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 [23–25], 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 Ginzburg–Landau 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 Ginzburg–Landau 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 – like the massive superconducting disk of the Pod- kletnov experiment [21,22] – 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 [1–20]. 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 Einstein–Hilbert 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 Ginzburg– 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 emission–absorption 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.

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