Gyroscope on polar orbit in the Kerr field

Transcription

1 Gyroscope on polar orbit in the Kerr field O. Semerák Abstract The description of the relativistic motion in terms of forces, proposed in [26], and its application to the problem of gyroscopic precession in [28] are used for a gyroscope on a zero-angular-momentum spherical polar orbit in the Kerr field. Whereas far from the centre the backward Thomas precession is dominant, below the photon geodesic spherical polar orbit the forward geodetic effect prevails. In this region (the θ-rotosphere ) the gyro s radial acceleration depends in a reverse manner on the orbital angular velocity [27]. Also the dragging term rotates the gyroscope in a way consistent with intuition. Running head: Gyroscope in the Kerr field Mailing address: KTF MFF UK, V Holešovičkách 2, CZ Praha 8, Czech Republic Department of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, CZ Praha 8, Czech Republic. semerak@hp03.troja.mff.cuni.cz 1

2 1 Introduction According to general relativity, mass currents lead to a differential dragging of surrounding inertial frames with respect to the rest frame at spatial infinity (represented by distant stars or by appropriate boundary conditions). This is accompanied by a whole menagerie of concomitant effects coming under the term frame dragging [7]. Within the often promoted gravitoelectromagnetic approaches which follow the analogy with classical electrodynamics, the dragging effects arise as a consequence of the existence of the gravitomagnetic (GM) field, while a pure attraction is provided by the gravitoelectric (GE) field, generated by mass density (see e.g. [13] and references therein). Both fields live in curved 3-space, which brings some extra terms (not apprehensible within the EM analogy) into the equations. The simplest generator of dragging is a rotating source and the simplest metric describing the field of such a source is the Kerr metric [19]. Leaving aside the most expressive manifestation of dragging the existence of the ergosphere with its peculiar properties, basically three types of GM effects are recognized in Kerr spacetime: various deviations (i) in the motion of test material (see e.g. [8]) and (ii) in the shape of test fields (e.g. [2]), and (iii) the additional GM precession 1 of test gyroscopes [15, 23, 17, 20, 22, 12], accompanied also by an additional force acting on their centre of mass [4]. The observational evidence of the geodetic precession 2 [29] the other general relativistic contribution to the gyroscopic precession (which is due to the GE field) refreshed the hope for a direct detection of the Earth s GM field. Accurate tracking of the orbits of satellites was proposed for this purpose [6], as well as the experiments with Foucault pendulum [5], with gravity gradiometers [16] and with gyroscopes [21, 9, 34] (cf. also [14]). The most awaited satellite gyroexperiment [9] is planned along a polar orbit. The spherical polar orbits are, after the circular orbits at r = const and θ = const, the simplest orbits in the Kerr field and are suitable for the demonstration of a rotational dragging [30, 3]. The gyroscopic precession along spherical polar geodesics was studied by [32, 33]. The orbits considered in the present note may have arbitrary acceleration. The arrows will denote the spatial parts of (contravariant) 4-vectors, signature is +2 and geometrized units (c = G = 1) are used. 2 Motion and precession of a relativistic gyroscope in classicallike terms In [28] I showed that the gyroscope precession is simple when studied with respect to the comoving Frenet triad (cf. [12]). This is defined along the projection of the gyro s trajectory onto a 3-space of the local hypersurface-orthogonal observer (HOO) and made up of its instantaneous tangent ˆv, (inward) normal n and binormal b. A clear interpretation of the precession can be obtained when employing the description of the relativistic motion in terms of forces proposed in [26]: the Thomas, the geodetic and the dragging (GM) components of the precessional angular velocity are respectively due to the centrifugal, the gravitational (GE) and the dragging (GM) 1 Alternatively it is named dragging, motional, hyperfine, or in the weak-field limit Lense-Thirring or Schiff precession. 2 Called also geodesic, de Sitter or Fokker precession. 2

3 plus Coriolis forces which all act at the gyroscope s centre of mass: 3 where = Ω G [Thom]+ Ω G [geod]+ Ω G [drag], (1) [Thom] = ˆv 2 ˆv acf, (2) [geod] = ˆv a g, (3) [drag] = ˆv a d ˆv 2 ˆv ac, (4) and the relevant components of the gyro s 4-acceleration a µ are given by a µ g = ˆγ 2 a µ HOO, (5) a µ d = ˆγ 2 ˆv ν ν u µ HOO, (6) a µ C = ˆγ 2 [u µ HOO a HOOνˆv ν +( Ω HOF ˆv) µ ], (7) a µ cf = ˆγˆvu ν νˆv µ 0 aµ C (8) [the remaining component of the 4-acceleration, a µ = ˆγ 3 (ˆv µ + ti 0 ˆvuµ )uν HOO νˆv (the tangent intrinsic inertial resistance ), does not enter the precession formula (1)]. In eqs. (5)-(8), the vector product in the HOO s 3-space is defined by ( A B) µ = u HOO νǫ νµ ρσa ρ B σ, u µ and HOO aµ HOO denote the 4-velocity and 4-acceleration of the local HOO; the HOO s privileged time t will be usedasatimecoordinate. Ω µ istheangularvelocity ofrotation ofthespatial vectors ofhoo s HOF local orthonormal frame (denoted by HOF) relative to a comoving Fermi-Walker transported triad; u µ is the 4-velocity of the gyroscope and ˆv µ its relative velocity with respect to the local HOO, given by the decomposition u µ = ˆγ(u µ + HOO ˆvµ ), where ˆγ = (1 ˆv 2 ) 1/2 = u ν u ν = HOO u t /u t, ˆv2 = ˆv HOO νˆv ν = ˆv ˆv and ˆv µ = 0 ˆvµ /ˆv. For different interpretations of gyroscope precession, following from the different decomposition of the 4-acceleration, see [1] and [18]. In [26] and [28] the simplest types of motion purely radial and purely azimuthal in the Kerr or simpler fields were treated in order to check that the general formulas obtained lead to plausible results in concrete situations. The present note shows that the same language yields a clear description also of zero-angular-momentum spherical polar trajectories and of gyroscopic precession along them. 3 Forces along polar orbits in the Kerr field Consider a gyroscope in the Kerr spacetime, in which HOOs are zero-angular-momentum observers (ZAMOs) and HOFs their locally non-rotating frames (LNRFs). The Boyer-Lindquist coordinates (t, r, θ, φ) will be used. Let the gyroscope move purely in latitudinal direction with respect to the ZAMO-congruence, i.e., with ˆv r = ˆv φ = 0, ˆv θ = u t ZAMO Ω, uµ = u t (1,0,Ω,ω K ), (u t ) 2 = Σ( /A Ω 2 ), (9) where Ω = dθ/dt stands for its latitudinal orbital angular velocity and, as usually, = r 2 2Mr +a 2, Σ = r 2 +a 2 cos 2 θ, A = (r 2 +a 2 ) 2 a 2 sin 2 θ, ω K = 2Mar/A. 3 To be precise: as was noted in [28], what is here called the Thomas part comprises, in fact, not only the pure special relativistic effect, but also a classical contribution (backward relative to the orbit s sense) accompanying the rotation of our reference Frenet basis with respect to an inertial frame at spatial infinity. 3

4 The contributions of the gravitational, the dragging, the Coriolis and the centrifugal forces to the gyroscope s 4-acceleration according to (5)-(8) become a g = ˆγ 2 M ( ) a ZAMO = Σ 2 A( AΩ 2 Σ(r 4 a 4 )+2 (rasinθ) 2, (r 2 +a 2 )ra 2 sin2θ, 0,(10) ) a d = a C = (u t ) 2 ΩLNRF ˆv = Ma3 sin2θ Σ( AΩ 2 ) b sinθ = Ma3 rωsin2θ Σ 2 ( AΩ 2 (0,0,1), (11) ) a cf = ˆγ 2ˆv 2 n = (u θ ) 2 (Γ r θθ,γθ θθ,0) = AΩ 2 Σ 2 ( AΩ 2 ) (r, a2 sinθcosθ, 0). (12) Hence, outside the black-hole horizon ( > 0) the gravitational force is always attractive (a r g > 0), while the centrifugal force repulsive (a r cf < 0); both pull towards the equatorial plane (aθ g,cf < 0). The same conclusions were obtained in [25] for a purely azimuthal motion. For a purely latitudinal (with respect to ZAMOs) motion around the Kerr source one can show [27] that a r / Ω < 0 M(r 2 a 2 ) r = r 2 (3M r) a 2 (r +M) < 0, (13) i.e., below the radius of the photon geodesic spherical polar orbit the radial acceleration a r depends on Ω in a counter-intuitive manner: a r increases with increasing Ω. (Note that on the axis and in the equatorial plane a r is the only non-zero component of a µ.) A similar behaviour (modified by dragging) was also found [11, 24, 25] for purely azimuthal orbits: in the equatorial plane the dependence of a r (also there the only non-zero component of a µ ) on the azimuthal orbital angular velocity ω is, close to the lower/upper limit of the permitted-ω range, usual above the outer/inner photon geodesic circular orbit and reverse below it. In [25] we interpreted this azimuthal effect (the occurrence of φ-rotospheres ) as a result of the relative weakening (and finally vanishing) of the dragging+coriolis and centrifugal forces as compared with the gravitational force when the horizon is approached. 4 On the basis of eqs. (10) and (12), we arrive at a similar explanation also for the latitudinal effect (the occurrence of θ-rotospheres ) considered here: by differentiation of a r = a r g +ar cf one finds that a r Ω = 2 Ω (r2 +a 2 ) Σ 2 ( AΩ 2 ) 2 [M(r2 a 2 ) r ]. (14) The first term within the bracket comes from the gravitational component of 4-acceleration and the second, which vanishes as 0, from the centrifugal component. 4 Gyroscope precession along a polar orbit in the Kerr field From eqs. (2)-(4) and (10)-(12) (with ˆv r = ˆv φ = 0) we find that Ω G [Thom] and Ω G [geod] have only φ-component and Ω G [drag] only r-component non-vanishing: gφφ Ω φ G [Thom] = A Σ 3 gφφ Ω φ G [geod] = A Σ r Ω AΩ 2, (15) Σa r ZAMO Ω AΩ 2, (16) grr Ω r G [drag] = Σ Ωr LNRF = Asinθ 2 Σ 3 ω K,θ. (17) 4 See [1] for an alternative interpretation in terms of differently defined forces, involving a reversal of the action of the centrifugal force. 4

5 Hence, if one imagines the gyroscope s spin momentarily aligned with its velocity, the Thomas and the geodetic effects are located within the instantaneous plane of the orbit, the former being backward and the latter forward with respect to the sense of the orbit. Both vanish for Ω = 0; a gyroscope is then at rest in the Boyer-Lindquist mesh, i.e., with respect to distant stars. The dragging shifts the gyroscope in the plane which is instantaneously tangent to the sphere r = const spanned by the orbit; the gyroscope is rotated in such a manner that its end which is closer to the equatorial plane is dragged (in the φ-direction) more rapidly than the opposite one. The results confirm the properties of the components (2)-(4) of relativistic precession which were demonstrated, on purely radial and purely azimuthal motions, in [28], and which are in a harmony with one s intuition: 5 outside black holes, the Thomas precession is backward, the geodetic precession forward, and the effect of dragging depends on the orientation of the gyroscope relative to the gradient of the dragging angular velocity ω K. Finally, the sum of the expressions (15) and (16) provides a simple answer to the question raised at the end of [27] whether and how is the behaviour of a r (Ω) on a (zero-angularmomentum spherical) polar orbit tied to the precession of orbiting gyroscopes: 6 The gyro precesses backward when it is above the photon geodesic spherical polar orbit (then the Thomas contribution due to the centrifugal force is dominant), whereas it precesses forward below this orbit, i.e., inside the (θ-)rotosphere where the geodetic term due to the gravitational force prevails. Acknowledgements I thank Prof. Jiří Bičák for a number of improvements in the text. I gratefully acknowledge the support from the grant GACR-202/96/0206 of the Grant Agency of the C.R. and from the grant GAUK-230/96 of the Charles University. References [1] Abramowicz, M. A., Nurowski, P., Wex, N. (1995). Class. Quantum Grav. 12, [2] Bičák, J., Karas, V. (1989). In Proc. of the 5th M. G. Meeting on Gen. Rel., edited by D. G. Blair and M. J. Buckingham (World Scientific, Singapore), p [3] Blockley, C. A., Stedman, G. E. (1990). Phys. Lett. A 147, 161. [4] Braginsky, V. B., Caves, C. M., Thorne, K. S. (1977). Phys. Rev. D 15, [5] Braginsky, V. B., Polnarev, A. G., Thorne, K. S. (1984). Phys. Rev. Lett. 53, 863. [6] Ciufolini, I. (1994). Class. Quantum Grav. 11, A73. [7] Ciufolini, I., Wheeler, J. A. (1995). Gravitation and Inertia (Princeton Univ. Press, Princeton). [8] Dymnikova, I. G. (1986). Sov. Phys. Usp. 29, 215. [9] Everitt, C. W. F. (1992). In Proc. of the 6th M. Grossmann Meeting on Gen. Rel., edited by H. Sato and T. Nakamura (World Sci., Singapore), p in particular, with the analogy representing the Kerr dragging as an effect similar to that of a viscous fluid which is dragged along into differential co-rotation by an immersed rotating ring. 6 A similar question concerning the purely azimuthal motion was studied by [10]. 5

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