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A new approximate metric representing the spacetime of a rotating deformed body is obtained by perturbing the Kerr metric to include up to the second order of the quadrupole moment. It has a simple form, because it is Kerr-like. Its Taylor expansion form coincides with second order quadrupole metrics with slow rotation already found. Moreover, it can be transformed to an improved Hartle-Thorne metric, which guarantees its validity to be useful in studying compact object, and it is possible to find an inner solution.

Nowadays, it is widely believed that the Kerr metric does not represent the spacetime of a rotating astrophysical object. It seems that there is no reasonable perfect fluid inner solution which serves as source of this spacetime [

The Ernst formalism [

To ensure the validity of a metric, the given metric is expanded to its post-linear form and compared with the post-linear version of the Hartle-Thorne (HT) spacetime [

This paper is organized as follows. Our perturbation method of the Kerr metric using the Lewis one is discussed in section 2. In section 3, it is shown that the application of this method leads to a new approximate solution to the EFE with rotation and quadrupole moment. It is checked by means of a REDUCE program that the resulting metric is a solution of the EFE [

First of all, we need a spacetime to work on. To this end, the Lewis metric is chosen and is given by [

where the chosen canonical coordinates are

The Ernst formalism and HKX transformation are based on this metric. Here, these formalisms are not employ to generate a new one. Rather, a new method to find a Kerr-like metric with quadrupole is developed. To this goal, we use the known transformation that leads to the Kerr metric [

where

Now, one chooses the Lewis potentials as follows

where

The cross term potential W is unaltered to preserve the following metric form

The so chosen potentials guarantee that one gets the Kerr metric if

Now, we have to solve the EFE perturbatively

where

Terms such as

(with

To solve the remaining terms of

where q represents the quadrupole parameter, and

From (5), the metric components reads

It was checked by means of a REDUCE program that the proposed metric is valid up to the order

In order to establish if our metric 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, we employ the exterior HT spacetime [

with metric components

where

The functions

A Taylor expansion of the metric components (12) up to the second order of J, M and q leads to

where we have added the second order terms of the quadrupole moment obtained by Frutos and Soffel [

Now, let us expand in Taylor series the metric components (10) up to the second order of a, J, M and q, the result is

Comparing these results with the ones obtained by Frutos and Soffel [

To compare our spacetime with the HT metric, we have to find a transformation that converts our metric (14) into the HT one (13). The following transformation converts the Kerr-like truncated metric (14) into the improved HT spacetime (13) changing

where

The constant are given by

Since our expanded Kerr-like metric can be transformed to the improved HT spacetime, it is possible to construct an interior metric that could be matched to our exterior spacetime. It can be considered as an improvement of the HT spacetime.

There are many other stationary metrics. We concentrate on the Quevedo-Mashhoon [

where

where

Taking the Killing vector as in the Kerr metric

This twist is the same as for the Kerr spacetime. Now, the Ernst function is [

One can show that this Ernst function and its inverse are solutions of the Ernst equation

For the sake of calculating the relativistic multipole moments, it is better to employ the inverse function [

where

The potential

where

The procedure to get the relativistic multipole moments is the following [

1) employ the inverse Ernst potential

2) set

3) change

4) expand in Taylor series of z the inverse Ernst potential, and finally,

5) use the Fodor-Hoenselaers-Perjés (FHP) formulae [

To obtain the multipole moment, we wrote a REDUCE program with the latter recipe. The first six mass and first five spin moments are

A direct comparison of these multipole moments with the corresponding ones of QM [

Our metric was obtained solving the EFE perturbatively. The Lewis metric with the modified potentials from the Kerr spacetime was used. This metric has three parameters m, a and q representing the mass, the rotation parameter and the quadrupole, respectively. It is valid until including

The form of our expanded metric suggests that it is possible to construct an interior solution, because it can be transformed to the improved HT spacetime. It is known that the approximate exterior HT metric is coupled to the interior HT one. This gives meaning to our results. Our spacetime may represent the approximative spacetime of a rotating deformed object. Moreover, we improved the HT metric including the second order of the quadrupole moment accuracy. Furthermore, it seems that by means of our perturbation procedure, one could improve our metric to include more terms to a desirable accuracy.

Moreover, the relativistic multipole moments were calculated to show that our spacetime was not isometric with the QM and the MN metrics. Our metric has a simple form and its multipole structure is Kerr-like, the only difference is that it has mass quadrupole.

This metric has potentially many applications because it could be employed as spacetime for real rotating astrophysical objects in a simple manner. Besides, it is easier to implement computer programs to apply this metric, because it maintains the simpleness of the Kerr metric. As an example of possible applications, the influence of the quadrupole moment in the light propagation and the light cone structure of this spacetime could be investigated using this Kerr-like spacetime.

We thank the Editor and the referee for their comments. Research of F. Frutos-Alfaro is funded by The Research Vice-Rectory of the University of Costa Rica. This support is greatly appreciated.

Francisco Frutos-Alfaro, (2016) Approximate Kerr-Like Metric with Quadrupole. International Journal of Astronomy and Astrophysics,06,334-345. doi: 10.4236/ijaa.2016.63028

The non-null Ricci tensor components for the metric (5) (here

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