Approximate Metric for a Rotating Deformed Mass

A new Kerr-like metric with quadrupole moment is obtained by means of perturbing the Kerr spacetime. The form of this new metric is simple as the Kerr metric. By comparison with the exterior Hartle-Thorne metric, it is shown that it could be matched to an interior solution. This approximate metric may represent the spacetime of a real astrophysical object with any Kerr rotation parameter a and slightly deformed.


Introduction
In 1963, R. P. Kerr [25] 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 has difficulties when matching it to a realistic interior metric according to [3]. However, there has been a considerable amount of efforts trying to match the Kerr metric with a realistic interior metric that represents a physical source, see for example [8,17,18,20,27]. For a concise and comprehensive review of the different methods that have been used in order to try and obtain an interior solution for the Kerr metric, see [26].
Krasiński [26] arrived to the conclusion that the most obvious approach for the interior metric is to consider the θ dependency. This is important because a lot of works like [19] try to match the interior metric in an oversimplified θ-dependency and then used a boundary sphere. Also, Krasiński in [26] and [27] exploit the possibility of an ellipsoidal shape in the interior metric, allowing the use of a boundary ellipsoid. In addition, Burghardt [4], using a geometrical method matches a differentially rotating fluid source to the Kerr solution with an ellipsoidal boundary that in the limit of no rotation is the Schwarzschild interior. Regarding disk-matches, in [17] the authors use a spheroidal fluid shell between two Kerr vacuum regions. Furthermore, Wahlquist [42] concludes that a prolate spheroid is an appropriate boundary and adds that this geometry is not a relativistic effect because it appears in the Newtonian limit. Cuchí et al. [8] used an approximation method to obtain the global solution for an asymptotically flat vacuum containing a source composed of two thick shells of perfect fluid and concluded that it is not a possible source for the Kerr metric; nonetheless, if they allowed a singular term in the inner region it was compatible with Kerr up to order O(λ, Ω 3 ), where λ is a parameter controlling the postminkowskian approximation and Ω is the rotational parameter. In [10] and [40], the Newman-Janis algorithm was applied to look for interior solutions. Drake and Turolla [10] also propose a general method for finding interior solutions with oblate spheroidal boundary surfaces and note that the boundary surfaces reduce to a sphere in the case with no rotation, however Vaggiu in [40] argues that it is more helpful to start with the Schwarzschild interior and then proceed to the Kerr interior. Vaggiu uses an anisotropic conformally flat static interior and is lead to interior Kerr solutions with oblate spheroidal boundary surfaces, additionally he points out that his procedure can be applied to find interior solutions matching with a general asymptotically flat vacuum stationary spacetime. Other multipole and rotating solutions to the Einstein field equations (EFE) were obtained by Castejón et al. (1990) [6], Manko & Novikov (1992) [29], Manko et al. (2000) [30], Pachón et al. (2006) [31], and Quevedo (1986) [32], Quevedo (1989) [33], Quevedo & Mashhoon (1991) [34], Quevedo (2011) [35]. In the first four articles, they used the Ernst formalism [11], while in the four last ones, the solutions were obtained with the help of the Hoenselaers-Kinnersley-Xanthopoulos (HKX) transformations [24]. 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. Furthermore, Quevedo in [36] not only presents an exact electrovacuum solution that can be used to describe the exterior gravitational field of a rotating charged mass distribution, but also considers the matching using the derivatives of the curvature eigenvalues, this leads to matching conditions from which one can expect to obtain the minimum radius at which the matching can be made. In Nature, it is expected that astrophysical objects are rotating and slightly deformed as is pointed out in [1] and in [39]. In addition, Andersson and Comer in [1] use a two fluid model for a neutron star one, layer with neutrons that has a differential rotation and another layer consisting of a solid crust with constant rotation. The aim of this article is to derive an appropriate analytical tractable metric for calculations in which the quadrupole moment can be 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 [1,13,22,39], relativistic magnetohydrodynamic jet formation [12], astrometry [16,38] and gravitational lensing [14]. Furthermore, software related with applications of the Kerr metric can be easily modified in order to include the quadrupole moment [9,15,41]. This paper is organized as follows. In section 2, we give a succinct explanation of the Kerr metric, and the weak limit of the Erez-Rosen metric is presented. In section 3, the Lewis metric is presented, and the perturbation method is discussed. 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 [23] that the resulting metric is a solution of the EFE. In section 4, we compare our solution with the exterior Hartle-Thorne metric in order to assure that our metric has astrophysical meaning. The approach for an interior matching used in [19] is modified in section 5 in order to show that it is possible to match this new metric with a fluid source. Forthcoming works with this metric are discussed in section 6.

The Kerr Metric
The Kerr metric represents the spacetime of a non-deformed massive rotating object. The Kerr metric is given by [25,5] 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.

The Erez-Rosen Metric
The Erez-Rosen metric [5,43,44,45] represents the spacetime of 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 [16] where dΣ 2 = dθ 2 + sin 2 θdφ 2 , and where P 2 (cos θ) = (3 cos 2 θ − 1)/2. The quadrupole parameter is given by , with Q representing the quadrupole moment. This metric is valid up to the order O(qM 4 , q 2 ).

The Lewis Metrics
The Lewis metric is given by [28,5] 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 = V Z + W 2 ). Choosing µ = ν and performing the following changes of potentials we get the Papapetrou metric

The Perturbation Method
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 [25], one particular choice is [5,7] where ∆ = r 2 − 2Mr + a 2 . From (6) we get If we choose the term (7) becomes From (5), we propose the following metric where where the potentials V, W, X, Y, Z, and ψ depend on x 1 = r and x 2 = θ.
The potential W = W is so chosen to maintain the same cross components of the Kerr metric. Now, let us choose The only potential we have to find is ψ. In order to obtain this potential, the EFE must be solved 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 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: The solution for this equation is where K is a constant. To determine this constant, we compare the weak limit of the metric (8) with the approximate Erez-Rosen metric (2). The result is K = 2qM 3 /15 (ψ = χ). Then, the new modified Kerr metric containing quadrupole moment is where the tilde over ρ is dropped. We verified that the metric (14) is indeed a solution of the EFE using RE-DUCE [23] up to the order O(qM 4 , q 2 ).

Comparison with the Exterior Hartle-Thorne Metric
In order to validate the metric (14) as representing the gravitational field of a real astrophysical object, one should show that it is possible to construct an interior solution, which can appropriately be matched with our exterior solution. To this aim, we employed the exterior Hartle-Thorne metric [21,2,3,16] where M, J , and Q are related with the total mass, angular momentum, and mass quadrupole moment of the rotating object, respectively. This approximation for the Hartle-Thorne metric (15) was obtained by Frutos-Alfaro et al. using a REDUCE program [16]. The spacetime (14) has the same weak limit as the metric obtained by Frutos-Alfaro et al. [16]. A comparison of the exterior Hartle-Thorne metric (15) with the weak limit of the metric (14) shows that upon defining both metrics coincide up to the order O(M 3 , a 2 , qM 4 , q 2 ). Hence, the metric (14) may be used to represent a compact astrophysical object.

A Possible Fluid Source for the Kerr-like Metric
As was justified in the previous section, (14) can be used to represent a compact astrophysical object. Nonetheless, we have not mentioned nothing about a possible interior matching of this Kerr-like metric. In order to address the problem of matching to a interior metric, we will follow the work done by Haggag [19], and describe the modifications to use it with our metric. The method used in [19] consists of splitting the spacetime in three regions, two with matter, one static and the other stationary; and a vacuum region described by the Kerr metric. The matching is done by introducing some appropriate boundary conditions in the two boundary spheres that separates the regions. The first modification is changing the boundary spheres for ellipsoids and modifying the interior metrics.

Stationary
Hence, according to Haggag [19], the spacetime is divided in three regions as follows: Let A(r < a, θ, φ), and B(a < r < b, θ, φ) denote the material regions (0 < a < b) , and C(r > b, θ, φ) denote the vacuum region; with the two ellipsoids E A (r ∼ a, θ, φ) and E B (r ∼ b, θ, φ) separating them (See Figure 1). The region A is described by the following metric In the region B, the metric components are constructed in analogy with [19]: where P (r, θ) satisfies the boundary conditions where the prime denotes differentiation with respect to r. Otherwise P (r, θ) is arbitrary. The region C is described by (14). Using (14), (17) and (18) one can obtain expressions for the energy density µ, the angular velocity Ω, and the principal pressures p (r) , p (θ) , p (θ) ; in the matter region A. Also, if we define a matching function P (r, θ) that satisfies (19) in analogy with the matching functions used by Ramadan in [37] we can derive the above parameters for the matter region B. A derivation of these parameters is the objective of a forthcoming work. Besides, we note that one could add a more general matching introducing another stationary matter metric to the above procedure, the physical reason behind this is to minimize the size of the static region and allow us to use the two fluid model presented in [1] for the interior matching using (14) as the exterior solution.

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 this 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, additionally we would like to point out that works in superfluid neutron stars can be repeated but using this new metric instead of the Hartle-Thorne metric as an exterior solution. Furthermore, the existing software with applications of the Kerr metric can be easily modified to include the quadrupole moment. The limiting cases for the new Kerr metric correspond to the Kerr metric, the Erez-Rosen-like metric, the metric obtained by Frutos-Alfaro et al. and the Schwarzschild metric as expected. Additionally, regarding the matching with an interior solution the Haggag algorithm was modified in order to allow us to explore in a future work the physical nature of the source in the matter regions. Note that this modification is actually more natural as can be observed by a lot of the works mentioned in the introduction that use a spheroidal boundary. Also, we suggest another modification of the Haggag method by introducing another stationary matter region in order to use it in two fluid models of neutron stars.

A Appendix
The Ricci tensor components for the metric (8) are given by (with the tilde over ρ dropped)