On relativistic multipole moments of stationary space–times

Among the known exact solutions of Einstein's vacuum field equations the Manko–Novikov and the Quevedo–Mashhoon metrics might be suitable ones for the description of the exterior gravitational field of some real non-collapsed body. A new proposal to represent such exterior field is the stationary q-metric. In this contribution, we computed by means of the Fodor–Hoenselaers–Perjés formalism the lowest 10 relativistic multipole moments of these metrics. Corresponding moments were derived for the static vacuum solutions of Gutsunayev–Manko and Hernández–Martín. A direct comparison between the multipole moments of these non-isometric space–times is given.


Introduction
In 1968, Ernst [1,2] developed a complex procedure to simplify the Einstein field equations (EFE) for a stationary Weyl-Lewis-Papapetrou (WLP) type metric by means of two complex potentials. After this seminal work, several methods to find new solutions of the EFE were developed by Hoenselaers, Kinnersley and Xanthopoulos (HKX) [3] and Belinsky & Zakharov [4,5], among others. The HKX method was employed by Quevedo and Mashhoon (QM) to find new stationary solutions from the Erez-Rosen space-time as seed metric [6][7][8]. The Erez-Rosen metric is an exact vacuum solution of EFE representing a static metric with a mass and a quadrupole parameter (M and Q) [9]. The QM metric is an axially symmetric stationary vacuum solution with parameters M, a and q n with n being an (even) integer (q 2 can be identified with a quadrupole parameter) [10].
2018 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

The stationary metrics
The stationary space-time is represented by the WLP metric in prolate spheroidal coordinates (t, x, y, φ) where σ is a constant. This metric was transformed from cylindrical coordinates to these prolate coordinates which are defined by and z = σ xy. 2) The functions f and ω are related to the twist scalar Ω through where φ as the azimuthal unit vector and ρ the cylindrical coordinate. The Ernst potential is E = f + iΩ, and the Ernst function is given by There are several techniques to solve this Ernst equation in these prolate coordinates; see for instance [3][4][5][12][13][14].
As examples, let us see what form the metric potentials (f , ω and γ ) take in these coordinates for the Kerr and Schwazschild metrics. The mapping from the Kerr spherical coordinates to the prolate coordinates is given by σ x = r − m and y = cos θ, (2.6) where Using this mapping, the Kerr metric potentials take the following form: where J = ma. For the Schwarzschild metric (set a = 0 in (2.7)) one has where σ = m.

The Manko-Novikov metric
This metric represents the space-time of a rotating massive object with mass and spin multipole moments. This metric has parameters: k, α and α n , n > 1. The parameters k and α are related to the mass and the rotation, respectively. It contains the following exterior metrics [15]: To compare it with the Kerr or QM metric one has to set If there is no rotation, α = 0, then k represents the mass. This solution has an infinite set of relativistic mass and spin multipoles. They correspond to the Kerr ones if α n = 0.
The Ernst function is where  [12], Cosgrove [13] and Dietz & Hoenselaers [14], they generated the following metric potentials: The functions η 1 , η 2 , χ 1 and χ 2 are with the following definitions: The γ function is not involved in the calculation of the relativistic multipole moments. The interested reader may consult the references. In §3, the Ernst function ξ will be employed to determine the mass and spin multipole moments of the MN space-time.
In our notation the parameters m ≡ GM/c 2 and a have the dimension of a length and the parameters q n are dimensionless. This solution is characterized by an infinite set of relativistic mass and spin multipoles. They correspond to the Kerr ones if q n = 0, and to the ER ones if a = 0 [8]. Boshkayev et al. [10] have shown how to get the HT from the QM metric. The HT metric is an approximate solution of the EFE for an object with three parameters, mass, spin and quadrupole moment. Although this solution is intended for slow rotation, it also has been employed to model fast rotating neutron stars. It has been argued (e.g. Boshkayev et al. [10]) that the QM space-time could be applied to model real astrophysical objects, such as neutron stars. From two HKX transformations on the static Ernst potential E 0 = e −2ψ for the ER space-time as seed metric, QM found the new Ernst potential, which is given by [6,8] From the Ernst function (2.16), they found the following functions [6][7][8]10]: The functions A, B and C are given by For the QM metric the potentials ψ and δ ± are of the form [30,31] ψ = ∞ n=1 (−1) n q n P n (y)Q n (x) (2.21) and The functions P l (y) and Q l (x) are Legendre polynomials of the first and second kind, respectively. The general form of the potential χ was determined by Quevedo [8,32]. To find the relativistic multipole moments, one does not need this χ function. The interested reader may consult the references. With the Computer Algebra System REDUCE [33], we were able to check the validity of the Ernst equation directly without employing HKX transformations. In the next section, the Ernst function ξ will be employed to determine the relativistic multipole moments of the QM space-time.

The stationary q-metric
The q-metric is a generalization of the Schwarzschild metric with quadrupole parameter. The static version in spherical coordinates is given by [17,34] From the parameters m and q, the mass and the quadrupole moment of the object are given by quadrupole parameters. The rotating version of this metric has the following Ernst potential in prolate spheroidal coordinates: and The parameter α is related to the rotation parameter a by (2.20). The prolate spheroidal coordinates are linked to the spherical coordinates through The rotating metric can be read off from the general QM metric with ZV parameter. The Papapetrou potentials are and and

Relativistic multipole moments
There are several methods to get the spin and quadrupole moments from a given metric [25,26,28]. In this section, we apply the FHP procedure to the QM metric. An excellent review of the FHP formalism was given by Filter [35] in his diploma thesis. Filter also found the S 11 component in the same way. The procedure to obtain the relativistic multipole moments is the following [26]: (i) employ the inverse Ernst potential ξ −1 , (ii) set y = cos θ = 1, and σ x → 1/z into ξ −1 , (iii) expand in Taylor series of z the inverse Ernst potential, and finally, (iv) use the FHP formulae [26].
To get the relativistic multipole moment, we wrote a REDUCE program with the latter recipe.

The Manko-Novikov multipole moments
Taking the first 10 even members of the ψ function ψ = n=10 n=1 α n P n R n+1 , (3.1) and defining we derive the lowest relativistic multipole moments for the MN metric

The Quevedo-Mashhoon multipole moments
In this case, the Ernst function take the following simple form: Taking the first 10 even members of the function ψ, ψ = n=10 n=1 q n P n Q n (P n (1) = 1).

(3.6)
From the ER metric, the static massive quadrupole is given by Q = 2qMm 2 /15. From the Kerr metric, one infers that the spin-dipole S = Mac. Using these relations, and β as in (3.2), one can write the relativistic multipole moments as follows:

The q-metric multipole moments
For this metric, the relativistic multipole moments are (S 0 = Mac)  again the higher order spin moments cannot be chosen freely. It is interesting to analyse some particular cases from the multipole structures for these space-times. For an extreme black hole, all these metrics are isometrics with relativistic multipoles of the form M 2n = ±Mm 2n and S 2n+1 = ±Mm 2n+1 as expected. For neutron stars, the forms of the first five multipole moments are [36][37][38] M 0 = M, where α, β, γ are parameters. For the MN metric, if we set a → −a, Q = (1 − α)S 2 /M and in (3.7), we get the values (3.12), except for S 3 = (1 − 2α)S 3 /M 2 . Finally, for the q-metric (see (3.9)), it is not possible to construct all values (3.12).
In the next section, we compare the static multipoles of the MN metric derived from (3.3) and the ER multipole moments deduced from (3.7) with those ones of different static metrics.

Comparisons with static metrics
For a comparison with other static metrics, we consider the Gutsunayev-Manko (GM) [39], the Hernández-Martín space-times (HM I and II) [40], and the static q-metric [17,34]. These metrics are static solutions of the EFE and might be related to the external gravitational field of a body with mass and quadrupole moment. The differences of the metrics are in the fields ψ and γ . To obtain the multipole moments of these metrics, the FHP procedure was also employed. In the case of the generalized ER metric, we have from (3.7) the following mass multipoles: Although the MN and the QM metrics have a large set of multipole moment parameters for the exterior gravitational field, their higher-order spin moments cannot be chosen freely, because they are totally determined by the set of metric parameters. The GH moments for the generalized ER metric and those of the static MN metric were compared with those of other metrics, namely the GM, HM (I and II), and q-metrics. All compared space-times are not isometric to each other. The generalized ER or QM and the MN metrics are more appropriate to describe the exterior gravitational field of a real static object due to their more general multipole structures.
Ethics. F.F.-A. received ethical approval from the Vice-Rectory of the University of Costa Rica (UCR) to realize this work. M.S. received ethical approval from the Technical University Dresden and Lohrmann Observatory to realize this work.
Data accessibility. This article has no additional data. Authors' contributions. M.S. proposed the idea of the study, supervised the project, and co-drafted the paper. F.F.-A.
contributed in all sections and co-drafted the paper. All authors regularly discussed the progress during the entire work.