General Relativity and Gravitation, VoL 22, No. 6, 1990

A New Approach to Studying Local Gravitomagnetic Effects on a Superconductor

Huel Peng I

Received August 9, 1988

In this paper we introduce gravitomagnetic field equations into the in- vestigation of gravitomagnetlc effects on a superconductor. We point out that in the absence of an applied magnetic field, an applied gravito- magnetic field will induce twin currents, gravitational and electric super- currents. The latter will create a magnetic field. The slightly modified Josephson, London, and London-type gravitomagnetic equations are ob- tained. Some applications of these equations are discussed.

1. I N T R O D U C T I O N

The predicted effects regarding moving sources, e.g. gravitomagnetic ef- fects and gravitat ional waves, have yet to be directly tested [1,2]. The orbiting gyroscope experiment currently planned could detect the global effect of the gravitomagnetic field [3,4]. I t is obviously important to t ry to design experiments tha t detect local gravitomagnetic effects (dragging of inertial frames). Several ground-based experiments which could test the local gravitomagnetic effect have been proposed [5], but the practical difficulties are very great.

One of the reasons for studying the effect of gravity on a superconduc- tor is tha t the extreme sensitivity of superconducting devices may enable

Plwsics Department, The University of Alabama in Huntsville, Huntsville, Alabama 35899, USA and Institute of Applied Mathematics, Academia Sinlca, Beijing, P. R. China

609 0001-7701/90[0600-0609506.00/0 ~ Z990 Plea-ore t:~zb]~s.~g Corpo~a~o~

610 Peng

one to test gravitational effects. Gravitomagnetic effects on a supercon- ducting circular ring were studied by DeWitt [6] and Anandan [7]. From the flux quantization condition, they calculated the magnetic field inside a superconducting ring, which is induced by an applied gravitomagnetic field. Papini [8] studied gravitomagnetic effects on a Josephson interferom- eter and designed an experiment to test the Machian principle. Cerdonio [9] discussed two kinds of detectors for the local gravitomagnetic effect of the earth, one of them based on the London moment phenomenon.

In this paper we propose a new approach to studying gravitomag- netic effects on superconductors. We derive the gravitational equivalent of the London equation and the generalized Josephson equation. Then we solve the London-type gravitomagnetic equation for a semi-infinite su- perconductor to find the penetration depth of a gravitomagnetic field into the superconductor. We show that in the absence of an applied magnetic field, an applied gravitomagnetic field induces twin currents in the bulk of a superconductor, the gravitational and electric supereurrents. The latter induces a magnetic field. Taking this fact into account, the coupled Lon- don and London-type gravitomagnetic equations are derived. Then the generalized Meissner effect, the distribution of the twin currents, and the effect of the applied gravitomagnetic field on Josephson junctions and the SQUID are discussed.

We will choose G = c = 1.

2. THE LONDON-TYPE GRAVITOMAGNETIC EQUATION

By analogy with the London theory of syperconductivity, it is nec- essary to introduce the gravitomagnetic field Bg and field equations for deriving the penetration depth of the gravitomagnetic field and the gravi- tational supercurrent.

The Einstein equations

R u~" _ �89 = 8~rT uu,

give the gravitoelectric and gravitomagnetic field equations (we use the notation of Ref. 1)

O G m , ~, Oz a - -47r(T"" + tu"), (1)

where (2)

and the gauge condition hUU,v = 0 has been used without loss of general- ity [10], The t uu represent all nonlinear terms. We refer to GUUa as the gravitational field strength.

Local Gravi tomagnet ic Effects on Superconductor 611

In the limit of weak gravity, the nonlinear terms can be neglected, t ~ _ 0. For small velocities, we take T oo "~ P,n, T ~ "" Pray i - j m ~, and

T ~j _'~ 0. Equations (1) and (2) give the gravitomagnetic field equations

V x B g - - 4 w j , n , (3)

V . Bg = 0, (4)

Bg = V x Ag, (5)

where Ag i - 1/4h ~ -- 1/4g ~ Equations (3)-(5) have also been derived in the framework on the PPN formalism [4].

In this paper 'we do not deal with effects which occur in the period of transition from normal to superconducting. We will restrict ourselves to the stationary situation in which we can ideMize the superconductor by assuming that it is rigid and at rest in a reference frame. 1 For simplicity, place a superconducting sample on a platform and choose a sample rest frame which does not rotate relative to the distant stars. The external magnetic fields are shielded. Then there is only the external gravitomag- netic field acting on the sample.

For the gravitomagnetic field induced by the rotation of the earth, the magnitude of the gravitomagnetic field in the frame is of the order, from eq. (3)

Bg _ 10 -14/second.

In the absence of an applied magnetic field, an applied gravitomag- netic field will induce a gravitational mass current, i.e. drive superelectrons to move, which creates an electric current since superelectrons also carry electric charges. This electric current in turn creates a magnetic field which will react on superetectrons. We will take into account the effect of the induced magnetic field on superelectrons in Section 4. In this and following sections, in order to derive the gravitational analogue of London equations and compare with London theory, we ignore the effect of the in- duced magnetic field on superelectrons, although we should take this effect into account.

The Harniltonian for the ensemble of free electrons inside a supercon- ductor in an applied weak gravitomagnetic field is

{_2_' H =: ~ 2 m e [Pi q - 4 m e A g ( r j ) f f } §

J (6)

1 I would like to thank the referee for pointing this out to me.

6 1 2 Peng

where VI contains the usual interaction of the BCS theory [11], Ag is the vector potential of the applied gravitomagnetic field.

The velocity of an electron is, from the ttamiltonian,

v = (p + 4meAa)/rne = ( - i h V + 4rneAa)/me. (7)

Because an electron pair carries the gravitational mass charge M (= 2me), the motion of electron pairs will cause the gravitational mass current den- sity

jm = M e * r e = n(liVO + 4MAa) , (8) where

r = nZ/2e ie, (9)

and n is the pair concentration. Taking the curl of both sides of eq. (8) gives the London-type gravit-

omagnetic equation V2Ba : 327rnme Ba, (10)

where eqs. (3)-(5) have been used.

3. THE PENETRATION DEPTH OF A GRAVITOMAGNETIC FIELD WITHOUT TAKING INTO ACCOUNT EFFECTS OF TWIN CUR- RENTS

For a simple case we let a semi-infinite superconductor occupy the space on the positive side of the x-axis. In the absence of an applied magnetic field, we let the applied gravitomagnetic field be parallel to the boundary, and Bg(0) = Ba0. Then the solution of eq. (10) is

Bg(z) = exp(-x/ g), (11)

where 1 ~ 1/2

Aa = \ 3 2 ~ m e J " (12)

For n ~_ 4 x 102S/m a, the penetration depth A 9 is of the order of 1013 meter. Actually, Ag might be temperature dependent. Following the

Loca l G r a v l t o m a g n e t i c Ef fec t s o n S u p e r c o n d u c t o r 613

usual definition of the penetration depth [12], the penetration depth of a gravitomagnetic field can be defined as

----- 1 / I B g o t / B g ( z ) A dz

2 dq (13) - - - - J o q2 7r + K(q)"

From eq. (13), we can compute )~ for any superconductor model which determines a response function K(q): For K(q) -- 1/~g 2, we obtain A = Ag, where ~g is given by eq. (12). For an appropriate choice of g(q), eq. (13) will give temperature dependent A(T).

Equation (11) might represent the gravitomagnetic field inside a semi- infinite superconductor. Equations (10)-(13) show that an applied grav- itomagnetic field as well as the induced gravitational mass current might penetrate into a superconductor about 1013 meter.

. TWIN CURRENTS AND COUPLING BETWEEN THE LONDON EQUATION AND THE LONDON-TYPE GRAVITOMAGNETIC EQUATION

Because an electron pair carries both the electric charge q (= -2e) and the gravitational mass charge M (= 2me), the motion of electron pairs will cause twin currents, the gravitational mass current and electrical current (i.e. the applied gravitomagnetic field induces the electrical current in the superconductor). The magnetic field created by this current will affect the twin currents. 2 Therefore the Hamiltonian becomes

{ 1 H --- ~.~ ~ [pj + eA(rj) + 4meAg(rj)] 2 } + Vx.

J

(14)

In the presence of both the applied magnetic and gravitomagnetic fields, a similar tIamiltonian was derived by DeWitt [6].

The velocity of an electron becomes

v = (p + eA + 4meAg)/m e = ( - i h V + eA + 4meAa)/m e. (15)

The gravitational mass current density due to the motion of electron pairs becomes

jm = 2mer162 = n(hVO + 2cA + 8meA,), (16)

2 I would like to thank the referee for point ing this out to me.

614 Peng

and an electrical current density arises

j , = -2er162 (17)

The relation between twin currents is

j , = ( i s )

The gravitational supercurrent is

j,ns = 2neA .q- 8nmeA 9. (19)

We must determine a unique gauge if we are to use expression (19) which is not manifestly gauge invariant. To ensure no charge buildup, V "jes = 0, thus

V . j ~ , = 0, (20)

and V �9 A 9 = 0, without any satisfactory gauge choice. By taking the curls of both sides of eqs. (16) and (17), the London

equation and London-type gravitomagnetic equation are modified slightly and coupled

V2B _LaB + bBg, (21)

V~B9 = d(aB + bBg), (22)

where eqs. (3)-(5) and Maxwell's equation

V x B = ~oj~

have been used, a =-- 2pone2/m~, b -- 8pone, and d ~ 47rme/(poe).

5. THE GENERALIZED MEISSNER EFFECT AND THE DISTRIBU- TION OF TWIN CURRENTS

Let us consider a consequence of eqs. (21) and (22). These equations yield that

V2(eB + 4meB~) = (32rcnrae + 2pone2/me) (eB + 4m~Bg). (23)

Since 32rnme << 2pone2/rne, eq. (23) reduces approximately to

V2(eB + 4m, Ba) _~ (2pone2/m,) (eB + 4m,Bg)

= (eB + 4m,Bg)/AL ~. (24)

Local Gravitomagnetic Effects on Superconductor 615

where AL is the London penetration depth. Equation (24) indicates the vanishing of the combination of the mag-

netic and gravitomagnetic fields,

eB + 4rn~Bg = 0, (25)

in the interior of a superconductor, i.e. the generalized Meissner effect. To discuss the distribution of twin currents in a superconductor, we

start with eq. (16). Taking the curl of both sides ofeq. (16) twice gives us

V2jm = (327rnme + 2yone2/m~)jm, (26)

where eqs. (3), (5) and (20) have been used. Equation (26) is approximately equivalent to

v jm (27) There are the same equations for Je-

Equation (27) shows that the twin currents induced by an applied gravitomagnetic field are distributed in a thin layer of the surface of a superconductor. That the twin currents disappear in the interior of the superconductor implies that

eB § 4meBg "- O,

which is consistent with the generalized Meisner effect, the result of eqs. (21) and (22). This relation agrees with the result obtained by DeWitt [6] and Anandan [7].

6. F•UXOID QUANTIZATION

The Cooper pairs in a superconductor are in a macroscopic quantum state described by a single wavefunction r of eq. (9). This macroscopic quantum state gives rise to several observable phenomena, such as flux quantization and Josephson tunneling. In the presence of both applied gravitomagnetic and magnetic fields, the requirement that r be single- values at any point in a closed superconducting ring is expressed by

where the subscript c indicates that the integration is along a closed path, v is the pair-velocity, h is Planck's constant, and n is an integer. We may choose the path of integration in a region of zero current, so that fc v . dl -- 0. The terms fe A �9 and fc Ag �9 are respectively the total magnetic and gravitomagnetic fluxes ~B mad ~g, in the ring. Equation (28) then reduces to the condition

(YPappl - " ~ e "J" 4rne/e ~g = n'~o, (29) which implies that the total applied flux (I'~ppl in the ring is quantized.

616 Peng

7. GRAVITOMAGNETIC EFFECT ON ]OSEPHSON JUNCTIONS AND SQUID

The SQUID is a class of devices which uses different types of junctions and responds to applied flux with period @0 (= h/2e). Let us consider a rf SQUID system. Of fundamental importance to understanding the oper- ation of the rf SQUID is the relation of the flux actually passing through the SQUID loop ~T to that applied externally to the loop r which includes the gravitomagnetic flux.

For simplicity, let us consider a supercurrent I passing through a Josephson junction. Then the generalized Josephson equations have the same form as the ordinary one

I = I0 sin(Ar (30a)

0(Ar = 2 Y/h, (30b)

wehre V is the instantaneous voltage across the junction. The gravitomag- netic effect is included in the Ar term. The phase difference Ar is

( 4 m ~ O ~ ) A4 ~ = 2~rO._..__~B 1 + . (31) ~o e~B

For the earth's gravitomagnetic field, and B ~- 10 - l~ (Tesla) which is the minimum value that a SQUID can measure,

( 4 r n ~ , ~ ,,~ 1 x 10 -15, (32 / eCB J -

therefore

Cappl ~ (]~B, (33)

Ar _ 21feB/C0. (34)

Equations (30)-(34) imply that the gravitomagnetic effect of the earth on the SQUID magnetometer can be neglected.

ACKNOWLEDGEMENTS

The author gratefully acknowledges the referee's very helpful com- ments and Dr. D. G. Torr for discussions. This work was supported in part by NASA/MNFC under grant NAG8-047.

Local Grav i t omagne t i c Effects on S u p e r c o n d u c t o r 617

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