Superconductor in a Weak Static Gravitational Field

Eur. Phys. J. C (2017) 77:549 DOI 10.1140/epjc/s10052-017-5116-y

Regular Article - Theoretical Physics

Superconductor in a weak static gravitational ﬁeld

Giovanni Alberto Ummarino1,2,a, Antonio Gallerati1,b 1 Politecnico di Torino, Dipartimento DISAT, Turin, Italy 2 National Research Nuclear University MEPhI-Moscow Engineering Physics Institute, Moscow, Russia

Received: 12 June 2017 / Accepted: 25 July 2017 © The Author(s) 2017. This article is an open access publication

Abstract We provide the detailed calculation of a gen- dence for a gravitational shielding in a high-Tc superconduc- eral form for Maxwell and London equations that takes into tor (HTCS) [21,22]. After their announcement, other groups account gravitational corrections in linear approximation. tried to repeat the experiment obtaining controversial results We determine the possible alteration of a static gravitational [23Ð25], so that the question is still open. field in a superconductor making use of the time-dependent In 1996, Modanese interpreted the results by Podkletnov GinzburgÐLandau equations, providing also an analytic solu- and Nieminem in the frame of the quantum theory of Gen- tion in the weak field condition. Finally, we compare the eral Relativity [26,27] but the complexity of the formalism behavior of a high-Tc superconductor with a classical low- makes it very difficult to extract quantitative predictions. Tc superconductor, analyzing the values of the parameters Afterwards, Agop et al. wrote generalized Maxwell equa- that can enhance the reduction of the gravitational field. tions that simultaneously treat weak gravitational and electromagnetic fields [28,29]. Superfluid coupled to gravity. It is well known that, Contents in general, the gravitational force is not influenced by any dielectric-type effect involving the medium. In the classi- 1 Introduction ...... cal case, this is due to the absence of a relevant number of 2 Weak field approximation ...... charges having opposite sign, which, redistributing inside the 2.1 Generalizing Maxwell equations ...... medium, might counteract the applied field. On the other side, 2.2 Generalizing London equations ...... if we regard the medium as a quantum system, the probability 3 Isotropic superconductor ...... of a graviton excitation of a medium particle is suppressed, 3.1 Time-dependent GinzburgÐLandau equations .. due to the smallness of gravitational coupling. This means 3.2 Solving dimensionless TDGL ...... that any kind of shielding due to the presence of the medium 3.2.1 Approximate solution ...... can only be the result of an interaction with a different state of 3.3 YBCO vs. Pb ...... matter, like a Bose condensate or a more general superfluid. 4 Conclusions ...... The nature of the involved field is also relevant for the Appendix: Sign convention ...... physical process. If the gravitational field itself is consid- References ...... ered as classical, it is readily realized that no experimental device Ð like the massive superconducting disk of the Pod- kletnov experiment [21,22] Ð can influence the local geome- 1 Introduction try so much as to modify the measured sample weight. This means that the hypothetical shielding effect should consist There is no doubt that the interplay between the theory of the of some kind of modification (or “absorption”) of the field in gravitational field and superconductivity is a very intriguing the superconducting disk. field of research, whose theoretical study has been involving Since the classical picture is excluded, we need a quan- many researchers for a long time [1Ð20]. Podkletnov and tum field description for the gravitational interaction [26,27]. Nieminem declared the achievement of experimental evi- In perturbation theory the metric gμν(x) is expanded in the standard way [30] a e-mail: [email protected] b e-mail: [email protected] gμν(x) = ημν + hμν(x) (1) 123 549 Page 2 of 12 Eur. Phys. J. C (2017) 77:549 as the sum of the flat background ημν plus small fluctuations regions, the gravitational field would tend to assume fixed encoded in the hμν(x) component. The Cooper pairs inside values due to some physical cutoff that prevents arbitrary the superconducting sample compose the Bose condensate, growth. The mechanism is similar to classical electrostatics described by a bosonic field φ with non-vanishing vacuum in perfect conductors, where the electric field is constrained expectation value φ0 =0|φ|0. to be globally zero within the sample. In the latter case, the The EinsteinÐHilbert Lagrangian has the standard form1 physical constraint’s origin is different (and is due to a charge redistribution), but in both cases the effect on field propaga- 1 tion and on static potential turns out to be a kind of partial Leh = (R − 2 ) , (2) 8πGn shielding. In accordance with the framework previously exposed, the where R is the Ricci scalar and is the cosmological con- superfluid density φ (x) is determined not only by the inter- stant. The part of the Lagrangian describing the bosonic field 0 nal microscopic structure of the sample, but also by the same φ coupled to gravity has the form magnetic fields responsible for the Meissner effect and the currents in the superconductor. The high-frequency compo- 1 μν ∗ 1 2 ∗ Lφ =− g ∂μφ ∂νφ + m φ φ (3) nents of the magnetic field can also provide energy for the 2 2 above gravitational field modification [26]. where m is the mass of the Cooper pair [26]. The previous calculation shows how Modanese was able ¯ If we expand the bosonic field as φ = φ0 + φ, one can to demonstrate, in principle, how a superfluid can determine a consider the v.e.v. φ0 as an external source, related to the gravitational shielding effect. In Sect. 3 we will quantify this structure of the sample and external electromagnetic fields, effect by following a different approach, as the GinzburgÐ while the φ¯ component can be included in the integration Landau theory for a superfluid in an external gravitational variables. The terms including the φ¯ components are related field. to graviton emissionÐabsorption processes (which we know to be irrelevant) and can safely be neglected in Lφ. Perturba- tively, the interaction processes involving the metric and the 2 Weak field approximation condensate are of the form Now we consider a nearly flat spacetime configuration, i.e. μν ∗ Lint ∝ h ∂μφ0 ∂νφ0 (4) an approximation where the gravitational field is weak and where we shall assume Eq. (1), that is, the metric gμν can be and give rise to (gravitational) propagator corrections, which expanded as: are again irrelevant. The total Lagrangian L = Leh + Lφ contains a further gμν ημν + hμν, (7) coupling between gμν and φ0, which turns out to be a contribution to the so-called intrinsic cosmological term given by where the symmetric tensor hμν is a small perturbation of . Explicitly, the total Lagrangian can in fact be rewritten as the flat Minkowski metric in the mostly plus convention,2 ημν = diag(−1, +1, +1, +1). The inverse metric in the lin- 1 ¯ ear approximation is given by L = Leh +Lφ = (R − 2 )+Lint +L0 +Lφ¯ , (5) 8πGn μν μν μν g η − h . (8) ¯ where Lφ¯ are the negligible contributions having at least one field φ¯ and where 2.1 Generalizing Maxwell equations

1 ∗ μ 1 2 2 If we consider an inertial coordinate system, to linear order L0 =− ∂μφ0 ∂ φ0 + m |φ0| , (6) 2 2 in hμν the connection is written as that is, a Bose condensate contribution to the total effec- λ 1 λρ tive cosmological term. This may produce slightly localized μν η ∂μhνρ + ∂ν hρμ − ∂ρ hμν , (9) 2 “instabilities” and thus an observable effect, in spite of the smallness of the gravitational coupling (4). The Ricci tensor (Appendix 1) is given by the contraction of The above instabilities can be found in the superconduc- the Riemann tensor tor regions where the condensate density is larger: in these σ Rμν = R μσν. (10) 1 We work in the “mostly plus” framework, η = diag(−1, +1, +1, +1), and set c = h¯ = 1 2 see Appendix 1 for definitions and sign conventions. 123 Eur. Phys. J. C (2017) 77:549 Page 3 of 12 549 and, to linear order in hμν, it reads We can impose a gauge fixing making use of the harmonic λ λ ¨ ¨ coordinate condition, expressed by the relation [30] Rμν ∂λ μν + ∂μ λν + ¨ − ¨ √ 1 ρ ρ μν μ = ∂μ∂ hνρ + ∂ν∂ hμρ ∂μ −gg = 0 ⇔ x = 0, (20) 2 1 ρ 1 − ∂ρ∂ hμν − ∂μ∂ν h where g ≡ det gμν ; it can be rewritten in the form 2 2 ρ 1 2 1 μν λ = ∂ ∂(μhν)ρ − ∂ hμν − ∂μ∂ν h, (11) g μν = 0, (21) 2 2 = σ wherewehaveusedEq.(9) and where h h σ . also known as the De Donder gauge. The Einstein equations have the form [30,31] Imposing the above condition and using Eqs. (7) and (9), in first-order approximation we find (E) 1 Gμν = Rμν − gμν R = 8πGn Tμν, (12) 2 1 μν λρ μν 1 ν μν 0 η η ∂μhνρ + ∂ν hρμ − ∂ρ hμν = ∂μh − ∂ h, and the term with the Ricci scalar R = g Rμν can be rewrit- 2 2 ten, in first-order approximation and using Eq. (11), as (22) 1 1 ρσ 1 ρ σ 2 that is, we have the condition gμν R ημν η Rρσ = ημν ∂ ∂ hρσ − ∂ h , 2 2 2 μν 1 ν μ 1 (13) ∂μh ∂ h ⇔ ∂ hμν ∂ν h. (23) 2 2 so that the l.h.s. of the Einstein equations in weak field Now, one also has approximation reads (E) 1 μ μ ¯ 1 μ ¯ 1 Gμν = Rμν − gμν R ∂ hμν = ∂ hμν + ημν h = ∂ hμν + ∂ν h, (24) 2 2 2 ∂ρ∂ − 1 ∂2 − 1 ∂ ∂ (μhν)ρ hμν μ ν h and, using Eq. (23), we find the so-called Lorenz gauge con- 2 2 1 ρ σ 2 dition: − ημν ∂ ∂ hρσ − ∂ h . (14) 2 ∂μ ¯ . If one introduces the symmetric tensor hμν 0 (25)

¯ 1 The above relation further simplifies Eq. (17)forGμνρ , which hμν = hμν − ημν h, (15) 2 takes the very simple form the above expression can be rewritten as ¯ Gμνρ ∂[ν hρ]μ , (26) (E) ρ ¯ 1 2 ¯ 1 ρ σ ¯ Gμν ∂ ∂(μhν)ρ − ∂ hμν − ημν ∂ ∂ hρσ 2 2 and verifies also the relation ρ ¯ ρ σ ¯ = ∂ ∂[ν hρ]μ + ∂ ∂ ημ[σ hν]ρ ρ ¯ σ ¯ = ∂ ∂[ν hρ]μ + ∂ ημ[ρ hν]σ , (16) ∂[λ|G0|μν] = 0 ⇒ G0μν ∝ ∂μAν − ∂νAμ, (27) where we have exchanged dumb indices in the last term of which implies the existence of a potential. the second line. Gravito-Maxwell equations. Now, let us define the If we now define the tensor fields3 [28] ¯ σ ¯ Gμνρ ≡ ∂[ν hρ]μ + ∂ ημ[ρ hν]σ , (17) 1 1 ¯ E ≡ Ei =− G i =− ∂[ hi] , (28.i) g 2 00 2 0 0 whose structure implies the property 1 ¯ G =−G , A ≡ Ai = h i , (28.ii) μνρ μρν (18) g 4 0 the Einstein equations can be rewritten in the compact form

3 = (E) = ∂ρG = π . For the sake of simplicity, we initially set the physical charge e Gμν μνρ 8 Gn Tμν (19) m = 1. 123 549 Page 4 of 12 Eur. Phys. J. C (2017) 77:549

2 ≡ = 1 ε jk G , 1 e Bg Bi i 0 jk (28.iii) εg = (35) 4 4πGn m2 = , , where obviously i 1 2 3 and and the vacuum gravitational permeability ¯ 1 ¯ ¯ 2 G0ij = ∂[i h j]0 = ∂i h j0 − ∂ j hi0 = 4 ∂[i A j]. (29) m 2 μ = 4πGn . (36) g c2 e2 One can immediately see that For example, on the Earth’s surface, Eg is simply the New- 1 jk jk B = εi 4 ∂[ j Ak] = εi ∂ j Ak =∇×A , tonian gravitational acceleration and the B field is related g 4 g g to angular momentum interactions [28,29,32Ð34]. The mass ⇒ ∇ · Bg = 0. (30) current density vector j can also be expressed as Then one also has g G = ρ , i i 00i T00 jg g v (37) ∇·E = ∂ Ei =−∂ =−8πGn = 4πGn ρ , g 2 2 g (31) where v is the velocity and ρg is the mass density. Gravito-Lorentz force. Let us consider the geodesic ρ ≡−T using Eq. (19) and having defined g 00. equation for a particle in the field of a weakly gravitating E If we consider the curl of g, we obtain object: G jk jk 00k ∇×Eg = εi ∂ j Ek =−εi ∂ j 2 λ μ ν 2 d x λ dx dx + μν = . 2 0 (38) 1 jk ¯ ds ds ds =− εi ∂ j ∂[ hk] 2 0 0 1 ∂Bg If we consider a particle in non-relativistic motion, the veloc- =− ∂ ε jk ∂ =−∂ =− . v i 4 0 i j Ak 0 Bi (32) i dx 4 ∂t ity of the particle becomes c dt .Ifwealsoneglect j vi v Finally, one finds for the curl of Bg terms in the form c2 and limit ourselves to static fields ∂t gμν = 0 , it can easily be verified that a geodesic equa- jk 1 jk m ∇×Bg = εi ∂ j Bk = εi εk ∂ j G0 m tion for a particle in non-relativistic motion can be written as 4 [35,36] 1 jm m j 1 j = δi δ − δi δ ∂ j G m = ∂ G ij 4 0 2 0 dv 1 μ 1 μ = Eg + v × Bg , (39) = ∂ G iμ + ∂ G i = ∂ G iμ − ∂ G i dt 2 0 0 0 0 2 0 0 00 ∂ 1 Ei which shows that the free fall of the particle is driven by = (8πGn T0i − ∂0G00i ) = 4πGn ji + 2 ∂t the analogous of a Lorentz force produced by the gravito- ∂ Eg Maxwell fields. = 4πGn jg + , (33) ∂t Generalized Maxwell equations. It is possible to define using again Eq. (19) and having defined jg ≡ ji ≡ T0i . the generalized electric/magnetic field, scalar and vector Summarizing, once defined the fields of (28) and having potentials containing both electromagnetic and gravitational restored physical units, one gets the field equations: term as 2 m m m m ρg E = E + E ; B = B + B ; φ = φ + φ ; ∇·E = 4πGn ρ = e g e g e g g 2 g ε e e e e g m ∇· = A = Ae + Ag, (40) Bg 0 e ∂B ∇×E =− g where m and e are the mass and electronic charge, respec- g ∂ t tively, and the subscripts identify the electromagnetic and 2 ∂ ∂ m 1 Eg 1 Eg gravitational contributions. ∇×Bg = 4πGn jg + = μg jg + , c2 e2 c2 ∂t c2 ∂t The generalized Maxwell equations then become (34) ∇· = 1 + 1 ρ formally equivalent to Maxwell equations, where E and E g εg ε0 B g are the gravitoelectric and gravitomagnetic field, respec- ∇·B = 0 tively, and where we have defined the vacuum gravitational ∂B ∇×E =− permittivity ∂t 123 Eur. Phys. J. C (2017) 77:549 Page 5 of 12 549

∂ 1 E fields and the vector potential the generalized form of defi- ∇×B = μg + μ0 j + , (41) c2 ∂t nition (40): where we have set m m e B = B + B , A = A + A , B =∇×A. (49) ρ = ρ, e e g e e g g m e j = j, (42) If A is minimally coupled to the wave function g m ε μ ψ = ψ iϕ,ψ2 ≡|ψ|2 = , and where 0 and 0 are the electric permittivity and mag- 0 e 0 ns (50) netic permeability in the vacuum. the second London equation can be derived from a quantum 2.2 Generalizing London equations mechanical current density The London equations for a superfluid in stationary state read i ∗ ∗ j =− ψ ∇˜ ψ − ψ∇˜ ψ , (51) [37Ð39]: 2m

m ∂j where ∇˜ is the covariant derivative for the minimal coupling: E = , e 2 (43.i) ns e ∂t ∇=∇−˜ i g˜ A, (52)

m B =− ∇×j, (43.ii) so that one has for the current e n e2 s i g˜ j =− ψ∗∇ψ − ψ∇ψ∗ − A |ψ|2 2m m where j = ns e vs is the supercurrent and ns is the superelec- 1 tron density. If we also consider Ampère’s law for a super- = |ψ|2 (∇ϕ −˜g A) . conductor in stationary state (no displacement current) m If we now take the curl of the previous equation, we ﬁnd

∇×Be = μ0 j, (44) m 1 B =− ∇×j =− ∇×j, (53) ˜ |ψ|2 ζ from (43.ii) and using vector calculus identities, we obtain g which is the generalized form of the second London Equation 2 2 ns e (43.ii). ∇×∇×Be =∇(∇·Be)−∇ Be =μ0 ∇×j=−μ0 Be, m To ﬁnd an explicit expression for ζ , we consider the case (45) Bg = 0 obtaining that is, m 1 B = B + @B =− ∇×j, (54) e e @g ζ 1 ∇2B = B , (46) e λ2 e e and, using (43.ii), (47) and (50), we ﬁnd where we have introduced the penetration depth ˜ = 2, 1 = μ λ2. g e 0 e (55) ζ λ = m . e 2 (47) = μ0 ns e Then we consider the case Be 0, so that we have

Using the vector potential Ae, the two London equations (43) m 2 2 m B = ZBZe + Bg =−μ0 λ ∇×j =−μ0 λ ∇×jg, can be summarized in the (not gauge-invariant) form e e e e (56) 1 j =− A B =∇×A . (48) μ λ2 e e e 0 e together with gravito-Ampère’s law (34) in the stationary state, Generalized London equations. If we now take into account gravitational corrections, we should consider for the ∇×Bg = μg jg, (57) 123 549 Page 6 of 12 Eur. Phys. J. C (2017) 77:549 so that, taking the curl of the above equation, we find GinzburgÐLandau equation for the superfluid order parame- ter in an external gravitational field. 1 ∇×∇×B =−∇2B = μ ∇×j =−μ B Let us restrict ourselves to the case of an isotropic super- g g g g g μ λ2 g 0 e conductor in the gravitational field of the earth and in absence 1 = = =− B , (58) of an electromagnetic field, we can take Ee 0 and Be 0. λ2 g g Moreover, Bg in the solar system is very small [40,41], there- E = m E B = where we have introduced the penetration depth fore e g and 0. Finally, we also have the relations φ = m φ = m e g and A e Ag , so we can write down our set of conditions: μ λ2 2 λ = 0 e = c . g μ π (59) m g 4 Gn mns E = 0, B = 0, B = 0 ⇒ E = E , B = 0; e e g e g Finally, using the stationary generalized Ampère’s law from (65.i) (41) and using Eq. (59) we find together with

λ2 e ∇×B = μ0 + μg j = μ0 1 + j, (60) m m λ2 φ = φg, A = Ag. (65.ii) g e e and taking the curl we obtain the general form The situation is not the same as the Meissner effect but, rather, as the case of a superconductor in an electric field. λ2 1 λ2 ∇2B =−μ + e ∇×j = μ + e B 0 1 2 0 2 1 2 λ μ0 λ λ g e g 3.1 Time-dependent GinzburgÐLandau equations 1 1 1 = + B = B, (61) λ2 λ2 λ2 Since the gravitoelectric field is formally analogous to an e g electric field we can use the time-dependent GinzburgÐ where we have defined a generalized penetration depth λ: Landau equations (TDGL) which, in the Coulomb gauge ∇· = A 0 are written in the form [42Ð48] λ λ λ λ = g e λ g 21 . e 10 (62) 2 λ2 + λ2 λe h¯ ∂ 2 ie g e + φ ψ − a ψ + b |ψ|2ψ (66.i) 2 m D ∂t h¯ 2 The general form of Eq. (48)is 1 2 e + ih¯ ∇+ A ψ = 0, 2 m c j =−ζ A B =∇×A , (63) 4π σ ∂A ihe¯ and, since charge-conservation requires the condition ∇×∇×A −∇×H =− + σ ∇φ + c c ∂t m ∇·j = 0, we obtain for the vector potential 4 e2 × ψ∗∇ψ − ψ∇ψ∗ + |ψ|2A , ∇·A = 0, mc where D is the diffusion coefficient, σ is the conductivity in that is, the so-called Coulomb gauge (or London gauge). the normal phase, H is the applied field and the vector field A is minimally coupled to ψ. The above TDGL equations for the variables ψ, A are derived minimizing the total Gibbs 3 Isotropic superconductor free energy of the system [37Ð39]. The coefficients a and b in (66.i) have the following form: In Sect. 1 we have shown how Modanese was able to theo- retically describe the gravitational shielding effect due to the a = a(T ) = a0 (T − Tc), presence of a superfluid. Now we are going to study the same b = b(T ) ≡ b(Tc), (67) problem with a different approach. Modanese has solved gravitational field equation where a0 , b being positive constants and Tc the critical temperature the contribution of the superfluid was encoded in the energy- of the superconductor. The boundary and initial conditions momentum tensor. In the following, we are going to solve the are 123 Eur. Phys. J. C (2017) 77:549 Page 7 of 12 549 ⎫ 2 e ⎪ ih¯ ∇ψ + A ψ · n = 0 ⎬⎪ and the boundary and initial conditions (68) become, in the c ∂ × ( , ); dimensionless form ∇× · = · ⎪ on 0 t A n H n ⎭⎪ ⎫ A · n = 0 (i ∇ψ + A ψ) · n = 0, ⎬ ψ( , ) = ψ ( ) ∇×A · n = H · n on ∂ × (0, t); x 0 0 x ⎭ ( , ) = ( ) on (68) A · n = 0 A x 0 A0 x ψ(x, 0) = ψ (x), 0 on . (72) where ∂is the boundary of a smooth and simply connected A(x, 0) = A0(x), domain in RN. Dimensionless TDGL. In order to write Eqs. (66) in a 3.2 Solving dimensionless TDGL dimensionless form, the following quantities can be introduced: If the superconductor is on the Earth’s surface, the gravitational field is very weak and approximately constant. This |a(T )| h 2(T ) = ,ξ(T ) = √ , (69.i) means that one can write b | ( )| 2 m a T φ =− , bmc2 g x (73) λ(T ) = , 4π |a(T )| e2 with

λ(T )κmg g = √ 1, (74) 2 ( ) D 4πμ0 |a(T )| h 2 e Hc T Hc(T ) = = √ , (69.ii) b 4 e 2πλ(T )ξ(T ) g being the acceleration of gravity. The corrections to φ in the superconductor are of second order in g and therefore they are not considered here. λ(T ) λ2(T ) 4πσD κ = ,τ(T ) = ,η= , Now we search for a solution of the form 2 (69.iii) ξ(T ) D ε0 c ψ(x, t) = ψ0(x, t) + g γ(x, t), where λ(T ), ξ(T ) and Hc(T ) are the penetration depth, A(x, t) = g β(x, t), coherence length and thermodynamic field, respectively. The φ(x) =−g x. (75) dimensionless quantities are then defined as At order zero in g,Eq.(71.i)gives x t ψ x = , t = ,ψ= , (70.i) ∂ψ0(x, t) λ τ + κ2 |ψ (x, t)|2 − 1 ψ (x, t) ∂t 0 0 ∂2ψ (x, t) and the dimensionless fields are written − 0 = 0, (76) ∂x2 A κ φκ H κ A = √ ,φ= √ , H = √ . (70.ii) with the conditions 2Hc λ 2Hc D 2Hc ψ0(x, 0) = 0,

Inserting Eqs. (70) in Eqs. (66) and dropping the prime gives ψ0(0, t) = 0, the dimensionless TDGL equations in a bounded, smooth ψ0(L, t) = 0, (77) and simply connected domain in RN [42,43]: where L is the length of the superconductor, here in units of ∂ψ λ, and t = 0 is the instant in which the material undergoes + i φψ + κ2 |ψ|2 − 1 ψ + (i ∇+A)2 ψ = 0, ∂t the transition to the superconducting state. (71.i) The static classical solution of Eq. (76)is κ x κ (x − L) ψ ( , ) ≡ ψ ( ) = √ √ , ∂A 0 x t 0 x tanh tanh ∇×∇×A −∇×H =−η +∇φ (71.ii) 2 2 ∂t (78) i − ψ∗∇ψ − ψ∇ψ∗ −|ψ|2A, 2 and, from (71.i), one obtains 123 549 Page 8 of 12 Eur. Phys. J. C (2017) 77:549 ∂γ(x, t) ∂2γ(x, t) and its explicit form reads − + κ2 3 |ψ (x)|2 − 1 ∂ ∂ 2 0 t x 1 γ(x, t) = ixψ0(x) (79) Eg(x, t) g at first order in g, with the conditions −P(x) t t −P( ) ∂ e P( ) = 1 − e x t − dtJ(x, t) e x t . γ(x, 0) = 0, ∂t η 0 γ(0, t) = 0, (89) γ( , ) = . L t 0 (80) The above formula shows that, for maximizing the effect of The first-order equation for the vector potential is written the reduction of the gravitational field in a superconductor, it is necessary to reduce η and have large spatial derivatives ∂β(x, t) of ψ0(x) and γ(x, t). The condition for a small value of η is η +|ψ (x)|2 β(x, t) + J(x, t) − η = 0, (81) ∂t 0 a large normal-state resistivity for the superconductor and a small diffusion coefficient, with the constraint vf D ∼ , (90) β(x, 0) = 0. (82) 3 v The second-order spatial derivative of β does not appear in where f is the Fermi velocity (which is small in HTCS) and

Eq. (81): this is due to the fact that, in one dimension, one is the mean free path: this means that the effect is enhanced has in “bad” samples with impurities, not in single crystals. If we consider the case J(x, t) = 0, given by the condition ∂ ∇2 A = ∇·A, (83) ∂x ψ0(x) = Im γ(x, t) ≡ Im γ(x) , (91) and therefore, in the Coulomb gauge we obtain the simpliﬁed equation

2 ∇×∇×A =∇(∇·A) −∇ A = 0. (84) ∂β(x, t) η +|ψ (x)|2 β(x, t) − η = 0, (92) ∂t 0 The quantity J(x, t) that appears in Eq. (81) is given by which is solved, together with the constraint (82), by the J(x, t) function ∂ ∂ = 1 ψ ( ) γ( , ) − γ( , ) ψ , 0 x Im x t Im x t 0 |ψ ( )|2 2 ∂x ∂x η − 0 x β(x, t) = − η t . 2 1 e (93) (85) |ψ0(x)| and the solution of Eq. (81)is Using Eqs. (88) and (75) we find 1 −P(x) t |ψ ( )|2 β(x, t) = 1 − e Eg − 0 x t P(x) = 1 − e η . (94) g −P(x) t t e P( ) − dtJ(x, t) e x t , (86) η 0 The above equation shows that, unlike the general case, in the absence of the contribution of J(x, t) the effect is bigger than with in the case of single crystal low-Tc superconductor, where η is large. |ψ (x)|2 P(x) = 0 . η (87) 3.2.1 Approximate solution Now, we have the form (78)forψ (x, t) and also the 0 From the experimental viewpoint, the greater are the length above (86)forβ(x, t) as a function of γ(x, t) through the and time scales over which there is a variation of E ,the definition of J(x, t): the latter can be used in (75) to obtain g easier is the observation of this effect. Actually, we started both ψ(x, t) and A(x, t) as functions of γ(x, t). from dimensionless equations and therefore the length and The gravitoelectric field can be found using the relation time scales are determined by λ(T ) and τ(T ) of Eqs. (69), ∂A which should therefore be as large as possible. In this sense, E =−∇φ − , (88) λ( ) g ∂t materials having very large T could be interesting for 123 Eur. Phys. J. C (2017) 77:549 Page 9 of 12 549 the study of this effect [49]. Moreover, Eq. (89)showsthe where 2 dependence of relaxation with respect to |ψ0(x)| through 1 ∂ ∂ the definition of P(x): one can see that |ψ0(x)| must be as J (x) = ψ (x) γ (x) − γ (x) ψ (x) , 0 κ2 0 ∂ 0 0 ∂ 0 small as possible and this implies that also κ must be small; 2 x x see Eq. (78). This also means that λ(T ) and ξ(T ) must both (98.i) be large. β( , ) Up to now we have dealt with the expression of x t ∂ as a function of γ(x, t). If we want to obtain an explicit R (x) = ω ψ (x) (ω x) − (ω x) ψ (x), n n 0 cos n sin n ∂ 0 expression for E ,wehavetosolveEq.(79)forγ(x, t):this x g (98.ii) is a difficult task which can be undertaken only in a numerical way. Nevertheless, if one puts ψ0(x) ≈ 1, which is a good approximation in the case of YBa2Cu3O7 (YBCO) in which −C2 −P( ) C2 e n t − P(x) e x t κ = 94.4, one can find the simple approximate solution: S (x, t) = n . (98.iii) n P( ) − C2 x n ∞ −C2 t By making the approximation γ(x, t) = i γ0(x) + i Qn sin (ωn x) e n , (95) n=1 ix γ(x) , (99) 2 κ2 with one finds the result x α α α γ (x) = 1 − cosh − x sech , (96.i) ( ) 0 κ2 1 −P(x) t J00 x 2 2 L 2 Eg(x, t) = 1 − e 1 − , (100) g η

where 1 L (−1)n Q = x γ (x) (ω x) = ∂ n d 0 sin n 2 (96.ii) 1 L 0 2 κ J (x) = ψ (x) − x ψ (x) . (101) 00 2 κ2 0 ∂x 0 (1) + (2) × 1 − ω Qn Qn , ω n C2 n n In spite of its crudeness, in the case of YBCO the above approximate solution (100) gives the same results of the and solution (97). Moreover, nothing changes signiﬁcantly if one neglects the ﬁnite size of the superconductor and uses √ C2 = ω2 + κ2,ω= π/ ,α= κ , √ n n 2 n n L 2 L (96.iii) ψ0(x) = tanh κx/ 2 (102)

instead of Eq. (78). 2 (1) 2 αω Q = (−1)n − cosh α + n sinh α, (96.iv) n C2 L n 3.3 YBCO vs. Pb

In the case of YBCO, the variation of the gravitoelectric field n E in time and space is shown in Figs. 1 and 2.Itiseasily (2) 2 α (−1) − cosh α g Q = (cosh α − 1) 1 + . seen that this effect is almost independent on the spatial coor- n L2 C2 sinh α n dinate. (96.v) The results in the case of Pb are reported in Figs. 3 and 4, which clearly show that, due to the very small value of κ, Taking into account Eq. (78) and inserting Eq. (95)inEq.(85) the reduction is greater near the surface. Moreover, in this and then in Eq. (86), we can find a new expression for the particular case, some approximations made in the case of E gravitoelectric field g: YBCO are no longer allowed: for example, the simplified ( ) relation (99) is not valid for small values of L. In fact, when 1 −P(x) t J0 x Eg(x, t) = 1 − e 1 − κ is small, the length L plays an important role and, in par- g η ∞ ticular, if L is small the effect is remarkably enhanced, as 1 + Q R (x) S (x, t), (97) shown in Fig. 5. In the same condition, a maximum of the η n n n = n=1 effect (and therefore a minimum of Eg) can occur at t 0, as 123 549 Page 10 of 12 Eur. Phys. J. C (2017) 77:549

x =10λ

x =2λ Eg g

x = λ

5 Fig. 1 The gravitational ﬁeld Eg/g as a function of the normalized 10 t/τ time and space for YBCO at T = 77 K Fig. 4 The gravitational ﬁeld as a function of the normalized time for increasing values of the x variable for Pb x = λ

x =10−2 λ E g 10−4 Eg g g L =10λ x =5· 10−3 λ

L =8λ

L =6λ

t/τ

Fig. 2 The gravitational ﬁeld as a function of the normalized time for t/τ increasing values of the x variable for YBCO Fig. 5 The gravitational ﬁeld Eg/g as a function of the normalized time in the case of Pb, for different values of L and x = 4 λ.The maximum of the shielding effect is evident

Table 1 YBCO vs. Pb YBCO Pb

Tc 89 K 7.2 K T 77 K 6.3 K − − ξ(T) 3.6 · 10 9 m1.7 · 10 7 m − − λ(T) 3.3 · 10 7 m7.8 · 10 8 m σ −1 4 · 10−7 m2.5 · 10−9 m [ T = 90 K ] [ T = 15 K ]

Hc(T) 0.2 Tesla 0.018 Tesla κ 94.4 0.48 − − τ(T) 3.4 · 10 10 s6.1 · 10 15 s Fig. 3 The gravitational ﬁeld E /g as a function of the normalized g η . · −2 . · 3 time and space for Pb at T = 6.3K 1 27 10 6 6 10

123 Eur. Phys. J. C (2017) 77:549 Page 11 of 12 549

Table 1 continued thank Fondazione CRT that partially supported this work for dott. A. Gallerati. YBCO Pb Open Access This article is distributed under the terms of the Creative D 3.2 · 10−4 m2/s1m2/s Commons Attribution 4.0 International License (http://creativecomm − − 6 · 10 9 m1.7 · 10 6 m ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 5 6 vf 1.6 · 10 m/s1.83 · 10 m/s and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. Table 2 YBCO and Pb

λτg i. YBCO Appendix: Sign convention T = 0K 1.7 · 10−7 m9.03 · 10−11 s2.6 · 10−12 T = 70 K 2.6 · 10−7 m2.1 · 10−10 s9.8 · 10−12 We work in the “mostly plus” convention, where T = 77 K 3.3 · 10−7 m3.4 · 10−10 s2· 10−11 η = diag(−1, +1, +1, +1). (A.1) T = 87 K 8 · 10−7 m2· 10−9 s2.8 · 10−7 ii. Pb We define the Riemann tensor as T = 0K 3.90 · 10−8 m1.5 · 10−15 s1· 10−17 σ σ σ σ ρ σ ρ − − − = ∂ − ∂ + − T = 4.20 K 4.3 · 10 8 m1.8 · 10 15 s1.4 · 10 17 R μλν λ μν ν μλ ρλ νμ ρν λμ σ σ ρ T = 6.26 K 7.8 · 10−8 m6.1 · 10−15 s8.2 · 10−17 = 2 ∂[λ ν]μ + 2 ρ[λ ν]μ, (A.2) = . . · −7 . · −14 . · −15 T 7 10 K 2 3 10 m53 10 s22 10 where λ λμ νρ = g μνρ, can be seen in the same figure. In the extreme case L = 6 λ, 1 we found that the system returns to the unperturbed value μνρ = ∂ρ gμν + ∂ν gμρ − ∂μgνρ . (A.3) 5 2 after a time t0 10 τ. Table 1 reports the values of the parameters of YBCO The Ricci tensor is defined as a contraction of the Riemann and Pb, calculated at a temperature T such that the quantity tensor T−Tc T is the same in the two materials. The calculated values σ c Rμν = R μσν , (A.4) of λ, τ and g at different temperatures are shown in Table 2. the Ricci scalar is given by

μν 4 Conclusions R = g Rμν , (A.5)

It is clearly seen that λ and τ grow with the temperature, (E) and the so-called Einstein tensor Gμν has the form so that one could think that the effect is maximum when the temperature is very close to the critical temperature T . c (E) 1 T Gμν ≡ Rμν − gμν R . (A.6) However, this is true only for low- c superconductors (LTSC) 2 because in high-Tc superconductors fluctuations are of pri- mary importance for some Kelvin degree around Tc.The The Einstein equations are written presence of these opposite contributions makes it possible ≤ (E) 1 that a temperature Tmax Tc exists at which the effect is Gμν ≡ Rμν − gμν R = 8πGn Tμν , (A.7) maximum. In all cases, the time constant tint is very small, 2 and this makes the experimental observation rather difficult. where Tμν is the total energy-momentum tensor. The cosmo- Here we suggest to use pulsed magnetic fields to destroy logical constant contribution can be pointed out splitting the and restore the superconductivity within a time interval of tensor Tμν into the matter and component: the order of tint. The main conclusion of this work is that the reduction of the gravitational field in a superconductor, if it (M) () (M) Tμν = Tμν + Tμν = Tμν − gμν, (A.8) exists, is a transient phenomenon and depends strongly on 8πGn the parameters that characterize the superconductor. so that the Einstein equation can be rewritten as Acknowledgements This work was supported by the Competitive- ness Program of the National Research Nuclear University MEPhI 1 (M) () Rμν − gμν R = 8πGn Tμν + Tμν , (A.9) of Moscow for the contribution of prof. G. A. Ummarino. We also 2 123 549 Page 12 of 12 Eur. Phys. J. C (2017) 77:549 or, equivalently, 25. C. Unnikrishnan, Does a superconductor shield gravity? Phys. C Supercond. 266(1Ð2), 133Ð137 (1996) 1 (M) 26. G. 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