against
local variations, as it contains a gradient squared - is not effective.

For this reason we believe that the exact value
of m^{2} is not important, as
long as it is larger than the threshold value |L|/8p*G*
and can thus trigger the instability. It follows that the computation
in Ref. GAU, exactly valid for a scalar field, represents in the
gravitational case only a useful model for a pinning of the field
following an instability, while the parameter g
must be fitted from the experimental data. As explained in Section
2b, g is the sum of the products of the
unknown runaway probabilities x_{i}
by the unknown strengths *h*_{i}^{2}.