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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}