Approximate spacetime for neutron stars

Abstract

An approximate realistic metric representing the spacetime of neutron stars is obtained by perturbing the Kerr metric. This metric has five parameters, namely the mass, spin or angular momentum, mass quadrupole, spin octupole and mass hexadecapole. Moreover, a version of the Hartle-Thorne metric containing these parameters is constructed by means of a series transformation between these spacetimes and solving the Einstein field equations. The form of the Pappas metric in Schwarzschild spherical coordinates is found. The three relativistic multipole structures are compared. IntroductionAmong compact objects are neutron stars (NS). These stars are relativistic rotating objects with high density, and strong gravitational and magnetic fields. The study of NS is relevant to understand the extrem conditions of matter in there, the behaviour of particles around them, and the structure of its spacetime [12, 3, 1]. The quest to find a realistic spacetime representation for neutron stars (NS) is an important task in astrophysics. Many attemps to obtain this spacetime have been done from approximate metrics until numerical metrics. The first attemp was made by Hartle-Thorne (HT). The relevance of the HT work was that they matched the interior solution with the exterior one [11]. Quevedo and Mashhoon [18] and Manko and Novikov [14] obtained exact solutions with charge and arbitrary mass multipoles. Later, more exact solutions containing other features, for instance magnetic dipole, were found [15]. With the advent of computer technology, the implementations of computer programs to find numerical solutions become a vogue [20]. However, approximate solutions are still important to extract astrophysical information from NS [17]. Moreover, a fourth order HT metric for the exterior of neutron stars was obtained in [21]. There are several techniques to find solutions of the Einstein field equations (EFE). Among them, the Ernst formalism has played an important role in finding new exact and approximate solutions. This formalism is employed in [18,14,17]. In this contribution, however, we include features like mass quadrupole, spin octupole and mass hexadecapole to the Kerr metric perturbatively. This is achieved easily by means of perturbing the Lewis metric form of the Kerr spacetime [7,8]. The form of the perturbations due to spin octupole and mass hexadecapole has the structure proposed by Ryan [19]. Then, one is certain that these features are introduced in the right manner. This metric has the advantage that it reduces to the Kerr metric which is an exact solution with mass and angular momentum. Moreover, it is simple to implement computationaly.

Approximate Kerr-Newman-like metric with quadrupole

“…where the quadrupole perturbations were obtained in [34]. The functions V µ , W µ , X µ , Y µ have a similar form as in (19).…”

“…To see which form has this metric in spherical-like coordinates, we use the Kerr mapping (Schwarzschild mapping) ρ = √ ∆ sin θ, z = (r −M) cos θ with a = 0 and q e = 0 [33,34]. Then, the function η is…”

“…In this section, we generate a perturbed metric using the Kerr-Newman metric as a seed metric. To find this approximate metric, we proceed as in [33,34], i.e. the mass quadrupole and the magnetic dipole are included perturbatively.…”

An approximate Kerr–Newman-like metric endowed with a magnetic dipole and mass quadrupole

“…In this section, we generate a perturbed metric using the Kerr-Newman metric as a seed metric. To find this approximate metric, we proceed as in [44,45], i.e. the magnetic dipole is included perturbatively.…”

“…the magnetic dipole is included perturbatively. The mass quadrupole was already added in [44], and in [45] other parameters like mass hexadecapole and spin octupole were taken into account. In this case, we have to solve the EME for the Lewis metric, see appendix A.…”