Ahmedov, B J

Karim, M A

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EXT-2000-06901/11/1999Available at: http://www.ictp.trieste.it/~pub off IC/99/157United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSGRAVITOMAGNETIC EFFECTS IN CONDUCTORIN APPLIED MAGNETIC FIELDB.J. Ahmedov1Institute of Nuclear Physics, Ulughbek, Tashkent 702132, UzbekistanandThe Abdus Salam International Centre for Theoretical Physics, Trieste, ItalyandM. KarimDepartment of Physics, St John Fisher College, Rochester, NY 14618, USA.AbstractThe electromagnetic measurements of general relativistic gravitomagnetic eects whichcan be performed within a conductor embedded in the space-time of slow rotating gravi-tational object in the presence of magnetic eld are proposed.MIRAMARE { TRIESTENovember 19991E-mail: ahmedov@suninp.tashkent.suThe general relativistic electromagnetic eects arising from the gravitomagnetic eld innoncurrent carrying (super-)conductors with no applied magnetic eld present have beendiscussed by several authors (see, for review, [1]). However, the general-relativistic eectscan be amplied by the interplay between gravitomagnetic eld and either electric currentor magnetic eld and in this respect, we discuss here a new test of gravitomagnetic eld ofEarth by using conductors embedded in an external magnetic eld while in the previouspaper [2] we have already shown that the interaction between the gravitomagnetic eldand electric current can lead to the galvanogravitomagnetic eect.Space-time outside a spherically symmetric mass M with angular momentum a isdescribed by the Kerr metric. This diers from the Schwarzschild solution for a staticbody by having non-diagonal terms, which imply a local inertial frame to be rotating withrespect to the distant stars at innity with the Lense-Thirring angular velocity [3] !(r) 2aM=r3. Then the metric of the reference frame corotating with the slowly rotatinggravitational object with mass M (in the linear angular momentum a approximation) isds2=  N2c2dt2+N 2dr2+ r2d2+ r2sin2d'2+ 2!r2sin2dtd'; (1)where N  (1  2M=r)1=2, !   !(r),  is angular velocity of rotation of gravitationalobject with respect to the distant stars.Suppose that the material relations between inductions and elds have linear characterH=1F+1  (uF  uF)u;F= H+  1(uH  uH)u(2)and the general relativistic Ohm's law for conduction current |isj= Fu RH(F+ Fuu)j+RggjA+  1=2?r(1=2e); (3)here FandHare the tensors of electromagnetic eld and induction, respectively,  and are the parameters for the conductor, eis the electrochemical potential per unit charge,Rgg= 2mc=ne2is the parameter for the conductor called as galvano-gravitomagnetic one,n is the concentration of the conduction electrons, obviously  is the electrical conduc-tivity, RHis the Hall constant, uis the four-velocity of the conductor, w= u;uisthe absolute acceleration, A= u;+ u[w]is the relativistic rate of rotation of theconductor,?rdenotes the spatial part of covariant derivative and [ ] represents anti-symmetrization. The gravitational eld is assumed to be stationary that is space-timemetric gadmits a timelike Killing vector (t)that is  Ltg= 0 ( Ltdenotes the Liederivative with respect to (t),  =  (t)(t)).2Then the general relativistic expression for the charge distribution inside conductor [4]0=RHc|2+14f(|);+ [2RH(1|2+  1=2|?r(1=2e))u]; RggAw|+ g(Rgg|A); w|  w 1=2?r(1=2e)+g( 1=2?r(1=2e));+H[A+ RHw|+ (RH|);]g (4)can be derived from the general relativisic Maxwell equationseF;= 0; H;=4cJ; J c0u+ |^: (5)by using material relationships (2) and (3).The charge density 0inside a conductor which has no conduction current | = 0 butembedded in an external magnetic eld B is0=14fAw2  (Aw);+ FAg (6)and has two contributions: the rst one is due to the absolute acceleration wand secondone is due to the relativistic rate of rotation of the conductor Aand can be adjusted andamplied by the magnetic eld penetrating inside the conductor. Here F= 2[E]+Bis the electromagnetic eld tensor, A is the parameter for the conductor, is four-velocity of observer, Eand Bare the electric and magnetic elds measured byobserver.In our approximation the charge density, inside a conductor at rest in the orbitingstation (1), is0=  12nF23A23+ F13A13o: (7)We do not consider the charge redistribution arising from the absolute acceleration of theconductor since it does not depend on electromagnetic eld characteristics.If the electromagnetic eld tensor components areF31=NBr sin ; F23=Brr2sin (8)and the nonvanishing components of the relativistic rate of rotation have formA13=r + !r=2cNsin2; A23=!r2cNsin  cos ; (9)then in our approximation, the space charge density inside the conductor at rest inthe frame of reference (1) is0=2c"Bsin   Brcos N#+!4c"2Brcos N+Bsin #; (10)3where the magnetic eld components are measured by zero angular momentum observerswith four-velocity  f N; 0; 0; 0g.The rst term in the right hand side of equation (10) results from angular velocity and last one is due to the gravitomagnetic eld of the Earth and has pure generalrelativistic nature.On Earth, the angular velocity of the conductor is given by [1] cond=    Th S  !, where Thand Sare the contributions of the Thomas precession arising fromnon-gravitational forces and of the de Sitter or geodetic precession. In order to measure! one should measure condand then substract from it the independently measured valueof  with Very Long Baseline Intererometry [5] and the contributions due to the Thomasand de Sitter precession.In contrast to (10), for a superconductor embedded in the gravitational eld (1) thespace charge density 0(sc) = 0, that is according to the solutions of the general-relativisticMaxwell equations and London equations, the magnetic eld penetrating superconductoris proportional to ! and consequently the charge density is at least of order of !2. There-fore, if the temperature T is increased then in the point of the phase transition T = Tctheapplied magnetic eld penetrates inside the sample and induces a nonvanishing chargedensity with the corresponding ow of charges.For the Earth with mass M = 0:44cm and radius R  6:37 108cm, !(r) =4M5R 0:6  10 9 near the surface. If the value of applied magnetic eld around conductoris 103G and the relaxation time trel= 10 8s then one can nd a typical value of chargeexchange current arising from gravitomagnetic Lense-Thirring frequency is of order 10 14Awhich is within capacity of modern technical measurements. However, in the presentcase, there are serious problems arising from environment and the design of proposedexperiment is under consideration.AcknowledgmentsBJA acknowledges the hospitality at the Max-Planck-Institute fuer Gravitationsphysik,Golm and the Abdus Salam International Centre for Theoretical Physics, Trieste, dur-ing his visit in Autumn 1999 and the generous nancial support from Friedrich-ShillerUniversitat, Jena, for his participation at the Journees Relativiste 1999.4References[1] I. Ciufolini and J.A. Wheeler, Gravitation and Inertia, Princeton Univ. Press, NewJersey 1995[2] B.J. Ahmedov, Phys. Lett. A 256 (1999) 9[3] B. Mashhoon, F.W. Hehl, and D.S. Theiss, Gen. Rel. Grav. 16 (1984) 711[4] B.J. Ahmedov, and L.Ya. Arifov, Gen. Rel. Grav. 26 (1994) 1187[5] J. Kovalevsky, Rep. Prog. Phys. 61 (1998) 775

Gravitomagnetic effects in conductor in applied magnetic field

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