We extend the analysis of Poynting-Robertson effect, i.e., the deviation from geodesic motion of test particles due to scattering by a superposed radiation field to the Schwarzschild background, to the case of spinning bodies. The extra contribution of the deviation due to spin can be relevant for astrophysical systems like the binary pulsar system PSR J0737-3039 orbiting Sgr A∗, but not for the Earth-Sun system

Bini, D.

Geralico, A.

English

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DOI 10.1393/ncc/i2013-11501-yColloquia: 12th Italian-Korean Symposium 2011IL NUOVO CIMENTO Vol. 36 C, N. 1 Suppl. 1 Gennaio-Febbraio 2013Scattering of spinning bodies by a radiation fieldin Schwarzschild spacetimeD. Bini(1)(2)(3)(∗) and A. Geralico(4)(2)(∗∗)(1) Istituto per le Applicazioni del Calcolo “M. Picone”, CNRvia Manzoni, 30 I-00185 Roma, Italy(2) ICRA, University of Rome “La Sapienza” - P.le A. Moro, 2, I-00185 Roma, Italy(3) INFN, Sezione di Firenze - via G. Sansone, 1, I-50019 Sesto Fiorentino (FI), Italy(4) University of Rome “La Sapienza” - P.le A. Moro, 2, I-00185 Roma, Italyricevuto il 9 Marzo 2012Summary. — We extend the analysis of Poynting-Robertson effect, i.e., the de-viation from geodesic motion of test particles due to scattering by a superposedradiation field to the Schwarzschild background, to the case of spinning bodies. Theextra contribution of the deviation due to spin can be relevant for astrophysicalsystems like the binary pulsar system PSR J0737-3039 orbiting Sgr A∗, but not forthe Earth-Sun system.PACS 04.20.Cv – Fundamental problems and general formalism.1. – IntroductionThe motion of classical spinning test particles in a given gravitational backgroundis described by the well-known Mathisson-Papapetrou (MP) model [1, 2]. Let Uα =dxα/dτ be the timelike unit tangent vector to the “center-of-mass line” CU of the spinningparticle used to perform the multipole reduction, parametrized by the proper time τ . Theequations of motion areDPμdτ= −12RμναβUνSαβ ≡ F (spin)μ,(1)DSμνdτ= PμUν − P νUμ,(2)(∗) E-mail: binid@icra.it(∗∗) E-mail: geralico@icra.itc© Societa` Italiana di Fisica 3738 D. BINI and A. GERALICOwhere Pμ is the total 4-momentum of the particle and the antisymmetric tensor Sμνdenotes the spin tensor (intrinsic angular momentum) associated with it; both fieldsare defined only along this center-of-mass world line. This system of 10 equations isconsistently completed by the supplementary conditions (see [3,4] and references therein)(3) SμνPν = 0.The spin structure of the particle should produce very small deviations from geodesicmotion; otherwise, the particle backreaction on the spacetime metric should be takeninto account.Let us consider a spinless test particle orbiting a star which emits radiation. Theradiation pressure of the light emitted by the star exerts a drag force on the particle’smotion, which usually causes the body to fall into the star unless it is so small thatthe radiation pressure pushes it away from the star: this phenomenon is called thePoynting-Robertson effect [5, 6] and has been fully investigated in the context of bothNewtonian and Post-Newtonian gravity. Recently the Poynting-Robertson effect for aspinless test particle orbiting a black hole was studied in both the Schwarzschild andKerr spacetimes [7,8] without the restriction of slow motion, but ignoring the finite sizeof the radiating body. Here we generalize the above discussion to the more realistic caseof a spinning test particle undergoing the Poynting-Robertson effect by including theradiation force in the Mathisson-Papapetrou model.2. – Schwarzschild spacetime and Poynting-Robertson effectConsider a Schwarzschild spacetime, whose line element written in standard coordi-nates is given by(4) ds2 = −N2dt2 + N−2dr2 + r2(dθ2 + sin2 θdφ2),where N = (1−2M/r)1/2 denotes the lapse function, and introduce the usual orthonormalframe adapted to the static observers (or Zero Angular Momentum Observers, ZAMOs)following the time lines(5) etˆ = N−1∂t, erˆ = N∂r, eθˆ =1r∂θ, eφˆ =1r sin θ∂φ,where {∂t, ∂r, ∂θ, ∂φ} is the coordinate frame.We limit our analysis to the equatorial plane θ = π/2.2.1. Test radiation field . – Let a pure electromagnetic radiation field be superposedas a test field on the gravitational background described by the metric (4), with theenergy-momentum tensor(6) Tαβ = Φ2kαkβ , kαkα = 0,where k is assumed to be tangent to an affinely parametrized outgoing null geodesic inthe equatorial plane, i.e., kα∇αkβ = 0 with kθ = 0.SCATTERING OF SPINNING BODIES BY A RADIATION FIELD IN SCHWARZSCHILD SPACETIME39We will only consider photons in the equatorial plane which are in outward radialmotion with respect to the ZAMOs, namely with 4-momentum(7) k = E(n)(n + erˆ),where n = etˆ is the ZAMO 4-velocity and E(n) = E/N is the relative energy of thephoton as seen by the ZAMOs. Here E = −kt is the conserved energy associated withthe timelike Killing vector field ∂t. Note also that in this case L = kφ = 0.Since k is completely determined, the coordinate dependence of Φ then follows fromthe conservation equations Tαβ ;β = 0. For photons on the equatorial plane we find(8) Φ =Φ0r.Consider now a test particle moving in the equatorial plane θ = π/2 accelerated bythe radiation field, i.e. with 4-velocity(9) U = γ(U, n)[n + ν(U, n)], ν(U, n) ≡ ν rˆerˆ + νφˆeφˆ = ν(sinαerˆ + cosαeφˆ),where γ(U, n) = 1/√1− ||ν(U, n)||2 ≡ γ is the Lorentz factor and the abbreviated nota-tion νaˆ ≡ ν(U, n)aˆ has been used. Similarly ν ≡ ||ν(U, n)|| and α are the magnitude ofthe spatial velocity ν(U, n) and its polar angle measured clockwise from the positive φdirection in the r-φ tangent plane, respectively.A straightforward calculation gives the coordinate components of U(10) U t ≡ dtdτ=γN, Ur ≡ drdτ= γNν rˆ, Uφ ≡ dφdτ=γνφˆr,where τ is the proper time parameter along CU , and Uθ ≡ dθ/dτ = 0. The forceassociated with the scattering of radiation is given by [5,6, 9](11) F (rad)α = −σP(U)αβ T βμ Uμ,where σ is the scattering cross section and P(U)αβ = δαβ + UαUβ projects orthogonallyto U . Explicitly(12) F (rad) =mAN2r2γ3(1− ν rˆ)[(ν rˆ − ν2)n +(1− ν rˆ − (νφˆ)2)erˆ − (1− ν rˆ)νφˆeφˆ],where we have used the notation σΦ20E2 = mA, as in ref. [7]. For a later use it isconvenient to introduce a friction parameter f ≡ A/M , which again should be smallenough not to perturb the spacetime. The motion is then described by the equations(13) ma(U)μ ≡ mDUμdτ= F (rad)μ,and has been studied in detail in ref. [7].40 D. BINI and A. GERALICO2.2. Spinning particles undergoing PR effect . – The most direct generalization ofeq. (13) to the case of spinning test particles consists in including the radiation forceterm in eq. (1), so that one has(14)DPμdτ= F (spin)μ + F (rad)μ,plus the additional relations (2) and (3) involving the spin. Let us proceed to analysethe motion of spinning particles undergoing Poynting-Robertson effect in the equatorialplane of the Schwarzschild spacetime.Let P = mu be the 4-momentum for motion in the equatorial plane, with(15) u = γu[n + νu(sinαuerˆ + cosαueφˆ)], γu = 1/√1− ν2u,and introduce the spin vector associated with Sμν by spatial duality(16) Sβ = uαηαβμνSμν ,where ηαβγδ =√−gαβγδ is the unit volume 4-form and αβγδ (0123 = 1) is the Levi-Civita alternating symbol. It is also useful to consider the scalar invariant s2 = 12SμνSμν ,constant along CU because of eqs. (2) and (3). Consistency of the model requires |sˆ| ≡|s|/(mM) 1 as stated above.From the evolution equations for the spin it follows that the spin vector has a singlenonvanishing and constant component along θ, i.e.,(17) S = −S θˆeθˆ = −seθˆ.In the absence of both spin and radiation we assume the geodesic motion of the testparticle to be circular at r = r0, i.e.,(18) U = UK = γK(n± νKeφˆ),where the Keplerian value of speed (νK) and the associated Lorentz factor (γK) andangular velocity (ζK) are given by(19) νK =√Mr0 − 2M , γK =√r0 − 2Mr0 − 3M , ζK =√Mr30.The parametric equations of UK are(20) tK = t0 + ΓKτ, r = r0, θ =π2, φK = φ0 ± ΩKτ,where now t0, r0 and φ0 are constants and(21) ΓK =√r0r0 − 3M , ΩK =1r0√Mr0 − 3M .SCATTERING OF SPINNING BODIES BY A RADIATION FIELD IN SCHWARZSCHILD SPACETIME41Corrections to geodesic circular motion are given by(22) U = UK + fUf + sˆUsˆ, u = U + fuf + sˆusˆ,to first order in both parameters sˆ and f . The geodesic value are thus modified as(23) t = tK + ftf + sˆtsˆ, r = r0 + frf + sˆrsˆ, φ = φK + fφf + sˆφsˆ.The solution is straightforward (see ref. [10] for details)tsˆ = ∓6M2r0Ω3KΩ3ep[sin(Ωepτ)− Ωepτ ],(24)tf = 4r0ζKν2KΩ3KΩ4ep{[cos(Ωepτ)− 1] + Ωep2r0ζKΩK [sin(Ωepτ)− Ωepτ ] +38γ2KΩ2epτ2},rsˆ = ±3r0ΩKζKΩ2ep[cos(Ωepτ)− 1],rf = −r0ζK ΩKΩ2ep{[cos(Ωepτ)− 1]− 2r0ζK ΩKΩep [sin(Ωepτ)− Ωepτ ]},φsˆ = ± ζKν2Ktsˆ, φf = ± ζKν2Ktf , Ωep =√M(r0 − 6M)r30(r0 − 3M),where Ωep is the well-known epicyclic frequency governing the radial perturbations ofcircular geodesics.As an application we can estimate the amount of variation of the radial distance bytaking the mean values over a period of the perturbed radius, i.e.,(25)〈δrr〉≡ r − r0r0= ΓKζ2KΩ2ep[(1− 2πr0ζK ΩKΩep)f ∓ 3MsˆγKζKN0].For instance, for the motion of the Earth about the Sun we find(26)〈δrr〉≈ 3× 10−15 ∓ 4× 10−15 ≈ 10−15,whereas for the binary pulsar system PSR J0737-3039 as orbiting Sgr A∗ [11](27)〈δrr〉≈ 7.6× 10−19 ∓ 1.8× 10−7 ≈ 10−7.Therefore, in the latter case the effect of the spin on the orbit dominates with respect tothe friction due to the radiation field.REFERENCES[1] Mathisson M., Acta Phys. Pol., 6 (1937) 167.[2] Papapetrou A., Proc. R. Soc. London, 209 (1951) 248.[3] Tulczyjew W., Acta Phys. Pol., 18 (1959) 393.42 D. BINI and A. GERALICO[4] Dixon W. G., Philos. Trans. R. Soc. London, Ser. A, 277 (1974) 59.[5] Poynting J. H., Philos. Trans. R. Soc., 203 (1903) 525.[6] Robertson H. P., Mon. Not. R. Astron. Soc., 97 (1937) 423.[7] Bini D., Jantzen R. and Stella L., Class. Quantum Grav., 26 (2009) 055009.[8] Bini D., Geralico A., Jantzen R., Semera´k O. and Stella L., Class. Quantum Grav.,28 (2011) 035008.[9] Abramowicz M. A., Ellis G. F. R. and Lanza A., Astrophys. J., 361 (1990) 470.[10] Bini D. and Geralico A., Class. Quantum Grav., 27 (2010) 185014 (see also arXiv:1107.2793 containing corrections of few misprints).[11] Lyne A. G. et al., Science, 303 (2004) 1153.

Scattering of spinning bodies by a radiation field in Schwarzschild spacetime

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Scattering of spinning bodies by a radiation field in Schwarzschild spacetime

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