Superconductor Meissner Effects for Gravito-Electromagnetic Fields in Harmonic Coordinates Due to Non-Relativistic Gravitational Sources

Abstract

There is much discrepancy in the literature concerning the possibility of a superconductor expelling gravito-electromagnetic fields just as it expels electromagnetic fields in the Meissner effect. Contradicting results are found in at least 18 papers written collectively by more than 20 authors and published over the course of more than 55 years (from 1966 to the present year of 2022). The primary purpose of this paper is to carefully explain the reason for the discrepancies, and provide a single conclusive treatment which may bring coherence to the subject. The analysis begins with a covariant Lagrangian for spinless charged particles (Cooper pairs) in the presence of electromagnetic fields in curved space-time. It is known that such a Lagrangian can lead to a vanishing Hamiltonian. Alternatively, it is shown that using a “space + time” Lagrangian leads to an associated Hamiltonian with a canonical momentum and minimal coupling rule. Discrepancies between Hamiltonians obtained by various authors are resolved. The canonical momentum leads to a modified form of the London equations and London gauge that includes the effects of gravity. A key result is that the gravito-magnetic field is expelled from a superconductor with a penetration depth on the order of the London penetration depth only when an appropriate magnetic field is also present. The gravitational flux quantum (fluxoid) in the body of a superconductor, and the quantized supercurrent in a superconducting ring, are also derived. Lastly, the case of a superconducting ring in the presence of a charged rotating mass cylinder is used as an example of applying the formalism developed.

1 Introduction

Over the course of many decades, researchers have investigated a wide variety of new theoretical gravitational effects and the possibility of experiments that can detect these effects. Many of these include the use of superconductors [1–3], [11–27], [30–35], [38–47], [49, 50, 58, 59, 62], [63–66], [71–78], [80], [83–91], [100, 101, 104, 105], [109–111]. Some of the earliest interest in this topic is exhibited by the founding of the Gravity Research Foundation (GRF) in 1949 by Roger Babson. “His views were reflected by the wording in the announcement of the first essay competition that said the awards were to be given for suggestions for anti-gravity devices, for partial insulators, reflectors, or absorbers of gravity, or for some substance that can be rearranged by gravity to throw off heat - although not specifically mentioned in the announcement, he was thinking of absorbing or reflecting gravity waves.” [1]

Yet by 1953, the winning GRF essay by Bryce DeWitt dispensed with such endeavors as unrealizable. DeWitt expressed doubt that a material which absorbs or reflects gravity exists [2]. He states, “... first fix our sights on those grossly practical things, such as ‘gravity reflectors’ or ‘insulators,’ or magic ‘alloys’ which can change ‘gravity’ into heat, which one might hope to find as the usual by-products of new discoveries in the theory of gravitation... Of primary importance is the extreme weakness of gravitation coupling between material bodies ... The weakness of this coupling has the consequence that schemes for achieving gravitational insulation, via methods involving fanciful devices such as oscillation or conduction, would require masses of planetary magnitude.”

However, in 1966 DeWitt’s viewpoint apparently changed when he published a highly influential paper predicting the gravito-magnetic field (also known as frame-dragging or Lense-Thirring field) is expelled from superconductors in a Meissner effect [3]. It is evident DeWitt considered quantum mechanical systems (such as superconductors) may possess properties that were not previously considered when the statement was made barring the possibility of gravitational reflectors or insulators.

In [3], DeWitt begins with the Lagrangian for a relativistic spinless charged particle in an electromagnetic field in curved space-time and develops the associated Hamiltonian. He identifies a minimal coupling rule involving a gravitational vector potential and concludes this result implies the associated gravitational field must be expelled from a superconductor, just as the magnetic field is expelled from a superconductor in the Meissner effect.

1.1 A Summary of the Sections in This Paper

Section 2 begins with a covariant Lagrangian for a relativistic spinless charged particle in an electromagnetic field in curved space-time. The Euler-Lagrange equation of motion leads to the geodesic equation of motion modified by the Lorentz four-force in curved space-time. Although the equation of motion correctly describes the dynamics of the particle, the associated Hamiltonian is identically zero and therefore cannot be used to describe a quantum mechanical system such as a superconductor.

Alternatively, a “space + time” Lagrangian is used to obtain a canonical three-momentum and Hamiltonian valid to all orders in the metric. The result is compared to DeWitt’s in Eq. 1. Some relevant metric relationships will be used to match the Hamiltonian with that of other authors for confirmation of its validity. The Hamiltonian is then expanded to first order in the metric to show the lowest order coupling of the momentum, electromagnetic fields, and gravity. Again, the result is compared to DeWitt’s result in Eq. 2. The Hamiltonian is further simplified by introducing the trace-reversed metric perturbation and assuming non-relativistic gravitational sources. Alternative Hamiltonians are also discussed as well as the conditions necessary for a Hamiltonian to be quantized.

In Section 3, gravito-electric and gravito-magnetic fields are defined in terms of the metric perturbation. Using the stress tensor of a non-relativistic ideal fluid, and the linearized Einstein field equation in harmonic coordinates, leads to gravito-electromagnetic field equations. In addition, the canonical momentum is used to develop constitutive equations for the supercurrent. These lead to a modified set of London equations describing the interaction of electromagnetic and gravito-electromagnetic fields with a superconductor. A modification to the London gauge condition is also identified.

In Section 4, the constitutive equations are used in the field equations to identify a penetration depth associated with the magnetic field and the gravito-magnetic field. It is found that the usual London penetration depth for the magnetic field is modified by the presence of a gravito-magnetic field, however, the modification is miniscule. It is also found that in the absence of a magnetic field, the superconductor demonstrates a paramagnetic effect rather than a diamagnetic (Meissner) effect for the gravito-magnetic field. In other words, the gravito-magnetic field is not expelled. However, when the magnetic field and the gravito-magnetic field are both present, it is possible for the gravito-magnetic field to be expelled with a penetration depth on the same order as the London penetration depth. However, it is demonstrated that the gravito-magnetic field is a coordinate-dependent quantity and therefore effects associated with it can be made to vanish with an appropriate coordinate transformation.

In Section 5, it is found that the usual London penetration depth for the electric field is modified by the presence of a gravito-magnetic field, however, again the modification is miniscule. In the process of developing these results, a penetration depth is also found for a field defined as the linear combination of the magnetic and gravito-magnetic vector potentials. However, since the gravito-magnetic vector potential is time-independent in this approximation, then it is shown not to have an associated penetration depth. This is expected since it is known that the Newtonian gravitational field generally cannot be shielded.

In Section 6, the new minimal coupling rule obtained in Section 2 will be used to write the Ginzburg-Landau supercurrent with coupling to electromagnetism and gravity. The complex order parameter must be single-valued around a closed path according to the Byers-Yang theorem. Then using the fact that all the fields vanish within the body of the superconductor leads a quantization condition for the flux of the magnetic and gravitational fields.

Lastly, in Section 7, the canonical momentum is used to develop an expression for the Ginzburg-Landau phase around a superconducting ring. Once again, using the fact that the wave function is single-valued around a closed path leads to a quantization condition involving the flux of the magnetic and gravitational fields, as well as the supercurrent around a superconducting ring. A charged, rotating mass cylinder is introduced as a source for electromagnetic and gravitational fields. The effect of placing the rotating cylinder coaxial with the superconducting ring is carefully analyzed. It is argued that the electric current predicted by DeWitt in Eq. 4 is not induced. Rather, the supercurrent in the ring is quantized along with the flux of electromagnetic and gravitational fields through the ring.

2 The “Space + Time” Hamiltonian for a Charged Spinless Particle in Curved Space-Time

Using an action of the form , where τ is proper time, leads to a Lagrangian for a relativistic spinless charged particle in an electromagnetic field in curved space-time that can be written as¹where u^μ = dx^μ/dτ is the four-velocity, is the electromagnetic four-potential, and m and q are the rest mass and charge of the test particle, respectively. It is well known that using Eq. 5 in the Euler-Lagrange equation of motionleads to the geodesic equation of motion modified by the Lorentz four-force in curved space-timewhere are the metric connections (Christoffel symbols).² This demonstrates that the Lagrangian in Eq. 5 correctly characterizes the dynamics in a covariant form. Evaluating a canonical four-momentum, , and using q = −e leads to p_μ = P_μ − eA_μ. This implies a covariant minimal coupling rule given by P^μ → P^μ − eA^μ.

However, it is known that using Eq. 5 in a covariant Legendre transformation, , leads to a Hamiltonian that is identically zero and therefore cannot have the interpretation of energy. This issue of a vanishing Hamiltonian is discussed in Jackson [7] and Barut [8] in the context of flat space-time, and by Bertschinger [9, 10] in curved space-time. The problem stems from the fact that u^μu_μ = −c² imposes an additional constraint on the Lagrangian in Eq. 5. This issue is discussed in further detail in the subsection, “Alternative methods of obtaining the Hamiltonian” found further below.

For the purposes of this paper, we will take an approach similar to [3, 11] by reparametrizing the action for the Lagrangian in Eq. 5 in terms of coordinate time rather than proper time.³ Note that the four-velocity can be written as u^μ = γv^μ, where the Lorentz factor is γ = dt/dτ, the coordinate velocity is , and t is the coordinate time. Then the action for the Lagrangian in Eq. 5 can be written as

Therefore the action can be written as , where the “space + time” Lagrangian is

This is the Lagrangian used by DeWitt [3] except he uses the notation and sets c = 1. DeWitt and other authors such as [11] leave the electromagnetic field in the Lagrangian Eq. 11 as A_μv^μ instead of qg_μνA^μv^ν which masks the explicit coupling of gravity to the electromagnetic field. Using g_μνu^μu^ν = −c², the Lorentz factor in curved space-time can be evaluated asThen the canonical three-momentum, P_i = ∂L/∂vⁱ, can be found from Eq. 11 to be

Note that p_μ = g_μνp^ν and lead to which is the first term on the right side of Eq. 13. Also note that A_μ = g_μνA^ν leads to

Therefore Eq. 13 can be written as simply p_i = P_i − eA_i which is consistent with the covariant canonical momentum relationship, p_μ = P_μ − eA_μ. Similarly, in [10] a covariant canonical momentum of the form P_μ = mg_μνv^ν + qA_μ appears. In [13], a canonical momentum is shown as P_i = mg_iju^j − eA_i, where u^j = γv^j. This form is missing the term involving γmcg_0i in Eq. 13 which may be because the canonical momentum in [13] is not formally derived from a Lagrangian but obtained by replacing the canonical momentum in flat space-time with a form proposed for curved space-time. Also [14, 15], have a result similar to Eq. 13 but without the terms involving A⁰ and Aⁱ. This is due to starting from a Lagrangian similar to Eq. 11 but without the electromagnetic field.

Using g_μν = η_μν + h_μν, where η_μν is the Minkowski metric of flat space-time, and h_μν is a perturbation, makes Eq. 13 become

In [16], a result similar to Eq. 15 is obtained but with h_0i = 0 and γ = 1 due to using transverse-traceless coordinates and remaining to lowest order in vⁱ. Likewise, a result similar to Eq. 15 appears in [17–27], but the terms involving h_ijv^j, h_0iA⁰, and h_ijA^j are all missing⁴.

Note that in the absence of electromagnetism and gravity, the canonical momentum reduces to P_i = γmv_i. Therefore, Eq. 15 implies that the presence of electromagnetism and gravity leads to a minimal coupling rule given by

For an electron this is the usual minimal coupling rule, P_i → P_i − eA_i. As previously stated, the first term on the right side of Eq. 13 can be identified as

Using a Legendre transformation, H = P_kv^k − L, requires solving Eq. 17 for vⁱ which requires constructing the inverse of g_ik. This is shown in of Supplementary Appendix SA to bewhich satisfies . Then the velocity and the Lorentz factor can be expressed as, respectively,

Expressions equivalent to Eq. 19 can also be found in [11, 14, 15]. Inserting Eq. 19 into Eq. 17, and making use of Eq. 13 with q = −e, makes the Hamiltonian become

This can be considered a “space + time” Hamiltonian for a relativistic spinless charged particle in an electromagnetic field in curved space-time⁵. The result is exact in the metric perturbation and particle velocity. To compare with DeWitt’s result in Eq. 1, we can use Eq. 14 to write the Hamiltonian as

This matches DeWitt’s result in Eq. 1 except that DeWitt uses g^jk instead of . The same issue also appears in [30] where a Legendre transformation is applied to the same Lagrangian used in Eq. 11. The Hamiltonian in Eq. 1 is also quoted and utilized in [31]. However, it is shown in Supplementary Appendix SA that is only true to first order in the metric perturbation. This issue is properly recognized in [11].

In the literature, the “space + time” Hamiltonian appears in several other forms. To compare with Eq. 20, the following metric relationships developed in Supplementary Appendix SA can be used:

Using Eq. 22 in Eq. 20 leads to a form that matches Cognola, Vanzo, and Zerbini [11].

It is stated in [11] that Eq. 23 and DeWitt’s result in Eq. 1 are not equivalent when there is a nonvanishing g_0i. However, it is demonstrated here that Eq. 20, which is the corrected form of DeWitt’s approach, is indeed equivalent to Eq. 23via the use of Eq. 22. Also, substituting Eq. 18 into Eq. 23 leads to a result that matches Bertschinger [9], except [9] does not include electromagnetic fields.

With some algebraic manipulation, this Hamiltonian also matches the result derived by Piyakis, Papini, and Rystephanick [32].

Therefore, the discrepancies between all of the authors stated above are resolved.

2.1 The Weak-Field, Low-Velocity Limit of the Hamiltonian

The lowest order expansion of Eq. 20 is now considered. As discussed below Eq. 7, the canonical four-momentum is found to be P_μ = p_μ + eA_μ. Using the metric perturbation leads to

This shows that to lowest order, . It can also be assumed that pⁱ ∼ eAⁱ since the vector potential drives the supercurrent. Then for approximation purposes, P_i can be treated the same as pⁱ.

Following the procedure of Supplementary Appendix SB, but remaining to first order in the perturbation and second order in momentum, leads to

To compare with DeWitt’s result in Eq. 2, the indices of the electromagnetic potentials can be lowered using Eq. 14. Remaining to first order in the perturbation leads to

Although Eq. 28 has the benefit of being more compact than Eq. 27, it can be misleading due to the fact that A_i and A₀ contain metric perturbation terms as demonstrated by Eq. 14. In fact, terms that are second order in could be falsely interpreted as containing the metric perturbation to second order which is inconsistent with the first order expansion used to obtain the Hamiltonian. Therefore, it can be argued that Eq. 27 is preferred since all metric perturbations are explicitly shown to first order

The Hamiltonian in Eq. 31 can also be written using the trace-reversed metric perturbation, , where and . It will be shown in the following section that using the harmonic coordinate condition, , leads to for non-relativistic gravitational sources. Then the components of the perturbation are

A gravito-scalar potential and gravito-vector potential can also be defined as, respectively,

For brevity, a minimally coupled canonical momentum can be defined as , where . Also using A⁰ = φ/c in Eq. 27 leads to

Terms are listed from largest to smallest magnitude based on the order-of-magnitude analysis found in Supplementary Appendix SB. Terms within an order of magnitude are grouped in parentheses. It is shown that using the maximum momentum permitted for Cooper pairs (which requires keeping the kinetic energy below the BCS energy gap), and estimating the largest value for hⁱ that can be reasonably produced in a lab, then hⁱP_i ∼ 10^–47 J while J. Therefore the last term in Eq. 31 can be dropped and still maintain consistency. However, it is also shown in Supplementary Appendix SB that an expansion to second order in the perturbation, and fourth order in the momentum, introduces seven more terms that are several orders of magnitude larger than hⁱP_i. Therefore, a consistent approximation leads to the following result for the weak-field, low-velocity limit of the Hamiltonian.where A² = AⁱAⁱ. Note that the terms involving , and (which come from a second order expansion) are all larger than hⁱP_i (which comes from a first order expansion). We can attempt to eliminate these terms to obtain a result similar to DeWitt’s Eq. 2. However, it is shown in Supplementary Appendix SB that keeping hⁱP_i and eφ, while eliminating other terms that do not appear in Eq. 2, is only possible if the electromagnetic potentials are reduced to absurdly small (but non-zero) values: A ∼ 10^–32 T⋅m and φ ≪ 10^–44 V. Then the second order Hamiltonian found in Supplementary Appendix SB is

Again, terms are listed from largest to smallest magnitude. Notice the term involving still remains since it is many orders of magnitude larger than hⁱP_i. This is due to the fact that φ_G is produced by earth and hence is outside experimental control. Note that the last three terms cannot be combined into a single expression of the form since 4ehⁱAⁱ does not appear in the Hamiltonian. Therefore Eq. 33 is arguably the closest to DeWitt’s Hamiltonian in Eq. 2 that can be obtained.

2.2 Alternative Methods of Obtaining the “Space + Time” Hamiltonian

The following is a discussion of the various covariant Hamiltonians that describe the same particle dynamics, and the differing ways that a “space + time” Hamiltonian has been derived by various authors. It is emphasized that all of these approaches found throughout the literature are shown here to be equivalent. The issue of a vanishing covariant Hamiltonian, and the issue of singular Lagrangians is also discussed.

It was stated after Eq. 7 that using a Lagrangian, , in the form shown in Eq. 5, and applying a covariant Legendre transformation, , leads to a vanishing Hamiltonian. This occurs due to the constraint u^μu_μ = −c². Therefore, cannot have the interpretation of energy.⁹ The problem is recognized in [11] where it is stated that instead of dealing with the constrained system, the choice is made to give up the general covariance of and instead use the reparametrized Lagrangian, L₁ in Eq. 11 which is claimed to be nonsingular. Then [11] obtains the “space + time” Hamiltonian in Eq. 23 and claims that it is a correction to DeWitt’s result in [3] .

However, it is demonstrated in [51] that even L₁ is singular due to the constraint u^μu_μ = −c². It is also shown in this paper that the Hamiltonians in Eq. 20 and Eq. 23 are equivalent. Therefore, contrary to the statement in [11], it is demonstrated here that both Eq. 20 and Eq. 23 are valid expressions of the “space + time” Hamiltonian despite the fact that they are both derived from a singular Lagrangian, L₁.

Furthermore [9], shows that a covariant Lagrangian of the form will lead to . This demonstrates that abandoning covariance is not necessary to obtain a non-vanishing Hamiltonian. Also, both and lead to equivalent covariant Hamilton canonical equations of motion. Therefore they are equivalent means of describing the same particle dynamics. It will also be shown below that leads to the same “space + time” Hamiltonian as .

As an extension to this approach, it was shown in [12] that electromagnetic fields can also be included by writing . Like the case of , this leads to a covariant minimal coupling rule given by p_μ → P_μ − eA_μ. However, unlike which vanishes, it is found that which further confirms the validity of the covariant minimal coupling rule.

This topic is also discussed in [8] where two covariant Hamiltonians are shown. One of them is and the other can be described as . They both obey the constraint , where f^μ is a function of the external fields and velocities.¹⁰ Furthermore, it is shown that and . Both Hamiltonians lead to the same canonical equations of motion but with the added feature of including f^μ for external fields.

Lastly, note that the method used in this paper to obtain the “space + time” Hamiltonian in Eq. 20 is different from the method used by [12] to obtain the “space + time” Hamiltonian in Eq. 23. In this paper, is used to obtain the canonical momentum. The velocity is also solved in terms of the inverse metric in Eq. 19. Then the velocity and canonical momentum are both substituted into a Legendre transformation¹¹ to obtain Eq. 20. However [12], starts from and reparametrizes in terms of coordinate time to obtain L₂ = mv^μp_μ + eA^μv_μ. A Legendre transformation, H₂ = P_ivⁱ − L₂, is again used. However, rather than substituting for the velocity, instead the Lagrangian and kinetic four-momentum, p_μ = P_μ − eA_μ, are used to write the Hamiltonian as H₂ = −cP₀. Then using , solving algebraically for P₀, and substituting back into H₂ leads to Eq. 23. The key feature of this approach is that the constraint p^μp_μ = −m²c² is utilized explicitly due to , versus the case of which vanishes.¹² All of these examples demonstrate that there are numerous ways of obtaining the same “space + time” Hamiltonian appearing in Eq. 20 and equivalently Eq. 23-Eq. 25.

2.3 Quantization of the Canonical Momentum and Hamiltonian

Concerning the quantization of a classical Hamiltonian, it is common to simply promote canonical quantities to quantum operators. Then xⁱ and P_j are replaced with and which satisfy the canonical quantization condition . However, the canonical quantization rule developed by Dirac [52–54] involves a formal procedure to go from Poisson brackets to commutators, .

In [8], it is shown that the general theory of canonical transformations and Poisson brackets in Hamilton-Jacobi theory can be put in covariant form for a single particle. However, it is noted that a relativistic theory of multiple interacting particles introduces the complication that we cannot assume a common proper time for all particles.

In the case of Eq. 32, the issue of proper time is avoided by using a “space + time” approach, as well as the Lorentz factor obtained in Eq. 19. The Lorentz factor essentially removes proper time in favor of coordinate time and the metric. Therefore, Eq. 32 can be generalized to a collection of particles which are assumed to be embedded in the same space-time metric, with velocities measured by an observer in terms of a single coordinate time. This is particularly relevant to the fact that DeWitt states in [3], “For the Hamiltonian of the ensemble of free electrons inside a superconductor,Eq. 2is replaced bywhere and are the canonical variables of the nth electron and V_int includes the electron-phonon interaction and the phonon energy.”

In [8], it is also pointed out that using versus in a quantum theory leads to the additional complication of taking the square root of operators, as in the case of Dirac theory (involving anti-commuting matrices). This would seem to favor H₂ for the purposes of describing a quantum system. However, using a “space + time” approach, bothH₂ and H₃ lead to Eq. 20 and Eq. 23, which still involve square roots. Therefore, dealing with the issue of square roots (without resorting to anti-commuting matrices) is generally achieved by expanding the Hamiltonian in powers of h_μν and P_i, as is done to obtain Eq. 28. It is assumed in the remainder of this paper that P_i and H can be promoted to the usual quantum mechanical operators, and , respectively. However, for a formal treatment of quantizing manifestly Lorentz-invariant Lagrangians, see [55, 56].

Lastly, a modified Schrödinger equation can be obtained from Eq. 33. Dropping the rest mass energy and acting the Hamiltonian on a wave function, , leads to

In the absence of gravity, this reduces to the usual Schrödinger equation in the presence of an electromagnetic field. In that case, the solution could be written as , where is the solution to the field-free Schrödinger equation, and the phase is . If the presence of gravity introduced a modification to the Hamiltonian given by , then the phase would becomesimilar to what is shown in [14, 15, 30, 35, 40, 42, 44, 47]. However, this is not how the presence of gravity appears in Eq. 35. Rather, gravity enters as a multiplicative factor to a first derivative term, hⁱ∂_i, not an additive factor inside the second derivative term. In fact, attempting to use ψ = e^iϕψ₀, with a phase given by Eq. 36, leads to and . Substituting these into Eq. 35 does not lead to the field-free Schrödinger equation.

This topic can be understood in the context of a covariant Ginzburg-Landau model [50, 57–63]. The transformations of a complex scalar field and the four-potential can be written, respectively, as and A_μ → A_μ + ∂_μχ, where χ is an arbitrary scalar function. In flat space-time, the gauge covariant derivative is which also transforms as . Therefore, the gauge freedom of ψ and A_μ can be unified in a fully covariant approach. Notice that the gauge freedom of A_μ leads effectively to a phase factor e^iqχ in D_μ. This is analogous to the presence of Aⁱ in Eq. 35 leading to a phase factor e^iϕ in .

In curved space-time, the gauge covariant derivative becomes , where ∇_μ is the covariant derivative involving Christoffel symbols [10, 11, 47, 58, 59, 62], [64–66]. However, the gauge covariant derivative still transforms as even in curved space-time. This means that the phase factor e^iqχ is unaffected by the presence of gravity. This is because in General Relativity (or any metric theory of gravity), the gravitational “field” is not an additional field on the same footing as electromagnetism, rather it introduces curvature to the otherwise flat space-time that all other fields live in.¹³ This is represented by the fact that ∂_μ → ∇_μ. Furthermore, the spatial part of the gauge covariant derivative is , where ∇_i is intended to denote a derivative involving Christoffel symbols, not just a gradient. Notice that the covariant derivative is not as stated in [25], [71–75].

In fact, since the gauge covariant derivative transforms as , and the wave function transforms as , then incorrectly assuming the solution to Eq. 35 is ψ = e^iϕψ₀, where the phase is Eq. 36, gives the false impression that transforms as . However, and do not have identical transformation properties. To linear order, the gauge (coordinate) freedom of the perturbation is h_μν → h_μν + ∂_μξ_ν + ∂_νξ_μ, where ξ^μ is associated with a coordinate transformation such as x^μ → x^μ + ξ^μ. This is contrary to what is shown in [40] where the gauge freedom of is written as , where μ is an arbitrary scalar function.¹⁴

A possible cause for confusion is the fact that quantizing the canonical momentum Eq. 16 leads to

This seems to imply that is the gauge covariant derivative to be used in the Schrödinger equation, . However, Eq. 35 demonstrates that this is not the case.

In fact, since Eq. 35 involves a second order spatial derivative, then this issue can be further understood by starting from a second order covariant derivative acting on a scalar wave function in the absence of electromagnetic fields: g^μν∇_μ∇_νψ. Since ψ is a scalar, then the first covariant derivative of ψ is just a partial derivative: ∇_νψ = ∂_νψ. However, since ∂_νψ is a vector, then acting the second covariant derivative brings in the Christoffel symbol: . Using g_μν = η_μν + h_μν, choosing harmonic coordinates, , and remaining to first order in the perturbation leads to

Similar to Eq. 35, it is found that h_0i multiplies a first derivative term, rather than appearing as an additive factor inside the second derivative term. Furthermore, Eq. 38 can be used to write the Klein-Gordon equation in curved space-time as , where k_c = mc/ℏ is the Compton wave number [15, 65, 66, 76]. Since the Klein-Gordon equation is , with , then the Schrödinger equation can be obtained by taking the non-relativistic limit. Using g_μν = η_μν + h_μν and leads to

Again, notice that h_0i is multiplying which is consistent with Eq. 35 and Eq. 38.

In summary, the distinction between the way gravity and electromagnetism couple to a quantum field is fundamentally the reason for the difference in the way they appear in the modified Schrödinger equation Eq. 35, and the reason why it is not the case that ψ = e^iϕψ₀, where ψ₀ is the solution to the field-free Schrödinger equation if the phase is assumed to be Eq. 36. However, it will be shown in Sections 6, 7 that the canonical momentum Eq. 13 can be used to obtain a quantum phase with terms that include those appearing in Eq. 36.

2.4 Alternative Hamiltonians and Methods of Coupling Gravity to a Superconductor

On a more fundamental level, and are derived respectively from Lagrangians, and , which are both associated with the geodesic equation of motion Eq. 7. However, the geodesic equation of motion is subject to the Equivalence Principle which makes it possible to transform into a frame of reference where all gravitational effects vanish. Stated another way, there is no unique absolute motion of a test particle since the observed motion of a particle depends on the frame of reference of the observer.

However, the relative motion between a particle and an observer is uniquely described by the geodesic deviation equation. Therefore, it is argued in [77, 78] that a Hamiltonian based on the geodesic deviation equation is preferred (which involves the Riemann curvature tensor) versus a Hamiltonian based on the geodesic equation (which involves the Christoffel symbols). A similar approach was used by Weber¹⁵ in the context of gravitational wave detection [79].

In fact, it is shown in [78] that for the case of gravitational waves (in the weak field, low velocity limit), the Lagrangian leading to the geodesic deviation equation is , where xⁱ is the coordinate distance of the test particle from the observer. The associated interaction Hamiltonian is . This result clearly differs from Eq. 31 which predicts the interaction Hamiltonian is . This discrepancy in Hamiltonians is directly parallel to the fact that the geodesic equation of motion (for the same conditions) is , while the geodesic deviation equation is . Note that the latter is what is physically observed by a gravitational wave detector such as LIGO [36]. The former gives the false impression that a detector must be in motion to interact with the gravitational wave. This further highlights the importance of carefully considering the approach to formulating a Hamiltonian that correctly describes the physically observed effects of gravity on a quantum system.¹⁶

Another approach is the use of Fermi normal coordinates¹⁷ which also expresses the Hamiltonian in terms of the Riemann curvature tensor (which is coordinate invariant) rather than Christoffel symbols (which are coordinate dependent) [16, 34], [83–86]. There are also other approaches that depart entirely from the use of a Lagrangian associated with either the geodesic equation of motion or the geodesic deviation equation. For example [87], uses a Ginzburg-Landau free energy density that includes a coupling to gravity via a term involving the Ricci scalar.

On the other hand [88], uses a Hamiltonian derived from an effective field theory describing a system of quantum oscillators coupled to a stochastic gravitational radiation background, where the coupling to gravity occurs via a parameter associated with an Ohmic bath spectral density. Although [88] does not consider a superconductor, it is possible that a similar treatment could be used to describe the coupling of the stochastic gravitational wave background to a superconductor. Fundamentally, the coupling of the quantum system to gravity is obtained via the action for a scalar field stress tensor coupled to the curved space-time background [89].

On a related note, it should be emphasized that the action in Eq. 8 only describes the particle dynamics but treats electromagnetism and gravity as fixed background fields with no dynamics. This means that any fields produced by the motion of test particles (which are Cooper pairs leading to a supercurrent) are neglected. However, in Section 4 it is evident that the supercurrent does indeed produce fields which can lead to a Meissner effect. Therefore, a complete action should also include the dynamics of the fields by including terms for electromagnetism and gravity which are, respectively, and , where R is the Ricci scalar, and g is the determinant of the metric [50, 74, 75]. In fact, the action associated with particle dynamics could be replaced by an action describing the current density, .

It is also shown in [10] that a Ginzburg-Landau Lagrangian density in curved space-time can be used to obtain a quantum current density and quantum stress tensor. The action has the form , where ϕ is a complex scalar field, and μ and λ are coupling parameters. This action is effectively “phi-fourth” field theory due to the quartic self-interaction term [60, 61]. Other treatments for embedding phi-fourth (covaraint Ginzburg-Landau) theory in curved space-time are found in [62, 64], [90–93]. All of the treatments described above demonstrate there is a great diversity in the approaches that can be taken to couple gravity to superconductors and quantum systems in general. However, the remainder of this paper will focus on the coupling described by quantizing the canonical momentum in Eq. 13.

3 Modified London Equations and Gauge Condition

In this section, gravito-electromagnetic field equations are introduced, and constitutive equations are developed from the canonical momentum and combined with the field equations to obtain a modified form of the London equations [94–96]. Using harmonic coordinates, , makes the linearized Einstein equation, G^μν = κT^μν, become , where κ = 8πG/c⁴. For a non-relativistic ideal fluid, the components of the stress tensor arewhere ρ_M is mass density, and Vⁱ is the velocity, of the gravitational sources. Note that in the presence of electromagnetic fields, the full stress tensor should also includewhere the components of the electromagnetic strength tensor are

To lowest order, using E = cB, the components of Eq. 41 are

Comparing Eq. 43 to Eq. 40, it is evident that the contribution of electromagnetic fields to the total stress tensor can be neglected provided B²/μ₀ ≪ ρ_McV. For a superconductor such as niobium, we can use ρ_M ∼ 10⁴ kg/m³. It is also shown in Supplementary Appendix SB that a maximum of v ∼ 10⁴ m/s will preserve the superconducting state¹⁸. This leads to B ≪ 10⁵ T which is certainly satisfied in a laboratory setting.

Since T^ij ≈ 0, then in this approximation. A time-independent gravito-electric field (the Newtonian gravitational field) and a gravito-magnetic (Lense-Thirring) field can also be defined respectively as¹⁹

Defining the mass current density as , leads to non-homogeneous field equations given bywhere and μ_G ≡ 4πG/c². The field equations in Eq. 45 can be described as a gravito-Gauss law (Newton’s law of gravitation), and a gravito-Ampere law, respectively.

To develop constitutive equations for a superconductor, we begin by promoting the canonical momentum in Eq. 13 to a quantum mechanical operator, , and act on the Ginzburg-Landau complex order parameter, , where ψ² = n_s is the number density of Cooper pairs. This gives

This can be considered a semiclassical approach where the gravitational field, h_μν, is still a classical field while , , and are quantum operators that act on the Cooper pair state, Ψ. Since the bulk of the superconductor is in the zero-momentum eigenstate, then . Then taking the expectation value gives

Applying Ehrenfest’s theorem allows this equation to return to a classical equation of motion once again. To first order in the metric perturbation, and first order in test mass velocity, Eq. 12 becomes γ ≈ 1 + h₀₀/2 + h_0jv^j/c. Then using Eq. 29 and Eq. 30 in Eq. 47, remaining to first order in the metric perturbation, and using q = −e and m = m_e for electrons, leads to

A similar expression appears in [19] but the scalar potentials, φ and φ_G, are absent. The charge and mass supercurrent densities are, respectively²⁰where n_s is the number density of Cooper pairs. Inserting Eq. 48 into Eq. 49andwhere Λ_L ≡ n_se²/m_e can be defined as the London constant, and

The expressions in Eq. 50 and Eq. 51 are the London constitutive equations for a non-relativistic supercurrent in the presence of electromagnetism and gravity (from non-relativistic gravitational sources in harmonic coordinates). Similar expressions can be found in [22–24], [26], however, with α = 1 and β = m_e/e which is a special case where φ_G = 0 and φ = 0. Notice that if , then Eq. 50 becomes the standard London constitutive equation, . Also notice that setting the charge to zero in Eq. 51 gives which is the constitutive equation for a neutral superfluid in the presence of a gravito-vector potential. Taking a time derivative of Eq. 50 and Eq. 51, and using the fact that ∇φ = 0 inside a superconductor²¹ and in this approximation, leads toand

Equations similar to Eq. 53 and Eq. 54 are also obtained in [17, 21, 22, 24, 71, 72], but with the absence of the gravitational and electric scalar potentials which means α = 1, , and . Also, these authors include a term involving which is not valid since it was shown that for non-relativistic gravitational sources. Notice that Eq. 53 is the usual electric London equation, , but with correction terms due to gravity. Also, Eq. 54 is a redundant equation since , however, for a neutral superfluid, only Eq. 54 would be relevant.

Taking the curl of Eq. 50 and Eq. 51, and using Eq. 44, leads to, respectively,and

Similar expressions can be found in [17, 20, 71, 72], however, with α = 1 and β = m_e/e. Notice that Eq. 55 is the usual magnetic London equation, , but with correction terms due to gravity. For the case of a neutral superfluid, setting the charge to zero makes Eq. 56 becomeThe same result is obtained in [26, 27, 62, 73, 100]. A result similar to Eq. 57 is also obtained in [74, 75], however, it will be shown later that there is a critical sign difference which impacts whether a gravito-magnetic Meissner effect is predicted to occur in the absence of a magnetic field.

Concerning the gauge condition, note that the usual London gauge, , follows from the London constitutive equation, , with the requirement . By the continuity equation, this means ∂_tρ_c = 0 which is consistent with a static Cooper pair density in the interior of the superconductor. Applying the same condition, , to the modified London constitutive equation in Eq. 50 leads to a new gauge condition given by . Using Eq. 52 and ∇φ = 0 yields

This is the modified gauge condition associated with the London equations developed above. It is similar to the gauge condition shown in [74, 75] as , except Eq. 58 has an additional term involving is absent. On the other hand [19, 20, 23], set and as independent gauge conditions. Note that due to the use of harmonic coordinates, , the last term in Eq. 58 can also be expressed using . This means that for a static gravitational scalar potential, the last term in Eq. 58 vanishes from the gauge condition.

Lastly, taking the curl of Eq. 55, using , where , and making use of Eq. 52 and Eq. 45, as well as ∇φ = 0 and leads to

It will be shown in the following section that αμ₀Λ_L ≫ βμ_Gn_se and α ≈ 1. Also, using makes Eq. 59 reduce to a Yukawa-like equation,where the London penetration depth is

Notice that Eq. 59 contains a quantity which can be defined as

As explained in the following section, this quantity can be understood as leading to a modified London penetration depth, .

4 Meissner Effects and Penetration Depths for Magnetic and Gravito-Magnetic Fields

The Maxwell equation (with sources) in curved space-time iswhere is the charge four-current [4]. The covariant derivative of the strength tensor is

Since is symmetric in νσ, and F^νσ is anti-symmetric in νσ, then the last term above is zero. The linearized Christoffel symbols are

Then Eq. 63 becomes

Setting μ = i, and using Eq. 30, Eq. 44, and Eq. 42 leads to

This is the usual Ampere law but with added corrections due to the presence of gravity.

Assume a steady-state supercurrent so that , and a static gravitational scalar potential so that . Taking the curl of Eq. 67, using the identity , where , and using Eq. 55 leads to

Since is primarily due to earth, then the spatial variation of over the dimensions of the superconductor is negligible. Therefore . Also, if , and the magnetic field is arranged in the x- or y-direction, then . Lastly using leads towhere

Notice that α′ differs from α in Eq. 52 by an additional factor given by , assuming the mass density of earth and the superconducting sample are  kg/m³, and m^-2, where λ_L ∼ 10⁻⁹ m is the London penetration depth of a superconductor such as niobium. Note that at the surface of earth, , where M_E and R_E are the mass and radius of earth, respectively. Since the last term in Eq. 70 is 31 orders of magnitude smaller than the second term, then α′ ≈ α.

A similar treatment can be applied to the gravito-Ampere law in Eq. 45 which is for the case of a steady-state supercurrent. Taking the curl, using , where , and using Eq. 56 leads to

Coupled differential equations similar to Eq. 69 and Eq. 71 can also be found in [18–22]. In the absence of gravity, α = 1 and . Then Eq. 69 becomes a Yukawa-like equation²², , where the London penetration depth is given in Eq. 61. The general solution for the magnetic field iswhere , , are constant vectors.

For a superconducting slab occupying the region z > 0, where z is the distance from the surface to the interior of the superconductor, then the boundary conditions for the magnetic field arewhere is the magnetic field at the surface of the superconductor. These conditions require , and . Then Eq. 72 reduces to . This is the standard Meissner effect which predicts that the magnetic field vanishes within the superconductor at depths beyond the London penetration depth.

4.1 The Presence of Only

If is the only field present (or equivalently, a neutral superfluid is used), then Eq. 71 becomes which is a Helmholtz-like differential equation (rather than a Yukawa-like differential equation), and therefore only allows sinusoidal solutions, not exponential solutions. Since there is no exponential decay of the field then there is no penetration depth and no associated Meissner effect. The reason can be traced back to the difference in the sign appearing in the magnetic field equation, , and the gravito-magnetic field equation, . The negative sign in the gravito-Ampere law eliminates a gravitational Meissner effect for a neutral superfluid. Physically speaking, this implies a paramagnetic effect instead of a diamagnetic (Meissner) effect.

In fact, for a maximum gravito-magnetic field at the surface of the superconductor , the solution to the gravito-Ampere field equation would have the form , where is the gravito-magnetic field at the surface of the superconductor, and the spatial periodicity in the field is

Note that Eq. 74 can also be written in the following alternative forms using μ_G = 4πG/c² and Eq. 61.

To obtain a numerical estimate for this quantity, consider a superconductor such as niobium which has a London penetration depth of λ_L ∼ 10⁻⁹ m. Then Eq. 75 gives approximately 10²²λ_L ∼ 10¹³ m which is clearly not observable on a terrestrial scale.

The absence of a gravito-magnetic Meissner effect (when ) is in agreement with [20, 49, 71, 72], but in disagreement with [19, 26, 27, 62], [73–75], [100]. In most of the papers that predict a gravito-magnetic Meissner effect (with ), it is due to a minus sign error somewhere in the calculation. In some cases, finding the error requires careful analysis, but in others it is straight forward. For example, in [100], the minus sign error occurred while using the vector identity .

However, in the case of [19], the issue is more subtle. The gravito-vector potential is defined as . Notice the metric perturbation has upper indices compared to the lower indices in Eq. 30. This leads to a sign difference and therefore the mass current density is written in [19] as rather than as obtained from Eq. 51. The use of a different convention is not problematic, however, the error occurs in the use of the Einstein equation, G^μν = κT^μν. Using the convention of [19] should lead to a gravito-Ampere law with a positive sign on the source term contrary to Eq. 45. However [19], has a negative sign. The net result is a Yukawa-like differential equation is incorrectly obtained for , rather than a Helmholtz-like differential equation. Also, an expression similar to Eq. 74 is obtained, with a value on the order of 10¹³ m, but is interpreted as a penetration depth rather than a spatial periodicity.

In the case of [74, 75], the gravito-vector potential is defined as similar to Eq. 30. The same approach as [71–73] is used with a “generalized vector potential” defined as , and an associated covariant derivative, , where . This leads to a current density given by which matches Eq. 51. Furthermore, it is shown that setting leads to which appears to match Eq. 57. However, the authors defined the mass current density as , where ρ_g ≡ − T₀₀. This introduces an additional minus sign which leads to . This is is in disagreement with Eq. 57 which can be written as . This difference in sign is the reason why [74, 75] obtain a Yukawa-like differential equation rather than a Helmholtz-like differential equation for . Then the gravito-magnetic penetration depth is given as λ_G ≈ 10²¹λ_L. This is similar to the result obtained from Eq. 75 but it is interpreted in [74, 75] as a penetration depth rather than a spatial periodicity.

In the case of [26, 27], the presence of a gravito-magnetic Meissner effect is argued on the basis of spontaneous symmetry breaking. A Lagrangian is written in the form , where and β > 0. The associated field equation is found to be , with . It is stated that α < 0 and α > 0 correspond to the presence and absence of spontaneous symmetry breaking, respectively. Therefore, the sign of α determines the sign of which determines the sign in the field equation for . This distinguishes whether the equation is a Yukawa-like or a Helmholtz-like differential equation. However, it is shown in 4.1 of Tinkham [94] that if α < 0, then , which means there is no superconducting state. Also, the minimum energy is only when α < 0. This means that is always positive, which corresponds to the fact that in Ginzburg-Landau theory, , the number density of Cooper pairs which can only be positive. Lastly, notice that if α < 0, then the field equation for is actually a Helmholtz-like differential equation, and the expression for λ² is negative which therefore cannot be interpreted as a penetration depth.

In the case of [73], there is discussion about whether the “covariant derivative” is or , where g is a coupling constant. Using leads to a Yukawa-like differential equation for , and an associated Meissner effect. Using leads to a Helmholtz-like differential equation for , and the absence of a Meissner effect. However, there is no statement concerning which is the correct approach. The reason may be because the canonical momentum Eq. 13, which determines the sign, is not formally derived in [73]. Instead, the discussion is based on the concept of spontaneous symmetry breaking. As previously explained, the Ginzburg-Landau theory of superconductivity does not permit to be negative which is associated with . Acting the canonical momentum Eq. 13 on Ψ leads to which [73] describes as the “classical” covariant derivative since it is associated with the absence of spontaneous symmetry breaking.

4.2 The Effect of on the Penetration Depth of

For a charged supercurrent in the presence of both magnetic and gravito-magnetic fields, the differential equations Eq. 69 and Eq. 71 need to be decoupled to obtain solutions. For simplicity, consider a system arranged such that . Then solving Eq. 69 for , substituting the result into Eq. 71, and canceling common terms gives²³where k matches Eq. 62 which is

The modified London penetration depth can be defined as . Using gives

Notice that the first term in Eq. 78 encodes a correction due to the gravitational scalar potential, while the second term encodes a correction due to the electric scalar potential. An order of magnitude can be calculated for each term in Eq. 78 using Eq. 52. The first term on the right side of Eq. 78 implies a correction to the London penetration depth given by . At the surface of earth, , where M_E and R_E are the mass and radius of earth, respectively. Then . For a superconductor such as niobium, the London penetration depth is λ_L ∼ 10⁻⁹ m. This means the correction due to the scalar potential of earth is m which would not be observable.

For the second term in Eq. 78, note that if m_ec² ≫ eφ, then β ≈ 4m_e/e ∼ 10⁻¹¹ kg/C. Also using n_s ∼ 10²⁶ m⁻³ means the second term in Eq. 78 is m⁻². Since the first term is m⁻², then the second term is completely negligible. Hence we find that the presence of a Newtonian and/or gravito-magnetic field will not likely have a measurable effect on the penetration depth of the magnetic field.

Since , then is positive and therefore the general solution to Eq. 76 is

Using the boundary conditions in Eq. 73 leads to , , , and . Therefore, the solution is reduced to

This is the standard Meissner effect but with a modified London penetration depth given by Eq. 78. Notice that the first term in Eq. 78 encodes a correction due to the gravitational scalar potential since α = 1 + φ_G/c². The second term in Eq. 78 encodes a correction due to the gravito-magnetic field since it can be traced back to the terms involving in Eq. 69 and Eq. 71.

4.3 The Effect of on the Penetration Depth of _G

Again, for simplicity, consider a system arranged such that . Solving Eq. 71 for , substituting the result into Eq. 69, and canceling common terms gives²⁴where k is given by Eq. 77. For a neutral superfluid (or in the absence of a magnetic field), Eq. 77 reduces to k² = −4μ_Gn_sm_e and therefore Eq. 81 leads to a paramagnetic effect, as stated before. However, for a charged supercurrent in the presence of both and , both terms in Eq. 77 must be considered. Since , then the first term in Eq. 77 can be expressed in terms of the London penetration depth which gives m⁻². An order of magnitude can be calculated for the second term in Eq. 77 using β ≈ 4m_e/e ∼ 10⁻¹¹ kg/C and n_s ∼ 10²⁶ m⁻³ which gives m⁻². Since αμ₀Λ_L ≫ μ_Gn_seβ, then k² is positive and the general solution to Eq. 81 is

The general form of the solutions in Eq. 79 and Eq. 82 are also found in [20]. Using the boundary conditions in Eq. 73 leads to , , , and . Therefore, the solution reduces to

This result predicts a diamagnetic (Meissner) effect for the gravito-magnetic field when a magnetic field is also present. In fact, the gravito-magnetic field is expelled with approximately the same penetration depth as the magnetic field. Defining the gravitational penetration depth as λ_G ≡ 1/k, and using Eq. 77 gives

Therefore, it is found that is expelled from the superconductor with a penetration depth on the order of the London penetration depth, provided a magnetic field is also present. This observation is being taken into account for guiding experimental work [101].

This effect can be understood by comparing how the physics contained in Eq. 77 applies to the magnetic field versus the gravito-magnetic field. As it applies to the magnetic field, Eq. 77 predicts a diamagnetic (Meissner) effect with only a miniscule modification due to the presence of the gravito-magnetic field. However, as it applies to the gravito-magnetic field, Eq. 77 predicts a paramagnetic effect which is drastically altered by the presence of the magnetic field. In fact, the alteration is so substantial that it switches a paramagnetic effect into a diamagnetic (Meissner) effect for the gravito-magnetic field.

This can be further understood by returning to Eq. 71 and noticing that when , or equivalently, , then the differential equation switches from a Helmholtz-like equation, to a Yukawa-like equation. Physically speaking, this mechanism can be understood by the following example. Consider a case where the superconductor is in the presence of a gravito-magnetic field but no magnetic field. According to Eq. 55, there will be a small supercurrent given by . Now introduce an opposing magnetic field, , which will cause an opposing supercurrent given by . Since the supercurrent will switch direction, then the gravito-Ampere law, , predicts that a gravito-magnetic field will be produced inside the superconductor which opposes the incident gravito-magnetic field and therefore cancels it. This is effectively a gravito-magnetic Meissner effect. The key feature of this concept is that the magnetic field and gravito-magnetic field must be generated by independent sources so that their strength and direction can be independently adjusted. This key feature may be missing from formulations such as [71, 72, 74, 75] which work in terms of a single field defined as or .

This description also demonstrates that there is a threshold value for the minimum magnetic field necessary to produce a Meissner effect for the gravito-magnetic field. It is given by . For a numerical estimate describing this condition, again use φ_G/c² ∼ 10^–9 so that α = 1 + φ_G/c² ≈ 1, and assume m_ec² ≫ eφ so that β ≈ 4m_e/e ∼ 10⁻¹¹ kg/C. Then . Therefore, it takes an extremely small magnetic field (relative to the gravito-magnetic field) to produce a gravito-magnetic Meissner effect.

These results demonstrate an important interaction between electromagnetism, gravitation, and a quantum mechanical system that only occurs when all three are present. The superconductor provides the quantum mechanical system which is necessary to have any kind of Meissner effect. The gravito-magnetic field is required to create a novel gravitational effect. Lastly, the magnetic field is necessary to mediate the interaction. In the absence of a magnetic field, the novel gravito-magnetic Meissner effect would not take place.

Note that the conclusion that both and penetrate together to the same depth given by Eq. 84 was also found in [18–22, 71, 72, 74, 75]. In the case of [19], the exponential decay solutions in Eq. 80 and Eq. 83 are also obtained, but with the assumption . In the case of [19, 20], the Meissner effect is shown for the combined field, . Similarly [21, 22], show a Meissner effect for . In [20], screening currents are taken into account by introducing the following boundary conditions in addition to those shown in Eq. 73.

Using Eq. 60 and leads to a solution for both J_c and J_m of the form , where J₀ is the current density at the surface. Therefore, Eq. 85 becomeswhere σ_c,0 ≡ λ_LJ_c,0 and σ_m,0 ≡ λ_LJ_m,0 are charge and mass surface current densities, respectively. Similarly, the solutions for and in [21, 22] contain σ_c,0 and σ_m,0. The combined “screening” current also decays exponentially so that for z ≫ λ_L.

The use of Eq. 86 leads to solutions for and in [20–22] that contain an exponential decay term plus a constant term. Due to the constant term, [22] states that a “constant residual magnetic and gravito-magnetic fields will exist within a pure superconductor.” When z ≫ λ, the residual fields are

Notice there are residual terms remaining when z ≫ λ_L. However, setting B₀ = μ₀σ_c,0 leads to the usual solution, . This corresponds to setting in Eq. 72. Therefore, it can be argued that the residual fields in Eq. 87 are an artifact of using boundary conditions that are not consistent with the standard Meissner effect.

In the case of [71, 72], a “generalized field” is defined as . The associated field equation is found to bewhere λ_e is the London penetration depth, and λ_g is essentially the expression found in Eq. 75. It is recognized that Eq. 91 becomes a Yukawa-like equation when λ_g > λ_e which leads to a “generalized Meissner effect.” Since λ_g ∼ 10¹³ m and λ_e ∼ 10⁻⁸ m, then the condition for a “generalized Meissner effect” is clearly satisfied. In fact, since λ_g ≫ λ_e, then and the “generalized penetration depth” from Eq. 91 becomes . This demonstrates that the penetration depth of the gravito-magnetic field becomes the same as the London penetration depth as indicated by Eq. 84.

In [74, 75], the same approach is used except the combined field is defined as . As previously explained, using , where ρ_g ≡ − T₀₀, leads to . This result for the gravito-magnetic case has the same sign as for the magnetic case. As a result, the “generalized penetration depth” becomes which has a plus rather than the minus seen in Eq. 91. Since λ_g ≫ λ_e, then the result is still λ ≈ λ_e for the generalized field. However, in the absence of a magnetic field, this difference in sign is the reason why [74, 75] still find a gravito-magnetic Meissner effect, while [71] does not.

A final important consideration is the issue of coordinate-freedom in linearized General Relativity. The gravito-magnetic field, , is a coordinate-dependent quantity which can be made to vanish by a linear coordinate transformation, x′^μ = x^μ − ξ^μ. Since the linearized metric perturbation transforms as , then and transform, respectively, as [12]

Therefore, the effects associated with and can be made to vanish by a coordinate transformation. Alternatively, a coordinate-invariant approach can be used which also applies to gravitational waves. This is discussed in [12, 38, 39].

5 Electric Field Penetration Depth, and Absence of Meissner Effect for the Gravito-Electric Field

Using in , and using in , leads to, respectively,On the left side of both equations, we can use the identity . On the right sides, we can use Eq. 50 and Eq. 51 in each equation respectively. This givesand

Since in this approximation, then taking a time derivative of Eq. 95 will have a vanishing result and therefore eliminates a Meissner effect for the time-independent gravito-electric field (which is the Newtonian field). This is expected since it is known that the Newtonian field generally cannot be shielded. This conclusion is in agreement with [20, 24], but in disagreement with [71–73].

In the case of [20, 24], the gravitational sources are non-relativistic and harmonic coordinates are used . However, it is not recognized that this leads to . Then [20] sets ∇φ_G = 0 (by assuming the Newtonian gravitational field of the superconductor is negligible) which leads to a “transverse gravito-electric field” written as . As a result of this error, coupled differential equations for and are obtained which have a similar form to Eq. 69 and Eq. 71. Also, solutions for and have the same form as Eq. 79 and Eq. 82. Once again, the requirement that the fields are finite (but not necessarily zero) for z → ∞ leads to solutions which contain an exponential decay term plus a constant term. Then it is stated that when z ≫ λ, the residual fields are given in a form identical to Eq. 87 but with and . It follows that the entire discussion concerning Eq. 87 would apply, including the claim that is satisfied for the residual fields. This would imply a Meissner effect for the combined field. However, since for non-relativistic gravitational sources, then the entire discussion becomes moot.

In the case of [71–73], the exact same problem is at the root of the analysis. Again the gravitational sources are non-relativistic . The use of harmonic coordinates is not explicitly stated, but it is clearly implied based on the Maxwell-like field equations appearing, and the references cited in the paper. Again it is overlooked that in this approximation. As discussed in [12, 28], this eliminates the gravito-Faraday law since . However [71–73], include the gravito-Faraday law and use it (in differing notation) to obtain . Then the usual approach of taking the time derivative of Eq. 60 and using Eq. 53 leads to a Yukawa-like equation. In the case of [73], only gravity is considered, so the result is , where λ_g is the gravitational penetration depth. In the case of [71–73], a generalized field is defined as , and a field equation identical to Eq. 91 is obtained. Again, it follows that a “generalized penetration depth” becomes . However, since for non-relativistic gravitational sources, then once again then the entire discussion becomes moot.

Returning to Eq. 94, an expression for the electric field can be obtained by taking a time derivative and using for a uniform charge density.²⁶

In the absence of gravity, Eq. 96 becomes . For a superconducting slab occupying z ≥ 0, the solution is which predicts the electric field vanishes within the superconductor at depths beyond the London penetration depth. However, if gravity is present, then the associated penetration depth must be modified. If φ_G and φ are static, then the solution to Eq. 96 is where

However, if φ_G and φ are time-dependent, then Eq. 96 requires obtaining solutions for and found in the coupled equations Eq. 94 and Eq. 95. We can use the gauge condition Eq. 58 in Eq. 94 to eliminate in favor of which gives

Multiplying Eq. 95 by β/α, adding the result to Eq. 98, and using a field defined as giveswhere k is given by Eq. 77. Consider a system where . Since ∇φ = 0, and φ_G due to earth has miniscule variation over the length scale of a superconductor, then the solution to Eq. 99 is , where is given by Eq. 78. This solution can be written as

Taking a time derivative gives

Inserting Eq. 101 into Eq. 96 giveswhere . The general solution iswhere are constant vectors. Using the same boundary conditions as Eq. 73 requires C₁, . Therefore, the solution reduces to . An explicit function for will depend on the particular time-dependence of and , however, the penetration depth for will still be .

6 Flux Quantum (Fluxoid) in the Body of a Superconductor

In Ginzburg-Landau theory, the minimal coupling rule, , makes the supercurrent becomewhere is the complex order parameter [94, 103]. Using Eq. 29 and Eq. 30 in Eq. 16, and promoting the canonical momentum to a quantum mechanical operator makes the minimal coupling rule become

Staying to first order in the metric perturbation, and first order in test mass velocity, requires using γ ≈ 1 in Eq. 105. For convenience, the entire coupling vector can be defined as

Then the corresponding supercurrent becomeswhich reduces to Eq. 104 in the absence of gravity. Using , where ϕ is the phase, leads to

An expression similar to Eq. 108 is found in [18, 19, 21, 23, 24, 26, 49], [71–75], [104, 105], however, the terms involving φ and φ_G in Eq. 106 are missing. In the previous section, it was shown that inside the body of the superconductor (much deeper than the London penetration depth), all the fields in Eq. 48 vanish and therefore the supercurrent velocity is zero. Therefore, using J_i = 0 in Eq. 108 and v_i = 0 in Eq. 106 makes Eq. 108 become

Integrating around a closed loop gives

Since the order parameter is single-valued, then it must return to the same value when the line integral returns to the same point. Therefore, the right side must be 2πn, where n is an integer.²⁷

Applying Stokes’ theorem on the left side giveswhere S is the surface bounded by the curve C. Using ∇φ = 0 within the body of the superconductor, and q = −2e and m = 2m_e for Cooper pairs, giveswhere Φ_B and are the flux of and , respectively. Since φ_G ≪ c² and eφ ≪ m_ec², then the result can be approximated to

In the absence of electromagnetism, the gravito-magnetic flux condition can be written as , where is a gravito-magnetic flux quantum (fluxoid). This result is also found in [27, 50, 73]. The total flux quantum in Eq. 113 is found in [17, 19, 62], and is consistent with DeWitt’s statement in [3] that “the total flux of linking a superconducting circuit must be quantized in units of ,” where . However, none of these authors have the additional term in Eq. 113 involving the flux of . This may be due to applying the approximation φ_G ≪ c² before taking the curl in Eq. 111.

In the absence of gravity, Eq. 113 gives the usual magnetic flux condition, Φ_B = nΦ_B,0, where . This has been experimentally verified, even for the n = 1 state [107, 108]. Using around a closed loop surrounding the fluxoid, implies that the n = 1 state leads towhere r = r₀ is the radius of a fluxoid. Then Eq. 113 in the n = 1 state can be written aswhere S₀ is the area of a single fluxoid. Since the dominant source of is due to earth, consider a system arranged so that is parallel to the surface of earth. Since is approximately uniform over the area of a fluxoid, then using Eq. 114 in Eq. 115, and integrating, leads to

Since , then which means Eq. 116 can also be expressed as

The two terms involving gravity are small corrections. Therefore, solving for Φ_B and keeping only lowest order terms involving E_G and B_G leads to

This result can be interpreted as a modified magnetic fluxoid in the presence of gravity. Since [107] used r₀ ∼ 10⁻⁵ m, then Eq. 114 gives A₀ ∼ 10⁻³⁰ T⋅m. Therefore, the last term in Eq. 118 becomes Wb which is hopelessly too small to be observed.

For the term involving to be comparable to Φ_B, we need Wb. It is shown in Supplementary Appendix SB that the gravito-vector potential of a large flywheel can be m/s. Using and r₀ ∼ 10⁻⁵ m gives Wb. Again, this is hopelessly too small. A more promising approach might be to use a low Earth orbit (LEO) satellite. It is found in Supplementary Appendix SB that the gravito-vector potential due to earth as observed by the satellite would be m/s. This leads to Wb which implies that an experiment would still need extreme sensitivity that can measure to demonstrate the presence of a gravito-magnetic fluxoid.

7 Supercurrent in a Closed Loop

Promoting the canonical momentum in Eq. 13 to a quantum mechanical operator and acting on the complex order parameter gives

Again use Ψ = ψe^iϕ, but now let be a uniform number density around a ring. Therefore, . Then using Eq. 119 and taking the expectation value gives

Applying Ehrenfest’s theorem allows this equation to return to a classical equation. Staying to first order in the metric perturbation and test mass velocity, and using Eq. 29 and Eq. 30 leads to

This expression can be written in terms of the supercurrent density, Jⁱ = −qn_svⁱ. Then integrating around a closed loop gives

Since the order parameter is single-valued, then it must return to the same value when the line integral returns to the same point. Therefore, the right side must be 2πn, where n is an integer. Applying Stokes’ theorem on the left side, and using q = −2e and m = 2m_e for Cooper pairs, gives

Since φ_G ≪ c² and eφ ≪ m_ec², then the expression can be approximated toThis result is similar to Eq. 113 which applies to the bulk of the superconductor. However, Eq. 124 contains additional terms involving and which can be non-zero on the surface of the ring (shallower than the penetration). An expression similar to Eq. 124 also appears in [17] [35, 109, 110], except the terms involving and are absent.

7.1 Supercurrent Quantization for a Superconducting Ring Coaxial With a Rotating Massive Cylinder

As a practical example, consider a superconducting ring in the presence of a rotating massive cylinder of length ℓ and radius R. The cylinder rotates at a constant non-relativistic angular velocity and hence has a stationary mass current (See Figure 1).

FIGURE 1

This is effectively the same system that was considered by DeWitt [3]. He states, “Now consider an experiment in which the superconductor is a uniform circular ring surrounding a concentric, axially symmetric, quasirigid mass. Suppose the mass, initially at rest, is set in motion until a constant final angular velocity is reached. This produces a Lense-Thirring field which, in a coordinate system for which the metric is time-independent, takes the formwhereκis the gravitation constant, andρandare, respectively, the mass density and velocity field of the rotating mass.” For a steady-state current, the gravito-Ampere equation in Eq. 45 can indeed be written as . This matches Eq. 125 up to a factor of 4 (which is due to DeWitt’s being related to used in this paper by .) Also note that DeWitt sets c = 1.

Neglecting terms that are in Eq. 124, and using DeWitt’s notation, , where , gives

DeWitt states, “Ifis initially zero then so is. Because of the flux quantization condition, the flux ofthrough the superconducting ring must remain zero. But sinceis nonvanishing in the final state, a magnetic field must be induced.” This implies that DeWitt is considering a system where every term in Eq. 126 is zero in the initial state. Then in the final state, he requires that and n are still zero, but the flux of is non-zero. Therefore, he concludes that the flux of must become non-zero as well.

A similar statement is found in Papini’s paper [111] which is a follow up to DeWitt’s paper [3]. Papini states, “The main result of this [DeWitt’s] work is that whenever a Lense-Thirring field is present, it is not the magnetic field which vanishes inside a superconductor, but a combination of magnetic and gravitational fields. Similarly, the flux which is quantized is the total flux of magnetic and gravitational fields... It is important that initially the total flux linking the loop be zero. The total flux is in fact quantized in units ofπand if it vanishes in the initial state, it also vanishes in the final state. It then follows that the magnetic flux equals in absolute value the flux of the gravitational field.” Using Eq. 124, this condition can be written as

DeWitt states, “Suppose the rotating mass is kept electromagnetically neutral... Then the magnetic field must arise from a current induced in the ring. The magnitude of this current will bewhereSis the area spanned by the ring,Lis its self-inductance, and the final integral is taken around the ring.” Since a static flux does not generate a current, and the presence of inductance in Eq. 128 implies a changing current, then Faraday’s law and Eq. 127 must have been used to obtain

Integrating with respect to time gives . Then letting the current and flux be initially zero, and using Eq. 125 leads to Eq. 128. Applying Stokes’ theorem to Eq. 128, integrating around the perimeter of the ring with a diameter of d = 2R, and using , where ℓ and M are the length and mass of the cylinder, respectively, gives

Evidently DeWitt assumes to obtain a result of²⁸except he sets c = 1. However, a more accurate estimate for Eq. 128 can be obtained using the gravito-magnetic flux evaluated in Supplementary Appendix SB aswhere since . For the case of a large flywheel (as described in Supplementary Appendix SB), we can use M ∼ 4 × 10³ kg, R ∼ ℓ ∼ 1 m, and ω ∼ 6 × 10² rad/s. For a 1 m length of 0.25 mm diameter superconducting wire well below its transition temperature, the inductance is on the order of L ∼ 10⁻¹⁰ H [112]. Then the order of magnitude predicted by Eq. 128 is²⁹

With these considerations in mind, it is suggested that a correct interpretation of Eq. 126 would be described as follows. Introducing a non-zero B_G can generate a mass current in the ring via a gravito-Faraday flux rule. Because the mass current consists of Cooper pairs, then there will be an associated electric current, . The electric current will produce an associated magnetic field and thereby a magnetic flux in the ring. The sum total of fluxes, , and current, , will be quantized according to Eq. 126. If the total on the left side of Eq. 126 is kept smaller than h/2, then the system will remain in the n = 0 state.

In this sense, Eq. 126 plays the role of a constraint equation describing the quantization of the current, rather than a field equation that can generate a flux. To elaborate on this, we can return to the more general expression Eq. 124 and write it aswhere the total flux through the ring is

If the current density is assumed to be uniform and only occupies the skin of the superconducting ring (no deeper than the penetration depth), then . Therefore, using Eq. 61 and Eq. 134 leads to a quantization condition on the electric current given bywhereis an “electric current quantum” analogous to the magnetic flux quantum, . Similar to DeWitt, the fluxes in Eq. 135 can be expressed in terms of the properties of the massive cylinder (such as mass density, rotation speed, etc). However, rather than applying ∇⁻² to both sides of the gravito-Ampere law as DeWitt did in Eq. 125, we can use the expressions derived in Supplementary Appendix SC for the potentials, fields, and associated fluxes. This makes Eq. 135 become³⁰

Here the relative electric permittivity and magnetic permeability of the massive cylinder are, respectively, ɛ_r = ɛ/ɛ₀ and μ_r = μ/μ₀. Also d is the diameter of the ring. Notice from Eq. 138 that F is a continuous function of ω, however, the entire left side of Eq. 136 is discretized and can only increase by increments of I₀. This means that even though F can change gradually, the smallest change in F that can change the state of the system is ΔF = μ₀eI₀d/2, and each change by ΔF will move the system between n states. (If F changes by an amount smaller than μ₀eI₀d/2, the quantum “rigidity” of the Cooper pair wave function will preserve the state of the system.) However, since m⋅A, then macroscopic values for d will make I₀ extremely small relative to the other terms in Eq. 136. Thus for macroscopic experiments, Eq. 136 becomes which does not involve Planck’s constant, in accordance with the Correspondence Principle.

For simplicity, consider if the diameter of the ring is similar to the diameter of the solenoid , and the solenoid is electrically neutral . This corresponds to setting the flux of and to zero in Eq. 135. Consider if increases with time due to either enlarging the ring with time, or spinning up the massive cylinder from rest. The changing gravito-magnetic flux will induce a current in the ring which points in the direction shown in Figure 1. This is in agreement with DeWitt [3] who states, “The currentIarises from an induced motion of electrons on the surface of the superconductor. This motion is in the same direction as the motion of the rotating mass.” This current will produce a magnetic field that points downward in Figure 1. Therefore, Φ_B will be subtracted from in Eq. 135. For simplicity, we can approximate over the entire area of the ring³¹ Then using Eq. 132 makes Eq. 135 becomeand Eq. 136 becomes

This is the quantization condition for the superconducting ring in the configuration of Figure 1. Again, it is emphasized that we cannot set n = 0 and claim thatis generated in the ring. Rather, Eq. 141 should be interpreted as simply the smallest non-zero current that can exist in the ring. However, next we proceed to calculate the current that will be generated.

7.2 Induced Supercurrent in the Ring Due to a Motional Gravitational emf

The current generated in the superconducting ring can be evaluated by developing a gravitational version of the Faraday flux rule. In Supplementary Appendix SD it is shown that the Lorentz four-force in curved space-time Eq. 7 can be evaluated to first order in the perturbation and test mass velocity to obtainwhere the electromagnetic force isthe gravitational force isand the electromagnetic force coupled to gravity is

The electromagnetic emf is given bywhich leads to the usual Faraday flux rule,where is the velocity of the moving boundary. In Supplementary Appendix SD, a “gravitational emf” is also defined aswhich leads to an associated “gravitational Faraday flux rule” given bywhere the notation “constant B_G” indicates that the term in parentheses involves a time-varying boundary, not a time-varying B_G (which is accounted for in the next term). The emf generated by a time-varying boundary can be referred to as a motional gravitational emf, while the emf generated by a time-varying B_G can be referred to as a transformer gravitational emf. Lastly, a “coupled emf” associated with Eq. 145 can be defined aswhich becomes

The total emf associated with the total force Eq. 142 is

For the system in Figure 1, there is no electric field. Also, the gravito-scalar potential is primarily due to earth, therefore . Lastly, points radially inward while circulates around the ring, so . It was also previously explained that Φ_B must be subtracted from . With all of these considerations, the total emf becomes

Since φ_G is primarily due to earth, then 1 − 3φ_G/c² ≈ 1 − 10⁻⁹ ∼ 1. For simplicity, again we can approximate over the entire interior area of the ring. Applying the definition of inductance, , and using Eqs 132, 153 giveswhere r is the time-varying radius of the ring. Consider if the ring is expanding at a constant rate, r = v_bt, and definewhere a characterizes electromagnetic effects, and b characterizes gravitational effects. Then Eq. 154 becomes

In the absence of gravity , the equation describes the induced changes in the current of a superconducting ring if the ring already has a current and the size of the ring changes at a constant rate. Therefore gravity introduces an additional driving term, bt, to the system. If the rotating cylinder is a large flywheel (as described in Supplementary Appendix SB), we can use M ∼ 4 × 10³ kg, R ∼ ℓ ∼ 1 m, and ω ∼ 6 × 10² rad/s. This leads to a characteristic value for the current given bywhich is indicative of the fact that gravity has a miniscule effect on the induced changes in the current. In superconductors, L ∼ 10⁻¹⁰ H at most [112]. Also, if v_b ∼ 1 m/s, then a ∼ 10⁴ s⁻². Therefore, 1 + 2at² ≈ 1 for s. In that case, if I = 0 at t = 0, the solution to Eq. 156 is

Note that the values of L and v_b are not relevant until t ≪ 10⁻² s is no longer satisfied.

7.3 Induced Supercurrent in the Ring Due to a Transformer Gravitational emf

As previously stated, Eq. 129 implies DeWitt was considering a situation associated with a transformer emf (not a motional emf) since he states, “Suppose the mass, initially at rest, is set in motion until a constant final angular velocity is reached.” It has been emphasized throughout this paper that the use of non-relativistic gravitational sources and harmonic coordinates leads to , as shown in Section 3. This implies the absence of a gravito-Faraday law and a gravitational transformer emf.

However, to treat the system DeWitt was considering, we can relax this requirement and permit . This has the effect of turning into . There are other modifications to the gravito-electromagnetic field equations which are discussed in [28], however, for our purposes here, the only relevant change is the presence of a gravito-Faraday law, . Therefore, the emf induced in the ring now involves the second term on the right side of Eq. 149. Using Stokes’ theorem on this term and including the electromagnetic flux leads to a total flux given bywhere the notation “constant boundary” indicates that the term in parentheses involves a time-varying B_G, not a time-varying boundary. Again, 1 − 3φ_G/c² ∼ 1. This leads to a result similar to Eq. 154 but this time r = R (which is constant) but ω varies with time. Therefore we havewhere . Definingleads to

Again a′ characterizes electromagnetic effects, and b′ characterizes gravitational effects. Letting I = 0 at t = 0, and t = T when the mass cylinder reaches ω_max, leads to the following solution to Eq. 162.

If α is constant, then ω_max = αT. For the case of a large flywheel, again we can use M ∼ 4 × 10³ kg, R ∼ ℓ ∼ 1 m, and ω_max ∼ 6 × 10² rad/s. Since L ∼ 10^–10 H at most in superconductors [112], then 1 + a′ ≈ a′ ∼ 10⁴ and Eq. 163 becomes

Note that the electric current produced by a motional gravitational emf Eq. 157 and transformer gravitational emf Eq. 164 are not unique to a superconductor since the London equations were not utilized anywhere in the analysis. The only feature in the calculation unique to superconductors is the value of the inductance, however, the inductance cancels in both calculations.

Alternatively, a treatment involving the London equations could begin with Eq. 53. When we permit and use for the non-conservative gravito-electric field, then Eq. 53 becomeswhere we set and φ = 0. An expression for can be obtained by using the gravito-Faraday law in integral form

Applying this to the ring and using Eq. 132 leads to a magnitude given bywhere . Likewise, an expression for can be obtained using Faraday’s law in integral form and approximating over the entire interior area of the ring. This leads to a magnitude given by

As previously stated, the effect of Eqs 167, 168 are in opposite directions. Therefore, using Eq. 165 as well as and Eq. 61, and approximating 1 + φ_G/c² ≈ 1 leads to

This has the same form as Eq. 162 and therefore has a solution similar to Eq. 163.

Assuming α is constant, and using the same parameters for a large flywheel again leads to

Again, the result is not dependent on the inductance, and the value is comparable to Eq. 164. It is arguable that the result obtained in Eq. 171 is more reliable since it is calculated using the London equation which is derived from the canonical momentum. Therefore, it stems from a quantum mechanical approach (since the canonical momentum operator acting on the Ginzburg-Landau order parameter leads to the London equations). In contrast, Eq. 171 is derived from a purely classical equation of motion (the geodesic equation of motion with a Lorentz four-force).³²

7.4 Gravitational Hall Effect in the Ring

Due to the current in the ring and the presence of magnetic and gravito-magnetic fields, there will be a corresponding Hall effect as shown in Figure 2.

FIGURE 2

Following the usual derivation of the Hall effect, we begin with the total force on the charge carriers. As previously stated, for the gravito-scalar potential of earth. However, now we let since charge carriers will accumulate on opposite sides of a cross-section of the ring, as shown in Figure 3. Therefore Eq. 142 becomeswhere v_s is the velocity of the supercurrent around the ring, and is the electric field produced due to the polarization across the cross-section of the ring. The current on one side of the ring will generate a magnetic field that points downward on the other side of the ring. This will generate a magnetic force that points radially inward on Cooper pairs which are negative charge carriers . Meanwhile, the gravito-magnetic field will produce a force that also points radially inward on Cooper pairs which are mass carriers . For simplicity, assume points directly down through the ring, and points directly up so that and .

FIGURE 3

The net effect of the magnetic and gravito-magnetic forces is to cause Cooper pairs to accumulate on the inner perimeter of the ring. As a result, there will be an electric field pointing radially inward. There will not be any in the radial direction since due to earth points downward, and due to is azimuthal. Also note that since is azimuthal but is radial. However, points in the radially inward direction. Therefore, the radial component of Eq. 172 becomes

Following the usual derivation of the Hall effect, the total radial force on the charge carriers will be zero for a steady-state current. Also using and , as well as Eq. 61, leads to

Then approximating 1 − 3φ_G/c² ∼ 1 makes Eq. 173 becomeWe can approximate and use and from Supplementary Appendix SB. Also writing the hall potential as V_H = wE_p, where w is the width of the cross-section of the ring, leads to

The first term is purely electromagnetic and quadratic in the current, while the second term involves gravitational effects and is linear in the current. In fact, for the case of a large flywheel (M ∼ 4 × 10³ kg, R ∼ ℓ ∼ 1 m, ω ∼ 6 × 10² rad/s), the two terms become comparable when

This is similar to the result obtained in Eq. 157. Using Eq. 177 and ω ∼ 10⁻⁴ m in Eq. 176 gives

A gravitational Hall effect for a superfluid is also considered in [27], however, no Hall potential value is obtained.

7.5 Induced Force on the Ring

Lastly, consider if the ring is raised some distance above the massive cylinder so that at the ring is not completely in the vertical direction, as shown in Figure 3.

A vertical force on the ring will exist due to the current in the ring and the gravito-magnetic field of the mass cylinder. This effect is analogous to the “jumping ring” experiment where a magnetic force is evaluated using and is found to be F = 2πRIB sin θ, where B is an external magnetic field, and θ is the angle shown in Figure 3. However, to deal with the gravitational case, we can start with the total force in Eq. 142 and set and . Neglecting the weight of the ring giveswhere is the gravito-electric field induced by the time-varying gravito-magnetic flux. Since is azimuthal, it only serves to drive the supercurrent and does not contribute to any vertical force on the ring. Likewise, and also do not contribute to a vertical force since points downward, but v_s and are azimuthal.

Recall that the differential magnetic force in a current segment, , is obtained from by using qv → Idl. Similarly, for the gravitational force, we can use mv → I_mdl, where I_m is the mass current which is related to the electric current by . Since is azimuthal and B_G has no azimuthal component, then . However, the component of dF in the vertical direction is dF sin θ, as can be seen from Figure 3. Therefore, the differential force in the vertical direction for each differential segment is

In Supplementary Appendix SB, the gravito-magnetic field along the axis of the mass cylinder is found as . Since the field near the end of a solenoid is approximately half of that at the center. Then the total force around the ring can be approximated as

Approximating θ ≈ 30⁰, using the case of a large flywheel (M ∼ 4 × 10³ kg, R ∼ ℓ ∼ 1 m, ω ∼ 6 × 10² rad/s), and I ∼ 10⁻²⁶ A obtained in Eq. 171 leads to F ∼ 10⁻⁵⁷ N which is completely negligible. To get a more substantial effect, an electric current can be driven in the ring prior to introducing B_G from the massive cylinder. It is shown in Supplementary Appendix SB that the maximum supercurrent velocity that preserves the superconducting state is v ∼ 10⁴ m/s. Then using Eq. 174 leads to

This is a factor of 10²⁵ larger than the current induced by B_G. Furthermore, it was previously mentioned that in a low Earth orbit (LEO) satellite, the gravito-vector potential due to earth (as observed by the satellite) is h_LEO ∼ 10⁻³ m/s. Compared to the large flywheel (h ∼ 10⁻²¹ m/s) this leads to another factor of 10¹⁸ increase for B_G. As a result, the force on a ring could be F ∼ 10⁻¹⁴ N.

On a related note, the factor of 10¹⁸ increase in would also increase the electric current produced by a motional emf Eq. 157 and by a transformer emf Eq. 171 so that they become I ∼ 10⁻⁶ A and I ∼ 10⁻⁸ A, respectively. Also, using h_LEO ∼ 10⁻³ m/s makes Eq. 177 become I ∼ 10⁻⁵ A and the Hall potential in Eq. 178 becomes V_H ∼ 10⁻¹ V.

Since the gravito-vector potential of the earth cannot be controlled (unlike that of a spinning flywheel), an option for controlling the exposure of the system to would be to place the system in an enclosure surrounded by superconducting material. The gravitational Meissner effect described in Section 4 could be used to shield the system from by controlling the temperature of the enclosure material such that it can transition above and below the critical temperature for the superconducting state.

8 Conclusion

The last three terms in Eq. 185 are missing from DeWitt’s result in Eq. 2 which is

This result depends on L ∼ 10⁻¹⁰ H for the inductance of the superconductor. In comparison, using a motional gravitational emf (a time-varying boundary of the ring) leads to an electric current in the ring given by Eq. 157 as

For a transformer gravitational emf (a time-varying ), using an inductance approach leads to Eq. 164 which gives

This result does not depend on inductance because it cancels out in the analysis. Alternatively, using a gravito-Faraday law and London equation approach leads to Eq. 171 which gives

References

• 1.

  GeorgeR. Gravity Research Foundation (1949). Available at: www.gravityresearchfoundation.org.

  • Google Scholar

• 2.

  DeWittB. New directions for research in the theory of gravitation. In: Reprinted in cécile DeWitt-morette, the pursuit of quantum gravity: Memoirs of Bryce DeWitt from 1946 to 2004. New York: Springer (2011). p. 61–8. 1953available in GRF Box 2, Folder 8, and at Available at: www.gravityresearchfoundation.org.

  • Google Scholar

• 3.

  DeWittB. Superconductors and gravitational drag. Phys Rev Lett (1966) 16:1092–3. 10.1103/physrevlett.16.1092

  • CrossRef
  • Google Scholar

• 4.

  MisnerCThorneKWheelerJ. Gravitation. San Francisco: Freeman (1972).

  • Google Scholar

• 5.

  WaldR. General relativity. Chicago: University of Chicago Press (1984).

  • Google Scholar

• 6.

  WeberJ. General relativity and gravitational waves. Dover Publications (2004).

  • Google Scholar

• 7.

  JacksonJ. Classical electrodynamics. 3rd ed.John Wiley & Sons (1998).

  • Google Scholar

• 8.

  BarutA. Electrodynamics and classical theory of fields and particles. New York: Macmillian (1964). Dover reprint.

  • Google Scholar

• 9.

  BertschingerE. Physics 8.962 notes, “Hamiltonian dynamics of particle motion (1999).

  • Google Scholar

• 10.

  BertschingerE. Physics 8.962 notes, “Symmetry transformations, the Einstein-Hilbert action, and gauge invariance (2002).

  • Google Scholar

• 11.

  CognolaGVanzoLZerbiniS. Relativistic wave mechanics of spinless particles in a curved space-time. Gen Relativ Gravit (1986) 18(9):971–82. 10.1007/bf00773561

  • CrossRef
  • Google Scholar

• 12.

  InanN. Formulations of General Relativity and their applications to quantum mechanical systems (with an emphasis on gravitational waves interacting with superconductors). Open Access Publications from the University of California (2018). PhD Dissertation, eScholarshipProQuest ID: Inan_ucmerced_1660D_10366, Merritt ID: ark:/13030/m5jb13w2.

  • Google Scholar

• 13.

  HirakawaH. Superconductors in gravitational field. Phys Lett (1975) 53A:number 5.

  • Google Scholar

• 14.

  PapiniG. In: BergmannPGde SabbataV, editors. Quantum systems in weak gravitational fieldsAdvances in the interplay between quantum and gravity physics, 60. Dordrecht: Springer (2002). NATO Science Series. 10.1007/978-94-010-0347-6_13

  • CrossRef
  • Google Scholar

• 15.

  LambiaseGPapiniG. The interaction of spin with gravity in particle physics. Heidelberg, Germany: Springer Nature (2021).

  • Google Scholar

• 16.

  RothmanTBoughnS. Can gravitons be detected. Found Phys (2006) 36:1801–25. 10.1007/s10701-006-9081-9

  • CrossRef
  • Google Scholar

• 17.

  RossD. The London equations for superconductors in a gravitational field. J Phys A: Math Gen (1983) 16:1331–5. 10.1088/0305-4470/16/6/026

  • CrossRef
  • Google Scholar

• 18.

  PengHTorrD. Effects of a superconductor on spacetime and the potential effect on the gyroscope experiment. Nucl Phys B - Proc Supplements (1989) 6:411–3. 10.1016/0920-5632(89)90485-4

  • CrossRef
  • Google Scholar

• 19.

  PengH. A new approach to studying local gravitomagnetic effects on a superconductor. Gen Relativ Gravit (1990) 22(6):609–17. 10.1007/bf00755981

  • CrossRef
  • Google Scholar

• 20.

  PengHLindGChinY. Interaction Between Gravity and Moving Superconductors. Gen Relativ Gravit (1991) 23(11):1231–50. 10.1007/bf00756846

  • CrossRef
  • Google Scholar

• 21.

  LiN. A magnetically induced gravitomagnetic field inside a superconductor,” Essays on Gravitation. Gravity Research Foundation (1990). arxiv.org/abs/2201.12969v1.

  • Google Scholar

• 22.

  LiNTorrD. Effects of a gravitomagnetic field on pure superconductors. Phys Rev D (1991) 43:457–9. Number 2. 10.1103/PhysRevD.43.457

  • CrossRef
  • Google Scholar

• 23.

  LiNTorrD. Gravitational effects on the magnetic attenuation of superconductors. Phys Rev B (1992) 46:5489–95. 10.1103/PhysRevB.46.5489

  • CrossRef
  • Google Scholar

• 24.

  TorrDLiN. Gravitoelectric-electric coupling via superconductivity. Found Phys Lett (1993) 6:371–83. 10.1007/BF00665654

  • CrossRef
  • Google Scholar

• 25.

  HoVMorganM. An experiment to test the gravitational Aharonov-Bohm effect. Aust J Phys (1994) 47(3):245–52. 10.1071/ph940245

  • CrossRef
  • Google Scholar

• 26.

  AgopMGh BuzeaCGrigaVCiubotariuCBuzeaCStanCet alGravitational paramagnetism, diamagnetism and gravitational superconductivity. Aust J Phys (1996) 49:1063–22. 10.1071/ph961063

  • CrossRef
  • Google Scholar

• 27.

  AgopM. Gravitomagnetic field, spontaneous symmetry breaking and a periodical property of space. Il Nuovo Cimento Gennaio (1998) 11.

  • Google Scholar

• 28.

  WilliamsLInanN. Maxwellian mirages in general relativity. New J Phys (2021) 23:053019. 10.1088/1367-2630/abf322

  • CrossRef
  • Google Scholar

• 29.

  LandauLLifshitzE. The classical theory of fields. 3rd ed., 2. Butterworth-Heinemann (2000).

  • Google Scholar

• 30.

  PapiniG. Particle wave functions in weak gravitational fields. Nuov Cim B (1967) 52:136–41. 10.1007/bf02710658

  • CrossRef
  • Google Scholar

• 31.

  CabreraBGutfreundHLittleW. Relativistic mass corrections for rotating superconductors. Phys Rev B (1982) 25:6644–54. 11. 10.1103/physrevb.25.6644

  • CrossRef
  • Google Scholar

• 32.

  PiyakisKPapiniGRystephanickR. Static paramagnetic response of thin soft superconductors. Can J Phys (1980) 58(6):812–9. 10.1139/p80-111

  • CrossRef
  • Google Scholar

• 33.

  DysonF. Seismic response of the Earth to a gravitational wave in the 1-Hz Band. Astrophys J (1969) 156:529. 10.1086/149986

  • CrossRef
  • Google Scholar

• 34.

  BoughnSRothmanT. Aspects of graviton detection: Graviton emission and absorption by atomic hydrogen. Class Quan Gravity (2006) 23:5839–52. 10.1088/0264-9381/23/20/006

  • CrossRef
  • Google Scholar

• 35.

  GalleratiAModaneseGUmmarinoG. Interaction between macroscopic quantum systems and gravity. Front Phys (2022) 10. 10.3389/fphy.2022.941858

  • CrossRef
  • Google Scholar

• 36.

  FlanaganÉHughesS. The basics of gravitational wave theory. New J Phys (2005) 7:204. 10.1088/1367-2630/7/1/204

  • CrossRef
  • Google Scholar

• 37.

  PoissonEWillC. Gravity: Newtonian, post-Newtonian, relativistic. Cambridge University Press (2014).

  • Google Scholar

• 38.

  InanNThompsonJChiaoR. Interaction of gravitational waves with superconductors. Fortschr Phys (2016) 65:1600066. 10.1002/prop.201600066

  • CrossRef
  • Google Scholar

• 39.

  InanN. A new approach to detecting gravitational waves via the coupling of gravity to the zero-point energy of the phonon modes of a superconductor. Int J Mod Phys D (2017) 26(12):1743031. 10.1142/S0218271817430313

  • CrossRef
  • Google Scholar

• 40.

  ChiaoRHaunRInanNKangBMartinezLMinterSet al “A gravitational Aharonov-Bohm effect, and its connection to parametric oscillators and gravitational radiation,” in Quantum theory: A two-time success story (yakir aharonov festschrift), 213–46. (Springer-Verlag Italia) (2014). 10.1007/978-88-470-5217-8

  • CrossRef
  • Google Scholar

• 41.

  PapiniG. London Moment of Rotating Superconductors and Lense-Thirring Fields of General Relativity. Nuov Cim B (1966) 45(1):66–8. 10.1007/bf02710584

  • CrossRef
  • Google Scholar

• 42.

  PapiniG. Detection of Inertial Effects with Superconducting Interferometers. Phys Lett A (1967) 24(1):32–3. 10.1016/0375-9601(67)90178-8

  • CrossRef
  • Google Scholar

• 43.

  PapiniG. Gravity-induced electric fields in superconductors. Nuov Cim B (1969) 63(2):549–59. 10.1007/bf02710706

  • CrossRef
  • Google Scholar

• 44.

  PapiniG. Inertial potentials in classical and quantum mechanics. Nuov Cim B (1970) 68:1–10. 10.1007/bf02710354

  • CrossRef
  • Google Scholar

• 45.

  LeungMPapiniGRystephanickR. Gravity-induced electric fields in metals. Can J Phys (1971) 49(22):2754–67. 10.1139/p71-334

  • CrossRef
  • Google Scholar

• 46.

  PapiniG. The transport of electricity in superconductors subject to gravitational or inertial forces. Phys Lett A (1975) 53(4):331–2. 10.1016/0375-9601(75)90089-4

  • CrossRef
  • Google Scholar

• 47.

  CaiYPapiniG. Particle interferometry in weak gravitational fields. Class Quan Gravity (1989) 6:407–18. 10.1088/0264-9381/6/3/017

  • CrossRef
  • Google Scholar

• 48.

  BertschingerE. Cosmological dynamics,” cosmology and large scale structure. In: SchaefferRSilkJSpiroMZinn-JustinJ, editors. Proc. Les houches summer school, session LX. Amsterdam: Elsevier Science (1995).

  • Google Scholar

• 49.

  CiubotariuCAgopM. Absence of a gravitational analog to the Meissner effect. Gen Relativ Gravit (1996) 28(4):405–12. 10.1007/bf02105084

  • CrossRef
  • Google Scholar

• 50.

  AgopMCiubotariuCGh BuzeaCBuzeaCCiobanuB. Physical implications of the gravitational fluxoid. Aust J Phys (1996) 49(3):613–22. 10.1071/PH960613

  • CrossRef
  • Google Scholar

• 51.

  Cisneros-ParraJ. On singular Lagrangians and Dirac’s method. Revista Mexicana de Fısica (2012) 58:61–8.

  • Google Scholar

• 52.

  DiracP. Generalized Hamiltonian dynamics. Can J Math (1950) 2:129–48. 10.4153/cjm-1950-012-1

  • CrossRef
  • Google Scholar

• 53.

  DiracP. The theory of gravitation in Hamiltonian form. Proc Roy Soc Lond A (1958) 246:326.

  • Google Scholar

• 54.

  DiracP. “Lectures on quantum mechanics,” in Belfer Graduate School of Science, Yeshiva University. New York, NY: Dover Publications Inc. (1964).

  • Google Scholar

• 55.

  CastellaniLDominionDLonghiG. Quantization rules and Dirac’s correspondence. Nuov Cim A (1978) 48A:359–68. N.1. 10.1007/bf02781602

  • CrossRef
  • Google Scholar

• 56.

  RobertsM. The quantization of geodesic deviation. Gen Relativ Gravit (1996) 28:1385–92. 10.1007/bf02109528

  • CrossRef
  • Google Scholar

• 57.

  NielsenHOlesenP. Vortex-line models for dual strings. Nucl Phys B (1973) 61(24):45–61. 10.1016/0550-3213(73)90350-7

  • CrossRef
  • Google Scholar

• 58.

  MeyerVSalierN. Generally relativistic phenomenological theory of superconductivity. Theor Math Phys (1979) 38:270–5. 10.1007/bf01018547

  • CrossRef
  • Google Scholar

• 59.

  DinarievOMosolovA. Relativistic generalization of the Ginzburg-Landau theory. Soviet Phys J (1989) 32:315–9. 10.1007/bf00897278

  • CrossRef
  • Google Scholar

• 60.

  PeskinMSchroederD. Introduction to quantum field theory. Boca Raton, FL: CRC Press (1995).

  • Google Scholar

• 61.

  RyderL. Quantum field theory. 2nd ed.Cambridge University Press (1996).

  • Google Scholar

• 62.

  LanoR. Gravitational phase transition in neutron stars (1996). arXiv:gr-qc/9611023v1.

  • Google Scholar

• 63.

  BertrandD. A relativistic BCS theory of superconductivity: An experimentally motivated study of electric fields in superconductors. Dissertation, Université catholique de Louvain (2005). Available at: http://hdl.handle.net/2078.1/5380

  • Google Scholar

• 64.

  CaiQPapiniG. Applying Berry's phase to problems involving weak gravitational and inertial fields. Class Quan Gravity (1990) 7:269–75. 10.1088/0264-9381/7/2/021

  • CrossRef
  • Google Scholar

• 65.

  PapiniG. Quantum physics in inertial and gravitational fields. In: Relativity in rotating frames (fundamental theories of physics), 135. Dordrecht: Springer (2004). 10.1007/978-94-017-0528-8_18

  • CrossRef
  • Google Scholar

• 66.

  PapiniG. On gravitational fields in superconductors. Front Phys (2022) 10. 10.3389/fphy.2022.920238

  • CrossRef
  • Google Scholar

• 67.

  BaryshevY. Einstein’s geometrical versus Feynman’s quantum-field approaches to gravity physics: Testing by modern multimessenger astronomy. Universe (2020) 6(11):212. 10.3390/universe6110212

  • CrossRef
  • Google Scholar

• 68.

  FeynmanR. Quantum theory of gravitation. Acta Phys Pol (1963) 24:697–722.

  • Google Scholar

• 69.

  FeynmanR. Lectures on Gravitation. Pasadena, CA, USA: California Institute of Technology (1971).

  • Google Scholar

• 70.

  FeynmanRMorinigoFWagnerW. Feynman Lectures on Gravitation. Boston, MA, USA: Addison-Wesley Publishing Company (1995).

  • Google Scholar

• 71.

  AgopMBuzeaCCiobanuB. On gravitational shielding in electromagnetic fields (1999). arXiv:physics/9911011.

  • Google Scholar

• 72.

  AgopMBuzeaCNicaP. Local gravitoelectromagnetic effects on a superconductor. Physica C: Superconductivity (2000) 339(2):120–8. 10.1016/S0921-4534(00)00340-3

  • CrossRef
  • Google Scholar

• 73.

  AgopMIaonnoiPDiaconuF. Some implications of gravitational superconductivity. Prog Theor Phys (2000) 104:733–42. 10.1143/ptp.104.733

  • CrossRef
  • Google Scholar

• 74.

  UmmarinoGGalleratiASuperconductor. Superconductor in a weak static gravitational field. Eur Phys J C (2017) 778:549. 10.1140/epjc/s10052-017-5116-y

  • CrossRef
  • Google Scholar

• 75.

  GalleratiAUmmarinoG. Superconductors and gravity. Symmetry (2022) 14(3):554. 10.3390/sym14030554

  • CrossRef
  • Google Scholar

• 76.

  CaiQPapiniG. The effect of space-time curvature on Hilbert space. Gen Relativ Gravit (1990) 22:259–67. 10.1007/bf00756276

  • CrossRef
  • Google Scholar

• 77.

  SpeliotopoulosA. Quantum mechanics and linearized gravitational waves. Phys Rev D (1995) 51:1701–9. 10.1103/physrevd.51.1701

  • CrossRef
  • Google Scholar

• 78.

  SpeliotopoulosAChiaoR. Differing calculations of the response of matter-wave interferometers to gravitational waves (2004). arXiv:gr-qc/0406096v1.

  • Google Scholar

• 79.

  WeberJ. Gravitational radiation and relativity. In: WeberJKaradeTM, editors. Proceedings of the sir arthur eddington centenary symposium, nagpur, India, 3. Singapore: World Scientific (1986). 1984.

  • Google Scholar

• 80.

  SpeliotopoulosAChiaoR. Coupling of linearized gravity to nonrelativistic test particles: Dynamics in the general laboratory frame. Phys Rev D (2004) 69:084013. 10.1103/physrevd.69.084013

  • CrossRef
  • Google Scholar

• 81.

  CarigliaMHouriTKrtousPKubiznakD. On integrability of the geodesic deviation equation. Eur Phys J C (2018) 78:661. 10.1140/epjc/s10052-018-6133-1

  • CrossRef
  • Google Scholar

• 82.

  KernerR. Generalized geodesic deviations: A lagrangean approach. Banach Cent Publications (2003) 59(N 1):173–88. 10.4064/bc59-0-9

  • CrossRef
  • Google Scholar

• 83.

  ParkerL. One-electron atom in curved space-time. Phys Rev Lett (1980) 44:1559–62. 10.1103/PhysRevLett.44.1559

  • CrossRef
  • Google Scholar

• 84.

  ParkerL. One-electron atom as a probe of spacetime curvature. Phys Rev D (1980) 22(8):1922–34. 10.1103/PhysRevD.22.1922

  • CrossRef
  • Google Scholar

• 85.

  ParkerLPimentelL. Gravitational perturbation of the hydrogen spectrum. Phys Rev D (1982) 25:3180–90. 10.1103/PhysRevD.25.3180

  • CrossRef
  • Google Scholar

• 86.

  MinterS. On the implications of incompressibility of the quantum mechanical wavefunction in the presence of tidal gravitational fields,” doctoral dissertation. Merced: Department of Physics, University of California (2010). ProQuest ID: 2010_sminter. Merritt ID: ark:/13030/m5f77sm2.

  • Google Scholar

• 87.

  AtanasovV. The geometric field (gravity) as an electro-chemical potential in a Ginzburg-Landau theory of superconductivity. Physica B: Condensed Matter (2017) 517(15):53–8. 10.1016/j.physb.2017.05.006

  • CrossRef
  • Google Scholar

• 88.

  BlencoweM. Effective field theory approach to gravitationally induced decoherence. Phys Rev Lett (2013) 111:021302. 10.1103/physrevlett.111.021302

  • CrossRef
  • Google Scholar

• 89.

  ArteagaDParentaniRVerdaguerE. Propagation in a thermal graviton background. Phys Rev D (2004) 70:044019. 10.1103/PhysRevD.70.044019

  • CrossRef
  • Google Scholar

• 90.

  GregorashDPapiniG. A unified theory of gravitation and electromagnetism for charged superfluids. Phys Lett A (1981) 82(2):67–9. 10.1016/0375-9601(81)90939-7

  • CrossRef
  • Google Scholar

• 91.

  CallanCJr.ColemanJackiwSR. A new improved energy-momentum tensor. Ann Phys (1970) 59(1):42–73. 10.1016/0003-4916(70)90394-5

  • CrossRef
  • Google Scholar

• 92.

  BirrellNDaviesP. Quantum fields in curved space. Cambridge University Press (1982).

  • Google Scholar

• 93.

  ParkerLTomsD. Quantum field theory in curved spacetime. Cambridge University Press (2009).

  • Google Scholar

• 94.

  TinkhamM. Introduction to superconductivity. 2nd ed.Mineola, N.Y.: Dover Publications (2004).

  • Google Scholar

• 95.

  LondonFLondonH. The electromagnetic equations of the supraconductor. Proc Roy Soc (London) (1935) 149:866. 10.1098/rspa.1935.0048

  • CrossRef
  • Google Scholar

• 96.

  LondonF. Superfluids, 1. Dover Publications (1950).

  • Google Scholar

• 97.

  BailinDLoveA. Superconductivity for Relativistic Electrons. J Phys A: Math Gen (1982) 15:3001–5. 10.1088/0305-4470/15/9/046

  • CrossRef
  • Google Scholar

• 98.

  GovaertsJBertrandD. Superconductivity and electric fields: A relativistic extension of BCS superconductivity. In: Proceedings of the fourth international workshop on contemporary problems in mathematical physics. World Scientific (2006). arxiv.org/pdf/cond-mat/0608084.

  • Google Scholar

• 99.

  LipavskyP. Bernoulli potential in superconductors. Springer (2007).

  • Google Scholar

• 100.

  LanoR. Gravitational meissner effect (1996). arXiv:hep-th/9603077v1.

  • Google Scholar

• 101.

  TajmarMNeunzigOKößlingM. Measurement of anomalous forces from a Cooper-pair current in high-T_c superconductors with nano-Newton precision. Front Phys (2022) 10. 10.3389/fphy.2022.892215

  • CrossRef
  • Google Scholar

• 102.

  GriffithsD. Introduction to electrodynamics. 3rd ed.. Upple Saddle River, NJ: Prentice-Hall (1999).

  • Google Scholar

• 103.

  GinzburgVLandauL. On superconductivity and superfluidity. Berlin, Heidelberg: Springer (2009). 10.1007/978-3-540-68008-6_4

  • CrossRef
  • Google Scholar

• 104.

  UmmarinoGGalleratiA. Exploiting weak field gravity-Maxwell symmetry in superconductive fluctuations regime. Symmetry (2019) 11(11):1341. 10.3390/sym11111341

  • CrossRef
  • Google Scholar

• 105.

  UmmarinoGGalleratiA. Possible alterations of local gravitational field inside a superconductor. Entropy (2021) 23(2):193. 10.3390/e23020193

  • CrossRef
  • Google Scholar

• 106.

  ByersNYangC. Theoretical considerations concerning quantized magnetic flux in superconducting cylinders. Phys Rev Lett (1961) 7:46–9. 10.1103/PhysRevLett.7.46

  • CrossRef
  • Google Scholar

• 107.

  DeaverBFairbankW. Experimental evidence for quantized flux in superconducting cylinders. Phys Rev Lett (1961) 7:43–6. 10.1103/PhysRevLett.7.43

  • CrossRef
  • Google Scholar

• 108.

  DollRNäbauerM. Experimental proof of magnetic flux quantization in a superconducting ring. Phys Rev Lett (1961) 7:51–2. 10.1103/PhysRevLett.7.51

  • CrossRef
  • Google Scholar

• 109.

  TajmarMde MatosC. Gravitomagnetic field of a rotating superconductor and of a rotating superfluid. Physica C: Superconductivity (2003) 385(4):551–4. 10.1016/S0921-4534(02)02305-5

  • CrossRef
  • Google Scholar

• 110.

  TajmarMde MatosC. Extended Analysis of Gravitomagnetic Fields in Rotating Superconductors and Superfluids. Physica C: Superconductivity its Appl (2005) 420(1–2):56–60. 10.1016/j.physc.2005.01.008

  • CrossRef
  • Google Scholar

• 111.

  PapiniG. A test of General Relativity by means of superconductors. Phys Lett (1966) 23:418–9. number 7. 10.1016/0031-9163(66)91071-7

  • CrossRef
  • Google Scholar

• 112.

  MeserveryRTedrowP. Measurements of the kinetic inductance of superconducting linear structures. J Appl Phys (1969) 40:2028–34. 10.1063/1.1657905

  • CrossRef
  • Google Scholar

• 113.

  GüllüIŞişmanTÇTekinB. Unitarity analysis of general Born-Infeld gravity theories. Phys Rev D (2010) 82:124023. 10.1103/PhysRevD.82.124023

  • CrossRef
  • Google Scholar

• 114.

  HinterbichlerK. Theoretical aspects of massive gravity. Rev Mod Phys (2012) 84:671–710. 10.1103/RevModPhys.84.671

  • CrossRef
  • Google Scholar

• 115.

  AtlasETekinB. Second order perturbation theory in general relativity: Taub charges as integral constraints. Phys Rev D (2019) 99:104078. 10.1103/PhysRevD.99.104078

  • CrossRef
  • Google Scholar