Possible alterations of local gravitational field inside a superconductor
PPossible alterations of local gravitational field inside asuperconductor
Giovanni Alberto Ummarino
Politecnico di Torino, Dipartimento di Scienza Applicata e Tecnologia, corso Duca degli Abruzzi 24, 10129 Torino, ItalyNational Research Nuclear University MEPhI, Kashirskoe hwy 31, 115409 Moscow, Russia [email protected]
Antonio Gallerati
Politecnico di Torino, Dipartimento di Scienza Applicata e Tecnologia, corso Duca degli Abruzzi 24, 10129 Torino, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Torino, via Pietro Giuria 1, 10125 Torino, Italy [email protected]
Abstract
We calculate the possible interaction between a superconductor and the static Earth’s gravitationalfields, making use of the gravito-Maxwell formalism combined with the time-dependent Ginzburg–Landau theory. We try to estimate which are the most favourable conditions to enhance theeffect, optimizing the superconductor parameters characterizing the chosen sample. We also givea qualitative comparison of the behaviour of high– T c and classical low– T c superconductors withrespect to the gravity/superfluid interplay. Contents a r X i v : . [ g r- q c ] F e b Introduction
The study of possible gravitational effects on superconductors is more than 50 years old and startedwith the seminal paper of DeWitt [1]. In the following years, there has been a fair amount of scientificliterature on the subject [2–21], but it was only after the 1992 Podkletnov’s reported effect [22, 23]that experimental, laboratory configurations were proposed to detect the interaction.Theoretical interpretations of the interplay between the condensate and the local gravitationalfield were produced in 1996 exploiting the framework of quantum gravity [24], showing how a suitableLagrangian coupling of the superfluid can determine a gravitational interaction with the condensateand consequent localized slight instabilities [25, 26]. Although being a solid and elegant formulationoffering a general, theoretical explanation for the described interplay, the quantum gravity approachinvolves a formalism that makes it hard to extract quantitative predictions.Parallel to DeWitt (and related) studies about gravity/supercondensate coupling, other theoret-ical [27, 28] and experimental [29–31] researches were conducted about electric-type fields induced inconductors by the presence of the gravitational field, analysing the importance of the internal struc-ture of special classes of solids and fluids when gravity is taken into account. Those researches alsoinspired other recent papers that focus on various relevant aspects of the behaviour of superconductorsinteracting with gravitational waves [32–34].One of the results of the above studies was the introduction of a fundamental, generalized electric-like field, featuring an electrical component and a gravitational one. In the following, we are goingto extend those results making use of the gravito-Maxwell formalism [35–39]. In particular, we willsee that the latter approach can provide a solid framework where to obtain a generalized form forthe electric/magnetic fields, involved in quantum effects originating from the interaction with theweak gravitational background. On the other side, the formalism also turns out to be powerful inthe study of gravity/superconductivity interplay, since the formal analogy between the Maxwell andweak gravity equations allows us to use the Ginzburg–Landau theory for the microscopic descriptionof the interaction. We will in fact analyse how the weak local gravitational field can be affected by thepresence of the superfluid condensate, writing explicit time-dependent Ginzburg–Landau equations forthe superconductor order parameter.With respect to our previous analysis [35], we will perform new calculations in a different gauge andthis will lead us to clearer and deeper conclusions on the interpretation of the conjectured effect. Wewill also analyse which parameters could be optimized to enhance the interaction, choosing appropriateconditions and sample characteristics.
Let us consider a nearly–flat spacetime configuration (weak, static gravitational field approximation)so that the metric can be expanded as: g µν (cid:39) η µν + h µν , (1)where the symmetric tensor h µν is a small perturbation of the constant, flat Minkowski metric in themostly plus convention, η µν = diag( − , +1 , +1 , +1). The inverse metric, in linear approximation, isgiven by g µν (cid:39) η µν − h µν . (2)2hile the metric determinant can be expanded as g = det [ g µν ] = ε µνρσ g µ g ν g ρ g σ (cid:39) − − h ⇒ √− g (cid:39) h , (3)where h = h σσ . If we consider an inertial coordinate system, to linear order in h µν the connection is expanded asΓ λµν (cid:39) η λρ ( ∂ µ h νρ + ∂ ν h ρµ − ∂ ρ h µν ) . (4)The Riemann tensor is defined as: R σµλν = ∂ λ Γ σµν − ∂ ν Γ σµλ + Γ σρλ Γ ρνµ − Γ σρν Γ ρλµ , (5)while the Ricci tensor is given by the contraction R µν = R σµσν , (6)and, to linear order in h µν , it reads R µν (cid:39) ∂ σ Γ σµν + ∂ µ Γ σσν + (cid:8)(cid:8) Γ Γ − (cid:8)(cid:8) Γ Γ = 12 ( ∂ µ ∂ ρ h νρ + ∂ ν ∂ ρ h µρ ) − ∂ ρ ∂ ρ h µν − ∂ µ ∂ ν h == ∂ ρ ∂ ( µ h ν ) ρ − ∂ h µν − ∂ µ ∂ ν h , (7)having used eq. (4).The Einstein equations have the form [40]: R µν − g µν R = 8 π G T µν , (8)where R = g µν R µν is the Ricci scalar. In first-order approximation, we can write12 g µν R (cid:39) η µν η ρσ R ρσ = 12 η µν (cid:0) ∂ ρ ∂ σ h ρσ − ∂ h (cid:1) , (9)having used eq. (7), and the left hand side of (8) turns out to be R µν − g µν R (cid:39) ∂ ρ ∂ ( µ h ν ) ρ − ∂ h µν − ∂ µ ∂ ν h − η µν (cid:0) ∂ ρ ∂ σ h ρσ − ∂ h (cid:1) . (10)Now, we introduce the symmetric traceless tensor¯ h µν = h µν − η µν h , (11)3o that the above (10) can be rewritten as R µν − g µν R (cid:39) (cid:0) ∂ ρ ∂ µ ¯ h νρ + ∂ ρ ∂ ν ¯ h µρ − ∂ ρ ∂ ρ ¯ h µν − η µν ∂ ρ ∂ σ ¯ h ρσ (cid:1) = ∂ ρ ∂ [ ν ¯ h ρ ] µ + ∂ ρ ∂ σ η µ [ σ ¯ h ν ] ρ == ∂ ρ (cid:0) ∂ [ ν ¯ h ρ ] µ + ∂ σ η µ [ ρ ¯ h ν ] σ (cid:1) . (12)We then define the tensor G µνρ ≡ ∂ [ ν ¯ h ρ ] µ + ∂ σ η µ [ ρ ¯ h ν ] σ , (13)so that the Einstein equations can be finally recast in the compact form: ∂ ρ G µνρ = 8 π G T µν . (14) Gauge fixing.
We now consider the harmonic coordinate condition , expressed by the relation [40]: ∂ µ (cid:0) √− g g µν (cid:1) = 0 ⇔ (cid:50) x µ = 0 , (15)that in turn can be rewritten in the form g µν Γ λµν = 0 , (16)also known as De Donder gauge . The requirement of the above coordinate condition (15) plays thenthe role of a gauge fixing. Imposing the above (16) and using eqs. (1) and (4), in linear approximationwe find: 0 (cid:39) η µν η λρ ( ∂ µ h νρ + ∂ ν h ρµ − ∂ ρ h µν ) = ∂ µ h µλ − ∂ λ h , (17)that is, we have the condition ∂ µ h µν (cid:39) ∂ ν h ⇔ ∂ µ h µν (cid:39) ∂ ν h . (18)Now, one also has ∂ µ h µν = ∂ µ (cid:18) ¯ h µν + 12 η µν h (cid:19) = ∂ µ ¯ h µν + 12 ∂ ν h , (19)and, using eq. (18), we find the so-called Lorentz gauge condition : ∂ µ ¯ h µν (cid:39) . (20)The above relation further simplifies expression (13) for G µνρ , which takes the very simple form G µνρ (cid:39) ∂ [ ν ¯ h ρ ] µ , (21)and verifies also the relation ∂ [ λ | G | µν ] = 0 ⇒ G µν ∝ ∂ µ A ν − ∂ ν A µ , (22)implying the existence of a potential (see next paragraph).4 ravito-Maxwell equations. Now, let us define the fields E g ≡ E i = − G i = − ∂ [0 ¯ h i ]0 , (23. i ) A g ≡ A i = 14 ¯ h i , (23. ii ) B g ≡ B i = 14 ε ijk G jk , (23. iii )where i = 1 , , G ij = ∂ [ i ¯ h j ]0 = 12 (cid:0) ∂ i ¯ h j − ∂ j ¯ h i (cid:1) = 4 ∂ [ i A j ] . (24)One can immediately see that B g = 14 ε ijk ∂ [ j A k ] = ε ijk ∂ j A k = ∇ × A g , = ⇒ ∇ · B g = 0 . (25)Then one also has ∇ · E g = ∂ i E i = − ∂ i G i − π G T π G ρ g , (26)using eq. (14) and having defined ρ g ≡ − T .If we consider the curl of E g , we obtain ∇ × E g = ε ijk ∂ j E k = − ε ijk ∂ j G k − ε ijk ∂ j ∂ [0 ¯ h k ]0 == −
14 4 ∂ ε ijk ∂ j A k = − ∂ B i = − ∂ B g ∂t . (27)Finally, one finds for the curl of B g ∇ × B g = ε ijk ∂ j B k = 14 ε ijk ε k(cid:96)m ∂ j G (cid:96)m = 14 (cid:16) δ i(cid:96) δ jm − δ im δ j(cid:96) (cid:17) ∂ j G (cid:96)m == 12 ∂ j G ij = 12 ( ∂ µ G iµ + ∂ G i ) = 12 ( ∂ µ G iµ − ∂ G i ) == 12 (8 π G T i − ∂ G i ) = 4 π G j i + ∂E i ∂t = 4 π G j g + ∂ E g ∂t , (28)using again eq. (14) and having defined j g ≡ j i ≡ T i .Summarizing, once defined the fields of (23) and having restored physical units, one gets the field for the sake of simplicity, we initially set the physical charge e = m = 1 ∇ · E g = 4 π G ρ g , ∇ · B g = 0 , ∇ × E g = − ∂ B g ∂t , ∇ × B g = 4 π G c j g + 1 c ∂ E g ∂t , (29)formally equivalent to Maxwell equations, where E g and B g are the gravitoelectric and gravitomagneticfield, respectively. For instance, on the Earth’s surface, E g corresponds to the Newtonian gravitationalacceleration while B g is related to angular momentum interactions [15, 41–43]. The mass currentdensity vector j g can also be expressed as: j g = ρ g v , (30)where v is the velocity and ρ g is the mass density. Gravito-Lorentz force.
Let us consider the geodesic equation for a particle in the presence of aweak gravitational field: d x λ ds + Γ λµν dx µ ds dx ν ds = 0 . (31)If we consider a non-relativistic motion, the velocity of the particle can be expressed as v i c (cid:39) dx i dt . Ifwe also neglect terms in the form v i v j c and limit ourselves to static metric configurations, we find thata geodesic equation for the particle in non-relativistic motion is written as [44, 45]: d v dt = E g + v × B g , (32)which shows that a free falling particle is governed by the analogous of a Lorentz force produced bythe gravito-Maxwell fields. Generalized Maxwell equations.
It is now straightforward to define generalized electric/magneticfields, scalar and vector potentials, containing both electromagnetic and gravitational contributions,as: E = E e + me E g ; B = B e + me B g ; φ = φ e + me φ g ; A = A e + me A g , (33)where m and e are the electron mass and charge, respectively.6he generalized Maxwell equations then become: ∇ · E = (cid:18) ε g + 1 ε (cid:19) ρ , ∇ · B = 0 , ∇ × E = − ∂ B ∂t , ∇ × B = ( µ g + µ ) j + 1 c ∂ E ∂t , (34)where ε and µ are the electric permittivity and magnetic permeability in the vacuum, and where wehave set ρ g = me ρ , j g = me j , (35) ρ and j being the electric charge density and electric current density, respectively. The introducedvacuum gravitational permittivity ε g and vacuum gravitational permeability µ g are defined as ε g = 14 π G e m , µ g = 4 π G c m e . (36)In this Section we have then shown how to define a new set of generalized Maxwell equations forgeneralized electric E and magnetic B fields, in the limit of weak gravitational field. In the following,we are going to use these results to analyse the interaction between a superconducting sample and theweak, static Earth’s gravitational field. Now we are going to study in detail the conjectured gravity/superconductivity interplay making useof the Ginzburg–Landau formulation combined with the described gravito-Maxwell formalism. In par-ticular, we write the Ginzburg–Landau equations for a superconducting sample in the weak, staticEarth’s gravitational field. The latter is formally treated as the gravitational component of a general-ized electric field, exploiting the formal analogy discussed in the previous Section 2.
Since the gravitoelectric field is formally analogous to a generalized electric field, we can use thetime-dependent Ginzburg–Landau equations (TDGL) written in the form [46–52]: (cid:126) m D (cid:18) ∂∂t + 2 i e (cid:126) φ (cid:19) ψ − a ψ + b | ψ | ψ + 12 m (cid:18) i (cid:126) ∇ + 2 ec A (cid:19) ψ = 0 , (37. i ) ∇ × ∇ × A − ∇ × H = − πc (cid:0) j n + j s (cid:1) , (37. ii )7here j n and j s are expressed as j n = σ (cid:18) c ∂ A ∂t + ∇ φ (cid:19) , j s = em (cid:18) i (cid:126) ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) + 4 ec | ψ | A (cid:19) , (38)and denote the contributions related to the normal current and supercurrent densities, respectively .In the above expressions, D is the diffusion coefficient, σ is the conductivity in the normal phase, H is the applied field and the vector field A is minimally coupled to ψ . The coefficients a and b in (37. i )have the following form: a = a ( T ) = a ( T − T c ) , b = b ( T c ) , (39) a , b being positive constants and T c the critical temperature of the superconductor. The boundaryand initial conditions are (cid:18) i (cid:126) ∇ ψ + 2 ec A ψ (cid:19) · n = 0 ∇ × A · n = H · nA · n = 0 on ∂ Ω × (0 , t ) , ψ ( x,
0) = ψ ( x ) A ( x,
0) = A ( x ) (cid:41) on Ω , (40)where ∂ Ω is the boundary of a smooth and simply connected domain in R N . Dimensionless TDGL.
In order to write eqs. (37) in a dimensionless form, the following expressionscan be introduced:Ψ ( T ) = | a ( T ) | b , ξ ( T ) = h (cid:112) m | a ( T ) | , λ ( T ) = (cid:115) b m c π | a ( T ) | e , κ = λ ( T ) ξ ( T ) ,τ ( T ) = λ ( T ) D , η = 4 π σ D ε c , H c ( T ) = (cid:115) π µ | a ( T ) | b = h e √ π λ ( T ) ξ ( T ) , (41)where λ ( T ), ξ ( T ) and H c ( T ) are the penetration depth, coherence length and thermodynamic criticalfield, respectively. We also define the dimensionless quantities x (cid:48) = xλ , t (cid:48) = tτ , ψ (cid:48) = ψ Ψ , (42)and the dimensionless fields are then written as: A (cid:48) = A κ √ H c λ , φ (cid:48) = φ κ √ H c D , H (cid:48) = H κ √ H c . (43) The TDGL equations (37) for the variables ψ , A are derived minimizing the total Gibbs free energy of the system[53–55]. R N [46, 47]: ∂ψ∂t + i φ ψ + κ (cid:16) | ψ | − (cid:17) ψ + ( i ∇ + A ) ψ = 0 , (44. i ) ∇ × ∇ × A − ∇ × H = − η (cid:18) ∂ A ∂t + ∇ φ (cid:19) − i κ ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) − | ψ | A , (44. ii )and the boundary and initial conditions (40) become, in the dimensionless form( i ∇ ψ + A ψ ) · n = 0 ∇ × A · n = H · nA · n = 0 on ∂ Ω × (0 , t ) ; ψ ( x,
0) = ψ ( x ) A ( x,
0) = A ( x ) (cid:41) on Ω . (45) Now we will study the possible local alterations of the Earth’s gravitational field (weak uniform field)inside a superconductor. Let us consider the dimensionless form of the time-dependent Ginzburg–Landau equations in the gauge of vanishing scalar potential φ = 0 [56]: ∂ψ∂t = − (cid:18) iκ ∇ + A (cid:19) ψ + (cid:16) − | ψ | (cid:17) ψ , (46. i ) η ∂ A ∂t = − ∇ × ∇ × A + ∇ × H − | ψ | (cid:18) A − κ ∇ θ (cid:19) , (46. ii )where ψ ≡ ψ ( x , t ) is a complex function that we express as ψ = | ψ | exp( i θ ) = Re ψ + i Im ψ = ψ + i ψ , (47)so that (46. i ) gives two distinct equations for the real and imaginary parts ψ and ψ . Let us now restrict to the 1-dimensional case (cid:0) ∇ ! ∂/∂x , A ! A x ≡ A (cid:1) . In thissituation, the above TDGL (46) give rise to the following equations: ∂ψ ∂t = 1 κ ∂ ψ ∂x + 2 Aκ ∂ψ ∂x + ψ κ ∂A∂x − ψ A + ψ − ψ (cid:0) ψ + ψ (cid:1) ,∂ψ ∂t = 1 κ ∂ ψ ∂x − Aκ ∂ψ ∂x − ψ κ ∂A∂x − ψ A + ψ − ψ (cid:0) ψ + ψ (cid:1) ,η ∂A∂t = − κ (cid:18) ψ ∂ψ ∂x − ψ ∂ψ ∂x (cid:19) − (cid:0) ψ + ψ (cid:1) A − πj n , (48)where j n indicates the normal current density. here we decide to use the most convenient option for subsequent calculations, since any gauge choice shall notinfluence any physical results, being the equations gauge-invariant. From a physical point of view, the choice is alsomotivated by the fact that there are no localized charges in the superconductor, while any contribution to the totalgravitational field coming from the superconductor mass is irrelevant and can be neglected (cid:126)x direction is perpendicularto superconductor surface (coinciding with the yz plane), i.e. we imagine that for x > x ≤
0. The system is immersedin a static, uniform gravitational field E ext g = − g (cid:126)u x , where g is the standard gravity acceleration.We are in the gauge where, in the dimensional form, we can write for the gravitoelectric field insidethe superconductor E g = − ∂ A g ( t ) ∂t , (49)while the external gravitational vector potential outside the superconductor is given by A ext g ( t ) = g ( C + t ) (cid:126)u x , (50)where C is a constant. In the 1-D dimensionless form, dropping the primes, we have A ext = me A ext g κ √ H c λ = g (cid:63) ( c + t ) , (51)with c = Cτ , g (cid:63) = m κ λ ( T ) g √ e D H c ( T ) (cid:28) . (52)having used relations (41).Next, we express the ψ , ψ and A fields as: ψ ( x, t ) = ψ ( x ) + g (cid:63) γ ( x, t ) , (53. i ) ψ ( x, t ) = ψ ( x ) + g (cid:63) γ ( x, t ) , (53. ii ) A ( x, t ) = g (cid:63) β ( x, t ) , (53. iii )where ψ and ψ represent the unperturbed system and satisfy0 = 1 κ ∂ ψ ∂x + ψ − ψ (cid:0) ψ + ψ (cid:1) , (54. i )0 = 1 κ ∂ ψ ∂x + ψ − ψ (cid:0) ψ + ψ (cid:1) . (54. ii )The ψ and ψ components satisfy the same kind of equation, and we choose to set ψ = 0( ψ = ψ + i ψ = ψ ∈ R ), so that ψ = tanh κx √ gives the standard solution for (54. i ) [54]. Weare then left with the following set of equations: ∂γ ∂t = 1 κ ∂ γ ∂x + (cid:0) − ψ (cid:1) γ , (55. i ) ∂γ ∂t = 1 κ ∂ γ ∂x + (cid:0) − ψ (cid:1) γ − βκ ∂ψ ∂x − ψ κ ∂β∂x , (55. ii ) η ∂β∂t = − κ (cid:18) γ ∂ψ ∂x − ψ ∂γ ∂x (cid:19) − ψ β , (55. iii )10here the last (55. iii ) implies that β ( x, t ) does not depend on γ ( x, t ). If we decide to put ourselvesaway from borders, we can set ψ (cid:39) ∂γ ∂t (cid:39) κ ∂ γ ∂x − γ , (56. i ) ∂γ ∂t (cid:39) κ ∂ γ ∂x − κ ∂β∂x , (56. ii ) η ∂β∂t (cid:39) κ ∂γ ∂x − β , (56. iii )that gives for β the explicit solution β ( x, t ) = e − tη (cid:18) b ( x ) + 1 κ η (cid:90) t dt e tη ∂γ ( x, t ) ∂x (cid:19) . (57)where b ( x ) = c , as it is implied by eq. (53. iii ) for t (cid:39) (cid:126)x axis (one-dimensional case) where the external vector potential isexpressed as: A ext ( t ) = ( c + t ) g (cid:63) . (58)At the time t = 0, the sample goes in the superconductive state, while we make the natural assumptionthat in the normal state ( t <
0) the material has just the standard (Newtonian) interaction with theEarth’s gravity, implying that the local gravitational field assumes the same values inside and outsidethe sample for t <
0. We then write the following boundary conditions: ψ (0 , t ) = 0 , ψ ( x,
0) = ψ ( x ) , ∂ψ ∂x ( x,
0) = 0 ,γ (0 , t ) = 0 , γ ( x,
0) = 0 , ∂γ ∂x ( x,
0) = 0 ,γ (0 , t ) = 0 , γ ( x,
0) = 0 , ∂γ ∂x ( x,
0) = 0 , (59)together with the condition lim t ! g (cid:63) ∂β∂t ( x, t ) = g (cid:63) . (60)implying that the effect takes place when the superconducting phase appears.Let us now fix the constant c . Using (55. iii ), we can express the relation between E g and β as E g g (cid:63) = − ∂β∂t = 1 κ η (cid:18) γ ∂ψ ∂x − ψ ∂γ ∂x (cid:19) + ψ η β . (61)Given the natural hypothesis that the affection of the gravitational field only exists when the materialis in the superconductive state ( t > t ! + E g g (cid:63) = 1 , (62)11hile from conditions (59) we also havelim t ! + γ ( x, t ) = 0 , lim t ! + ∂γ ∂x ( x, t ) = 0 , (63)from which we get in turn1 = ψ η β ( x, + ) = ψ η A ext (0 + ) g (cid:63) = ψ η c = ⇒ c = ηψ . (64)This constant is ineffective in the empty space, while it determines physical effects in the supercon-ductive state. The above formulation shows how the described interplay should work: the externalgravitational field is affected by the presence of the sample only when it goes in the superconductivestate (when the vector potential starts to “feel” the presence of the superfluid). From the other side,the external gravitational vector potential seems involved in the material superconductive transition,since the external constant c tends to assume a fixed value related to the properties of the superfluidentering the superconducting state.Now we can rewrite the explicit solution for β ( x, t ) away from borders ( ψ (cid:39) β ( x, t ) = e − tη (cid:18) η + 1 κ η (cid:90) t dt e tη ∂γ ( x, t ) ∂x (cid:19) , (65)from which we get the ratio E g g (cid:63) = − ∂β ( x, t ) ∂t = 1 η e − tη (cid:18) η + 1 κ η (cid:90) t dt e tη ∂γ ( x, t ) ∂x (cid:19) − κ η ∂γ ( x, t ) ∂x . (66) Given the explicit expression (66) for the ratio E g /g (cid:63) , we can estimate, for t (cid:39) + , the value ofgravitational field inside the superconductor: t (cid:39) + : E g g (cid:63) = 1 − tη − κ η ∂γ ( x, + ) ∂x . (67)In the superconductive state, the gravitational field is modified in a way that depends on physicalcharacteristic of the particular material. We can see from the above (67) that the involved quantitiesare η , κ and the spatial derivative of γ .Let us discuss which should be the most favourable choices for the parameters to enhance thedesired interaction. First of all, we would like to maximize ∂γ ∂x : to do this, it is sufficient to introducedisorder in the material, induced, for instance, by means of proton irradiation or chemical doping.Then, we also want a small η parameter: being the latter proportional to the product of the diffusioncoefficient times the conductivity just above T c , it is necessary to have materials that in the normal stateare bad conductors and have low Fermi energies, such as cuprates. The last parameter to optimize is areduced value for κ , which is usually small in low– T c superconductors and high in cuprates. Clearly, wecan see that optimizing at the same time last two parameters gives rise to contrasting effects; however,analysing the involved values, the better choice is to maximize η , thus using a superconducting cupratewith high disorder. 12t is also very important to maximize the time scale ( τ = λ / D ) in order to better observe theeffect. This is achieved by increasing the penetration length and reducing the diffusivity coefficient,just as it occurs in superconducting cuprates with disorder.In Tables 1 and 2 it is possible to see typical parameters of low (Pb) and high (YBCO) T c superconductors, some of which calculated at a temperature T (cid:63) such that the quantity T c − T (cid:63) T c is thesame in the two materials. If we go closer to T c , it is possible to increase the effect: for example,in the case of YBCO, at T = 87 K the τ parameter is of the order of 10 − s and the reduction ofthe gravitational field is of the order of 10 − , having neglected the last term in eq. (67) (in high– T c superconductors not irradiated, we usually have low disorder, so that the spatial derivative of γ is small; moreover, there is an additional reduction of order 10 coming from the κ parameter atdenominator). We have shown how the gravito-Maxwell formalism can be instrumental in describing a grav-ity/superfluid interplay, when combined with the condensed matter formalism of the time-dependentGinzburg–Landau equations. Our analysis suggests that a non-negligible interaction could be present,despite the experimental detection difficulties that may arise, especially in relation to the short timeintervals in which the effect occurs. In particular, the dimensionless TDGL can provide qualitativeand quantitative suggestion about the magnitude of the interaction, once chosen appropriate boundaryconditions.Clearly, proper arrangement of the experimental setup is crucial to maximize the effect. In partic-ular, the focus should be on suitable sample geometry, material parameters and laboratory settings, soas to enhance the interaction in workable time scales [37–39]. It is also possible that a significant im-provement comes from the presence of external electric and magnetic fields, since the latter determinethe presence of moving vortices, giving rise to a possible additional affection of the local gravitationalfield.
Acknowledgments
This work was supported by the MEPhI Academic Excellence Project (contract No. 02.a03.21.0005)for the contribution of prof. G. A. Ummarino. We also thank Fondazione CRT that partiallysupported this work for dott. A. Gallerati. 13BCO Pb T c
89 K 7 . T (cid:63)
77 K 6 . ξ ( T (cid:63) ) 3 . · − m 1 . · − m λ ( T (cid:63) ) 3 . · − m 7 . · − m σ − . · − Ω m ( ∗ ) . · − Ω m ( ∗∗ ) H c ( T (cid:63) ) 0 . .
018 Tesla κ . . τ ( T (cid:63) ) 3 . · − s 6 . · − s η . · − . · g (cid:63) . · − . · − D . · − m / s 1 m / s (cid:96) . · − m 1 . · − m v f . · m / s 1 . · m / s ( ∗ ) T = 90 K ( ∗∗ ) T = 15 K Table 1: YBCO vs. Pb.
YBCO λ τ g (cid:63) T = 0 K 1 . · − m 9 . · − s 2 . · − T = 70 K 2 . · − m 2 . · − s 9 . · − T = 77 K 3 . · − m 3 . · − s 2 · − T = 87 K 8 · − m 2 · − s 2 . · − Pb λ τ g (cid:63) T = 0 K 3 . · − m 1 . · − s 1 · − T = 4 .
20 K 4 . · − m 1 . · − s 1 . · − T = 6 .
26 K 7 . · − m 6 . · − s 8 . · − T = 7 .
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