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A metric representing a slowly rotating object with quadrupole moment is obtained using a perturbation method to include rotation into the weak limit of the Erez-Rosen metric. This metric is intended to tackle relativistic astrometry and gravitational lensing problems in which a quadrupole moment has to be taken into account.

The first quadrupole solution to the Einstein field equations (EFE) was found by Erez and Rosen [

The aim of this article is to derive an appropriate analytical tractable metric for calculations in astrometry and gravitational lens theory including the quadrupole moment and rotation in a natural form. For this new rotating metric, it is not necessary for a multipolar expansion in the potential to include the multipolar terms because the seed metric already has a quadrupole term, that is to say this metric is multipolar intrinsically.

This paper is organized as follows. In Section 2, we get the weak limit of the Erez-Rosen metric. The Lewis metric is presented in Section 3. The perturbation method is discussed in Section 4. 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 [

The Erez-Rosen metric [3,4,14,15] 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. This metric is given by

where M is the mass of the object and

with λ = r/M - 1 and μ = cosθ. From now on, we will keep in the derivations terms up to order^{ }M^{2} and qM^{3}. The approximate forms of ψ and γ are

where with Q representing the quadrupole moment, and is the second Legendre polynomial.

We define the following variables

and

where

If we substitute the former definitions into (1), the metric takes the form

where and the inverse of F is written as

It is interesting to note that the spherical symmetry is not presented in the weak limit.

The Lewis metric is given by [14,16]

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.

we get the Papapetrou metric

To include slow rotation into the Erez-Rosen metric we use the Lewis-Papapetrou metric (5). First of all, we choose expressions for the canonical coordinates ρ and z. For the Kerr metric [

where

From (6) we get

where we have expanded up to M^{2} order.

Now, let us choose V = f = F and neglect the second order in ω (ω^{2} ≈ 0, W^{2} ≈ 0). Then, we have

If we choose

the term (7) becomes

This term appears in the approximate Erez-Rosen metric (3). From (5), let us propose the following metric

where X = 1/V, Y = r^{2}e^{2χ}, and Z = r^{2}e^{2χ}sin^{2}θ.

We see that to obtain a slowly rotating version of the metric (3) the only potential we have to find is W. Then, 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.

Fortunately, the Ricci tensor components R_{00}, R_{11}, R_{12}, R_{22}, R_{33} and the curvature scalar R_{s} depend upon the potentials V, X, Y, Z and not on W. Therefore, these components vanish (see Appendix). The only remaining equation we have to solve is R_{03} = 0, because it depends upon W. The equation for this component up to the order O(M^{3}, a^{2}, qM^{4}, q^{2}) is

The solution for this equation is

where K is a constant that we have to find. This constant can be found from the Lense-Thirring metric which can be obtained from the Kerr metric, i.e.

where J = Ma is the angular momentum and a is the rotation parameter.

Comparing the second term of the latter metric with the corresponding of the metric (8), i.e. W, we note that K= -2J = -2Ma.

Then, the new rotating metric with quadrupole moment written in standard form [

We verified that the metric (10) is indeed a solution of the EFE using REDUCE [^{2}, qM^{4}, q^{2}). Hence, one does not need to expand the term in a Taylor series.

In order to establish whether our metric does really represent the gravitational field of an astrophysical object, one should show that it is possible to construct an interior solution, which can appropriately be matched with the exterior solution. For this purpose, we employ the exterior Hartle-Thorne metric [13,20], which is given by

where M_{0}, J_{0}, and Q_{0} are related with the total mass, angular momentum, and mass quadrupole moment of the rotating object, respectively.

Now, comparing the exterior Hartle-Thorne metric with our expression (10), it can easily be seen that upon defining

both metrics coincide up to the order O(M^{3}, a^{2}, qM^{4}, q^{2}). Our approximate expression for the Hartle-Thorne metric (11) was obtained by means of a REDUCE program using the expressions from Abramowicz et al. [^{4}), which we neglected, because it is beyond the order we are working with. Additional differences are that our metric parameters (M, J = Ma, qM^{3}) are distinct and our expressions (12) are simpler than those of Boshkayev et al. [

In some cases, the metric (10) has to be transform from spherical (r, θ, φ) into Cartesian coordinates (x, y, z). For example, if a comparison with a post-Newtonian (PN) metric is made, we have to transform the metric (10) by using one of the following radial coordinates transformation: the harmonic or the isotropic coordinates of Schwarzschild metric. The first one is r = R + M, and the second one is r = R(1 + M/2R)^{2}, where R is a new radial coordinate [

where

and

Noting that C = exp(-χ) is still given by (2) with r changed by R. Now, transforming the metric (13) into the Cartesian coordinates which are given by the usual relations

The resulting metric has the following form

where X = (X, Y, Z)

The former metric can be generalized as follows

where

and the vector V is defined as

with J = Je_{j} (e_{j} is an unit vector in the direction of J).

In this paper, we include the rotational effect using the weak limit of the Erez-Rosen metric as seed metric into the Lewis-Papapetrou metric. Thus, a new metric with quadrupole moment and rotation in the weak limit is obtained. Generally speaking, the quadrupole moment is included in the metric, for instance, in gravitational lensing, through the expansion of the gravitational potential in a power series [

The Ricci tensor components are

The scalar curvature is