Interaction between superconductors and weak gravitational field

Abstract

We consider the interaction between the Earth gravitational field and a superconductor in the fluctuation regime. Exploiting the weak field expansion formalism and using time dependent Ginzburg–Landau formulation, we show a possible short-time alteration of the gravitational field in the vicinity of the superconductor.

1. Introduction

The study of the interaction between superconductors and the gravitational field has received great attention in the last decades, due to its possible applications in both theoretical and applied physics. The seminal paper of DeWitt laid the foundation of the search field, that later led to the Podkletnov and Nieminen pioneering experiment, in which they reported having observed a gravitational shielding effect. Since no such effect can occur in the classical framework, several subsequent theoretical papers tried to clarify the possible origin of the gravity/superconductivity interplay in the frame of a quantum field formulation.

Another step towards the construction of a consistent theory came from the introduction of generalized electric-type fields induced by the presence of a gravitational field, the generalized field having the form of an electric component plus a gravitational one weighted by the ratio of the electron mass to the electron charge. Inspired by these experimental researches, we describe below how the same results can be formally obtained using the gravito-Maxwell formalism.

2. Weak field expansion

Here we consider a nearly flat space-time configuration (weak gravitational field), where the metric can be expanded as the flat Minkowski metric in the mostly plus convention plus a small perturbation, equation 1. Introducing the trace-reversed perturbation, equation 2, it can be easily demonstrated that the Einstein equations in the harmonic De Donder gauge can be rewritten, in first-order approximation, as equation 3: the Einstein tensor equals the divergence of a three-index tensor built from derivatives of the trace-reversed perturbation, defined in equation 4, and this equals eight pi times Newton’s constant times the stress-energy tensor.

2.1. Gravito-Maxwell formulation

We then define the fields of equation 5: a gravitoelectric field, a gravitational vector potential and a gravitomagnetic field, each built from components of the trace-reversed metric perturbation. For these, restoring physical units, we obtain the set of equations 6: the divergence of the gravitoelectric field equals four pi times Newton’s constant times the mass density; the divergence of the gravitomagnetic field vanishes; the curl of the gravitoelectric field equals minus the time derivative of the gravitomagnetic field; and the curl of the gravitomagnetic field equals four pi times Newton’s constant over the speed of light squared times the mass current density, plus the time derivative of the gravitoelectric field over the speed of light squared. Here the mass density is minus the time-time component of the stress-energy tensor and the mass current density is its time-space component.

The above equations have the same structure as the Maxwell equations, with the gravitoelectric and gravitomagnetic fields in the roles of the electric and magnetic fields respectively.

2.2. Generalized fields and equations

Now let us consider generalized electric and magnetic fields, scalar and vector potentials, having both electromagnetic and gravitational contributions, equation 7: in each case the electromagnetic quantity plus the corresponding gravitational quantity multiplied by the ratio of the electron mass to the electron charge. The generalized Maxwell equations for the above fields then become equations 8: the divergence of the generalized electric field equals the charge density multiplied by the sum of the reciprocals of the vacuum electric permittivity and a gravitational permittivity; the divergence of the generalized magnetic field vanishes; the curl of the generalized electric field equals minus the time derivative of the generalized magnetic field; and the curl of the generalized magnetic field equals the sum of the vacuum magnetic permeability and a gravitational permeability, multiplied by the current density, plus the time derivative of the generalized electric field over the speed of light squared.

In the above expressions the mass density and the mass current density have been expressed in terms of the electric charge density and current density by the same mass-to-charge ratio, equation 9, while the vacuum gravitational permittivity and permeability have the form of equation 10: the permittivity is the electron charge squared divided by four pi times Newton’s constant times the electron mass squared, and the permeability is four pi times Newton’s constant times the electron mass squared divided by the speed of light squared times the electron charge squared.

3. The quantum model

Let us now consider a superconductor in the vicinity of its critical temperature. The sample behavior is characterized by thermodynamic fluctuations of the order parameter creating superfluid regions of accelerated electrons, causing in turn an increase of the resistivity for temperatures above the critical temperature. This regime can be well described using the time-dependent Ginzburg-Landau formulation and, if we suppose we deal with sufficiently dirty materials, the effects of the fluctuations can be observed over a sizable range of temperature.

The time-dependent Ginzburg-Landau equation characterizing the system, for temperatures larger than the critical temperature, has the gauge-invariant form of equation 11, relating the gauge-covariant time derivative of the order parameter to its gauge-covariant Laplacian and to the Ginzburg-Landau coefficient. We make the ansatz of equation 12, writing the order parameter as a real amplitude multiplied by a complex exponential of a real phase, and one then finds for the superfluid speed and the associated current density the results of equation 13: the superfluid speed is the gradient of the phase plus the vector potential term, scaled by the charge-to-mass ratio, and the supercurrent density is minus twice the charge times the squared amplitude times that speed.

The latter can be explicitly calculated from equation 14, an integral over wavenumber of the fluctuation propagator weighted by an exponential decay in time. Equation 15 defines the quantities involved: the temperature difference from the critical temperature, the reduced temperature, the Ginzburg-Landau coefficient written in terms of the BCS coherence length, and the relaxation coefficient. Equation 16 gives the vector potential as the usual volume integral of the supercurrent density over distance, and the generalized electric field of equation 7 is then written as equation 17: minus the time derivative of the vector potential over the speed of light, plus the Earth-surface gravitational field weighted by the mass-to-charge ratio — where the first term factorises into the time derivative of the supercurrent multiplied by a geometrical factor whose expression depends on the shape of the superconducting sample.

4. Experimental predictions

Let us now consider the case of a superconducting sample, at a temperature very close to its critical temperature, that is put in the normal state with a weak magnetic field. The latter is then removed at time zero, so that the system enters the superconducting state. Using the described quantum model, we can calculate the variation of the gravitational field in the vicinity of the sample, in the fluctuation regime.

In particular, let us consider a superconducting disk with bases parallel to the ground. Figure 1 plots the variation of the gravitational field as a function of time, measured along the axis of the disk at a fixed distance of 0.25 centimetres above the base surface, for a niobium sample — a low critical temperature superconductor with a coherence length of 39 nanometres, a critical temperature of 9.250 kelvin and a temperature offset of one thousandth of a kelvin — having a radius of 15 centimetres and a thickness of 2 centimetres. We can appreciate that the gravitational field is initially reduced with respect to its unperturbed value, then subsequently increases up to a maximum value at a characteristic time, and finally relaxes to the standard external value. The maximum excess is 1.82 times ten to the minus seven metres per second squared, reached at a characteristic time of 7.45 nanoseconds. Figure 2 shows the field variation as a function of distance from the base surface, measured along the axis of the disk at that fixed characteristic time.

In figures 3 and 4 the same calculations are performed using a mercury-barium-calcium-copper-oxide sample, a high critical temperature superconductor with a coherence length of 230 nanometres, a critical temperature of 126 kelvin and a temperature offset of one tenth of a kelvin. The maximum variation is 1.14 times ten to the minus four metres per second squared, reached at a characteristic time of 7.5 times ten to the minus three nanoseconds.

It is easily shown that the maximum value for the variation of the external field is proportional to the reciprocal of the coherence length, implying a larger effect in high critical temperature superconductors, the latter having a small coherence length.

It is also possible to demonstrate that the characteristic time is proportional to the reciprocal of the difference between the temperature and the critical temperature, which in turn means that the time range in which the phenomenon takes place can be extended if the system is very close to its critical temperature.

5. Conclusions and future developments

As can be seen from the results obtained, the field variation is in principle perceptible (especially in high critical temperature superconductors), while the very short time intervals in which the effect occurs complicate direct measurements. In order to obtain non negligible experimental evidence of gravitational perturbations in workable time scales, a careful choice of parameters must be made. First of all, a large superconducting sample of dirty material is needed, so that the effects of fluctuations can be enhanced over a wider temperature range. Then, the best option currently is to choose a high critical temperature superconductor (short coherence length increases the intensity of the phenomenon) at a temperature very close to the critical temperature (increase in the time interval where the effect occurs).

Possible future developments of the described formalism derive from the application to different physical situations where generalized electric-magnetic fields of the form of equation 7 are induced by the presence of a weak gravitational field. An example of application to the Josephson junction physics of superconductors can be found in the work of Ummarino and Gallerati in Classical and Quantum Gravity.

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