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Relativistic models of magnetars: Nonperturbative analytical approach

In the present paper we focus on building simple nonperturbative analytical relativistic models of magnetars. With this purpose in mind we first develop a method for generating exact interior solutions to the static and axisymmetric Einstein-Maxwell-hydrodynamic equations with anisotropic perfect fluid and with pure poloidal magnetic field. Then using an explicit exact solution we present a simple magnetar model and calculate some physically interesting quantities as the surface elipticity and the total energy of the magnetized star.


Introduction
The Soft Gamma Repeaters are spectacular phenomena occurring in the visible Universe. The giant flares detected so far show that the peak luminosities are of order 10 44 −10 46 erg/s. One of the most promising and widely accepted explanations are the magnetars [1]. Magnetars are believed to be neutron stars with ultra strong magnetic field responsible for the observed giant flares. The huge amount of energy released in the giant flares can be explained by the existence of ultra strong magnetic fields with strength of the order (or larger than) 10 14 − 10 15 Gauss [2,3]. The giant flares SGR 0526-66, SGR 1900+14 and SGR 1806-20 detected so far reveal the existence of characteristic quasi periodic oscillations in the range of tenths of Hz to kHz [4]. These oscillations are believed to be seismic vibrations of the magnetars. If the hypotheses is true this will provide us with a tool to investigate the stellar interior. That is why the quasi periodic oscillations were intensively studied in the past years [5] - [7] (and references therein).
The study of the stelar interior by the quasi periodic oscillations require adequate models of the internal structure of the magnetars. In general, our understanding of magnetars as Soft Gamma Repeaters is intimately related to the understanding their internal structure and the construction of adequate models within General Relativity. Clearly, the building of completely realistic magnetar models is a formidable task. However, various simple relativistic models, more or less realistic, could be built and these models provide us with valuable physical insight into the internal structure of magnetars [8] - [17]. The existing simple magnetar models are based on Einstein-Maxwell equations coupled to the perfect fluid hydrodynamical equations. In modeling magnetar equilibrium configurations two main approaches have been followed so far. The first approach is to numerically solve the coupled systems of equations [8], [9], [12], [14]. The second approach is perturbative -magnetar equilibrium configurations are studied by using perturbative techniques, i.e. the Einstein-Maxwell-hydrodynamic equations are solved by linearizing them about a known static and spherically symmetric background solution of Einstein-hydrodynamic equations and then expanding the perturbed equations in tensor harmonics [10], [11], [13], [15] - [17]. Due to the linear character of the perturbative equations one can consider in a relatively simple manner more complicated magnetic field configurations as the simultaneous presence of poloidal and toroidal magnetic fields.
In the present paper we also address the problem of constructing equilibrium configurations of neutron stars with ultra strong magnetic fields within the framework of General Relativity. Contrary to the previous approaches mentioned above, our approach here is fully analytical and nonperturbative and based on exact solutions. Exact solutions provide a route to better and deeper understanding of the inherent nonlinear character of gravity and its interaction with matter. On the other hand, the exact solutions could serve as tests for checking the computer codes which is important for the advent of numerical relativity. More precisely, in this paper we find exact interior solutions to the coupled Einstein-Maxwell-hydrodynamic equations describing static (nonrotating) equilibrium configurations of strongly magnetized neutron stars. The interaction of the neutron star fluid with the magnetic field is also taken into account to some extent.

Setting of the problem and exact solutions
Our starting point is the coupled Einstein-Maxwell-hydrodynamic equations where T µν and T EM µν = 1 4π F µα F α ν − 1 4 F 2 g µν are the energy-momentum tensors of the neutron matter and the electromagnetic field, respectively. J µ is the current which sources the electromagnetic field.
Analytically solving of the coupled Einstein-Maxwell-hydrodynamic equations in the general case is a desperate task and therefore we need some simplifying assumptions. We will assume that the configurations (and the spacetime itself) are strictly static (nonrotating) and axially symmetric. In mathematical terms our assumptions mean that there exist one (hypersurface orthogonal) timelike Killing vector ξ and one spacelike axial Killing vector η, commuting with ξ and with closed periodic orbits shrinking down to zero on the axis of symmetry. In adapted coordinates the Killing vectors can be written in the usual form ξ = ∂/∂t and ξ = ∂/∂φ where t is the time coordinate and φ is the azimuthal angle around the axis of symmetry. Our geometrical assumptions impose restrictions on the possible configurations of the electromagnetic field and the energy momentum tensor of the neutron star matter. More precisely, they require the absence of meridional convective currents and electric field. The geometric assumptions require also the 4-velocity of the neutron matter to be aligned with the timelike Killing vector ξ.
The invariance of the Maxwell 2-form F under the axial Killing field η and the absence of meridional currents allow us to introduce a magnetic potential Φ defined by dΦ = i η F . The Maxwell 2-form then is given by where e 2u = g(η, η). The magnetic field B measured by a comoving observer In the models studied so far the neutron matter has been described by an isotropic perfect fluid with T µν = (ρ + p)v µ v ν + pg µν where ρ, p and v µ are the energy density, the pressure and the 4-velocity of the fluid. The description of the neutron star matter as an isotropic perfect fluid is not completely satisfactory because it neglects the interaction of the neutron matter with the ultra strong magnetic field. This problem is highly nontrivial and extremely difficult to be solved completely. From first principles it is clear that the strong magnetic field yields anisotropy in the neutron star matter and this should be taken into account in the energy-momentum tensor of the matter. Indeed, since the neutron has anomalous magnetic momentum the neutron matter will react to the strong magnetic field by polarizing itself due to the coupling of the neutron spin to the magnetic field. The things can get even more complicated if we take into account the possible manifestation of some quantum effects like the spin-spin interactions which can drive the system to some kind of feromagnetic-like state. In this context we should also note that the origin of the ultra strong magnetar magnetic fields is not completely clear and some sort of feromagnetic-like phase transition could give contribution. Even more, the estimated magnetic field on the magnetar surfaces mentioned above exceeds in fact the QED critical magnetic field value B c ≈ 10 13 G which shows that the nonlinear Euler-Heisenberg electrodynamics should be probably used instead of the linear Maxwell electrodynamics. The above arguments show that the proper description of the strong magnetic fields in the magnetars and the properties of the neutron matter require subtle and extremely complicated microscopic theory. The microscopic description of the magnetars is far beyond the scope of this paper where we are interested in the averaged macroscopic description which is astrophysically relevant.
From a macroscopic point of view we can describe the interaction (response) of the neutron matter with (to) the ultra strong magnetic field by adding an anisotropic term to the energy-momentum tensor of the isotropic perfect fluid. The anisotropy will manifest itself in different pressures along the meridional planes of magnetic field and in transverse direction. Indeed, according to the statistical physics [18], the pressure along the magnetic field is p = −Ω while in transverse direction p tr = −Ω − Bµ = p − Bµ where Ω is the grand canonical potential and µ is the megnetization. In general the dependences Ω(B) and µ(B) should be highly nonlinear and can be determined only by the microscopic theory. In ultra strong magnetic field, as we mentioned, most of the neutron spins should be oriented in the direction of the magnetic field which means that µ > 0. This shows that p tr < p in ultra strong magnetic field. As we will see later the exact solutions predict the same behaviour for the transverse pressure for realistic equations of state.
The only anisotropic term which we can add and which is orthogonal to the meridional planes and consistent with the geometrical symmetries we imposed, is of the form σe(η) µ e(η) ν where σ is a scalar and e(η) µ is the unit vector along the axial Killing field η. In other words we consider the following neutron star matter energy-momentum tensor The energy-momentum tensor (5) can be also written in the form which shows that p is the fluid pressure in the meridional planes where the magnetic field lays and p tr = p + σ is the pressure in direction orthogonal to the meridional planes and therefore orthogonal to the magnetic field. Armed with the energy-momentum tensor (5) we can write down the dimensionally reduced equations. Here we will perform the dimensional reduction with respect to the spacelike axial Killing vector η. For this purpose we need to introduce the 3-dimensional Lorentzian metric where e 2u = g(η, η). The covariant derivative associated with the metric H will be denoted by D i . Then for the reduced system of equations we obtain along with the contracted Bianchi identity Here R(H) ij is the Ricci tensor with respect to the 3-metric H ij , e 2U = −g(ξ, ξ) and J φ = η µ J µ . Our main task now is to solve the system of coupled partial differential equations (8)-(11). Our strategy for solving (8)-(11) is to "add nonlinearly" magnetic field to a known static and axisymmetric solution to Einstein-hydrodynamic equations (i.e. without magnetic field) described by the set {ρ 0 , p 0 , v 0 i , u 0 , H 0 ij }. In order to do so we partially follow [19] where a method for generating exact charged interior solutions was developed. We shall assume that u and Φ depend on the space coordinates through one function χ, i.e. u = u(χ) and Φ(χ). Substituting into eq.(9) we find If we impose the relations and we obtain that eq.(9) is automatically satisfied since {ρ 0 , p 0 , v 0 i , u 0 , H 0 ij } is a solution to the static, axisymmetric Einstein-hydrodymanic equations by definition. Then we can use eqs. (8) and (10) to find σ and J φ : It can be checked that eq. (11) is automatically satisfied. Let us summarize the results in the following  (13), (14), (15) and (16) form a solution to the Einstein-Maxwell-hydrodynamic equations (8)- (11).
This proposition allows to construct exact interior solutions with arbitrary equation of state for the background solution. The only exception is the case with stiff equation of state ρ 0 = p 0 which is very special and will not be considered here.
The 4-dimensional metric can be easily recovered form the data we have. Namely, if is the spacetime metric of the Einstein-hydrodynamic solution, then is the spacetime metric of the Einstein-Maxwell-hydrodynamic solution, i.e of the magnetized solution. From a physical point of view we have to impose some restrictions on the functional dependence u = u(χ). More precisely, in order for the new solution to possess a well defined axis of symmetry the function u(χ) should be of the form where f (χ) is a regular function with f (0) = 0. In this way the new solution will inherit the axis of symmetry from the background solution used for its generation.

Explicit exact solution
Now we consider a physically interesting and realistic explicit solution with σ and J φ vanishing on the star surface. The solution is obtained by requiring u(χ) and Φ(χ) to satisfy the equations of the affinely parameterized geodesics of the 2-dimensional metric dl 2 = du 2 + e −2u dΦ 2 , i.e. the equations This requirement considerably simplifies equations (15) and (16). The solution of the above equations is where b is an arbitrary parameter * . One can see that u(χ) is of the form (19) and therefore the solution has a well defined axis of symmetry. The physical meaning of the parameter b can be uncovered as follows. For the strength of the magnetic field we have Taking into account the space is locally Euclidian in small neighborhood of the axis and the fact that e 2χ | axis = 0, it is not difficult to find the strength B 0 of the magnetic field on the axis In fact B 0 is also the strength of the magnetic field on the north or south pole of the star surface. So the parameter b can be interpreted as being one half of the north pole magnetic field strength, i.e. b = 1 2 B 0 . In order to be more specific we will consider a spherically symmetric background solution. Also we will present the background solution in the widely used Schwarzschild coordinates r and θ with * The other parameter has been appropriately chosen in order to have a well defined axis. g 0 θθ = r 2 , g 0 φφ = e 2χ = r 2 sin 2 θ.
Then our magnetized solution is as follows where The nonzero components of the magnetic field are We see that when the background solution has a well defined boundary at r = R corresponding to the star surface where p 0 (R) = 0, the same is true for the magnetized solution, i.e. p(R) = 0 since p = Λ 2 p 0 . Moreover, if ρ 0 also vanishes on the star surface the same holds for σ and J φ according to (30) and (31). As we should expect the anisotropy pressure σ is yielded by the magnetic field and vanishes for zero magnetic field. Also, as we discussed in Section 2, the transverse pressure p tr = p + σ should be smaller than p in strong magnetic field. Indeed we see that for realistic equations of state for the background solutions, i.e. for ρ 0 ≥ p 0 we have σ ≤ 0.
In order to describe the way in which the magnetic field deforms the star we consider the space metric on the star surface, namely where Λ s = 1 + 1 4 B 2 0 R 2 sin 2 θ. The circumference about the equator (θ = π/2) is while for the polar circumference (φ = const) we have The surface elipticity ε surf is given by and ε surf < 0 for B 0 = 0. Therefore, for the solution under consideration the magnetic field elongates the star along the magnetic field -the star is prolate in shape † . For small Here we should note that the numerical and perturbative models with pure poloidal magnetic field predict positive surface elipticity. The reason for that discrepancy is the fact that the perturbative and numeric models consider the neutron star matter as pure isotropic perfect fluid without taking into account the anisotropy caused by the interaction with magnetic field.
The next physical quantity we shall consider is the total energy M concentrated in the star Taking into account (13) and (15) we find 41) † It is worth noting that the negative surface elipticity is a characteristic of the specific solution we consider. In principle the more general solutions which can be generated via our proposition may give positive elipticity. In those cases, however,the current J φ and anisotropy pressure σ do not vanish on the star surface is the total energy of the background solution. The explicit form of M depends of the background solution but we can give a good approximation by using the interior Schwarzschild solution as a representative example of a background solution. The interior Schwarzschild solution is characterized by a constant energy density ρ 0 = 3M 0 /4πR 3 and the metric and pressure are given by The mass-radius ratio satisfies the inequality 2M 0 /R < 8/9. Substituting the interior Schwarzschild solution into (40) and performing calculations up to terms in the order B 2 0 R 2 , we find

Conclusion
In this paper we presented a simple method for generating exact interior solutions to the static and axisymmetric Einstein-Maxwell-hydrodynamic equations with anisotropic perfect fluid. On this base we can build simple nonperturbative analytical relativistic models of the magnetars. To the best of our knowledge these are the first nonperturbative analytical relativistic models of the magnetars with arbitrary equation of state. As an illustration we gave an explicit realistic exact interior solution for the magnetars and on its base we calculated the suraface elipticity of the star and its energy. The present work could be extended in several directions. It would interesting and important more general configurations of the magnetic field, i.e. a mixture of poloidal amd toroidal magnetic fields to be investigated. The next interesting extension is to add rotation to the star. The mentioned possible extensions are very challenging due to the highly nonlinear character of the Einstein equations. We hope, however, that some progress could be made.