Gravitational Anomalies by HTC superconductors: a 1999 Theoretical Status Report. Part 3

G. Modanese

The Gravity Society -


(5) The threshold problem.

We mentioned earlier the condition defining a critical region in the condensate of the superconducting carriers: a certain function m2(x) of the condensate density and gradient must be larger than the "natural" cosmological term L/8pG. Therefore this term represents a threshold value for the density.

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

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

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

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

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

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


(1) Discussion - Relevance and innovation of the effect. Impossibility of an explanation within General Relativity.

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

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

Given the mass and proximity of the earth, it would seem impossible that any laboratory experiment could produce even a local modulation of the earth's gravity sufficient to be detected. Any object or physical system available on a laboratory scale, irrespective of its chemical composition or microscopic structure, generates gravitational fields of exceedingly small strength. These fields can be detected through very sensitive instruments, but they are typically of the order of 10-9 g or less (g9.8 m/s2 is the field generated by the earth at its surface).

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


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

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

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


(2a) Discussion - Coherent coupling between a Bose condensate and the gravitational field.

Quantum nature of the effect. Differences with the classical case.

A Bose condensate is formally described by a classical field f0(x), like a sort of ideal fluid. We assume that the lagrangian of this field has the standard scalar form and that external conditions define the field density, so that f0 behaves in the functional integral (or "quantum partition function") of the system like an external field and not like an integration variable.

At the classical level, in order to find the gravitational effects of f0 we take the first variation of the action and obtain the Einstein equations with a small cosmological term originating from the coupling to the fluid. This gives an extremely small correction to the vacuum Einstein equations.

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

Tmn = mf0*nf0 - gmnL(f0) =

= mf0*nf0 - gmn (1/2 af0*af0 + 1/2 m2|f0|2) =

=mf0*nf0 - gmn (1/2 m2)

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

L = L(g) + 8pG Tmn hmn;

Tmnhmn = hmnmf0*nf0 - (Tr h)(1/2 m2)

and the corresponding Einstein equations are

dL/dhmn = Rmn - 1/2 R gmn + 8pG Tmn

= Rmn - 1/2 R gmn + 8pG (mf0*nf0 - 1/2 gmnm2) = 0

The last term is of the form gmnL, corresponding to a "local" cosmological constant. It is also interesting to take the trace of these equations. We obtain

R = 8pG Tr T = 8pG(-mf0*mf0 - 2m2 |f0|2)

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

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


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

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

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

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

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


It is important to stress that in a quantum mechanical context the quantum coherence of the fluid described by f0 is essential. An incoherent fluid is not represented by a field like f0, but by an ensemble of pointlike particles whose energy-momentum tensor is of the form

Tmn = Si dsipi,m pi,nd(x-xi)

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

Finally we observe that also an electromagnetic field in a coherent state might play a role comparable to that of f0. This point deserves further investigation, even though a magnitude order estimate shows that a coherent e.m. field cannot achieve the mass-energy density present in a Bose condensate.


Meaning of f0.

In the works where the coherent coupling is discussed (SHI, LCC, BOS) the classical field f0(x) is introduced as the mean value of a quantum field: f0=<0|f(x)|0>. This definition raises formal problems for the definition of the state |0>, because this should be at the same time a relativistic invariant state, and the ground state of a condensate of massive particles.

It is therefore more convenient to work just with the functional integral of the system, without mentioning the states. We assume that f(x)=f0(x)+f'(x), with f0(x) a relativistic (this is necessary for the gravitational coupling) classical field describing the condensate.

While f' is a quantum field, f0 is an external classical field, and not an integration variable in the functional integral. We assume this to be a natural description of the condensate in the present context. We shall identify better f0 a posteriori; in particular, we are interested in its relation to the density of superconducting charge carriers.

The relativistic hamiltonian, or energy density, of this field isequal to m2(x) as given in eq. (1). This hamiltonian does not take into account the external fields (mainly the magnetic field) and the boundary conditions (normal regions of the superconductor) which let f0(x) take its particular value at every point x. It just accounts for the mechanical energy of the fluid: the rest energy + the term due to the 4-gradient (which reduces in practice to the spatial gradient, because the time derivative is divided by c).

Since in natural units the mass of the charge carriers is of the order of 1010 cm-1 and usually the spatial variations of f0 take place on a scale much larger than 10-10 cm, we can at first disregard the gradient term. Then, consistently, we can estimate m2mNm2V|f0|2 (compare BOS), from which we get the relation between f0 and the energy density. If the coherence length of the superconductor is very small, it may be necessary to introduce a small correction to this relation, but the magnitude order of the total energy will be basically unaffected, and the same holds for |f0|.



(2b) Discussion - Induced instability and "runaway" of zero-modes.

"Pinning" or "runaway" of the field?

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



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


The most typical example of this kind of potential is the double well potential (Fig. 8). Actually, in the Ref. GAU we used a potential of this kind in order to compute the effects of a local pinning on the field propagation in all space. That computation describes exactly the case of a scalar massless field with local pinning. We recall that the correction to the propagator is proportional in that case to the parameter g=xf2, in turn proportional to m2 - the local imaginary mass term.

But in the gravitational case, even though we have a term m2 (the "positive cosmological term") in the lagrangian comparable to the imaginary mass term, it is not the value of m2 which enters directly in the correction to the propagator. In other words, g is not proportional to m2. If it was so, than the shielding effect would be irrelevant, because m2 is extremely small.

We are not able, in the gravitational case, to relate directly m2 (which depends on the condensate density in the critical regions) to g (which is evaluated experimentally, through eq. (2), from the observed shielding strength). This is because the instability induced by the m2 term in the gravitational lagrangian is much worse than even that due to a double well potential in the scalar case. This is because there exist gravitational modes for which the kinetic term in the action - usually stabilizing against local variations, as it contains a gradient squared - is not effective.

For this reason we believe that the exact value of m2 is not important, as long as it is larger than the threshold value |L|/8pG and can thus trigger the instability. It follows that the computation in Ref. GAU, exactly valid for a scalar field, represents in the gravitational case only a useful model for a pinning of the field following an instability, while the parameter g must be fitted from the experimental data. As explained in Section 2b, g is the sum of the products of the unknown runaway probabilities xi by the unknown strengths hi2.