We investigate the influence of the quadrupole moment of a rotating source on
the motion of a test particle in the strong field regime. For this purpose the
Hartle-Thorne metric, that is an approximate solution of vacuum Einstein field
equations that describes the exterior of any slowly rotating, stationary and
axially symmetric body, is used. The metric is given with accuracy up to the
second order terms in the body's angular momentum, and first order terms in its
quadrupole moment. We give, with the same accuracy, analytic equations for
equatorial circular geodesics in the Hartle-Thorne spacetime and integrate them
numerically.Comment: 6 pages, 8 figures, published in the proceedings of the 12th
  Italian-Korean Sympothium on Relativistic Astrophysic

Bini, Donato

Boshkayev, Kuantay

Ruffini, Remo

Siutsou, Ivan

English

arXiv

D. Bini

K. Boshkayev

R. Ruffini

I. Siutsou

CiteSeerX

Equatorial circular geodesics in the Hartle-Thorne spacetime

We investigate the influence of the quadrupole moment of a rotating source on the motion of a test particle in the strong-field regime. For this purpose the Hartle-Thorne metric, that is an approximate solution of vacuum Einstein field equations that describes the exterior of any slowly rotating, stationary and axially symmetric body, is used. The metric is given with accuracy up to the second-order

terms in the body’s angular momentum, and first-order terms in its quadrupole moment. We give, with the same accuracy, analytic equations for equatorial circular geodesics in the Hartle-Thorne spacetime and integrate them numerically

Bini, D.

Boshkayev, K.

Ruffini, R.

Siutsou, I.

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DOI 10.1393/ncc/i2013-11483-8Colloquia: 12th Italian-Korean Symposium 2011IL NUOVO CIMENTO Vol. 36 C, N. 1 Suppl. 1 Gennaio-Febbraio 2013Equatorial circular geodesics in the Hartle-Thorne spacetimeD. Bini(1)(2)(3), K. Boshkayev(2)(4), R. Ruffini(2)(4)(5) and I. Siutsou(2)(4)(1) Istituto per le Applicazioni del Calcolo M. Picone, CNR - I-00185 Rome, Italy(2) Dipartimento di Fisica and ICRA, Universita` di Roma SapienzaP.le Aldo Moro 5, I-00185 Roma, Italy(3) INFN, Sezione di Firenze - I-00185 Sesto Fiorentino (FI), Italy(4) ICRANet - Piazzale della Repubblica 10, I-65122 Pescara, Italy(5) Universite de Nice Sophia Antipolis - Cedex 2, Grand Chateau Parc Valrose, Nice, Francericevuto il 9 Marzo 2012Summary. — We investigate the influence of the quadrupole moment of a rotatingsource on the motion of a test particle in the strong-field regime. For this purposethe Hartle-Thorne metric, that is an approximate solution of vacuum Einstein fieldequations that describes the exterior of any slowly rotating, stationary and axiallysymmetric body, is used. The metric is given with accuracy up to the second-orderterms in the body’s angular momentum, and first-order terms in its quadrupolemoment. We give, with the same accuracy, analytic equations for equatorial circulargeodesics in the Hartle-Thorne spacetime and integrate them numerically.PACS 04.20.-q – Classical general relativity.PACS 95.30.Sf – Relativity and gravitation.1. – IntroductionAstrophysical objects in general are characterized by a non-spherically symmetricdistribution of mass. In many cases, like ordinary planets and satellites, it is possible toneglect the deviations from spherical symmetry: it seems instead reasonable to expectthat deviations should be taken into account in case of strong gravitational fields. Themetric describing the exterior field of a slowly rotating slightly deformed object wasfound by Hartle and Thorne [1, 2]. However in this work we use the form of the metricc© Societa` Italiana di Fisica 3132 D. BINI, K. BOSHKAYEV, R. RUFFINI and I. SIUTSOUpresented in [3]. In geometrical units it is given byds2 = −(1− 2Mr)[1 + 2k1P2(cos θ) + 2(1− 2Mr)−1J2r4(2 cos2 θ − 1)]dt2(1)+(1− 2Mr)−1 [1− 2k2P2(cos θ)− 2(1− 2Mr)−1J2r4]dr2+r2[1− 2k3P2(cos θ)](dθ2 + sin2 θdφ2)− 4Jrsin2 θdtdφ ,wherek1 =J2Mr3(1 +Mr)− 58Q− J2/MM3Q22( rM− 1), k2 = k1 − 6J2r4,k3 = k1 +J2r4− 54Q− J2/MM2r(1− 2Mr)−1/2Q12( rM− 1), P2(x) =12(3x2 − 1),Q12(x)=(x2−1)1/2[3x2lnx + 1x− 1−3x2 − 2x2 − 1], Q22(x) = (x2−1)[32lnx + 1x− 1−3x3 − 5x(x2 − 1)2].Here P2(x) is Legendre polynomials of the first kind, Qml are the associated Legendrepolynomials of the second kind and the constants M , J and Q are the total mass,angular momentum and quadrupole parameter of a rotating star, respectively(1). Theapproximate Kerr metric [4] in the Boyer-Lindquist coordinates (t, R, Θ, φ) up to secondorder terms in the rotation parameter a can be obtained from (1) by setting(2) J = −Ma, Q = J2/M,and making a coordinate transformation given byr = R +a22R[(1 +2MR)(1− MR)− cos2 Θ(1− 2MR)(1 +3MR)],(3)θ = Θ +a22R2(1 +2MR)sinΘ cosΘ.2. – The domain of validity of the Hartle-Thorne approximationHaving on mind an application of the metric (1) to the exterior of a compact object,we demand that the energy-momentum tensor, which follows from (1), be much smallerthan the corresponding tensor of the source object. A correct comparison of these tensorsshould be performed in terms of eigenvalues. Consider a surface of the object whichgenerates the metric under consideration. According to the Einstein equations Gαβ =8πTαβ , where α, β = (t, r, θ, φ), the eigenvalues of the Einstein tensor inside the matterare equal to its density and pressure multiplied by 8π [5]. Due to the inequality ρ > p(1) We note here that the quadrupole parameter Q is related to the mass quadrupole momentdefined by Hartle and Thorne [2] through Q = 2J2/M −QHT .EQUATORIAL CIRCULAR GEODESICS IN THE HARTLE-THORNE SPACETIME 33which holds for all known types of matter, the maximum of eigenvalues can be estimatedas 8πρ, where ρ represents the average density of the body:(4) |Gαβ |  8πρ = 8πM4πr3/3 =6Mr3.On the other hand, the first non-vanishing terms in the expansion of the Einstein tensorof the Hartle-Thorne metric in powers of J and Q are G04 and G40. Then the Einsteintensor has two purely imaginary eigenvalues different from zero λ1,2 = 0 and two exactlyzero eigenvalues λ3,4 = 0. The first pair is diverging as r → 2M . Near this radius wehave, for δr = r − 2M approaching 0, the leading terms equal to(5) λ1,2 → ±i15JQ(1− 3 cos2 θ) sin θ32√2M11/2δr3/2.Finally, by comparing the absolute values of (4) and (5) for r → 2M , taking into accountthat 0 ≤ (1 − 3 cos2 θ)2 sin2 θ ≤ 1, we obtain the following inequality, describing thedomain of validity of the Hartle-Thorne metric around the gravitating body:(6) δr3  25J2Q2128M7.If we take the extreme values of the parameters for neutron stars such as J  M2,Q  10−2M3, we obtain δr  3 × 10−2M , that is certainly true for the exterior ofneutron stars while their radii are more than 2.5M [6], i.e. δr > 0.5M .3. – Equations for the equatorial circular geodesics3.1. The orbital angular velocity . – The 4-velocity U of a test particle on a circularorbit can be parametrized by the constant angular velocity with respect to infinity ζ(7) U = Γ[∂t + ζ∂φ],where Γ is a normalization factor which assures that UαUα = −1. From the normalizationand the geodesics conditions we obtain the following expressions for Γ and ζ = Uφ/U t:gtt + 2ζgtφ + ζ2gφφ = −1/Γ2, gtt,r + 2ζgtφ,r + ζ2gφφ,r = 0,(8)where gαβ,r = ∂gαβ/∂r. Hence, ζ, the solution of (8)2, is given by(9) ζ±(u) = ±ζ0(u)[1∓ jf1(u) + j2f2(u) + qf3(u)],where (+/−) stands for co-rotating/contra-rotating geodesics, j = J/M2 and q = Q/M3are the dimensionless angular momentum and quadrupole parameter and u = M/r. The34 D. BINI, K. BOSHKAYEV, R. RUFFINI and I. SIUTSOUrest quantities are defined as follows:ζ0(u) =u3/2M, f1(r) = u3/2,f2(u) =48u7 − 80u6 + 4u5 + 42u4 − 40u3 − 10u2 − 15u + 1516u2(1− 2u) − f(u),f3(u) = −5(6u4 − 8u3 − 2u2 − 3u + 3)16u2(1− 2u) + f(u),f(u) =15(1− 2u3)32u3ln(11− 2u).In fig. 1(a) we show the differences between the geodesics with the same initial conditionsarising due to the rotation of the central body i.e. the frame dragging effect in the strong-field regime. The solid line for J = 0 corresponds to equatorial circular geodesics in theSchwarzschild spacetime. The dashed line for J > 0 corresponds to co-rotating andthe dotted line for J < 0 corresponds to contra-rotating orbits. In fig. 1(b) we showthe differences between the geodesics with the same initial conditions arising due to thedeformation of the source i.e. the oblateness of the central body. The solid line for Q = 0corresponds to equatorial circular geodesics in the Schwarzschild spacetime. The dashedline for Q < 0 corresponds to the geodesics in the field of oblate and the dotted line forQ > 0 corresponds to the geodesics in the field of the prolate central body. It is easyto see that varying the quadrupole parameter Q one can recover the deviations fromthe Schwarzschild spacetime geodesics analogous to those caused by the frame draggingeffect. By selecting the values of J and Q one can recover the circular orbits as in fig. 1(c).In figs. 1(d), (e) and (f) we consider the geodesics with the same initial conditions inthe field of non-rotating bodies with the increasing values of Q. As a result, we obtaindifferent spiraling and bound trajectories of the test particle. For details see fig. 1.3.2. Radius of marginally stable, marginally bound and photon orbit . – The conditionε = −Ut = 1 gives the radius of the marginally bound orbit rmb, where ε is the conservedspecific energy per unit mass of the particle and the normalization condition PαPα = 0gives the photon orbit radius rph, where P = Γph[∂t +ζph∂φ] is the photon 4-momentum.Note, that the normalization condition PαPα = 0 gives the orbital angular velocity forthe photon ζph, however Γph remains arbitrary. In order to define the photon orbit radiusrph, first, one has to define ζph and evaluate the expression for the 4-acceleration aα. Forthe circular geodesic the condition aα = 0 is enough to find rph. In addition, by settingdl/dr = 0 one can find the radius of the marginally stable orbit rms, where l = −Uφ/Utis the specific angular momentum per unit energy of the particle.rmb = 4M[1∓ 12j +(8033256− 45 ln 2)j2 +(−100532+ 45 ln 2)q],rph = 3M[1± 2√39j +(1751324− 7516ln 3)j2 +(−6512+7516ln 3)q],rms = 6M[1± 23√23j +(−2519032592+ 240 ln32)j2 +(932596− 240 ln 32)q].EQUATORIAL CIRCULAR GEODESICS IN THE HARTLE-THORNE SPACETIME 35)c()b()a()f()e()d(Fig. 1. – (a) One revolution of the test particle in the field of a rotating central body for Q = 0.(b) The motion of the test particle in the field of non-rotating deformed object for J = 0.(c) Circular orbit with J = −0.05, Q = −0.1575. The other parameters for (a), (b) and (c) arer = 7, dφ/ds = 0.07145. (d) Spiral orbit for Q = −0.16. (e) Spiral orbit for Q = 0. (f) Boundorbit for Q = 0.16. For (d), (e) and (f) the other parameters are J = 0, r = 10, dφ/ds =0.035355. For all panels these parameters are common: M = 1, φ = π/2, dr/ds = 0.36 D. BINI, K. BOSHKAYEV, R. RUFFINI and I. SIUTSOUIt is clear that the presence of both the rotation and quadrupole parameters can increaseor decrease the values for rmb, rph and rms. For the sake of comparison, if one writesthese radii in the Boyer-Lindquist coordinates using the reverse of (3) for θ = π/2, andthe relation (2), then it is easy to obtain the following expressions for the Kerr solutionwith accuracy up to second-order terms in the rotation parameter a:Rmb = 4M[1± a2M− a216M2], Rph = 3M[1∓ 2√39aM− 2a227M2],Rms = 6M[1∓ 23√23aM− 7a2108M2].These radii are exactly those radii, expanded in terms up to second order in a, given inthe work of Bardeen et al. [7] for the Kerr solution.4. – ConclusionIn this work we have explored the domain of validity of the Hartle-Thorne solution aswell as the geodesics in this spacetime. We considered equatorial circular geodesics andinvestigated the role of the quadrupole parameter in the motion of a test particle. Besides,we have shown that the effects arisen from the rotation of the source can be balanced(increased or decreased) by its oblateness. It would be also interesting to investigate thetidal effects in the Hartle-Thorne spacetime. This task will be treated in a future work.REFERENCES[1] Hartle J. B., Astrophys. J., 150 (1967) 1005.[2] Hartle J. B. and Thorne K. S., Astrophys. J., 153 (1968) 807.[3] Bini D. et al., Class. Quantum Grav., 26 (2009) 225006.[4] Kerr R. P., Phys. Rev. Lett., 11 (1963) 237.[5] Synge J. L., Relativity: The general theory (North-Holland Publ. Co., Amsterdam) 1964.[6] Haensel P., Potekhin A. Y. and Yakovlev D. G., Neutron Stars 1 Equation of Stateand Structure, Vol. 326 (Astrophysics and Space Science Library) 2007, p. 619.[7] Bardeen J. M., Press W. H. and Teukolsky S., Astrophys. J., 178 (1972) 237.

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Equatorial Circular Geodesics in the Hartle-Thorne Spacetime

Equatorial Circular Geodesics in the Hartle-Thorne Spacetime

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