Spin-geodesic deviations in the Kerr spacetime

Abstract

The dynamics of extended spinning bodies in the Kerr spacetime is investigated in the pole-dipole particle approximation and under the assumption that the spin-curvature force only slightly deviates the particle from a geodesic path. The spin parameter is thus assumed to be very small and the back reaction on the spacetime geometry neglected. This approach naturally leads to solve the Mathisson-Papapetrou-Dixon equations linearized in the spin variables as well as in the deviation vector, with the same initial conditions as for geodesic motion. General deviations from generic geodesic motion are studied, generalizing previous results limited to the very special case of an equatorial circular geodesic as the reference path.Comment: 19 pages, 6 figures; published versio

Gyroscope precession along bound equatorial plane orbits around a Kerr black hole

“…This discussion holds in general off the equatorial plane, completing the explanation of Ref. [30]. The relation (A13) of Marck has a simple interpretation as the transformation law for an electric field given as Eq.…”

“…Now further decompose this Carter relative velocity vector parallel and perpendicular to the Carter radial direc-tion [30] ν (car) = νr (car) E(u (car) )r , ν ⊥ (car) = νθ (car) E(u (car) )θ + νφ (car) E(u (car) )φ = ||ν ⊥ (car) ||ν ⊥ (car) , (A7) and the useful cross product quantity (90 degree rotation of ν ⊥ (car) )…”

Gyroscope precession along general timelike geodesics in a Kerr black hole spacetime

“…(42)-(44) with u = n) is given byF (G) (fw,U,n) = −∇ U n = −γ(U, n)[a(n) + θ(n) ν(U, n)] . (A18)Appendix B: Marck's frame, tidal matrices and diagonalization propertiesThe nonvanishing components of the electric part of the Riemann tensor E(U ) αβ = R αµβν U µ U ν in the parallel propagated frame {E i } computed explicitly in Ref [41]. are given byE(U ) 11 = − 3M r Σ 3 K J 3 sinh 2 β cosh 2 β cos 2 Ψ + M r Σ 3 J 5 , E(U ) 12 = − 3M a cos θ Σ 3 K sinh β cosh β(J 1 cosh 2 β − 4r 2 J 4 ) cos Ψ , E(U ) 13 = − 3M r Σ 3 K J 3 sinh 2 β cosh 2 β cos Ψ sin Ψ , E(U ) 22 = M r Σ 3 K [3J 3 cosh 4 β − cosh 2 β(J 1 − 8a 2 cos 2 θJ 2 ) + 2r 2 J 5 ] , E(U ) 23 = − 3M a cos θ Σ 3 K sinh β cosh β(J 1 cosh 2 β − 4r 2 J 4 ) sin Ψ , E(U ) 33 = 3M r Σ 3 K J 3 sinh 2 β cosh 2 β cos 2 Ψ − M r Σ 3 K [3J 3 cosh 4 β −4 cosh 2 β(J 3 + 2a 2 cos 2 θJ 4 ) + r 2 J 5 ] ,(B1)where J 1 = 5r 4 − 10r 2 a 2 cos 2 θ + a 4 cos 4 θ ,J 2 = 3r 2 − a 2 cos 2 θ , J 3 = r 4 − 10r 2 a 2 cos 2 θ + 5a 4 cos 4 θ = J 1 − 4J 4 , J 4 = r 2 − a 2 cos 2 θ , J 5 = r 2 − 3a 2 cos 2 θ .…”

The Spinning Equations of Motion for Objects in AP-Geometry

“…Equations of spinning motion ,the case of P α = mU α can be related to geodesic if one follows the following transformation [22]…”