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π 4 so the superfluid velocity in the rim lags the rotation of Ie = nsmehR (9b) the body by the small amount 2 where h is the height of the cylinder. I is the moment ∆vs = −2λLω. (6) e of inertia of the superfluid electrons. As the system goes The full behavior of superfluid velocity versus radius is superconducting, electrons in the rim slow down accord- given by Laue [16] for a cylindrical geometry and by ing to Eq. (5), and the electronic angular momentum Becker et al [1] and London [2] for a spherical geome- decreases by try. The London penetration depth is given by [4] 2 ∆Le = (2πRλLhns) × (me∆vsR) (10) 1 4πnse 2 = 2 (7) λL mec with ∆vs given by Eq. (6). In Eq. (10), the first factor is with ns the superfluid density. the number of superfluid electrons in the surface rim, and For a perfect conductor that is set into rotation, Eq. the second factor is the change in angular momentum of (1) follows from Maxwell’s equations, as discussed by one electron. Using Eq. (7), Eq. (10) can be rewritten Becker et al [1], assuming the superfluid electrons are as completely detached from the lattice. As the body is set mec 2 2 ∆Le = −( ) R hω. (11) into rotation the moving ions generate an electric current e and hence a time-dependent magnetic field that generates a Faraday electric field that pushes the superfluid elec- The kinetic energy of the superfluid electrons in the trons to follow the motion of the ions, albeit with a small normal state due to the body rotation is lag near the surface that gives rise to the surface current 2 1 2 Le that generates the magnetic field Eq. (1). The derivation Ke = Ieω = (12) is reviewed in Appendix A. 2 2Ie In this paper we explain the behavior of rotating su- and when the system goes superconducting and the elec- perconductors using the theory of hole superconductivity trons in the rim slow down it decreases by the amount [17], and we argue that the conventional theory of super- conductivity is inconsistent with the physics of rotating Le∆Le ∆Ke = = ∆Leω (13) superconductors. Ie or, using Eqs. (11) and (1)

II. THE CONVENTIONAL VIEW 2 B 2 ∆Ke = − (πR h). (14) We consider a normal metallic cylinder rotating around 4π its axis with angular velocity ~ω that is cooled into the su- For the ions, the kinetic energy of rotation is perconducting state. We assume the cylinder is floating 2 in a gravitation-free environment and there is no friction. Li Ki = (15) Therefore, total angular momentum is conserved. 2Ii In the normal state, electrons and ions rotate together at the same speed. When the cylinder becomes super- with Ii the moment of inertia for the ions. From Eq. (8) conducting, electrons near the surface have to sponta- and conservation of total angular momentum we deduce neously slow down to the velocity Eq. (5) upon entering the superconducting state. The conventional theory of ∆Li = −∆Le, (16) superconductivity does not explain the dynamics of this therefore, angular momentum has to be transferred from process, which London himself considered to be “quite the electrons to the body. This gives rise to a change in absurd” from the perfect conductor viewpoint [18] and the kinetic energy of the ions, did not provide an explanation for. Furthermore, the body has to spontaneously speed up to compensate for Li ∆Ki = ∆Li = ∆Liω = −∆Ke (17) the electronic angular momentum change. The conven- Ii tional theory of superconductivity does not explain what is the physical process that causes the body to increase hence its rotation velocity. The total mechanical angular momentum of electrons ∆Ke + ∆Ki =0. (18) plus ions is Therefore, conservation of angular momentum requires L~ = L~ e + L~ i. (8) that the change in ionic and electronic kinetic energies of rotation exactly compensate each other. The body’s The electronic angular momentum in the rotating normal rotation speed slightly increases to compensate for the state is slowing down of the rim electrons, and its kinetic energy Le = Ieω (9a) of rotation slightly increases. The conventional theory of 3 superconductivity does not explain how the kinetic en- when the system becomes superconducting, because the ergy saved by the lagging electrons is transferred to the energy saved in the slowing down of the rim electrons was body to make it speed up. The frequency of rotation entirely used up in speeding up the body to satisfy an- when the rotating normal metal becomes superconduct- gular momentum conservation. Therefore, we argue that ing will be larger than ω by the amount the conventional theory cannot account for the physics of

2 rotating superconductors without violating either conser- ∆Li me ns 8λL vation of energy or conservation of angular momentum. ∆ω = = 2 ω (19) Ii mpA n R The implausibility of the conventional picture also fol- lows from the following argument. From the Meissner where mp is the nucleon mass, A the atomic weight and effect we learned that the final state of a superconductor n the ionic number density. in a magnetic field B is unique, independent of history. For a type I superconductor it is expected [19] that Whether we apply a magnetic field B to a normal metal when the body is rotating at frequency ω the transition and then cool it to the superconducting state, or we apply to the superconducting state will occur when the system the same magnetic field B to a metal already supercon- is cooled below Tc to the temperature T for which ducting, the final state of the system is exactly the same. Similarly here, the rotation frequency ω plays the role 2mec Hc(T )= − ω = B. (20) that B plays in the Meissner effect. Whether we apply e the rotation ω to the normal metal and then cool it to with Hc(T ) the thermodynamic critical field. Now the the superconducting state, or instead apply the same ro- magnetic field energy associated with the magnetic field tation ω to a metal already superconducting, the final Eq. (1) for a cylinder with zero demagnetizing factor is state of the system should be exactly the same, with the given by same ω [18]. This implies that the rotation speed of the body should not change when the rotating metal enters the 2 B 2 1 superconducting state, contrary to the prediction of the EB = (πR h)= |∆Ke|. (21) 8π 2 conventional theory that the rotation frequency should change by the amount given by Eq. (19). The condensation energy for a superconducting cylinder Finally, as reviewed in Appenix A, for the case of a su- at rest is given by perconductor at rest that is set into rotation, the deriva- 2 Hc(T ) 2 tion of Eq. (1) requires that there is no direct interaction Econd = ns(πR h). (22) between the superfluid electrons and the lattice. If that 8π is the case, how can there be a transfer of angular mo- In the Meissner effect, the condensation energy provides mentum from electrons to the body as a rotating normal the energy required to expel the magnetic field. Here, metal becomes superconducting? the condensation energy Eq. (22) provides the energy to generate the magnetic field Eq. (21). Therefore, conser- vation of energy requires that the condensation energy III. ALTERNATIVE VIEW Eq. (22) is the same for rotating and non-rotating su- perconductors. Instead, we propose that when the rotating cylinder However, we argue that for a superconductor rotating becomes superconducting its angular velocity does not at high speeds it is inconsistent to assume that the bind- change and hence that the ionic angular momentum ing energy of Cooper pairs, that determines the conden- doesn’t change. This then implies that the total elec- sation energy, would be independent of rotation speed. tronic angular momentum does not change either. According to the conventional theory this should be true The angular momentum of electrons in the rim de- for arbitrarily high frequencies, even for frequencies giv- creases by the amount Eq. (10). Hence the angular mo- ing rise to a magnetic field B larger than the thermody- mentum of electrons in the bulk has to increase by the namic critical field at zero temperature. If we consider same amount: two electrons in a Cooper pair separated by a typical bulk distance ξ ∼ λL, the energy associated with rotation at ∆Le + ∆Le =0 (24a) frequency ω is bulk 2 2 2 2 2 ∆L = me(2λL) ns(πR h)ω (24b) ǫrot = meλLω ∼ B /(8πns) (23) e i.e. the magnetic field energy per electron. We argue that Since the number of electrons in the bulk is it is inconceivable that the binding energy of Cooper pairs 2 Ne = nsπR h (25) would not be lowered by the rotation energy ǫrot for any magnitude of B. this implies that each electron in the bulk acquires an However, if the condensation energy is decreased at additional ‘intrinsic’ angular momentum finite rotation frequency, there is not enough energy to 2 account for the creation of the magnetic field Eq. (1) ℓ = me(2λL) ω. (26) 4

Eq. (26) says that when the rotating metal becomes which is half of the decrease in rim kinetic energy Eq. superconducting there is an additional contribution to (31). The other half goes into paying the cost in magnetic its electronic angular momentum that comes from the energy Eq. (1). This then implies that the condensation electron mass being spread out in a ring of radius 2λL. energy of the rotating superconductor at temperature T Equivalently, that the electron’s orbit expands from a is, instead of Eq. (22) microscopic radius to radius 2λ . Such physics was pre- L 2 2 dicted by the theory of hole superconductivity [20, 21] to Hc (T ) B (ω) 2 Econd(T,ω) = ( − )(πR h) (33) explain the dynamics of the Meissner effect: as an elec- 8π 8π tron expands its orbit from point-like to radius 2λL in the presence of a magnetic field, azimuthal current of the with B(ω) given by Eq.(1). The transition occurs at the magnitude required to expel the magnetic field gets gen- frequency or temperature where the condensation energy erated by the Lorentz force. Expansion of the electronic vanishes. wavefunction to radius 2λL is an indispensable element in the dynamical explanation of the Meissner effect within IV. KINETICS OF THE TRANSITION this theory [22]. In Eq. (26) we find a direct confirma- tion of this essential part of the theory. It implies that when electrons enter the superfluid state they acquire an Let us consider the process by which the magnetic field intrinsic moment of inertia attains the value Eq. (1). Consider a point at distance r from the origin. From Faraday’s law 2 iel = me(2λL) . (27) 1 ∂φ E~ · dl~ = − (34) so the total electronic moment of inertia increases by I c ∂t 2 2 ∆Ie = me(2λL) ns(πR h) (28) and assuming cylindrical symmetry we have and the electronic angular momentum increase in the 1 1 ∂φ(r, t) bulk Eq. (24b) is EF (r, t)= − (35) 2πr c ∂t bulk ∆Le = ∆Ieω (29) where EF (r, t) is the induced Faraday electric field and ′ Using Eq. (7), we can write the increase in moment of φ(r, t) the magnetic flux through r < r, at time t. Inte- inertia as grating over time, ∞ V B me ∆Ie = me (30) dtEF (r, t)= = ωr. (36) πrc Z0 2cr e 2 where V = πR h is the volume of the body and rc = 2 2 If the location r is superconducting, a superfluid electron e /mec is the classical electron radius. Because Eq. (30) at r obeys the equation of motion no longer depends on the geometry of the body we believe it is very likely that it is a general result for a body of dvs e = E(r, t) (37) arbitrary shape and for any rotation axis. dt me Note the interesting fact that Eq. (30) is also indepen- dent of the superfluid density and the London penetra- where vs and E are in the azimuthal direction. Assuming tion depth. Thus it expresses a qualitative difference be- the point r was superconducting during the entire process tween tne normal and superconducting states of matter. we have upon integrating Eq. (37) and using Eq. (36) Presumably the volume factor in Eq. (30) corresponds to ∞ dvs the volume of the sample that is in the superconducting ∆vs = = ωr. (38) state. Z0 dt How does the energetics work in this scenario? We Therefore, this equation describes the process shown in have from Eqs. (13) and (14) that the decrease in elec- Fig. 2 (a), where the body is initially at rest in the tronic kinetic energy because of the rim slowing down superconducting state and the electron initial speed is is vs(R,t = 0) = 0, and attains final speed ωr when the 2 2 B 2 body acquires angular velocity ω. Eq. (38) does not ∆Ke = ∆Leω = −∆Ieω = − (πR h) (31) 4π apply when r is within λL of the surface because the magnetic field does not acquire its full bulk value Eq. The ions do not acquire extra kinetic energy since the (1) in that region. frequency of rotation ω doesn’t change. The bulk elec- This reasoning also shows that if we are considering the tronic kinetic energy is given by Eq. (12), so when the process where the rotating normal metal is cooled into moment of inertia increases it increases by the superconducting state, an electron at radius r has to 2 bulk 1 2 B 2 be in the normal state during the entire time where the ∆K = ∆Ieω = (πR h) (32) e 2 8π magnetic flux φ(r, t) changes. This is because initially 5

ω=0 ω ' (a) ω ω FC F E r r EF r B=0 B B' N Fon-latt S electron S S backflow λL FE F r0 C Flatt B r0 EF ω ω (b) ω ion super Coriolisis current sE F r r EF r B=0 B B S S S N r0 r0 r0 N λL

FIG. 2: Growth of the magnetic field in rotating supercon- ductors (from left to right). (a) A superconductor at rest starts rotating until it reaches frequency ω. (b) A normal metal (N) enters the superconducting state (S) upon cooling, while rotating at frequency ω . Dots indicate magnetic field pointing out of the paper, their density indicates the strength of the field. At a given point r in the interior, the change in ′ FIG. 3: Rotating normal metal becoming superconducting. magnetic flux through the region r < r between the initial 2 The phase boundary at radius r0 expands outward. Black and final states is given by ∆φ = πr B and determines the dots indicate magnetic field pointing out of the paper, grey time integral of the Faraday electric field at point r, Eq. (36). crosses indicated demagnetizing field pointing into the paper (see Sect. VI). Electrons at the phase boundary expand their orbits to radius 2λL, and a backflow of normal electrons takes electrons at radius r in the normal state rotate together place to compensate for the radial charge imbalance. with the body with azimuthal speed ωr. If at any time while φ is changing Eq. (37) was valid, the electron at r would attain a final velocity different from ωr, however we know that in the final state electrons in the interior rotate together with the body with azimuthal speed ωr. We conclude that when a rotating metal is cooled into the superconducting state the superconducting region neces- body FE rotation sarily expands from the inside out, as shown schemati- B Fon-latt cally in Fig. 2 (b). The Faraday field acts on electrons N electron at point r in the normal state while the magnetic field is Flatt ′ backflow growing in the region r < r, during this time it generates Lorentz Joule heat but does not change the electron’s azimuthal λL FE speed. r0 FB super current EF EF V. DYNAMICS OF THE TRANSITION S ion λL Next we analyze the dynamics of the transition when r0 the rotating normal metal becomes superconducting, r0 r0 along the same lines that we analyzed the dynamics of the Meissner effect in Refs. [23, 24]. There are many similarities but some important subtle differences. Fig- ures 3 and 4 show schematically both processes, where r0 denotes the radius of the expanding phase boundary. The magnetic field pointing out of the paper is denoted by black circles, with their diameter illustrating the in- tensity. The driving force for the generation of the surface cur- FIG. 4: Meissner effect. See text and ref. [23] for details. rent in both cases is expansion of the electronic orbits from a microscopic radius to radius 2λL when the elec- 6 trons enter the superconducting state. In the Meissner above, but would cause dissipation and entropy produc- case, this expansion in the presence of magnetic field B tion rendering the transition irreversible in contradiction produces the Meissner counterclockwise current through with theory and experiment [23, 25]. the action of the magnetic Lorentz force. Similarly, in the rotating case the Coriolis force acts e VI. NON-ZERO DEMAGNETIZING FACTOR F~C =2me~v × ~ω = − ~v × B~ (ω) (39) c For samples other than long cylinders a demagnetiz- imparting clockwise speed (relative to the body) to the ing magnetic field will exist, and the above considered outgoing electron with radial velocity ~v. simplest situation needs to be modified. Consider for ex- In the Meissner effect, the body acquires momentum ample the case of a rotating sphere [1, 2]. The lagging in the opposite direction through the backflow process velocity of electrons in the rim is not constant as given by illustrated in Fig. 4: backflowing electrons with nega- Eq. (6) but rather depends on the location of electrons tive effective mass are subject to a force from the ions relative to the equator [2]: in the counterclockwise direction (Flatt), causing a clock- wise reaction force to be exerted on the body F − on latt ∆v =3λ (sinθ)ω (40) that makes the body turn. That force is partially com- s L pensated by the counterclockwise force exerted on the where θ is the angle between the position vector and the positive ions by the Faraday field. The quantitative anal- ysis is given in ref. [23]. axis of rotation. So at the equator, the lagging speed of electrons is larger than for the cylinder by a factor The momentum transferred by the backflowing elec- 3/2. How can this be understood within our scenario, trons to the body in the Meissner effect can be thought and what are its consequences? of as momentum that was stored in the electromagnetic field when electrons expanded their orbits and acquired For the case of the Meissner effect (Fig. 4), it is imme- azimuthal momentum through the Lorentz force [23]. diately clear why the electrons in the Meissner current at Therein lies the essential difference with the case of the the equator acquire the higher speed: the magnetic field rotating superconductor: there, when the orbit expands that imparts the azimuthal speed through the Lorentz and acquires azimuthal momentum through the Coriolis force acting on the expanding orbits is larger than the force (Fig. 3), no momentum is stored in the electromag- applied field precisely by the factor 3/2, due to demag- netic field because there is no magnetic field in the lightly netization. Recall that the critical magnetic field for a shaded region in Fig. 3 for a long cylinder. Instead, the sphere is (2/3)Hc rather than Hc [4]. same Coriolis force that deflects the electron expanding Similarly we can understand the larger lagging speed its orbit in the clockwise direction transfers momentum for the spherical rotating superconductor. In Fig. 4, to the ions in counterclockwise direction. That momen- in the lightly shaded region where the electron orbit ex- tum is cancelled by the clockwise momentum transferred pands and is deflected clockwise by the Coriolis force, by the Faraday electric field EF to the ions in that region. there is now a magnetic field pointing into the paper be- The balance of forces for the backflowing electrons for cause of demagnetization (indicated by crosses in Fig. the rotating superconductor is shown in Fig. 3. Coriolis 3). The magnetic Lorentz force on the expanding orbit and electric forces act counterclockwise, exactly balanced provides additional azimuthal momentum in the clock- by the clockwise force exerted on the electrons by the wise direction in addition to the one imparted by the Coriolis force. It also stores some momentum in the elec- lattice of ions (Flatt) so that backflowing electrons flow radially in, just like for the Meissner effect. However un- tromagnetic field. The angular momentum density in the like for the Meissner case, here the backflowing electrons electromagnetic field is do not transfer net momentum to the body because the 1 reaction to the force exerted by the ions on the electrons, L~ em = ~r × (E~ × B~ ). (41) Fon−latt, is cancelled by the sum of electric force FE and 4πc the reaction to the Coriolis force on the electrons FC (Fig. 3). The electric field E~ created by the outflow of negative In summary, for the reasons explained above, unlike for charge resulting from orbit enlargement points radially the Meissner effect there is no net transfer of angular mo- outward, the demagnetizing field B~ points into the pa- mentum to the body for the rotating superconductor as per, hence L~ em points parallel to ~ω (out of the paper) the superconducting region expands. The fact that back- and will increase the angular velocity when transferred flowing electrons have negative effective mass of course to the body by the backflowing electrons. Thus, unlike still plays an essential role: if they had positive effective the case of the long cylinder where there is no demag- mass, backflowing electrons would be deflected counter- netizing field, here (and for any sample with non-zero clockwise and transfer their momentum to the body by demagnetizing factor) there will be a small change (in- collisions. This would of course not alter the angular crease) in the velocity of rotation of the body when the momentum balance compared to the scenario described rotating normal metal becomes superconducting. 7

For example, for the case of the sphere of radius R the state. However, again the predicted change is extremely angular momentum of the rim current is small:

mec 2 3 ∆I ∆ω ∆Le = 2( ) R ω (42) = − (47) e I ω and the increase in the moment of inertia that we predict so that its detection is probably beyond the capabilities is, from Eq. (30) of even the most sensitive torsional oscillators. We also predict that at the transition point the con- 4 mec 2 3 ∆Ie = ( ) R (43) densation energy goes to zero (Eq. 33), while the conven- 3 e tional theory predicts it is finite (Eq. 22). Correspond- so it only accunts for 2/3 of the angular momentum ingly, we predict no latent heat and the conventional change Eq. (42). Thus the angular velocity will increase theory predicts finite latent heat. Within the two-fluid slightly by ∆ω given by model the latent heat per unit volume at the transition temperature T according to the conventional theory is mec 2 3 2( e ) R ∆ω = ω (44) H0 T 2 H0B(ω) 3I L(T )= ( ) Hc(T ) ∼ (48) 2π Tc 2π with I the moment of inertia of the body, while in the conventional theory the increase in rotation frequency is with H0 the zero temperature critical field. For H0 = given by 500G, −5 3 2( mec )2R3 L(T )=0.91 × 10 ω(rad/s)erg/cm (49) ∆ω = e ω (45) I which is extremely small even for very high rotation fre- i.e. a factor of 3 larger. We explained above how the quencies, hence very difficult to measure. body acquires the additional rotation speed within our theory. VIII. DISCUSSION

VII. EXPERIMENTAL CONSEQUENCES In this paper we have proposed that when a rotat- ing normal metal becomes superconducting the rotation Our theory predicts that upon cooling a rotating long speed of the body does not change, contrary to what the cylinder into the superconducting state, its rotation fre- conventional theory predicts, when the electrons near the quency should not change. Instead, the conventional surface slow down relative to the body motion and create theory predicts the body’s rotation frequency should in- the interior magnetic field Eq. (1). This unexpected ef- crease by the amount given by Eq. (19). For Al, with 3 fect occurs because electrons increase their contribution A = 27, density ρ = 2.7g/cm , λL(T = 0) = 500A˚, this to the bulk moment of inertia of the body when they enter is the superconducting state according to the theory of hole superconductivity, thus increasing the bulk electronic an- −5 λL 2 ∆ω =3.05 × 10 ( ) ω (46) gular momentum and thereby compensating for the de- R crease in the surface electrons angular momentum. Each Unfortunately, even for very thin cylinders with radius superfluid electron acquires an intrinsic angular momen- approaching λL, this small change would be extremely tum Eq. (26) that adds to the total angular momentum difficult to detect both by direct measurement of the fre- of the body without increasing its angular velocity. quency or by measuring the resulting very small change Our scenario certainly resolves the question of how to in the magnetic field, ∆B =1.137 × 10−7∆ω.. explain a speed-up of the rotational velocity of a normal What about directly measuring the change in moment metallic cylinder when it becomes superconducting, for of inertia between normal and superconducting states? which the conventional theory has provided no mecha- Using a sensitive torsional oscillator, experiments at- nism: there is no speedup. We also explain the mech- tempting to measure ‘non-classical rotational inertia’ in anism that causes the rim electrons to slow down when solid 4He were performed to study possible ‘supersolid’ the rotating normal metal becomes superconducting. In behavior [26]. For rotating superconductors we predict a nutshell, the Coriolis force acting on expanding orbits. that the total moment of inertia should not change be- The conventional theory provides no explanation, it sim- tween normal and superconducting states, because the ply postulates that it happens. increase in bulk moment of inertia is compensated by In fact, we found that except in the case of zero de- lowering of the moment of inertia of the rim. Instead, magnetizing factor our theory also predicts a speedup of the conventional theory predicts an overall decrease in rotation when the normal metal becomes superconduc- the moment of inertia upon entering the superconducting ing, albeit by a smaller amount than predicted by the 8

susceptibility. In other work we have pointed out [30] 2 that the same physics discussed here, increase in elec- r0 tronic moment of inertia when a normal metal becomes superconducting, should lead to an increase in both its 1 diamagnetic susceptibility (as observed) and in its mean r inner potential, which can be measured by electron holog- raphy [31] (not yet observed). Within the theory of hole superconductivity, supercon- ductors have a macroscopically inhomogeneous charge distribution in their ground state, with more negative charge near the surface [32]. One may wonder whether this will give an additional effect at sufficiently low tem- peratures where the resulting electric field is not screened by normal quasiparticles (that would also cancel the mass FIG. 5: Geometric illustration of increase in moment of inertia imbalance). This was suggested in Ref. [5]. In fact, it will caused by orbit enlargement (see text). not. The expelled electrons will rotate together with the body and the speed of body rotation will not change. The extra angular momentum of the expelled electrons will be conventional theory. We explained how the demagnetiz- exactly compensated by angular momentum stored in the ing field leads to a larger decrease in the speed of surface electromagnetic field and the angular momentum balance electrons and to transfer of angular momentum to the discussed in this paper will not change. body, using the same concepts that we recently used to We have pointed out that there is an inconsistency in explain the dynamics of the Meissner effect, in particu- the conventional theory, that assumes that there is no di- lar the fact that normal state charge carriers have to be rect interaction between electrons and the body when a hole-like [23]. superconductor at rest is set into rotation (Appendix A), Within the theory of hole superconductivity, electrons yet requires a transfer of momentum between electrons −1 and the body when a rotating normal metal becomes expand their orbits from microscopic radius kF (kF = Fermi momentum) to radius 2λL when theyR-2 pairλ up and superconducting. How is that inconsistency resolved in become superconductingR [20]. This explains why the our theory? In our theory, there can be a net momen- diamagnetic susceptibility grows from the normal metal tum transfer between electrons and the body when there is radial charge flow. Such charge flow occurs in the Landau susceptibilityr to −1/(4π)[27r], since the Larmor diamagnetic susceptibility is proportional to the square process of the normal metal becoming superconducting, of the radius of the orbit. In addition this provides a pic- but not when a superconductor is set into rotation. We torial understanding of the development of macroscopic can explain both situations where there is no momentum phase coherence due to overlapping orbits [27]. The fact transfer and where there is, provided in the latter case 2 that the moment of inertia of the body increases by mer0 there is also radial charge flow. Within the conventional per electron when the orbit of the electron increases from theory, there never is radial charge flow. essentially zero radius to radiusB r0 = 2λL follows from We point out that the physics discussed here is closely simple geometry illustrated in Fig. 5. Fig, 5 shows a related to physics of superfluid 4He. 4He has maximum point-like particle labeled 1 at radius r from the center, density at the superfluid transition temperature [33]. Be- and another point-like particle labeled 2 that moves in low the superfluid transition the system expands when an orbit of radius r0 centered at distance r from the cen- cooled further. As a consequence its bulk moment of ter. Simple geometry shows that the moment of inertia inertia will increase as it enters the superfluid state, as of particle 1 relative to an axis going through the cen- found here for superconductors. We have argued else- ter of the large circle is mr2 and that of particle 2 is where that this commonality between superconductors 2 2 4 m(r + r0) (m=mass of the particles) assuming particle and superfluid He derives from the fact that both super- 2 is uniformly distributed along its orbit of radius r0. conductivity and superfluidity are kinetic energy driven, The reason why the increase in the bulk moment of originating in expansion of the wavefunction driven by inertia of the body Eq. (30) is independent of the direc- quantum pressure [34, 35]. The physics of the Meissner tion of the rotation axis is easy to understand. When the effect discussed in refs. [22, 23] involving flow of super- body rotates, the expanded orbits will orient themselves fluid and counterflow of normal fluid is also closely related 4 so that they lie on planes perpendicular to the rotation to physics found in He that gives rise to the fountain axis to minimize their energy, as classical particles would. effect [36]. We conjecture that the concepts discussed It was pointed out by Bethe long ago [28] that the here may be relevant to the understanding of rotation 4 moment of inertia of the electronic charge distribution experiments in superfluid He [37–39]. in a solid determines its mean inner electric potential. In summary we find that superconducting matter has Shortly thereafter, Rosenfeld [29] pointed out that the a new distinct property. In addition to its electrical con- mean inner potential is proportional to the diamagnetic ductivity becoming infinite and its magnetic susceptibil- 9 ity becoming that of a perfect diamagnet, a metal be- superfluid electrons, and me is the bare electron mass. coming superconducting will increase its bulk moment The electric field E is determined by Faraday’s law Eqs. 2 of inertia per unit volume by the amount (mec/e) /π = (34) and (35), which yield at radial position r 1.03 × 10−15g/cm. This extra moment of inertia arises from the development of an intrinsic moment of inertia 1 dB(r, t) E(r, t)= − r . (A2) for each superfluid electron. Superfluid electrons behave 2c dt as an extended rim of mass me, radius 2λL, intrinsic 2 moment of inertia me(2λL) , and intrinsic orbital angu- In the interior, superfluid electrons rotate together with lar momentum ~/2 [21] rather than as point particles. the body, i.e. with azimuthal velocity vs = ωr. Using This leads to a dynamical understanding of the Meiss- this and combining Eqs. (A1) and (A2) then yields ner effect and the London moment, and follows from the quantization of orbital angular momentum in the pres- dω e dB(r, t) mer = −− r + Flatt. (A3) ence of Dirac’s spin orbit interaction predicted by the dt 2c dt theory [21]. Unfortunately as we saw in Sect. VII it appears very difficult in practice to test the different pre- Under the assumption that Flatt = 0, Eq. (A3) integrates dictions of our theory vis-a-vis the conventional theory to for rotating superconductors experimentally. e meω = − B (A4) 2c Appendix A: Derivation of Eq. (1) from Maxwell’s equations i.e. Eq. (1), assuming the initial conditions are ω = B = 0. In other words, Eq. (1) is obtained for a superconduc- When a superconducting body with cylindrical sym- tor set into rotation under the assumption that there is no metry is set into rotation, the superfluid electrons obey direct force F acting between the ions and electrons, the equation of motion latt i.e. that the electrons are perfectly free from interactions

dvs with the ions. This is precisely what was assumed in me = eE + Flatt (A1) the original work by Becker et al [1]: that “die mittlere dt Geschwindigkeit der Elektronen nur unter der Wirkung where the first term is the force on the electrons from eines elektrischen Feldes ¨andern” (“the mean velocity of the induced azimuthal electric field, the second term is the electrons only changes under the effect of an electric a direct force that may be exerted by the ions on the field”).

[1] R. Becker, G. Heller und F. Sauter, ‘Uber¨ die Physics, edited by J.G. Daunt et al. Plenum, New York, Stromverteilung in einer supraleitenden Kugel’, 1965, p. 459. Zeitschrift f¨ur Physik 85, 772 (1933). [11] M. Bol and W.M. Fairbank, in Proceedings of the 9th [2] F. London, “Superfluids’, Volume I, Wiley, New York, International Conference on Low Temperature Physics, (1950). edited by J.G. Daunt et al. Plenum, New York, 1965, p. [3] J.R. Schrieffer, “Theory of Superconductivity”, West- 471. view Press, Boulder, 1999. [12] J. Tate, B. Cabrera, S. B. Felch and J.T. Anderson, [4] M. Tinkham, “Introduction to Superconductivity”, 2nd Phys. Rev. Lett. 62. 845 (1989). ed, McGraw Hill, New York, 1996. [13] D. Hipkins, W. Felson and Y.M. Xiao, [5] J. E. Hirsch, “The London moment: what a rotat- Czech. J. Phys. 46, Suppl. S5, 2871 (1996). ing superconductor reveals about superconductivity”, [14] A.A. Verheijen et al., Physica B 165-166, 1181 (1990). Physica Scripta 89, 015806 (2013). [15] M. A. Sanzari,H. L. Cui, and F. Karwacki, [6] M. Tajmar, “Electrodynamics in super- Appl. Phys. Lett. 68, 3802 (1996). conductors explained by Proca equations”, [16] M. von Laue, “Theory of Superconductivity”, Academic Phys. Lett. A 372, 3289 (2008). Press, New York, 1952. [7] A. K. T. Assis and M. Tajmar, “Supercon- [17] References in http://physics.ucsd.edu/∼jorge/hole.html. ductivity with Weber’s electrodynamics: The [18] London’s remarks [2]: There is an implication which London moment and the Meissner effect”, might be worth mentioning since it would appear quite Annales de la Fondation Louis de Broglie 42, 307 (2017). absurd from the viewpoint of the perfect conductor con- [8] A. F. Hildebrandt, ‘Magnetic Field of a Rotating Super- cept. The uniqueness properties of the present theory pro- conductor’, Phys. Rev. Lett. 12, 190 (1964). vide for only one current distribution independent of the [9] N. F. Brickman, “Rotating Superconductors”, way in which the superconducting state is reached. Con- Phys. Rev. 184. 460 (1969). sequently we have to conclude that the same state as has [10] A.F. Hildebrand and M.M. Saffren, in Proceeding of been calculated above must also be obtained if the sphere the 9th International Conference on Low Temperature is cooled from the normal into the superconducting state 10

while it is rotating. The perfect conductor theory would, [28] H. Bethe, “Theorie der Beugung von Elektronen an of course, furnish no reason for the electrons near the Kristallen”, Ann. Physik 392, 85 (1928). surface of the metal to lag suddenly behind when the ro- [29] L. Rosenfeld, “Brechungsindex der Elektronen und Dia- tating sphere goes into the superconducting state. It would magnetismus”, Naturwissenschaften 17, 49 (1929). simply lead to a state of zero magnetic moment in which [30] J. E. Hirsch, “Superconductivity, diamag- all charges move in phase - the same below as above the netism, and the mean inner potential of solids”, transition point. Annalen der Physik 526, 63 (2013). [19] E. Babaev and B. Svistunov, “Rotational re- [31] A. Tonomura, “Electron Holography”, Springer, Berlin, sponse of superconductors: Magnetorotational 1999, and references therein. isomorphism and rotation-induced vortex lattice”, [32] J.E. Hirsch, ‘Charge expulsion and electric field in super- Phys. Rev. B 89, 104501 (2014). conductors’, Phys.Rev. B 68, 184502 (2003). [20] J.E. Hirsch, “Spin Meissner effect in supercon- [33] H. K. Onnes, “Further experiments with liquid helium. ductors and the origin of the Meissner effect”, A”, in: KNAW, Proceedings, 13 II, 1910-1911, Ams- Europhys. Lett. 81, 67003 (2008). terdam, 1911, pp. 1093-1113; H. K. Onnes and J. D. [21] J. E. Hirsch, “The Bohr superconductor”, A. Boks, “The variation of density of liquid helium be- Europhys. Lett. 113, 37001 (2016). low the boiling point”, Reports and Communications of [22] J. E. Hirsch, “On the dynamics of the Meissner effect”, the Fourth International Congres of Refrigeration, June Physica Scripta 91, -05801 (2016). 1924, Comm. 170b, p 18-23. [23] J.E. Hirsch, “Momentum of superconducting elec- [34] J.E. Hirsch, “Kinetic energy driven superconductivity trons and the explanation of the Meissner effect”, and superfluidity”, Mod. Phys. Lett. B 25, 2219 (2011). Phys. Rev. B 95, 014503 (2017). [35] J. E. Hirsch, “Kinetic energy driven superfluidity and [24] J.E. Hirsch, “Spinning Superconductors and Ferromag- superconductivity and the origin of the Meissner effect”, nets”, Acta Physica Polonica A 133, 350 (2018). Physica C 493, 18 (2013). [25] J.E. Hirsch, “Entropy generation and momentum trans- [36] F. London, “Superfluids”, Vol. II, Dover, New York, fer in the superconductor to normal phase transformation 1964. and the consistency of the conventional theory of super- [37] D. V. Osborne, Proc. Phys. Soc. A 63, 909 (1950). conductivity”, . [38] E. L. Andronikashvili and Y. G. Mamaladze, [26] D. Y. Kim and M. H. W. Chan, Rev. Mod. Phys. 38, 567 (1966). Phys. Rev. Lett. 109, 155301 (2012). [39] G. B. Hess and W. M. Fairbank, [27] J.E. Hirsch, ‘Electromotive Forces and the Meissner Ef- Phys. Rev. Lett. 19, 216 (1967). fect Puzzle’, J. Sup. Nov. Mag. 23, 309 (2010).