COMMENT: from the physics e-print server at 
         http://xxx.lanl.edu/abs/gr-qc/9612022/
         Postscript is available from that URL.

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\hfill \parbox{45mm}{{UTF-391/96} \par November 1996
\par gr-qc/9612022} 

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\begin{center}
{\LARGE Possible quantum gravity effects in a charged Bose}

\smallskip

{\LARGE condensate under variable e.m.\ field.}
        
\vspace{22mm}

{\large Giovanni Modanese}
\footnote{e-mail: modanese@science.unitn.it}

\medskip

{\em I.N.F.N. -- Gruppo Collegato di Trento \par
Dipartimento di Fisica dell'Universit\`a \par
I-38050 Povo (TN) - Italy}

\medskip        

{\em and}
        
{\large John Schnurer}

\medskip

{\em Director Applied Sciences \par
Physics Engineering \par
P.O. Box CN 446, Yellow Springs, Ohio 45387-0466 - U.S.A.} 
        

\end{center}

\vspace*{10mm}

\begin{abstract}
We prove that in Euclidean quantum gravity, in the weak field 
approximation, a ``local" positive cosmological term $\mu^2(x)$ 
induces localized gravitational instabilities. Such a term can be 
produced by the coupling to an external Bose condensate. We study 
the static classical limit of the functional integral in the presence of 
the (regulated) instabilities. This model is applied to a 
phenomenological analysis of recent experimental results. 
A new demonstration experiment is described.

\medskip
\noindent 04.20.-q Classical general relativity.

\noindent 04.60.-m Quantum gravity.

\noindent 74.72.-h High-$T_c$ cuprates.

\bigskip 

\end{abstract}

The behavior of a Bose condensate -- or more specifically of a 
superconductor -- in an external gravitational field has been the 
subject of some study in the past \cite{ana}. The presence in a 
superconductor of currents flowing without any measurable resistance 
suggests that it could be used as a sensitive detection system, in 
particular for gravitational fields. The possible back-reaction of 
induced supercurrents on the gravitational field itself has been studied 
too, in analogy with the familiar treatment of the Meissner effect. As 
one can easily foresee, it turns out that the ``gravitational Meissner 
effect" is extremely weak: it was computed for instance that in a 
neutron star with density of the order of $10^{17} \ kg/m^3$ the 
London penetration depth is ca.\ 12 $km$ \cite{lano}.

The reason for the extremely weak coupling of the supercurrents to the 
classical gravitational field is very general and originates of course 
from the smallness of the coupling between gravity and the 
energy-momentum tensor of matter $T_{\mu \nu}$. One might wonder 
whether in a quantum theory of gravity -- or at least in an 
approximation of the theory for weak fields -- the Bose condensate of 
the Cooper pairs, due to its macroscopic quantum character, can play 
a more subtle role than a simple contribution to the energy-momentum 
tensor. 

In a quantum-field representation the condensate is described by a 
field with non-vanishing vacuum expectation value $\phi_0$, possibly 
depending on the spacetime coordinate $x$. It is interesting to insert 
the action of this field, suitably ``covariantized", into the functional 
integral of gravity, expand the metric tensor $g_{\mu \nu}$ in weak 
field approximation and check if  the only effect of $\phi_0(x)$ is to 
produce gravity/condensate interaction vertices proportional to 
powers of $\kappa=\sqrt{16\pi G}$. One finds that in general this is 
not the case; the quadratic part of the gravitational action is modified 
too, by receiving a negative definite contribution. It can thus be 
expected that the condensate induces localized instabilities of the 
gravitational field, in a sense which we shall precise in Section 
\ref{role}.

The present paper is based on the letter \cite{s} and originates in part 
from previous theoretical work \cite{m1} and in part from recent 
experimental results (\cite{pk,js}; see Section \ref{exp}) which show 
the possibility of an anomalous interaction, in special conditions, 
between the gravitational field and a superconductor. We have 
developed a theoretical model (Section \ref{mod}) which on the 
basis of the general results of Section \ref{role} allows to interpret in 
a consistent way the main reported experimental observations. In 
Section \ref{bal} we collect some general considerations  concerning 
the total energetic balance of the process described in Section 
\ref{mod}. In Section \ref{hyp} we analyze arguments for and against 
the hypothesis of a ``threshold density" for the condensate density and 
finally Section \ref{conc} comprises some conclusive remarks. 
Sections \ref{exp}, \ref{mod} (in part) and \ref{bal} have a less 
complex formal content than Section \ref{role} and are readable 
without a detailed knowledge of quantum field theory. 

\section{Effect of a local cosmological term in Euclidean quantum 
gravity.}
\label{role}

\subsection{Global cosmological term.}
\label{glo}

Let us consider the action of the gravitational field $g_{\mu \nu}(x)$ 
in its usual form:
\begin{equation}
S_g = \int d^4x \, \sqrt{g(x)} \left[ \frac{\Lambda}{8\pi G} -
\frac{1}{8\pi G}R(x) \right] ,
\label{azione}
\end{equation}
where $-R(x)/8\pi G$ is the Einstein term and $\Lambda/8\pi G$ is 
the cosmological term which generally can be present.

It is known that the coupling of the gravitational field with another 
field is formally obtained by ``covariantizing" the action of the latter; 
this means that the contractions of the Lorentz indices of the field must 
be effected through the metric $g_{\mu \nu}(x)$ or its inverse 
$g^{\mu \nu}(x)$ and that the ordinary derivatives are transformed 
into covariant derivatives by inserting the connection field. Moreover, 
the Minkowskian volume element $d^4x$ is replaced by $d^4x \, 
\sqrt{g(x)}$, where $g(x)$ is the determinant of the metric. The 
insertion of the factor $\sqrt{g(x)}$ into the volume element has the 
effect that any additive constant in the Lagrangian contributes to the 
cosmological term $\Lambda/8\pi G$. For instance, let us consider a 
Bose condensate described by a scalar field $\phi(x)=\phi_0 + 
\hat{\phi}(x)$, where $\phi_0$ is the vacuum expectation value and 
$m_\phi |\phi_0|^2$ represents the particles density of the ground state 
in the non-relativistic limit (compare Section \ref{hyp}). The action 
of this field in the presence of gravity is
\begin{equation}
S_\phi = \frac{1}{2} \int d^4x \, \sqrt{g(x)} \left\{ [\partial_\mu 
\hat{\phi}(x)]^* [\partial_\nu \hat{\phi}(x)] g^{\mu \nu}(x) + m^2_\phi 
|\hat{\phi}(x)|^2 + m^2_\phi [ \phi_0^* \hat{\phi}(x) + \hat{\phi}^*(x) 
\phi_0] + m^2_\phi |\phi_0|^2 \right\}.
\end{equation}
One can easily check that in the total action $(S_g+S_\phi)$ the 
contribution $\frac{1}{2} m^2_\phi |\phi_0|^2 8\pi G$ is added to the
``intrinsic" gravitational cosmological constant $\Lambda$. (Note that 
in the covariantized action above the derivatives are unchanged, since 
$\hat{\phi}(x)$ is a scalar quantity.)

The astronomical observations impose a very low limit on the total 
cosmological term present in the action of the gravitational field. The 
presently accepted limit is of the order of $|\Lambda| G < 10^{-
120}$, which means approximately for  $\Lambda$ itself $|\Lambda| 
< 10^{-54}\ cm^{-2}$ (we use natural units, in which $\hbar=c=1$ 
and thus $\Lambda$ has dimensions $cm^{-2}$, while $G$ has 
dimensions $cm^2$; $G \sim L_{Planck}^2 \sim 10^{-66}\ cm^2$). 
This absence of curvature in the universe at large scale rises a 
paradox, called ``the cosmological constant problem" (see 
\cite{wei}). In fact the Higgs fields of the standard model as well as 
the zero-point fluctuations of any quantum field including the 
gravitational field itself generate huge contributions to the 
cosmological term, which however appear to be somehow ``rescaled" 
to zero at macroscopic distances. In order to explain how this can 
occur, several quantum field theoretical models have been proposed 
\cite{gre1}. No definitive and  universally accepted solution of the 
paradox seems to be at hand yet, since that would require in fact a 
complete non-perturbative treatment of gravity which appears not 
feasible up to now. 

A model in which the large scale vanishing of the effective 
cosmological constant is reproduced in a natural way through 
numerical simulations is the Euclidean quantum gravity on the Regge 
lattice \cite{ham1}. From this model emerges a property, which could 
turn out to be more general than the model itself: if we keep the 
fundamental length $L_{Planck}$ in the theory, the vanishing of the 
effective cosmological constant $|\Lambda|$ in dependence of the 
energy scale $p$ follows a law of the form $|\Lambda|(p)\sim G^{-1} 
(L_{Planck} \, p)^\gamma$, where $\gamma$ is a critical exponent 
\cite{ham2,m2}. It is not excluded that this behavior of the effective 
cosmological constant may be observed in certain circumstances (see 
\cite{c} and Section \ref{hyp}). Furthermore, the model predicts that 
in the large distance limit $\Lambda$ goes to zero while keeping 
negative sign. Also this property has probably a more general 
character, since the weak field approximation for the gravitational 
field is ``stable" in the presence of an infinitesimal cosmological term 
with negative sign, while on the contrary it becomes unstable in the 
presence of a positive cosmological term (see Section \ref{quag}).

\subsection{Local cosmological term.}
\label{loc}

Summarizing, independently of the model there appears to exist a 
dynamical mechanism which ``rescales to zero" any contribution to 
the cosmological term and fortunately makes the gravitational field 
insensitive to any constant term in the action of other fields coupled to 
it. Nevertheless, let us go back to the previously mentioned example 
of a Bose condensate described by a scalar field $\phi(x)=\phi_0 + 
\hat{\phi}(x)$. If the vacuum expectation value $\phi_0$ is not 
constant but depends on the spacetime coordinate $x$, in the 
gravitational action $S_g$ appears a positive ``local" cosmological 
term which can have interesting consequences. Let us suppose that 
$\phi_0(x)$ is fixed by external factors and let us decompose the 
gravitational field $g_{\mu \nu}(x)$ as usual in the weak field 
approximation, that is, $g_{\mu \nu}(x)=\eta_{\mu \nu}+\kappa 
h_{\mu \nu}(x)$, where $\kappa=\sqrt{8\pi G}\sim L_{Planck}$. The 
total action of the system takes the form
\begin{equation}
  S=\int d^4x \, \sqrt{g(x) } \left\{ \left[ \frac{\Lambda}{8 \pi G} + 
\frac{1}{2} \mu^2(x) \right] -\frac{1}{8 \pi G} R(x) \right\} + S_{h 
\phi_0}+S_{\hat{\phi}} ,
\label{usu}
\end{equation}
where
\begin{eqnarray}
  \frac{1}{2} \mu^2(x) & = & \frac{1}{2}
  [\partial_\mu \phi_0^*(x)] [\partial^\mu \phi_0(x)] +
  \frac{1}{2} m^2_\phi |\phi_0(x)|^2 ; \label{isu} \\
  S_{h \phi_0} & = & \frac{1}{2} \int d^4x \, \sqrt{g(x) }
  \, \partial^\mu \phi_0^*(x) \partial^\nu \phi_0(x) \kappa
  h_{\mu \nu}(x) ; \\
  S_{\hat{\phi}} & = & \frac{1}{2} \int d^4x \, \sqrt{g(x) }
  \Bigl\{ m^2_\phi |\hat{\phi}(x)|^2 + m^2_\phi
  \left[ \phi_0^*(x) \hat{\phi}(x) + \phi_0(x) \hat{\phi}^*(x)
  \right] + \nonumber \\
  & & + \left[ \partial_\mu \hat{\phi}^*(x) 
  \partial_\nu \hat{\phi}(x) + \partial_\mu \phi_0^*(x)
  \partial_\nu \hat{\phi}(x) + \partial_\mu \hat{\phi}^*(x)
  \partial_\nu \phi_0(x) \right] g^{\mu \nu}(x) \Bigr\}.
\label{fit}
\end{eqnarray}

In the action above the terms $S_{h \phi_0}$ and $S_{\hat{\phi}}$ 
represent effects of secondary importance. The term $S_{h \phi_0}$ 
describes a process in which gravitons are produced by the ``source" 
$\phi_0(x)$. The term $S_{\hat{\phi}}$ contains the free action of the 
field $\hat{\phi}(x)$ describing the excitations of the condensate, and 
several vertices in which the graviton field $h_{\mu \nu}(x)$ and 
$\hat{\phi}(x)$ interact between themselves and possibly with the 
source. All these interactions are not of special interest here and are 
generally very weak, due to the smallness of the coupling $\kappa$. 
The relevant point (eq.s (\ref{usu}), (\ref{isu})) is that the purely 
gravitational cosmological term $\frac{\Lambda}{8 \pi G}$ receives 
a local positive contribution $\frac{1}{2}\mu^2(x)$ which depends 
on the external source $\phi_0(x)$. 

We shall call ``critical regions" the regions of spacetime in which the 
following condition is satisfied: 
\begin{equation}
\left[ \frac{\Lambda}{8 \pi G} + \frac{1}{2} \mu^2(x) \right]>0.
\label{crit}
\end{equation}
Since we know that the intrinsic cosmological term $\Lambda/8 \pi 
G$ is very small, we expect that these regions are essentially 
determined by the source term $\mu^2(x)$ and thus, through 
(\ref{isu}), by the vacuum expectation value $\phi_0(x)$. We shall 
discuss later (Section \ref{hyp}) whether there might be a competition 
between the two terms in (\ref{crit}) and therefore a ``threshold" 
effect.

\subsection{Euclidean theory for weak fields.}
\label{euc}

It is more convenient at this point to carry on our analysis in the 
Euclidean formalism, in which the Minkowski metric $\eta_{\mu 
\nu}$ is replaced by the four-dimensional Euclidean metric 
$\delta_{\mu \nu}$. This amounts to replace the time variable with an 
imaginary variable and requires that the theory behaves regularly with 
respect to a rotation of the time axis in the complex plane. For the 
familiar quantum field theories in flat spacetime this requirement is 
usually satisfied, but in the case of quantum gravity the situation is in 
general much more complicated since the metric itself belongs to the 
dynamic variables of the theory. The equivalence of the gravitational 
Euclidean theory \cite{haw} with the theory in real spacetime has not 
been proved yet, in spite of the considerable efforts in this direction 
\cite{gre2}. Anyway, such an equivalence would be mainly formal, as 
neither theory is well defined in a general sense. 

We should also mention that in general the Euclidean Einstein action 
(the term $\frac{-1}{8\pi G}\int \sqrt{g}R$ in eq.\ (\ref{azione})) is 
not bounded from below \cite{haw}. Thus it is not possible to obtain 
the vacuum state of the quantum theory (flat space) by minimizing the 
action in an elementary way, like in the usual Euclidean theories. 
Several solutions to this problem have been proposed which exploit 
the freedom in the choice of the functional integration measure, the 
stochastic regularization or the regularization through an $R^2$ term, 
effective only at very small distances
\footnote{See for instance \cite{ham1,gre2}. The $R^2$ term makes 
the theory renormalizable but non-unitary at perturbative level. 
However, it is believed that the unitarity problem should be solved by 
the complete theory.}.

Nevertheless, since in our case the gravitational field is always 
regarded as weak (small fluctuations around a flat background), it is 
not necessary in fact to use Euclidean quantum gravity in its most 
general form. We may treat it, ``{\it a la} particle physics", like a 
normal quantum field theory (or possibly an effective low-energy 
theory \cite{don}) in which the background metric is fixed and the 
analytical continuation between the Euclidean and Minkowskian case 
is well defined. According to this approximation and to the physical 
reality we shall suppose that in the absence of external sources the 
ground state of gravity is flat spacetime, at least at macroscopic scale. 
We do not specify the dynamical mechanism through which this 
ground state emerges from the complete theory, although we regard 
the mentioned non-perturbative Regge calculus simulations as 
particularly instructive in this sense. 

The quadratic part of the Euclidean Einstein action is positive-definite 
on the average (see Section \ref{qua}) and thus effectively 
stable with respect to weak fluctuations. In any case, as we shall see, 
we are not really interested here in the stability of the $R$-term in the 
action, but into that of a term of the form $\Lambda_{eff} \sqrt{g}$ 
(see eq.\ (\ref{root})). If necessary we can admit that the Euclidean 
Einstein action has been suitably modified in order to allow a correct 
analytical continuation and to make it bounded from below also 
beyond the weak field approximation (compare \cite{gre2} and 
references) and this will not affect our conclusions. 

\subsection{Quadratic part of the action in harmonic gauge.}
\label{qua}

Let us consider the Einstein action with cosmological term 
(\ref{azione}) in the approximation of a weak Euclidean field 
$g_{\mu \nu}(x)=\delta_{\mu \nu}+\kappa h_{\mu \nu}(x)$, where 
$\kappa=\sqrt{8\pi G}$. We first observe \cite{vel} that after the 
addition of a harmonic gauge-fixing the quadratic part of the action 
takes the form
\begin{equation}
  S^{(2)}_g=\int d^4x \, {h}_{\mu \nu}(x) V^{\mu \nu \alpha \beta} (-
\partial^2 -\Lambda) {h}_{\alpha \beta}(x) ,
\label{azq}
\end{equation}
where $V^{\mu \nu \alpha \beta}=\delta^{\mu \alpha}\delta^{\nu 
\beta} + \delta^{\mu \beta}\delta^{\nu \alpha} -\delta^{\mu 
\nu}\delta^{\alpha \beta}$. In momentum space (\ref{azq}) becomes
\begin{eqnarray}
S^{(2)}_g&=&\int d^4p \, \tilde{h}^*_{\mu \nu}(p) V^{\mu \nu \alpha 
\beta} (p^2 -\Lambda) \tilde{h}_{\alpha \beta}(p) \label{ooo} \\
&=& \int d^4p \, (p^2 -\Lambda) \left\{ 2{\rm Tr} [\tilde{h}^*(p) 
\tilde{h}(p)] - |{\rm Tr} \tilde{h}(p) |^2 \right\}.
\label{azqp}
\end{eqnarray}
From (\ref{ooo}) one often desumes that in this approximation a 
cosmological term amounts to a mass term for the graviton, positive if 
$\Lambda<0$ and negative (with consequent instability of the theory; 
see also \cite{c} and references therein) if $\Lambda>0$. Since in the 
presence of a condensate the sign of the total cosmological term 
depends on the coordinate $x$ (eq.\ (\ref{usu})) we might expect 
some ``local" instabilities in the critical regions (\ref{crit})
in that case.

We make now a short digression from our main argument and check the 
positivity of the quadratic part of the Euclidean Einstein action
in harmonic gauge. Note 
that since $h$ is a symmetric tensor, we can diagonalize it 
at any point before extracting its trace. The quadratic form $2{\rm 
Tr}(\tilde{h}^*\tilde{h})-|{\rm Tr}\tilde{h}|^2$ in (\ref{azqp}), 
expressed in terms of the diagonal $\tilde{h}$, namely
\begin{equation}
Q = 2 \sum_{\alpha} |\tilde{h}_{\alpha \alpha}|^2 -|\sum_{\alpha} 
\tilde{h}_{\alpha \alpha}|^2 = \sum_{\alpha} |\tilde{h}_{\alpha 
\alpha}|^2 -\sum_{\alpha \neq \beta} \tilde{h}^*_{\alpha \alpha} 
\tilde{h}_{\beta \beta}
\label{xsx} 
\end{equation}
has negative as well as positive eigenvalues. This should be 
expected, as the scalar curvature $R$ has no definite sign, and means 
that the quadratic part of the action has no minimum for $h=0$. 
Remember however that we are working in the Euclidean functional 
formulation of the theory, so the field $h$ is supposed to ``fluctuate 
thermally" with temperature $\Theta=\hbar/k_B$ (compare also eq.s 
(\ref{ciao}), (\ref{ciaobis})). Also note that having already 
diagonalized $h$ at any point, we cannot do any further linear 
transformation on it. Taking the mean value of $Q$ and assuming that 
the correlations $\langle h_{\alpha \alpha} h_{\beta \beta} \rangle$ 
vanish by isotropy for $\alpha \neq \beta$ we obtain $\langle Q 
\rangle = \sum_{\alpha} \langle |h_{\alpha \alpha}|^2 \rangle$. We 
conclude that the quadratic part of the Einstein action is stable ``on the 
average" at $h=0$. For an effective theory (compare our discussion in 
Section \ref{euc}) this should be sufficient.

\subsection{Quadratic part of $g^{1/2}$ and instabilities.}
\label{quag}

We can prove the appearance of instabilities in the presence of a 
positive cosmological term at a more general level. Before 
introducing any gauge-fixing for the gravitational field and 
disregarding the Einstein action for a moment, let us study the stability 
of the cosmological term $\frac{1}{8\pi G}\sqrt{g}$. The expansion 
of the determinant $g$ gives to first order in $\kappa$ 
\begin{equation}
g^{(1)} = \kappa {\rm Tr} {h}
\end{equation}
and to second order
\begin{equation}
  g^{(2)} = \frac{1}{2} \kappa^2 \left[ ({\rm Tr} {h} )^2 -{\rm Tr} 
({h}^2) \right].
\label{secondo}
\end{equation}
Recalling the expansion $\sqrt{1+\delta} =1+\frac{1}{2} \delta-
\frac{1}{8} \delta^2+...$ and rearranging the second order terms one 
finds
\begin{equation}
\left[ \sqrt{g} \right]^{(2)} = \frac{1}{8}\kappa^2 \left[ ({\rm Tr} 
{h} )^2 -2{\rm Tr} ({h}^2) \right].
\label{root}
\end{equation}
(By the way, we have checked in this fashion that the quadratic part of 
$\sqrt{g(x)}$ has the same algebraic structure in ${h}$ as the 
Einstein term in harmonic gauge.) 

Now we can impose the condition ${\rm Tr} h=0 \,$
\footnote{We can always impose ${\rm Tr} h=0$ in the Euclidean 
weak field approximation, since the gauge transformations have the 
form
\begin{equation}
h_{\mu \nu}(x) \to h_{\mu \nu}(x) + \partial_\mu f_\nu(x) + 
\partial_\nu f_\mu(x)
\end{equation}
with $f_\mu(x)$ arbitrary and thus the condition on the transformation 
is
\begin{equation}
\partial^\mu f_\mu(x)=-\frac{1}{2} {\rm Tr} h(x).
\end{equation}
By choosing $f_\mu(x)$ of the form $\partial_\mu F(x)$ we obtain the 
condition
\begin{equation}
\partial^2 F(x)=-\frac{1}{2} {\rm Tr} h(x),
\end{equation}
which in Euclidean spacetime is easily solved.}
and look at the stability of the total action $\frac{1}{8\pi G} \int d^4x 
\, \sqrt{g} (\Lambda_{eff}-R)$. If $\Lambda_{eff}>0$ the 
cosmological term is clearly unbounded from below with respect to 
any ``zero mode" $h(x)$ which leaves unchanged the rest of the Euclidean 
action -- that is, the Einstein term $\frac{-1}{8\pi G} \int d^4x 
\sqrt{g} R$. This requirement is satisfied by fields $h$ which are 
solutions of Einstein equations
\begin{equation}
R_{\mu \nu}(x)-\frac{1}{2} g_{\mu \nu}(x) R(x) = -8 \pi G T_{\mu 
\nu}(x)
\label{ei2}
\end{equation}
with $T_{\mu \nu}$ satisfying the condition
\begin{equation}
\int d^4x \sqrt{g(x)} \, {\rm Tr}\,  T(x) = 0
\label{nonf}
\end{equation}
(note that for solutions of (\ref{ei2}) one has $R(x)=8 \pi G \, {\rm 
Tr} \, T(x)$). 

In Minkowski space one can easily exhibit zero modes of Einstein 
action: namely gravitational waves satisfy eq.\ (\ref{ei2}) with 
$T_{\mu \nu}(x)=0$ and in particular they satisfy a linear wave
equation in the weak-field approximation. In Euclidean space the 
linearized Einstein vacuum equations take the form of four-dimensional 
Poisson equations, as can be seen most easily in harmonic gauge. 
Thus they do not admit nontrivial solutions for $T(x)=0$ everywhere.  
However condition (\ref{nonf}) can be also satisfied by energy-momentum
tensors which are not identically zero but may have negative and positive
sign, in such a way that their total integral is zero. Of course, they
do not represent any acceptable physical source, but the
corresponding solutions of (\ref{ei2}) exist anyway and are 
zero modes of the Euclidean action. One can consider, for instance, 
the static field produced by a ``mass dipole" (which eventually we
imagine as centered in a critical region), with behaviour $h\sim r^{-2}$, 
etc.

In conclusion, in the regions where $\Lambda_{eff}>0$ the zero modes
of ${h}_{\mu \nu}(x)$ tend to grow without restriction. In the case of the
interaction with a Bose condensate, such regions are the critical 
regions (\ref{crit}). Since we are considering a 
weak-field approximation, we shall assume that in fact in those 
regions the field oscillates between extremal values, with null 
average. With an expression borrowed from experimental physics, we 
might say that $h^2$ ``saturates" in those regions. The extremal 
values will be determined by some physical cut-off and are not 
relevant if we are concerned only with the average (compare Section 
\ref{mod}).

\subsection{Comparison with the classical case.}
\label{com}

The instability effect described above is of quantum nature. In 
General Relativity the consequences of a positive effective 
cosmological term $\Lambda_{eff}(x)$ are not quantitatively 
different from those of an ordinary mass-energy density $T_{00}(x)$ 
and we do not see any hint of instability in the corresponding solutions 
of classical Einstein equations. For instance,
the trace of the Einstein vacuum equations derived from the 
action (\ref{azione}) is $R(x)=\Lambda_{eff}(x)$. On the 
other hand, taking the trace of the equations without 
cosmological term but in the presence of matter we obtain as 
mentioned
\begin{equation}
R(x)=8 \pi G \, {\rm Tr} \, T(x).
\end{equation}
This means that in the regions in which $\Lambda_{eff}(x)$ or 
$T_{00}(x)$ are different from zero, the curvature radius $\rho$ of 
spacetime is of the order of $\rho \sim 1/\sqrt{\Lambda_{eff}(x)}$ or 
$\rho \sim 1/\sqrt{T_{00}(x)}$. For an ordinary density $\rho$ is 
very large, say $\rho \sim 10^{16} \ cm$ at least (see Section 
\ref{hyp}). 

One key difference between a classical ``geometrodynamical" view
of General Relativity and the quantum field theory on a flat
background lies in the interpretation of the factor $\sqrt{g}$
in the action. In order to derive the classical equations (\ref{ei2}) 
from the total action of the [gravity+matter] system one defines the 
energy-momentum tensor $T^{\mu \nu}(x)$ in such a way that a 
variation $\delta g_{\mu \nu}(x)$ of the metric affects the action as 
follows:
\begin{equation}
\delta S_{matter} = \frac{1}{2} \int d^4 x \, \sqrt{g(x)} \, T^{\mu 
\nu}(x) \delta g_{\mu \nu}(x).
\end{equation}
In the quantum theory of gravity on a flat background ``{\it a la} 
particle physics", which we are adopting, the point of view is 
different. The factor $\sqrt{g(x)}$ does not have the classical 
geometrical meaning of four-volume measure anymore. It is expanded 
in a power series of $\kappa$ and the contributions obtained in this way are 
added to the rest of the covariantized action, giving rise as we saw also to 
a negative quadratic term in $h$ (eq.\ (\ref{root})). Thus the 
instability which follows from this term for $\Lambda_{eff} > 0$ 
should be regarded as a quantum gravity effect, connected to the 
perturbative theory around a flat background.

Finally we recall that in several cosmological theories 
\cite{giap} one assumes that at some 
stage of the evolution of the universe there can be large negative 
contributions to $T^{00}$, due to the instability of certain fields or to 
the formation of a condensate. Our model does not have anything in 
common with these (essentially classical) theories.


\subsection{General functional integral for the static potential.}
\label{hei}

In the following we shall be interested in the influence of the induced 
cosmological term $\mu^2(x)$ on the gravitational interaction of two 
masses $m_1$ and $m_2$ at rest. This influence can be computed in 
principle inserting $\mu^2(x)$ in the general formula for the static 
potential in Euclidean quantum gravity \cite{m1,ham2,mv}
\begin{equation}
U(L) = \lim_{T \to \infty} -\frac{\hbar}{T} \log \frac{1}{Z} \int d[g] 
\, \exp \left\{ -\hbar^{-1} \left[   S_g + \sum_{i=1,2} m_i \int_{-
\frac{T}{2}}^{\frac{T}{2}} dt \,  \sqrt{g_{\mu \nu}[x_i(t)] 
\dot{x}_i^\mu(t) \dot{x}_i^\nu(t)}\right] \right\}
\label{ciao}
\end{equation}
where $S_g$ is the gravitational action (\ref{azione}) and $Z$ a 
normalization factor. The trajectories $x_i(t)$ of the two masses 
$m_1$ and $m_2$ are parallel with respect to the metric $g$. $L$ is 
the distance between the trajectories, corresponding to the spatial 
distance of the two masses. The interaction energy $U(L)$ of the two 
masses depends on the correlations between the values of the 
gravitational field on the ``Wilson lines" $x_1(t)$ and $x_2(t)$. This can be 
verified explicitly in the weak field approximation or through 
numerical simulations.

We can rewrite eq.\ (\ref{ciao}) in the presence of the Bose 
condensate as
\begin{eqnarray}
U[L,\mu^2(x)] &=& \lim_{T \to \infty} -\frac{\hbar}{T} \log 
\frac{1}{Z} \int d[g] \int d[\hat{\phi}] \nonumber \\ 
& & \exp \left\{ -\hbar^{-1} \left[   S[g,\hat{\phi},\phi_0] + 
\sum_{i=1,2} m_i \int_{-\frac{T}{2}}^{\frac{T}{2}} dt \,  
\sqrt{g_{\mu \nu}[x_i(t)] \dot{x}_i^\mu(t) \dot{x}_i^\nu(t)}\right] 
\right\},
\label{ciaobis}
\end{eqnarray}
where $S[g,\hat{\phi},\phi_0]$ is the total action defined in 
(\ref{usu}). We have seen that in the weak field approximation  
$h^2$ ``saturates" in the critical regions. As a consequence, the 
field correlations present in the functional integral are modified. We 
showed earlier with a simple numerical model \cite{c} that in general 
this reduces $|U|$. Also from an intuitive point of 
view it is quite clear that local constraints on the field damp its 
correlations. An explicit evaluation of the functional integral 
(\ref{ciaobis}) is however quite difficult. In Section \ref{mod} we 
shall work out a simpler model based on a classical limit. 

\medskip
Two final remarks are in order:

\begin{enumerate}

\item 
Notice that according to the general definition of the center of mass of 
a system in the presence of gravity eq.\ (\ref{ciao}) holds, more 
generally, when $x_i(t)$ represent the trajectories of the centers of 
mass of two extended bodies. This property can explain why the 
observed height dependence of the shielding effect is so weak 
(compare Section \ref{exp} and \cite{up,unni}) although the disk 
apparently should ``shield only part of the Earth". 

\item
We observe that in order to affect the quadratic part of the 
gravitational action the local cosmological term $\mu^2(x)$ must 
contain only fields, like $\phi_0(x)$, which do not belong to the 
functional integration variables. For instance, the terms containing 
$\hat{\phi}(x)$ in eq.\ (\ref{fit}) are not included in $\mu^2(x)$ (eq.\ 
(\ref{isu})), because in perturbation theory they represent simply  
interaction vertices. This remains true even if $\hat{\phi}(x)$ is 
coupled in turn to a further external source. 

This reasoning can be generalized to other fields possibly present in 
the total action. One concludes that a cosmological contribution like 
$\mu^2(x)$ can only originate from a field with non-vanishing vacuum 
expectation value. In our case the vacuum expectation value 
$\phi_0(x)$ is the result of a number of external physical factors: the 
action of the e.m.\ field (see Section \ref{exp}), the equilibrium of the 
thermodinamic potentials of the condensate in the given conditions of 
temperature, the microscopic structure of the superconducting material etc.

\end{enumerate}

\section{Experimental evidences.}
\label{exp}

\subsection{Summary.}
\label{sum}

A recent experiment \cite{pk} has shown an unexpected interaction 
between the gravitational field and a superconductor subjected to 
external e.m.\ fields. In this Section we summarize the main reported 
observations, trying to focus on the essential elements, since the 
experimental situation is quite complex. We add a few qualitative 
remarks on the possible theoretical interpretations of these 
observations according to the model with ``anomalous" coupling 
between $h(x)$ and $\phi_0(x)$ introduced in the previous Sections. 
A more quantitative application of the model is given in Section 
\ref{mod}.

The core of the experimental apparatus is a toroidal disk of diameter 
27 $cm$ made of high critical temperature (HTC) superconducting 
material. The disk is kept at a temperature below 70 $K$; it levitates 
above three electromagnets and rotates (up to ca.\ 5000 
$rpm$) due  to the action of additional lateral magnetic fields. All 
electromagnets are supplied with AC current with variable frequency. 
Within certain frequency ranges one observes a slight decrease in the 
weight of samples hung above the disk, up to a maximum ``shielding" 
value of ca.\ 1\%. A smaller effect, of the order of 0.1\% or less, is 
observed if the disk is only levitating but not rotating. 

The percentage of weight decrease is the same for samples of 
different masses and chemical compositions. One can thus describe 
the effect as a slight diminution of the gravity acceleration $g_E$ 
above the disk. The dependence of the effect on the height above the 
disk is very weak: there appears to be in practice a ``shielding 
cylinder" over the disk (compare also \cite{up,unni}), extending for 3 
meters at least. The resulting field configuration is clearly non
conservative. An horizontal force at the border of the shielding
cylinder has occasionally been observed, but it is far too small to
restore the usual ``zero circuitation" property of the static field.
No weight reduction is observed under the disk.

The disk has a composite microscopic structure: the upper part is 
treated with a thermal process which melts partially the grains of the 
HTC material, while the lower part is more granular and has a lower 
critical temperature. This double structure aims at obtaining good 
levitation properties of the disk, while leaving also a layer in which 
considerable resistive effects can arise. In general both requirements 
appear necessary for the effect to take place: (1) the disk must be 
able to support intense super-currents; (2) a granular structure with 
pinning centers must be present, such to oppose resistance to 
variations of the super-currents pattern while the disk is subjected to 
alternate e.m.\ fields.

\subsection{Interpretation.}
\label{int}

We have already stressed in our analysis \cite{s} that an 
interpretation of the reported effect in the framework of General 
Relativity, as due to repulsive post-newtonian fields produced by the 
super-currents \cite{li}, is untenable, since the magnitude order of the 
effect is far too large. The work \cite{unni} shows that the 
contribution from the super-currents to the static component $g_{00}$ 
of the post-newtonian gravitational field over the disk is not only 
much smaller than the observed effect (by several magnitude orders), 
but it is attractive like the newtonian field of the Earth. Even taking 
into account perturbative quantum corrections to the Newton potential 
one reaches the same negative conclusion. 

Our interpretative model of the experimental results \cite{s} is based 
on the ``anomalous" coupling between Bose condensate and 
gravitational field described by eq.s (\ref{usu}), (\ref{isu}). In this 
model the essential ingredient for the shielding is the presence of 
strong variations of the Cooper pairs density in the disk: {\em we assume 
that such variations produce small regions with higher density, where 
the criticality condition (\ref{crit}) is satisfied and thus an additional
boundary condition is imposed on the gravitational field}.
\footnote{Clearly the local condensate density can never exceed the 
total electronic density. The average condensate density also increases
at lower temperatures but the power transfer process becomes more
difficult in that case (compare Section \ref{ver}).}

In this view the disk's rotation in the external magnetic field plays the 
role of forcing the pattern of the super-currents and thus the local
condensate density. It is known that in general by moving a type II 
superconductor (of which the HTC are one example, that is, with a 
structure of superconducting and non-superconducting regions) in an 
external field or by applying to it an AC, one causes resistive 
phenomena, since the super-currents' pattern is unable to follow the fields 
variations. In the present case it has been in fact observed that while 
the peak shielding values are produced, the disk tends to heat up. 

This point of view -- suggested by a long analysis of the experimental 
results in search of a consistent interpretation -- allows one to regard 
the experimental conditions reported in \cite{pk} as a particular case, 
open to changes and simplifications. The levitation of the disk does 
not appear to be {\em per se} necessary, but just convenient in order 
to rotate it. In turn, the rotation in the external field aims essentially, 
as mentioned, at forcing the currents' pattern. Thus one might perhaps 
simply rotate the disk mechanically in a fixed external field, or 
rapidly vary the fields direction and strength while leaving the disk at 
rest. The latter technique has been recently employed 
with positive results (Section \ref{demo}).

Even if we stick to the relatively simple picture above, according to 
which the essential are the variations of the Cooper pairs density, 
there remains for the theory the general task of predicting such 
variations in dependence of the disk structure, of the external fields 
etc. This is clearly a very difficult task, especially for an HTC 
superconductor. It seems more likely at present that the optimization 
of the parameters mentioned above (disk structure, external fields, 
etc.) will be first approached in a semi-empirical way.

Another crucial feature of the experimental apparatus is the frequency 
spectrum of the applied e.m.\ field. Independently of the reason for 
which the external e.m.\ field was originally employed, it is clear in 
our opinion that it plays a fundamental role in supplying the energy 
necessary for the ``absorption" of the gravitational field in the critical 
regions. A simple model which describes this mechanism is presented 
in Sections \ref{mod}, \ref{bal}. Experimentally one observes 
\cite{pk} that the maximum shielding value is obtained when the coils 
are supplied with high frequency current (of the order of 10 $MHz$).

In general, the power transfer to the disk has necessarily a limited 
efficiency. This represents one of the most serious problems to 
overcome in order to obtain the shielding effect in stable form, 
especially for heavy samples, without using an excessive amount of 
refrigerating fluid to avoid the heating of the disk and the ensuing loss 
of its superconducting properties. One should also remind that the 
maximum shielding value was observed in conditions close to the 
resistive transition. This makes clear why it is important to use a disk 
made of HTC material: in a low-temperature superconductor, 
admitted one can reach the critical density conditions, the specific 
heat is probably too small to maintain them in a stable way and to 
allow any power transfer.

\subsection{A new demonstration experiment.}
\label{demo}

One of us (J.S.) has recently succeded \cite{js} in reproducing the weak 
gravitational shielding effect for a short time interval (up to 5 seconds). 
The experimental setup was designed in such a way to eliminate as far as
possible any non-gravitational disturbance and to show a precise 
temporal correspondence between actions taken on the HTC disk and the
weight reduction of the samples. Although the observed weight reduction
was quite large (of the order of 5\%) this experiment should be regarded
just as a demonstration experiment. In fact, many actions had to be taken
literally by hand, the samples employed were in all cases very light
and the short duration of the effect did not allow any precise spatial
mapping of the field. 

It is very remarkable that in this experiment the effect was obtained 
without subjecting the disk to any rotation. This supports our 
interpretation of the effect and rises hopes that the original experimental 
setup of Podkletnov and co-workers could be substantially simplified.
However, at the present there is no evidence that the effect can be
obtained in stable form without rotation
\footnote{According to public releases, the NASA group in Huntsville,
Alabama, is cloning Podkletnov's experiment. This is a difficult task,
especially for the sophisticated technology involved in the construction
of the large HTC disk and in the control of its rotation. We are also aware,
though still at un-official level, of other groups working at the experiment
with smaller disks.}.
Also, it should be mentioned that similar temporary effects were observed by 
Podkletnov et al.\ since the first measurements, too \cite{pk}.
                             
The experimental setup consists of (a) a hexagon-shaped YBCO 1" (2.5 $cm$)
superconducting disk, 6 $mm$ thick; (b) a magnetic field generator 
producing a 600 $gauss$/60 $Hz$ e.m.\ field; (c) a beam balance with 
suspended sample.
          
The beam is made of bamboo, without any metal part, coming to a point on 
one end (24.6 $cm$ long, weight 1.865 $g$). The sample is made as follows. 
A cardboard rectangle (16 $mm$ by 10 $mm$ by 0.13 $mm$) is suspended from the 
balance with 2.8 $cm$ of cotton string. A polystyrene ``pan" (7.2 $cm$ by 
8.7 $cm$ by 1.7 $mm$) is attached with paper masking tape to the cardboard 
rectangle. The total sample assembly (with string, cardboard, tape) weighs 
1.650 $g$. 
   
The balance is suspended from the end of a 150 $cm$ wood crossbeam by ca.\ 
30 $cm$ of monofilament fishing line (8 lb.\ test) attached to the balance's 
center of mass (5.5 $cm$ from the end where the sample is attached). The 
other end of the crossbeam is firmly anchored by a heavy steel tripod. 
Thermal and e.m.\ isolation is provided by a glass plate (15 $cm$ by 30 $cm$, 
0.7 $cm$ thick) 
with a brass screen attachment. This plate-and-brass-screen assembly is 
held about 4.5 $cm$ below the sample by a ``3-finger" ring stand clamp.
A straightsided, 6" diameter, 10" deep dewar with 3-4" of liquid nitrogen 
is used to cool the superconducting disk below its critical temperature, 
and is removed from the experiment area before the trial.
 
The experimental procedure comprises the following steps.

\begin{enumerate}

\item The YBCO superconductor is placed in a liquid nitrogen bath and allowed 
    to come to liquid nitrogen temperature (as indicated when the boiling of 
    the liquid nitrogen ceases). The superconductor will remain below its 
    critical temperature (about 90 $K$) for the duration of the trial (less 
    than 20 seconds).

\item The disk is then removed from the bath and placed on a strong NdFeB
    magnet to induce a supercurrent. The Meissner effect is counteracted
    by a wooden stick. The superconducting disk has a cotton string attached 
    to it to assist handling.

\item The disk and wooden stick assembly is placed on the AC field generator, 
    about 33 $cm$ below the isolation plate and about 40 $cm$ below the 
    sample. 
    The AC field generator is then cycled for ca.\ 5 seconds with 0.75 $sec$ 
    equal-time on/off pulses. Prior to a run the sample is centered to be 
    over the middle of where the disk will finally be, on the AC field 
    generator. The idea is that the ``column" of modified gravity has to hit 
    the sample somewhere as the disk is only 1 inch in diameter and the 
    sample is much larger.

\end{enumerate}

One observes that while AC current is flowing through the generator 
the balance pointer dips 2.1 $mm$ downward. When the AC field generator
is pulsed with no superconductor present, there is no measurable pointer 
deflection. Also air flows do not cause any measurable deflection. The
whole procedure is well reproducible.

The weight difference required to raise the sample by 2.1 $mm$ was then found 
to be 0.089 $g$. This was measured taking advantage of the fact that the 
suspension wire produces a small torque on the balance beam toward the 
equilibrium position: the balance pointer was found to raise 2.1 $mm$ upward 
when a weight of 0.089 $g$ was placed above the sample.
 
An improved version of the experiment is being developed. Details will
appear elsewhere.


\section{The modified field to lowest order.}
\label{mod}

\subsection{Static classical limit of the functional integral.}
\label{sta}

In this Section we study in a suitable approximation the consequences 
of the anomalous coupling between the gravitational field and the 
Bose condensate taking place in the critical regions (i.e., where 
condition (\ref{crit}) is satisfied). We aim at verifying in this way 
whether from our theoretical hypoteses follow plausible 
phenomenological consequences and at composing a picture of the 
shielding phenomenon which may help in understanding various 
aspects: in which sense a constant gravitational field is slightly 
``absorbed" in the disk and turns out to be weaker above it; if the field 
modified in this way is conservative; which is the global energy 
balance of the process etc.

We have seen that in the presence of strong variable e.m.\ fields, in 
the superconducting disk small regions can appear in which the 
Cooper pairs density is particularly high. We shall discuss later 
whether there is necessarily a ``threshold" density which must be 
exceeded for the  anomalous coupling to be possible. The distribution 
of these regions varies with time, as the superconductor moves in the 
external e.m.\ field, but for our reasoning we can consider the 
situation at a fixed instant. We can also suppose that the number of 
singular regions for unit volume remains almost constant until the 
external conditions are modified (rotation frequency, field 
parameters, temperature). The size of the critical regions is of the 
order of the coherence length $\xi$, that is, of the scale at which 
typically the variations of the order parameter take place in the given 
material.

As explained in Section \ref{role}, let us suppose that inside the 
singular regions the gravitational field is ``forced" by the interaction 
with the condensate to oscillate around zero. We want to see how this 
can influence a pre-existing field configuration. Let us then consider, 
as done in Section \ref{hei}, the functional integral of Euclidean 
gravity and add to the action a static cosmological source term 
$\mu^2({\bf x})$. Let us also add a static source $T^{\mu \nu}({\bf 
x})$ which generates a constant background field of strength $g_E$. 
We can write the averaged Euclidean field in weak field 
approximation as follows:
\begin{equation}
\langle 0 | h_{\mu \nu}({\bf x}) | 0 \rangle = \frac{1}{Z} \int d[h] \, 
h_{\mu \nu}(x) \exp \left\{ -\hbar^{-1} \left[ \int d^4x \, \sqrt{g} \left( 
\frac{\Lambda}{8\pi G} + \frac{1}{2} \mu^2 -\frac{R}{8\pi G} + 
\kappa h_{\mu \nu} T^{\mu \nu} \right) \right] \right\}.
\label{media}
\end{equation}
where $g$ and $R$ must be expressed in terms of $h_{\mu \nu}$. The 
functional average is dominated by the functions $h$ which minimize 
the action. But we know from the analysis of Section \ref{role} that 
these functions are those which solve the field equation in the 
presence of the source $T^{\mu \nu}$ and of the constraints 
corresponding to the ``saturation" of $|h|^2$ in the critical regions 
(\ref{crit}). Thus $h$ is zero on the average in the critical regions. 
Since $\mu^2$ and $T^{\mu \nu}$ do not depend on time, we expect 
that the average field does not depend on time either. It follows that the 
analytical continuation to imaginary time necessary to translate back 
the expectation value into Minkowski space is trivial.

\subsection{Constrained field equation and its solution.}
\label{con}


In order to guess the solution of the static field equation in the 
presence of the given source and constraints, let us now follow an 
analogy with an electrostatic field. In that case the regions in which 
the electric field and potential are forced to zero could be realized by 
very small perfect conducting grounded spheres or plaquettes.

We must check that the gravitational field equation in the case we are 
considering is analogous to that of the electrostatic field. We first 
write the equation of the trajectory $x^\alpha(\tau)$  of a particle in 
free fall in a given gravitational field $g_{\mu \nu}(x)$, called the 
geodesic equation:
\begin{equation}
  \frac{d^2x^\alpha(\tau)}{d\tau^2}+\Gamma^\alpha_{\mu \nu} 
[x(\tau)] \frac{dx^\mu(\tau)}{d\tau} \frac{dx^\nu(\tau)}{d\tau} = 0,
\label{a1}
\end{equation}
where $\tau$ is the proper time, with differential $d\tau=\sqrt{dx^\mu 
dx^\nu g_{\mu \nu}}$ and $\Gamma^\alpha_{\mu \nu}$ is the 
Christoffel connection
\begin{equation}
  \Gamma^\alpha_{\mu \nu}=\frac{1}{2} g^{\alpha \beta} \left( 
\partial_\mu g_{\beta \nu} + \partial_\nu g_{\beta \mu}-\partial_\beta 
g_{\mu \nu} \right).
\label{a2}
\end{equation}
In this equation and in the following we omit for simplicity the 
arguments of the fields.

We now specialize to the static case, supposing the particle initially 
at rest, and compute its acceleration. For the spatial components $x^i$ 
eq.\ (\ref{a1}) takes the form
\begin{equation}
  \frac{d^2 x^i}{d\tau^2}+\Gamma^i_{00} \left( \frac{dx^0}{d\tau} 
\right)^2 = 0.
\label{a3}
\end{equation}
In the following we work in the weak field approximation and 
consider only terms of lowest order in $\kappa$. For the connection 
(\ref{a2}) we have
\begin{equation}
  \Gamma^{(1) \, \alpha}_{\mu \nu}=\frac{1}{2} \kappa 
\delta^{\alpha \beta} \left( \partial_\mu h_{\beta \nu} + \partial_\nu 
h_{\beta \mu}-\partial_\beta h_{\mu \nu} \right)
\label{a4}
\end{equation}
and in particular, in the static case 
\begin{equation}
  \Gamma^{(1) \, i}_{00}= -\frac{1}{2} \kappa \partial^i h_{00} .
\label{a5}
\end{equation}

In eq.\ (\ref{a3}) the proper time $\tau$ differs from the coordinate time
only for terms of order $\kappa$, thus to lowest order the acceleration 
is given by
\begin{equation}
  \frac{d^2 x^i}{dt^2}=G^i, \qquad {\rm with \ } G^i= -\Gamma^{(1) \, i}_{00}
  =\frac{1}{2} \kappa \partial^i h_{00}.
\label{a6}
\end{equation}

On the other hand the field must satisfy Einstein vacuum equations 
$R_{\mu \nu}=0$. Disregarding the quadratic terms in the connection 
we can write them as
\begin{equation}
 \partial_\nu \Gamma^\rho_{\mu \rho} -\partial_\rho 
\Gamma^\rho_{\mu \nu} = 0.
\label{a7}
\end{equation}
For the 00 component in the static case one has
\begin{equation}
  \partial_i \Gamma^i_{00}=0.
\label{a8}
\end{equation}

In conclusion, the gravitational acceleration of a particle at rest is 
given to lowest order in $\kappa$ by a vectorial field $G^i$ with zero 
divergence in the vacuum which is the gradient of the potential $-
\frac{1}{2} \kappa h_{00}$. This justifies, in that approximation, an 
analogy with the electrostatic field and the electrostatic potential.

Thus we can represent also the gravitational field in this case with 
field lines. Still following the electrostatic analogy (justified by the 
fact that the field equations and the boundary  conditions for the two  
fields are the same), we observe that when a field line meets a 
singular region it is intercepted. The field lines are just conventional 
objects and the number of lines which cross the unity surface is 
proportional to the field intensity, the proportionality constant being 
arbitrary. Thus the final effect must not depend on such constant, as 
can be easily verified. 

The fraction of intercepted field lines, corresponding to the shielding 
factor $\alpha$, is approximately equal to the ratio between the total 
cross section of the singular regions and the area of the disk. Let us 
suppose that in the presence of shielding the gravity acceleration over 
the disk is $g_E(1-\alpha)$; we have in this simplified model 
\begin{equation}
\alpha \sim \frac{N \sigma}{S_{disk}}=n \sigma ,
\label{reg}
\end{equation}
where $\sigma$ is the average cross section of a singular region, $N$ 
is the total number of singular regions in the disk and $n$ is the 
number of singular regions for unit disk surface.


\subsection{Discussion.}
\label{disc}

The average value for $h_{\mu \nu}({\bf x})$ obtained in the previous
Section is unsatisfactory for two reasons: (1) it does not account for the
observed non conservative character of the field in the presence of
shielding (compare \ref{sum}); (2) it does not reproduce the observed
null effect below the disk. 

In other words, while the observations indicate that there is a kind of 
``absorption" of the gravitational field in the disk, with a long 
cylindrical shielding region above it and no effect below, the field 
solution mentioned above has the typical features of an electric shielding: 
namely, a grounded conducting plaquette placed across a constant 
electrostatic field projects a ``+/- shadow" in the field with a 
length of the same order of its width. (In the electric case, this is 
due to the superposition of the constant field and of the field produced 
by the electric charges induced on the plaquette.) 

In fact we cannot be sure that averaging $h$ as done above is correct.
We know that in the quantum theory the gravitational force between two
masses at rest is given in principle by eq.\ (\ref{ciaobis}) but that
equation does not give us enough information to compute the average
field, nor it ensures that in the quantum case an average field is well 
defined at all. For instance, we would not be able to predict through
eq.\ (\ref{ciaobis}) if the gravitational red-shift is affected
by the shielding. (This would actually be an interesting experimental
check.)

Thus an important task for the theory would be that of
evaluating eq.\ (\ref{ciaobis}) in a suitable approximation in order
to check if the resulting {\it effective} field corresponds to the 
observed configuration. At present we shall limit ourselves to the following
consideration. We observe that in practice only the gravitational
acceleration of the samples is measured in the experiment, i.e., the 
connection $\Gamma^{(1) \, i}_{00}$ (compare (\ref{a6}). We can admit 
that due to the quantum instability effect $\Gamma$ vanishes in the
critical regions. This perturbs slightly the pre-existing field
configuration and produces a cylindrical shadow as observed. One
can easily verify that a $\Gamma$-configuration of this kind still
satisfies eq.\ (\ref{a8}) outside the disk and thus Einstein equations to 
lowest order. The energetic balance is ensured by the external 
non-gravitational
source which constrains $\Gamma$ (compare also the next Section).


\section{Energetic balance.}
\label{bal}

After having introduced in the preceding Section a model which 
describes in an approximate way the variation of the gravitational 
field in the presence of the disk, it is necessary now to discuss some 
issues of elementary character, but important from the practical point 
of view, concerning the overall energetic balance of the shielding 
process.

In general one will have to supply energy in order to reduce the 
weight of an object, because the potential gravitational energy of the 
object has negative sign and is smaller, in absolute value, in the 
presence of shielding. Nevertheless, since the field is not 
conservative, it is certainly wrong to compute the 
difference in the potential energy of an object between the interior and 
the exterior of the shielding cone by evaluating naively the difference 
(which turns out to be huge) between an hypothetical ``internal potential" 
\begin{equation}
U=-\frac{GMM_E(1-\alpha)}{R_E}=-Mg_E R_E(1-\alpha),
\end{equation}
where $M$ is the mass of the object, $R_E$ the Earth radius and 
$M_E$ the Earth mass, and an ``external potential" $U=-Mg_E R_E$.

Moreover, the gravitational fields with which we are 
most familiar, being produced by very large masses, are relatively 
insensitive to the presence of light test bodies and thus it makes sense 
in that case to speak of a field in the usual meaning: while a body falls 
down, we do not usually need to worry about its reaction on the Earth. 
On the contrary, in the present case the interaction between the 
shielded object and the external source (that is, the system [Bose 
condensate+external e.m.\ field]) which by fixing the constraints on 
the gravitational potential $h$ produces the shielding, is very 
important.
\footnote{Because of this, also considerations involving the energy 
density of the gravitational field, which can be properly defined for 
weak fields, are not helpful in the present case.}

Let us then ask one ``provocative" question, suggested by the 
experimental reality: if the superconducting disk is in a room and the 
shielding effect extends up to the ceiling, should we expect that the 
disk and all the shielding apparatus feel a back reaction? (And 
possibly an even stronger one if the ceiling is quite thick or if there 
are more floors above?) The most reasonable answer is, that since the 
ceiling is very rigid, the experimental apparatus is not able to exert 
any work on it and thus does not feel its presence.

To further clarify this point, suppose that we hang over the 
superconducting disk, before the shielding is produced, a spring of 
elastic constant $k$, holding a body of mass $M$ at rest. Then we 
operate on the disk with proper e.m.\ fields and produce the shielding 
effect with factor $\alpha$, that means, the gravity acceleration over 
the disk becomes $g_E(1-\alpha)$. If the shielding effect is obtained 
quickly, the mass will begin to oscillate, otherwise it will rise by an 
height $\Delta x=\alpha g_E M k^{-1}$, while remaining in equilibrium. 
In any case, since for a harmonic oscillator in motion  the kinetic 
energy and the potential energy have the same mean value, it is 
legitimate to conclude that the shielding apparatus has done on the 
system [mass+spring] a work of the order of 
\begin{equation}
  \Delta E \sim k (\Delta x)^2 \sim (\alpha g_E M)^2 k^{-1}.
\label{aas}
\end{equation}
This example shows that the work exerted by the apparatus on a 
sample in order to ``shield it" will depend in general of the response 
of the sample itself, being larger when such response is large itself.

At this point we can estimate how much energy is needed in this case to bring 
over the critical density one region of the condensate of cross section 
$\sigma$ (compare eq.\ (\ref{reg})). If $\sigma_{sample}$ is the 
projection of the sample on the disk, this energy is given by
\begin{equation}
  \Delta E_\sigma = \frac{\sigma}{\sigma_{sample}} \Delta E \sim 
\frac{\sigma}{\sigma_{sample}} (\alpha g_E M)^2 k^{-1}.
\label{ela}
\end{equation}
This energy must be supplied by the external variable e.m.\ field. 

In conclusion, we must expect in general an interaction between the partially
shielded samples and the shielding apparatus. The energy needed to 
shield a sample depends on the mass of the sample itself and on the 
way it is constrained to move. In particular, we deduce from eq.\ 
(\ref{aas}) that if we want to detect the shielding effect by measuring 
the deformation of a spring, in order to do this with the smallest 
influence on the shielding apparatus we should use, as far as allowed 
by the sensitivity of the transducer, a spring with high rigidity 
coefficient $k$.

\section{The ``threshold" hypothesis.}
\label{hyp}

\subsection{Estimate for $\mu^2(x)$.}
\label{est}

A local cosmological term can induce gravitational instabilities in 
those regions where its total sign is positive. We have shown that the 
contribution of a Bose condensate to the cosmological term is positive 
(in Euclidean spacetime) and equal to $\frac{1}{2} \mu^2(x) = 
\frac{1}{2} [\partial_\mu \phi_0^*(x)] [\partial^\mu \phi_0(x)] + 
\frac{1}{2} m^2_\phi |\phi_0(x)|^2$ (see eq.s (\ref{usu}), 
(\ref{isu})). It is important to give a numerical estimate of the 
magnitude order of this contribution in the case of a superconductor.

To this end we recall that the Hamiltonian of a scalar field $\phi$ of 
mass $m_\phi$ is given by
\begin{equation}
H=\frac{1}{2} \int d^3x \left\{ \left| \frac{\partial \phi(x)}{\partial t} 
\right|^2 + \sum_{i=1}^3  \left| \frac{\partial \phi(x)}{\partial x^i} 
\right|^2 + m_\phi^2 |\phi(x)|^2 \right\}.
\label{hami}
\end{equation}
In our case $\phi$ describes a system with a condensate and its 
vacuum expectation value is $\langle 0 | \phi(x) | 0 \rangle=\phi_0(x)$. 
We then have
\begin{equation}
\langle 0 | H | 0 \rangle = \frac{1}{2} \int d^3x \mu^2(x).
\end{equation}
In the non-relativistic limit, appropriate in our case, the energy of the 
ground state is essentially given by ${\cal N}Vm_\phi$, where 
$m_\phi$ is the mass of a Cooper pair (of the order of the electron 
mass; in natural units $m_\phi \sim 10^{10} \ cm^{-1}$), $V$ is a 
normalization volume and ${\cal N}$ is the number of Cooper pairs 
for unit volume. Assuming ${\cal N}\sim 10^{20} \ cm^{-3}$ at least 
we obtain
\begin{equation}
\mu^2 \sim {\cal N} m_\phi > 10^{30} \ cm^{-4} \qquad {\rm (In \ a \ 
superconductor.)}
\label{mu}
\end{equation} 
We also find in this limit $|\phi_0| \sim {\cal N}/\sqrt{m_\phi}$.

As we saw in Section \ref{role}, a typical upper limit on the intrinsic 
cosmological constant observed at astronomical scale is $|\Lambda|G 
< 10^{-120}$, which means $|\Lambda|/8\pi G<10^{12} \ cm^{-4}$. 
This small value, compared with the above estimate for $\mu^2(x)$, 
supports our hypothesis that the total cosmological term can assume 
positive values in the superconductor and the criticality condition can 
be satisfied in some regions. But in fact the positive contribution of 
the condensate is so large that one could expect the formation of 
gravitational instabilities in any superconductor, subjected to external 
e.m.\ fields or not -- a conclusion which contrasts with the 
observations. 

We wonder if the value of $\Lambda/8\pi G$ at small scale could be 
larger than that observed at astronomical scale and negative in sign, in 
such a way to represent a ``threshold" of the order of $\sim 10^{30} \ 
cm^{-4}$ for anomalous gravitational couplings. As we recalled in 
Section \ref{role}, a negative intrinsic cosmological constant can be 
present in models of quantum gravity containing a fundamental length. 
With a magnitude as mentioned above, it would not affect any other 
known physical process.

\subsection{Threshold versus power transfer efficiency.}
\label{ver}

In our opinion the hypothesis of a threshold accounts quite well for the 
features of the observed effect and in fact we have implicitly accepted 
it in several qualitative points of our analysis, especially in Section 
\ref{exp}. As we pointed out throughout the paper, all evidences 
show that a proper ``driving" and forcing of the supercurrents in the 
disk and thus of the Cooper pairs density are essential in order to 
obtain the shielding effect and to improve it. 

However, the available experimental data are not sufficient yet to 
decide whether the hypothesis of a threshold for the Cooper pairs 
density is necessary, or it is only a helpful schematic representation. 
In fact, as we stressed in Section \ref{bal}, a global energy balance 
must be respected in the shielding process. This energetic requirement 
might be very important in determining the critical regions.

Let us consider for instance another system in which a Bose 
condensate, described by a macroscopic wavefunction, is present: 
superfluid helium. From eq.\ (\ref{mu}) we see that in that case 
$\mu^2$ is ca.\ $10^3$ times larger than in an electronic Bose 
condensate. But superfluid helium does not show, according to 
common knowledge (although specific data are not available), any 
anomalous gravitational effects. To explain this we can observe that 
unlike an electronic condensate, superfluid helium is neutral and thus 
cannot absorb energy from an external e.m.\ field. Moreover, its 
specific heat is so low, that in general any power transfer process 
would be severely constrained. It is thus possible in our opinion that 
although its density is much larger than that of an electronic 
condensate, superfluid helium does not cause any appreciable 
shielding effect, since a modification of the field $h_{\mu \nu}(x)$ as 
described in Section \ref{mod}, with ensuing reduction of the 
samples' weight, would not be sustained energetically. In other words, 
the local ``saturation" of the field $h_{\mu \nu}(x)$ represents an 
energetically less favoured state and in the case of superfluid helium 
there is no suitable external energy source to allow the transition to 
this state.

Summarizing, the following reasoning holds, at least qualitatively, and 
might be applied to the next available experimental data. We have 
seen that the shielding phenomenon involves a power transfer 
process, which in general can be more or less efficient. There are two 
limiting cases:

\begin{enumerate}

\item
 The power transfer is very efficient. Then the shielding factor 
$\alpha$ is fixed by the number of critical regions and by their 
average cross section (compare eq.\ (\ref{reg})), independently of the 
energy transferred to the samples.

\item
 The power transfer is inefficient. In this case the shielding factor 
$\alpha$ can depend on the energy transferred to the samples. Thus 
$\alpha$ can depend on the mass of the samples. For samples with the 
same mass, it can depend on their cross section and possibly on the 
``rigidity" parameter $k$ (compare eq.\ (\ref{ela})).

\end{enumerate}

Suppose now to be in the limiting case (1), as experimentally it 
appears to be the case, at least for samples whose mass does not 
exceed $\sim 100 \ g$. If in these conditions the shielding factor 
$\alpha$ depends strongly on the rotation speed of the disk, on the 
applied e.m.\ field and in general on those factors which imply the 
generation of density variations in the condensate, this means that the 
number of critical regions and their average cross section depend 
strongly on such variations. In turn, the latter means that the existence 
of a threshold is very likely.

\section{Concluding remarks.}
\label{conc}

The analysis of Section \ref{role} allows us to conclude that there is 
broad evidence for instability of the quadratic part of the Euclidean 
gravitational action in the presence of a Bose condensate. We recall 
our starting hypotheses:

\begin{enumerate}

\item
Validity of the Euclidean formalism in the context of weak-field 
approximation was assumed. As discussed in Section \ref{euc} there 
are no reasons to doubt of such validity in our case.

\item 
We admitted (compare also Section \ref{hei}) that the vacuum 
expectation value $\phi_0(x)$ of the bosonic field 
is determined by external factors: e.m.\ 
field, temperature, microscopic structure of the material etc. From a 
phenomenological point of view this approach is completely justified 
and allows to divide the problem in two parts: the dynamics of the 
source $\phi_0(x)$ and the effects of the source on the gravitational 
field. This approximation is not adequate when the back-reaction of 
the gravitational field on the source cannot be disregarded (compare 
Section \ref{bal}).

\end{enumerate}

The description of the effects of the instability in terms of a classical 
``modified field" (Section \ref{mod}) is physically helpful, even though it 
leads to only partially correct consequences. 

In general, in the energetic balance for heavy samples one should keep 
into account the back-reaction mentioned above.

The existence of a critical threshold for the value of the induced 
cosmological term $\mu^2(x)$ is theoretically appealing and in reasonable 
agreement with the experimental observations.

\subsection{Acknowledgments.}

We are very grateful to E. Podkletnov for advice and encouragement.
We are very much indebted to J.R.\ Gaines, 
Vice President of Superconductive Components, Columbus, Ohio, USA,
who supplied the HTC disk.
We would like to thank G.\ Fiore, N.\ Oberhofer and B.\ Spires for invaluable 
advice and assistance. 


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\end{document}