Interaction between superconductors and weak gravitational field
IInteraction between superconductors and weakgravitational field
Antonio Gallerati
Politecnico di Torino, Dipartimento di Scienza Applicata e Tecnologia, corso Duca degliAbruzzi 24, 10129 Torino, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Torino, via Pietro Giuria 1, 10125 Torino,ItalyE-mail: [email protected]
Abstract.
We consider the interaction between the Earth’s gravitational field and asuperconductor in the fluctuation regime. Exploiting the weak field expansion formalism andusing time dependent Ginzburg–Landau formulation, we show a possible short-time alterationof the gravitational field in the vicinity of the superconductor.
1. Introduction
The study of the interaction between superconductors and the gravitational field has receivedgreat attention in the last decades, due to its possible applications in both theoretical andapplied physics. The seminal paper [1] laid the foundation of the search field, that later led tothe Podkletnov and Nieminen pioneering experiment [2], in which they claimed to have observeda gravitational shielding effect. Since no such effect can occur in the classical framework, severalsubsequent theoretical papers tried to clarify the possible origin of the gravity/superconductivityinterplay in the frame of a quantum field formulation [3–5].Another step towards the construction of a consistent theory came from the introductionof generalized electric-type fields induced by the presence of a gravitational field [6–8], thegeneralized field having the form E = E e + me E g and therefore characterized by an electriccomponent E e and a gravitational one E g , m and e being the electron mass and charge. Inspiredby these experimental researches, we describe below how the same results can be formallyobtained using the gravito-Maxwell formalism.
2. Weak field expansion
Here we consider a nearly flat space-time configuration (weak gravitational field), where themetric g µν can be expanded as g µν (cid:39) η µν + h µν , (1)where η µν = diag( − , +1 , +1 , +1) is the flat Minkowski metric in the mostly plus conventionand h µν is a small perturbation. If we introduce the tensor¯ h µν = h µν − η µν h , (2) a r X i v : . [ g r- q c ] J a n t can be easily demonstrated that the Einstein equations in the harmonic De Donder gauge ∂ µ ¯ h µν (cid:39) R µν − g µν R = ∂ ρ G µνρ = 8 π G T µν , (3)having defined the tensor G µνρ ≡ ∂ [ ν ¯ h ρ ] µ + ∂ σ η µ [ ρ ¯ h ν ] σ (cid:39) ∂ [ ν ¯ h ρ ] µ . (4) We then define the fields E g = − G i = − ∂ [0 ¯ h i ]0 , A g = 14 ¯ h i , B g = 14 ε ijk G jk , (5)for which we obtain, restoring physical units, the set of equations [9, 10]: ∇ · E g = 4 π G ρ g , ∇ · B g = 0 , ∇ × E g = − ∂ B g ∂t , ∇ × B g = 4 π G 1 c j g + 1 c ∂ E g ∂t , (6)having introduced the mass density ρ g ≡ − T and the mass current density j g ≡ T i .The above equations have the same structure of the Maxwell equations, with E g and B g gravitoelectric and gravitomagnetic field, respectively. Now let us consider generalized electric/magnetic fields, scalar and vector potentials, havingboth electromagnetic and gravitational contributions: E = E e + me E g , B = B e + me B g , V = V e + me V g , A = A e + me A g , (7)where m and e identify the mass and electronic charge, respectively. The generalized Maxwellequations for the above fields then become [9–11]: ∇ · E = (cid:18) ε + 1 ε g (cid:19) ρ , ∇ · B = 0 , ∇ × E = − ∂ B ∂t , ∇ × B = ( µ + µ g ) j + 1 c ∂ E ∂t , (8)where ε and µ are the vacuum electric permittivity and magnetic permeability. In the aboveexpression, ρ and j are the electric charge density and electric current density, respectively, whilethe mass density and the mass current density vector have been expressed in terms of the latteras ρ g = me ρ , j g = me j , (9)while the vacuum gravitational permittivity ε g and permeability µ g have the form ε g = 14 π G e m , µ g = 4 π G c m e . (10) . The quantum model Let us now consider a superconductor in the vicinity of its critical temperature. The samplebehavior is characterized by thermodynamic fluctuations of the order parameter creatingsuperfluid regions of accelerated electrons, causing in turn an increase of the resistivity fortemperatures
T > T c . This regime can be well described using time-dependent Ginzburg-Landau formulation [12] and, if we suppose we deal with sufficiently dirty materials, the effectsof the fluctuations can be observed over a sizable range of temperature.The time-dependent Ginzburg-Landau equations characterizing the system, for temperatureslarger than T c , have the gauge-invariant form [13, 14]:Γ ( (cid:126) ∂ t − i e φ ) ψ = 12 m ( (cid:126) ∇ − i e A ) ψ + α ψ . (11)We make the following ansatz for the solution ψ ( x , t ) = f ( x , t ) exp (cid:0) i g ( x , t ) (cid:1) , (12)and one then finds for the superfluid speed and the associated current density v s = 1 m (cid:16) (cid:126) ∇ g + 2 ec A (cid:17) , j s = − em | ψ | (cid:16) h ∇ g + 2 ec A (cid:17) = − e f v s . (13)The latter can be explicitly calculated from [10] j s ( t ) = 2 e m E t k b T π ∞ (cid:90) dk π k (cid:18) α + (cid:126) k m (cid:19) − exp (cid:18) − (cid:126) Γ (cid:16) α + (cid:126) k m (cid:17) t − (cid:126) Γ e m E t (cid:19) , (14)having defined the quantities∆ T = T − T c , (cid:15) ( T ) = (cid:114) ∆ TT c , α = (cid:126) m ξ (cid:15) ( T ) , Γ = α(cid:15) ( T ) π b T c , (15)where ξ is the BCS coherence length. The potential vector A ( x, y, z, t ) is given by: A ( x, y, z, t ) = 14 π (cid:90) j s ( t ) dx (cid:48) dy (cid:48) dz (cid:48) (cid:112) ( x − x (cid:48) ) + ( y − y (cid:48) ) + ( z − z (cid:48) ) , (16)and the generalized electric field (7) is then written as E ( x, y, z, t ) = − c ∂ t A ( x, y, z, t ) + me g = − c ∂ t j s ( t ) C ( x, y, z ) + me g , (17)featuring the contribution coming from the Earth-surface gravitational field g , while C ( x, y, z )is 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 T c ,that is put it in the normal state with a weak magnetic field. The latter is then removed at thetime t = 0, so that the system enters the superconducting state. Using the described quantummodel, we can calculate the variation of the gravitational field in the vicinity of the sample, inthe fluctuation regime.n particular, let us consider a superconducting disk with bases parallel to the ground. InFigure 1 is plotted the variation of the gravitational field as a function of time, measured alongthe axis of the disk at fixed distance d = 0 .
25 cm above the base surface, for a Nb sample (low- T c superconductor, ξ = 39 nm, T c = 9 .
250 K, ∆ T = 10 − K [15]) having radius R = 15 cm andthickness h = 2 cm. We can appreciate that the gravitational field is initially reduced withrespect to its unperturbed value, then subsequently increases up to a maximum value for t = τ and finally relaxes to the standard external value. In Figure 2 we show the field variation as afunction of distance from the base surface, measured along the axis of the disk at the fixed time t = τ = 7 .
45 ns that maximizes the effect.In Figures 3 and 4 the same calculations are performed using a HgBaCaCuO (HBCCO)sample, an high- T c superconductor ( ξ = 230 nm, T c = 126 K, ∆ T = 0 . ´ - ´ - ´ - ´ - PSfrag replacements (cid:0) m / s (cid:1) ( s ) g + ∆ g Nb ∆ = 1 . · − m / s Figure 1:
The gravitational field variation as afunction of time for a Nb samplemeasured along the axis of the disk atfixed distance d above the base surface.
10 40 70 100 130 1602. ´ - ´ - ´ - ´ - ´ - PSfrag replacements ∆ (cid:0) m / s (cid:1) ( cm ) Nb t = τ = 7 .
45 ns
Figure 2:
The gravitational field variation as afunction of distance from the basesurface for a Nb sample, measured at thefixed time t = τ . It is easily shown that the maximum value ∆ for the variation of the external field isproportional to ξ − , implying a larger effect in high- T c superconductors, having the lattersmall coherence length. It is also possible to demonstrate that τ ∝ ( T − T c ) − , that in turn ´ - ´ - ´ - ´ - PSfrag replacements (cid:0) m / s (cid:1) ( s ) g + ∆ g HBCCO ∆ = 1 . · − m / s Figure 3:
The gravitational field variation as afunction of time for a HBCCO samplemeasured along the axis of the disk atfixed distance d above the base surface.
10 40 70 100 130 1601. ´ - ´ - ´ - ´ - ´ - PSfrag replacements ∆ (cid:0) m / s (cid:1) ( cm ) HBCCO t = τ = 7 . · − ns Figure 4:
The gravitational field variation as afunction of distance from the basesurface for a HBCCO sample, measuredat the fixed time t = τ . means that the time range in which the phenomenon takes place can be extended if the systemis very close to its critical temperature. . Conclusions and future developments As can be seen from the results obtained, the field variation is in principle perceptible (especiallyin high- T c superconductors), while the very short time intervals in which the effect occurscomplicate direct measurements. In order to obtain non negligible experimental evidence ofgravitational perturbations in workable time scales, a careful choice of parameters must bemade. First of all, a large superconducting sample of dirty material is needed, so that theeffects of fluctuations can be enhanced over a wider temperature range. Then, the best optioncurrently is to choose an high- T c superconductor (short coherence length increases the intensityof the phenomenon) at a temperature very close to T c (increase in the time interval where theeffect occurs).Possible future developments of the described formalism derive from the application todifferent physical situation where generalized electric-magnetic fields of the form (7) are inducedby the presence of a weak gravitational field. An example of application to the Josephsonjunction physics of superconductors can be found in [16]. References [1] DeWitt B S 1966
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