Perturbation of the Kerr Metric

Transcription

1 Perturbation of the Kerr Metric arxiv: v1 [gr-qc] 5 Jan 2014 Francisco Frutos-Alfaro Abstract A new Kerr-like metric with quadrupole moment is obtained by means of perturbing the Kerr spacetime. By comparison with the exterior Hartle-Thorne metric, it is showed that it could be matched to an interior solution. This metric may represent the spacetime of an astrophysical object. 1 Introduction In 1963, R. P. Kerr [17] proposed a metric that describes a massive rotating object. Since then, a huge amount of papers about the structure and astrophysical applications of this spacetime appeared. Now, it is widely believed that this metric does not represent the spacetime of an astrophysical rotating object. This is because the Kerr metric cannot be matched to a realistic interior metric [2]. Other multipole and rotating solutions to the Einstein field equations EFE) were obtained by Castejón et al. 1990) [4], Manko & Novikov 1992) [19], Manko et al. 2000) [20], Pachon et al. 2006) [21], and Quevedo 1986) [22], Quevedo 1989) [23], Quevedo & Mashhoon [24], Quevedo 2011) [25]. In the four first articles, they used the Ernst formalism [7], while in the four last ones, the solutions were obtained with the help of the Hoenselaers- Kinnersley-Xanthopoulos HKX) transformations [16]. These authors obtain new metrics from a given seed metric. These formalisms allow to include other desirable characteristics rotation, multipole moments, magnetic dipole, etc.) to a given seed metrics. In Nature, it is expected that astrophysical objects are rotating and slightly deformed. The aim of this article is to derive an appropriate analytical tractable metric for calculations in which the quadrupole moment can be 1

2 treated as perturbation, but for arbitrary angular momentum. Moreover, this metric should be useful to tackle astrophysical problems, for instance, accretion disk in compact stellar objects [9, 14], relativistic magnetohydrodynamic jet formation [8], astrometry [26, 12] and gravitational lensing [10]. Furthermore, software related with applications of the Kerr metric can be easily modified in order to include the quadrupole moment [6, 27, 11]. This paper is organized as follows. In section 2, we give a succinct explanation of the Kerr metric. The weak limit of the Erez-Rosen metric is presented in section 3. In section 4, the Lewis metric is presented. The perturbation method is discussed in section 5. The application of this method leads to a new solution to the EFE with quadrupole moment and rotation. It is checked by means of the REDUCE software [15] that the resulting metric is solution of the EFE. In section 6, we compare our solution with the exterior Hartle- Thorne metric in order to assure that our metric has astrophysical meaning. Forthcoming works with this metric are discussed in section 7. 2 The Kerr Metric The Kerr metric represents the spacetime of a non-deformed massive rotating object. The Kerr metric is given by [17, 3] ds 2 = ρ 2[dt asin2 θdφ] 2 sin2 θ ρ 2 [r 2 +a 2 )dφ adt] 2 ρ2 dr2 ρ 2 dθ 2, 1) where = r 2 2Mr + a 2 and ρ 2 = r 2 + a 2 cos 2 θ. M and a represent the mass and the rotation parameter, respectively. The angular momentum of the object is J = Ma. 3 The Erez-Rosen metric The Erez-Rosen metric [3, 28, 29, 30] represents a body with quadrupole moment. The principal axis of the quadrupole moment is chosen along the spin axis, so that gravitational radiation can be ignored. Here, we write down an approximate expression for this metric obtained by doing Taylor series [12] 2

3 ds 2 = 1 2M r ) e 2χ dt 2 1 2M r ) 1 e 2χ dr 2 r 2 e 2χ dθ 2 +sin 2 θdφ 2 ), 2) where dσ 2 = dθ 2 +sin 2 θdφ 2, and χ = 2 15 qm3 r P 2cosθ). 3) 3 The quadrupole parameter is given by q = 15GQ/2c 2 M 3 ), with Q representing the quadrupole moment. This metric is valid up to the order OqM 4, q 2 ). 4 The Lewis Metrics The Lewis metric is given by [18, 3] ds 2 = Vdt 2 2Wdtdφ e µ dρ 2 e ν dz 2 Zdφ 2 4) where we have chosen the canonical coordinates x 1 = ρ and x 2 = z, V, W, Z, µ and ν are functions of ρ and z ρ 2 = VZ + W 2 ). Choosing µ = ν and performing the following changes of potentials V = f, W = ωf, Z = ρ2 f ω2 f and e µ = eγ f, we get the Papapetrou metric ds 2 = fdt ωdφ) 2 eγ f [dρ2 +dz 2 ] ρ2 f dφ2. 5) 5 Perturbing the Kerr Metric To include a small quadrupole moment into the Kerr metric we will modify the Lewis-Papapetrou metric 5). First of all, we choose expressions for the canonical coordinates ρ and z. For the Kerr metric [17], one particular choice is [3, 5] where = r 2 2Mr +a 2. ρ = sinθ and z = r M)cosθ 6) 3

4 From 6) we get ) dr dρ 2 +dz 2 = [r M) 2 sin 2 θ + cos 2 2 θ] +dθ2. 7) If we choose e µ = ρ 2 [r M) 2 sin 2 θ + cos 2 θ] 1, the term 7) becomes ) dr e µ [dρ 2 +dz 2 ] = ρ 2 2 +dθ2, where ρ 2 = r 2 +a 2 cos 2 θ. From 5), we propose the following metric where ds 2 = Vdt 2 2Wdtdφ Xdr 2 Ydθ 2 Zdφ 2, 8) V = Ve 2ψ W = W X = Xe 2ψ 9) Y = Ye 2ψ Z = Ze 2ψ, where the potentials V, W, X, Y, Z, and ψ depend on x 1 = r and x 2 = θ. Now, let us choose V = f = 1 ρ 2[ a2 sin 2 θ] W = ã ρ 2[ r2 +a 2 )]sin 2 θ = 2Jr ρ 2 sin2 θ X = ρ2 Y = ρ 2 Z = sin2 θ ρ 2 [r 2 +a 2 ) 2 a 2 sin 2 θ]. 10) 4

5 The only potential we have to find is ψ. In order to obtain this potential, the EFE must be solved G ij = R ij R 2 g ij = 0 11) where R ij i, j = 0, 1, 2, 3) are the Ricci tensor components and R is the curvature scalar. The Ricci tensor components and the curvature scalar R for this metric can be found in the Appendix. In our calculations, we consider the potential ψ as perturbation, i.e. one neglects terms of the form = Terms containing factors of the form = ψ ψ 0. a ψ x = m ψ 0 i = 1, 2) i xi are also neglected. Substituting the known potentials V, W, X, Y, Z) into the expressions for the Ricci tensor and the curvature scalar see Appendix), it results only one equation for ψ that we have to solved: sinθ r 2 ψ ) + sinθ ψ ) = 0 12) The solution for this equation is ψ = K r 3P 2cosθ), 13) where K is a constant. To determine this constant, we compare the weak limit of the metric 8) with the Erez-Rosen metric 2), i.e. ψ = χ. The result is K = 2qM 3 /15. Then, the new modified Kerr metric containing quadrupole moment is 5

6 ds 2 = e 2χ ρ 2 [ a2 sin 2 θ]dt 2 + 4Jr ρ 2 sin2 θdtdφ ρ2 e 2χ dr2 ρ 2 e 2χ dθ 2 e2χ sin 2 θ ρ 2 [r 2 +a 2 ) 2 a 2 sin 2 θ]dφ 2 = dt ae χ sin 2 θdφ] 2 sin2 θ [r 2 +a 2 )e χ dφ ae χ dt] 2 ρ 2[e χ ρ ) 2 ρ e 2χ 2 dr2 +ρ 2 dθ 2, 14) where the tilde over the ρ is dropped. We verified that the metric 14) is indeed a solution of the EFE using RE- DUCE [15] up to the order OqM 4, q 2 ). 6 Comparison with the Exterior Hartle-Thorne Metric In order to establish whether the metric 14) does really represent the gravitational field of an astrophysical object, we should show that it is possible to construct an interior solution, which can appropriately be matched with the exterior solution. For this purpose, Boshkayev et al. [2] and Frutos-Alfaro et al. [12] employed the exterior Hartle-Thorne metric [13, 1] ds 2 = 1 2M ) + 2QM3 P r r 3 2 cosθ) dt M ) + 4M2 2QM3 P r r 2 r 3 2 cosθ) dr 2 15) ) r 2 1 2QM3 P 2 cosθ) dσ 2 + 4J r sin2 θdtdφ, r 3 where M, J, and Q are related with the total mass, angular momentum, and mass quadrupole moment of the rotating object, respectively. The spacetime 14) has the same weak limit as the metric obtained by Frutos et al. [12]. A comparison of the exterior Hartle-Thorne metric [13] with the weak limit of the metric 14) shows that upon defining 6

7 M = M, J = J, 2QM 3 = 4 15 qm3, 16) both metrics coincide up to the order OM 3, a 2, qm 4, q 2 ). Hence, the metric 14) may be used to represent a compact astrophysical object. 7 Conclusions The new Kerr metric with quadrupole moment was obtained by solving the EFE approximately. It may represent the spacetime of a rotating and slightly deformed astrophysical object. This is possible, because it could be matched to an interior solution. We showed it by comparison of our metric with the exterior Hartle-Thorne metric. Moreover, the inclusion of the quadrupole moment in the Kerr metric does it more suitable for astrophysical calculations than the Kerr metric alone. There are a large variety of applications which can be tackled with this new metric. Amongst the applications for this metric are astrometry, gravitational lensing, relativistic magnetohydrodynamic jet formation, and accretion disks in compact stellar objects. Furthermore, the existing software with applications of the Kerr metric can be easily modified to include the quadrupole moment. 7

8 A Appendix R 00 = e 2ψ 4ρ 2 VX 2 Y 2 ψ 4ρ 2 X 2 Y VW2 X 2 Y 2ρ 2 VXY ψ X +2VX2 Y ψ ρ 2 4ρ2 X 2 Y ψ V 4W 2 X 2 Y ψ V +2ρ2 VX 2 ψ Y 4V 2 X 2 Y ψ Z ) 2 4ρ 2 VXY 2 2 ψ ψ 2 +8VW2 XY 2 +2ρ 2 VY 2 ψ X + 2VXY 2 ψ ρ 2 4ρ2 XY 2 ψ V 4W2 XY 2 ψ V 2ρ 2 VXY ψ Y 4V 2 XY 2 ψ Z +ρ2 XY X V ρ 2 Y 2 X V X2 Y ρ2 V 2 ρ2 V XY + 2ρ 2 X 2 Y 2 V 2 ρ2 X 2 V Y +2VX2 Y V Z + 2ρ 2 XY 2 2 V 2 +ρ2 XY V Y +2VXY 2 V Z ) 2 ) ) 2 W W + 2VX 2 Y +2VXY 2 R 01 = 0 R 02 = 0 8

9 R 03 = e 2ψ 8ρ 2 WX 2 Y 4ρ 2 X 2 Y 2 8W 3 X 2 Y 4WX 2 Y ψ ρ 2 +8W2 X 2 Y ψ W +8VWX2 Y ψ ) 2 ψ + 8ρ 2 WXY 2 8W 3 XY 2 4WXY 2 ψ + 8W 2 XY 2 ψ W +8VWXY 2 ψ Z ρ2 XY X W + ρ 2 Y 2 X W +X2 Y ρ2 W 2 ρ2 W +XY 2WX 2 Y V Z 2WXY 2 V Z 2ρ2 X 2 Y 2 W ) 2 2 W 2WX 2 Y +ρ 2 X 2 W Y 2ρ2 XY 2 2 W 2 ) ) 2 W 2WXY 2 ρ 2 XY W Y Z ρ 2 9

10 1 R 11 = 4ρ 4 X 2 Y 2 ψ 4ρ 4 XY 2 2 2ρ4 XY ψ X 2ρ 2 X 2 Y ψ ρ 2 +2ρ4 X 2 ψ Y 4ρ4 XY 2 2 ψ 2 8ρ 4 XY 2 ψ 2 +8ρ 2 W 2 XY 2 +2ρ 4 Y 2 ψ X + 6ρ 2 XY 2 ψ ρ 2 8ρ2 WXY 2 ψ W 2ρ4 XY ψ Y ) 8ρ 2 VXY 2 ψ 2 Z 2ρ4 XY 2 X X 2 +ρ4 Y ρ 2 XY X ρ 2 +ρ4 X X Y +ρ2 Y 2 X ρ 2 + ρ 4 Y X ) Y 2ρ2 XY 2 2 ρ 2 ρ +XY VXY 2 ρ2 4VWXY 2 W 2V 2 XY 2 Z Z +2W2 XY 2 V Z +2ρ2 XY 2 Z 2ρ4 XY 2 Y 2 +ρ4 X ) ) 2 ) 2 Y W ) 2 10

11 R 12 = 1 4ρ 4 XY 8ρ 4 XY ψ ψ +8ρ2 W 2 XY ψ W + 4ρ 2 XY ψ ρ 2 4ρ2 WXY ψ 4ρ2 VXY ψ Z + 4ρ 2 XY ψ ρ 2 4ρ2 WXY ψ W 4ρ2 VXY ψ Z + ρ 2 Y X ρ 2 2ρ2 XY 2 ρ 2 +W2 XY 2 ρ 2 + XY ρ2 ρ 2 +ρ2 X ρ2 Y ρ2 Z +VXY + VXY ρ2 Z W2 XYZ 2 V 2W3 XY 2 W + 2ρ 2 XY W W 2W2 XY W W W Z 2VWXY 2VWXY W Z VW2 XY 2 Z 2V 2 XY Z ) Z R 13 = 0 ψ 11

12 ) 2 1 R 22 = 4ρ 4 X 2 Y 2 ψ ψ 4ρ 4 X 2 Y 2 8ρ4 X 2 Y + 8ρ 2 W 2 X 2 Y 2ρ 4 XY ψ X +6ρ2 X 2 Y ψ ρ 2 8ρ 2 WX 2 Y ψ W +2ρ4 X 2 ψ Y 8ρ2 VX 2 Y ψ Z 4ρ 4 XY 2 2 ψ 2 +2ρ4 Y 2 ψ X 2ρ2 XY 2 ψ ρ 2 2ρ 4 XY ψ ) 2 Y 2ρ4 XY 2 X X 2 +ρ4 Y + ρ 4 X X Y +ρ4 Y X ) ρ + X Y +ρ 2 X 2 ρ2 ρ 2 XY ρ2 Y +2W2 X 2 Y V Y 2ρ2 X 2 Y 2 ρ 2 2 Y +2VX2 Y ρ2 Z Z W +2ρ2 X 2 Y ) 2 Y 4VWX 2 Y W Z 2ρ4 XY 2 Y 2 +ρ4 X ) ) 2 Z 2V 2 X 2 Y R 23 = 0 ) 2 12

13 R 33 = 1 4ρ 2 X 2 YZ 2 ψ 4ρ 2 X 2 Y 2 2 8W2 X 2 YZ 2ρ 2 XYZ ψ X 2YX2 Z ψ ρ 2 +8WX2 YZ ψ W + 2ρ 2 X 2 Z ψ Y 8W2 X 2 Y ψ Z 4ρ2 XY 2 Z 2 ψ ) 2 2 ψ 8W 2 XY 2 Z +2ρ 2 Y 2 Z ψ X 2XY 2 Z ψ ρ 2 + 8WXY 2 Z ψ W 2ρ2 XYZ ψ Y 8W2 XY 2 ψ Z ρ 2 XY X Z +ρ2 Y 2 X Z X2 Y ρ2 Z ) 2 XY 2 ρ2 Z W 2X2 YZ +4WX 2 Y W Z ) 2 W 2XY 2 Z +4WXY 2 W Z +ρ2 X 2 Y Z ρ 2 XY Y Z 2ρ2 X 2 Y 2 Z 2ρ 2 XY 2 2 Z 2 +2VXY VX2 Y ) ) 2 Z Z ) 2 13

14 Calculation of the scalar curvature e 2ψ R = 4ρ 4 X 2 Y 2 ψ 2ρ 4 X 2 Y ρ4 X 2 Y 4ρ 2 W 2 X 2 Y +2ρ 4 XY ψ + 4ρ 2 WX 2 Y ψ W 2ρ4 X 2 ψ Y + 4ρ 4 XY 2 2 ψ ψ 2 +4ρ4 XY 2 References X 2ρ2 X 2 Y ψ ρ 2 +4ρ2 VX 2 Y ψ Z ) 2 4ρ 2 W 2 XY 2 2ρ 4 Y 2 ψ X 2ρ2 XY 2 ψ ρ 2 +4ρ2 WXY 2 ψ + 2ρ 4 XY ψ Y +4ρ2 VXY 2 ψ Z +2ρ4 XY 2 X ) 2 2 X ρ 4 Y +ρ 2 XY X ρ 2 ρ4 X X Y ρ 2 Y 2 X ρ 2 ρ4 Y X Y +2ρ2 X 2 Y 2 ρ 2 ) 2 ρ X Y ρ 2 X 2 ρ2 Y +2ρ2 XY 2 2 ρ 2 2 ) ρ XY ρ 2 XY ρ2 Y ρ2 X 2 Y V Z ρ 2 XY 2 V ) 2 Z W W ρ2 X 2 Y ρ 2 XY 2 ) ) 2 + 2ρ 4 XY 2 Y Y 2 ρ4 X [1] Berti, E., White, F., Maniopoulou, A. & Bruni, M Rotating neutron stars: an invariant comparison of approximate and numerical spacetime models. MNRAS, 358, W ) 2 14

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