Exact Relativistic Magnetized halos around Rotating Disks

The exact relativistic treatment of a rotating disk surrounded by a magnetized material halo is presented. The features of the halo and disk are described by the distributional energy-momentum tensor of a general fluid in canonical form. All the relevant quantities and the metric and electromagnetic potentials are exactly determined by an arbitrary harmonic function only. For instance, the generalized Kuzmin-disk potential is used. The particular class of solutions obtained is asymptotically flat and satisfies all the energy conditions. Moreover, the motion of a charged particle on the halo is described. As far as we know, this is the first relativistic model describing analytically the magnetized halo of a rotating disk.

The exact relativistic treatment of a rotating disk surrounded by a magnetized material halo is presented. The features of the halo and disk are described by the distributional energy-momentum tensor of a general fluid in canonical form. All the relevant quantities and the metric and electromagnetic potentials are exactly determined by an arbitrary harmonic function only. For instance, the generalized Kuzmin-disk potential is used. The particular class of solutions obtained is asymptotically flat and satisfies all the energy conditions. The motion of a charged particle on the halo is described. As far as we know, this is the first relativistic model describing analytically the magnetized halo of a rotating disk.

I. INTRODUCTORY REMARKS
In the observational context, many ambiguities still exist about the main constituents, geometry and dynamics (thermodynamics) of the galactic disk-halos. However, there are several different observations which probe the galactic and surrounding galactic magnetic field. A current revision of the status of our knowledge about the magnetic fields in our Milky Way and in nearby star-forming galaxies is summarized in [1]. Additionally, a study of the disk and halo rotation are reported in [2], whereas the possibility of magnetic fields can be generated in the outskirts of disks is studied in [3]. Solutions for the Einstein and Einstein-Maxwell Field Equations which are consistently applicable to the context of astrophysics remains a topical problem. Nevertheless, the effects of magnetic fields on the physical processes in galaxies and their disk-halo interaction have been scarcely considered in the past. Similarly, the relevance of relativistic models of disks around black holes in a magnetic field is discussed in [4]. The presence of the electric field on the dark matter halo models has been considered in [5], whereas the presence of electromagnetic field in the halo-disk system has been studied in [6,7]. In the latter mentioned works the gravitational sources are statics. In [6] we provided an detailed overview of the research in the relativistic disks, accordingly we shall not repeat them here.
In the present paper we considered the conventional treatment of galaxies modeled as a stationary thin disk and, correspondingly, we associate the halo with the region surrounding the disk. We present the conformastationary version of the static thin disk-halo systems studied in [6]. Therefore, we take the definition in Ref. [8] as standard, following the original terminology by Synge [9]: conformastationary are those stationary spacetimes with a conformally flat space of orbits.
Accordingly, we show that the rotating disk-halos with isotropic pressure, stress tensor and heat flow generalize the static disk-halos obtained in such reference. Our results are compatibles with the presented in [5] on possible features of galactic halo. Moreover, the description of the motion of charged particles on disk is deduced and in agreement with the results of the similar analysis discussed in [10]. As far as we know, this is the first relativistic model describing analytically the relativistic magnetized halo of a rotating disk.
The paper is organized as follows. In Section II, the distributional Einstein-Maxwell equations for halos surrounding thin disks are obtained. In Section III we obtain expressions, in terms of an arbitrary harmonic function, for the most important quantities characterizing the dynamic of the disk and halo. In Section IV we first calculate quantities for an harmonic function described by the generalized Kuzmin-disk potential. Then, we analyze graphically the obtained results and calculate the constants of motion of the disk. Moreover, the description of the motion of a charged particle on the halo is shown in Section V. Finally, we complete the paper with a discussion of the results in Section VI.

II. THE EINSTEIN-MAXWELL EQUATIONS FOR HALOS SURROUNDING THIN DISKS
In this section we consider the conventional treatment of rotating galaxies modeled as a stationary thin disk and, correspondingly, we associate the magnetized halo with the region surrounding the disk. To do so, we formulate the distributional Einstein-Maxwell field equations assuming axial symmetry [11]. We also suppose that the derivatives of the metric and electromagnetic potential across the disk space-like hyper-surface are discontinuous. To formulate the corresponding distributional form of the Einstein-Maxwell field equations, we introduce the usual cylindrical coordinates x α = (t, r, z, ϕ) and assume that there exists an infinitesimally thin disk located at the hypersurface z = 0, so that the metric and the electromagnetic potential can be written as respectively. Accordingly, the Ricci tensor reads where θ(z) and δ(z) are, respectively, the Heaviside and Dirac distributions with support on z = 0. Here g ± αβ and R ± αβ are the metric tensors and the Ricci tensors of the z ≥ 0 and z ≤ 0 regions, respectively, and with γ αβ = 2g αβ,z and all the quantities are evaluated at z = 0 + . In agreement with (2) the energy-momentum tensor and the electric current density are expressed as where T ± αβ and J ± α are, respectively, the energy-momentum tensors and electric current density of the z ≥ 0 and z ≤ 0 regions (halo). Moreover, Q αβ and I α represent the part of the energy-momentum tensor and the electric current density of the z = 0 region (disk). The energy-momentum tensor T ± αβ in (4a) is taken to be the sum of two distributional components, the purely electromagnetic (trace-free) part and a "material" (trace) part, where E ± αβ is the electromagnetic energy-momentum tensor with F αβ = A β,α − A α,β and M ± αβ is an unknown "material" energy-momentum tensor to be obtained. Accordingly, the Einstein-Maxwell equations, in geometrized units such that c = 8πG = µ 0 = 0 = 1, are equivalent to the system of equations where H ≡ g αβ H αβ . The square brackets in expressions such as [F αβ ] denote the jump of F αβ across of the surface z = 0 and n β denotes a unitary vector in the direction normal to it. In the appendix (A) we give the corresponding field equations and the energy-momentum of the halo and disk for a sufficiently general metric. To obtain a solution of the distributional Einstein-Maxwell describing a system composed by a magnetized halo surrounding a rotating thin disk we shall restrict ourselves to the case where the electric potential A t = 0. Conveniently, we also assume the existence of a function φ depending only on r and z in such a way that the conformastationary metric (A1) can be written in the form with β an arbitrary constant. Accordingly, for the non-zero components of the energy-momentum tensor of the halo (EMTH) we have whereas the non-zero components of the electric current density on the halo have the form where all the quantities depending on r and z.
The discontinuity in the z-direction of Q αβ and I α defines, respectively, the surface energy-momentum tensor (SEMT) and the surface electric current density (SECD) of the disk S αβ , more precisely where ds n = √ g zz dz is the "physical measure" of length in the direction normal to the z = 0 surface. Accordingly, for the metric (8), the non-zero components of S αβ and J α are given by and respectively. Note that all the quantities are evaluated on the surface of the disk.

A. Relativistic exact solutions for magnetized disk-halos
In order to obtain the metric an electromagnetic potential compatibles with the last equation systems, we consider the astrophysical interesting case in which there is no electric current in the halo, i. e., we assume that Hence the system of equations (10) is equivalent to the very simple system where F ≡ e (1+β)φ . Ifê ϕ is a unit vector in azimuthal direction and λ is an arbitrary function independent of the azimuthal coordinate ϕ, then one has the identity The identity (16) may be regarded as the integrability condition for the existence of the function λ defined by Accordingly, the identity (16) implies the equation for the "auxiliary" potential λ(r, z). In order to have an explicit form of the metric function φ and magnetic potential A ϕ we suppose that φ and A ϕ depend explicitly on λ. Consequently the equation (18) implies where Let us assume the very useful simplification with k an arbitrary constant. Then, we have F = k 3 e kλ and − k∇λ · ∇λ + ∇ 2 λ = 0, where k 3 is an arbitrary constant. Furthermore, if we assume the existence of a function with k 4 and k 5 arbitrary constants, then Accordingly, λ can be represented in terms of solutions of the Laplace's equation: Hence, the metric potential φ can be written in terms of U as To obtain the metric function ω we first note that from (17) we have the relationship between A ϕ and λ: Then we have, A ϕ,r = −rF −1 λ ,r and A ϕ,z = rF −1 λ ,r , or, in terms of U where k 6 = 1/(kk 3 k 4 ). Furthermore, with (28) into (15a) we obtain which admits the solution ω = k ω U + k 8 , with k ω and k 8 arbitrary constants. As we know, the line element (8) must reduce to the Minkowski metric at spatial infinity. This means that the gravitational and magnetic fields vanish at large distances from the gravitational source, i.e., it is asymptotically flat. This requires that the constants k 3 k 4 = −k 5 = −1 and k 8 = 0.

III. EXACT RELATIVISTIC MODEL FOR MAGNETIZED DISK-HALOS
So far, by using the inverse method and the distributional formulation of the Einstein-Maxwell equations, we have obtained the separate energy-momentum tensors of the disk and halo. Now, the behavior of the energy-momentum tensors obtained must be investigated to find what conditions must be imposed on the solutions and the parameters that appear in the disk-halos models in such a way that it can describe reasonable physical sources. We shall now study the possible features of the disk by assuming that its possible express its energy-momentum tensor in the canonical form where Q α V α = Q α V α = 0, α = (t, r, ϕ) and all the quantities are evaluated in z = 0 + . Similarly, we assume that its possible to express the energy-momentum tensor of the halo in the canonical form where Q ± α V α = Q ±α V α = 0, α = (t, r, z, ϕ) and all the quantities depend on r and z. Consequently, we can say that the disk and halo are constituted by a some mass-energy distributions described by the energy-momentum tensors (30) and (31), respectively. The V α is the four velocity of certain observer. Correspondingly, µ, P , Q α and Π αβ are then the energy density, the isotropic pressure, the heat flux and the anisotropic tensor on the surface of the disk. Analogously, µ ± , P ± , Q ± α and Π ± αβ are then the energy density, the isotropic pressure, the heat flux and the anisotropic tensor on the halo, respectively. Thus, it is straightforward to see that for the halo we have where the projection tensor is defined by H µν ≡ g µν + V µ V ν and all the quantities depending on r and z. Whereas , for the disk we have where all the quantities are evaluated in z = 0 + . It is easy to note that by choosing the angular velocity to be zero in (B7) we have then a fluid comoving in our coordinates system. Hence, we may introduce a suitable reference frame in terms of the local observers tetrad (B3) and (B4) in the form with the corresponding dual tetrad The SECD of the disk J α can be also written in the canonical form then σ can be interpreted as the surface electric charge density and j as the "current of magnetization" of the disk. A direct calculation shows that the surface electric charge density σ = −V α J α = 0, whereas the "current of magnetization" of the disk is given by where, as above, [A ϕ,z ] denotes the jump of the z−derivative of the magnetic potential across of the disk and, all quantities are evaluated on the disk. By using the results obtained in (II A), we can write the surface energy density, the pressure, the heat flux, the non-zero components of the anisotropic tensor and the current of magnetization on the surface of the disk, respectively, as where, as we know, U (r, z) is an arbitrary suitable 2-dimensional harmonic function in cylindrical coordinates and [U ,r ] denotes of the jump of the r-derivative of the U across of the disk. All the quantities are evaluated on the surface of the disk, whereas, by using (31) and (32) we obtain the energy density and the pressure of the halo such as and respectively. Similarly, by inserting (31) into (32) we obtain for the heat flux of the halo Moreover, it is easy to see that the anisotropic tensor reads where and Notice that P ± ≡ P ± r + P ± z + P ± ϕ = 0 and, consequently, the trace Π ±α α = 0. We have obtained expressions for the energy, pressure and the other quantities characterizing the dynamic of the halo. All the dynamic quantities have been expressed in terms of an arbitrary U (r, z) harmonic function.
It is important to remark that k ω is a defining constant in (38c) and (41). Indeed, when k ω = 0 the heat flux functions Q α and Q ± α vanish, a feature of the static systems. Due to we used the inverse method, no "a priori" restriction are imposed on the physical properties of the material constituting the disk and halo. The non-zero components of the energy-momentum tensors of the disk and halo result of "the nature" of the chosen metric and the corresponding solutions. So, in our case, the non-zero component S rr and S tϕ are conditioned by the parameter β and the metric function ω in such a way that when β = 1 the component S rr vanishes, whereas S tϕ = 0 when ω vanishes. The decomposition of the energy-momentum tensor of the disk-halo system into (30) and (31) were chosen with the aim to describe the SEMT and EMTH by the more general fluid model. Hence, the heat flux appear here in a "natural" way as a function determined by the metric function ω and, consequently, by the "rotation". Unfortunately, as we can see from (33c) and (32c), this function is oriented along the closed circular orbits and thus its physical interpretation is unclear. It is an issue that remains unanswered in this manuscript, but should be addressed in the future.

IV. A PARTICULAR FAMILY OF CONFORMASTATIONARY MAGNETIZED DISK-HALOS SOLUTIONS
In precedent works [6,7] we have presented a model for a conformastatic relativistic thin disk surrounded by a material electromagnetized halo from the Kuzmin-disk potential in the form As it is well known, ∇ 2 U K must vanish everywhere except on the plane z = 0. At points with z < 0, U K is identical to the potential of a point mass m located at the point (r, z) = (0, −a), and when z > 0, U K coincides with the potential generated by a point mass at (0, a). Accordingly, it is clear that U K is generated by the surface density of a Newtonian mass ρ K (r, z = 0) = am 2π(r 2 + a 2 ) 3/2 .
In this work, we present a sort of generalisation of the Kuzmin-disk potential by considering a solution of the Laplace's equation in the form [8], where P n = P n (z/R) is the Legendre polynomials in cylindrical coordinates that was derived in the present form by a direct comparison of the Legendre polynomial expansion of the generating function with a Taylor of 1/R [12], being R 2 ≡ r 2 + z 2 and b n arbitrary constant coefficients. The corresponding magnetic potential, obtained from (28), is where, we have imposed A ϕ (0, z) = 0 in order to preserve the regularity of the axis of symmetry. Next, to introduce the corresponding discontinuity in the first-order derivatives of the metric potential and the magnetic potential required to define the disk we perform the transformation z → |z| + a. Thus, taking account of (38), the surface energy density of the disk, the heat flux and the non-zero components of the anisotropic tensor are and respectively. In the above expressions, we denote R 2 a ≡ r 2 + a 2 . As we know, the another quantities are P = (1 − β)µ/(3β) and Π ϕϕ = r 2 Π rr . The current of magnetisation is where we have used (38f) and we first assumed that the z-derivative of the magnetic potential present a finite discontinuity through the disk. In fact, as we have said above, the derivatives of U and A ϕ are continuous functions across of the surface of the disk. We artificially introduce the discontinuity through the transformation z → |z| + a . In order to illustrate the last solution we consider particular solutions with N = 0 and N = 1. Then we have U N for the two first members of the family of the solutions as follows, whereb 0 = b 0 /a andb 1 = b 1 /a 2 whereasr = r/a andz = z/a. For the corresponding magnetic potentials we have thenÃ whereÃ ϕ = A ϕ /a, whereas for the surface energy density we get For the two first members of the family we have the heat flux as and the corresponding anisotropic tensor Finally, for the two first members of the family we have the current of magnetization as In the last expressions we have used the dimensionless expressionsμ = aµ,Π ϕϕ = aΠ rr andj = aj. In Fig. 1, we show the dimensionless surface energy densitiesμ as a function ofr. In each case, we plotμ 0 (r) [ Fig.  1(a)] andμ 1 (r) [ Fig. 1(b)] for different values of the parameter β withb 0 = 1 andb 1 = 0.5. It can be seen that the surface energy density is everywhere positive fulfilling the energy conditions. It can be observed that for all the values of β the maximum of the surface energy density occurs at the center of the disk and that it vanishes sufficiently fast asr increases. It can also be observed that the surface energy density in the central region of the disk increases as the values of the parameter β increase.
In Fig. 2, we show the dimensionless current of magnetizationj as a function ofr. In each case, we plotj 0 (r) [ Fig. 2(a)] andj 1 (r) [ Fig. 2(b)] for different values of the parameter β withb 0 = 1 andb 1 = 0.5. It can be seen that the current of magnetization is everywhere positive. It can be observed that for all the values of β the current of magnetization is zero at the center of the disk, increases rapidly as one moves away from the disk center, reaches a maximum and later decreases rapidly. It can also be observed that the maximum of the current of magnetization increases as the values of the parameter β decrease.
We also computed the functionsμ andj for other values of the parameters and, in all the cases, we found the same behavior. We do not plot the heat flux, as it shows a similar behavior to that of the surface energy density. To illustrate the results corresponding to the principal quantities describing the halo in Fig. 3, we show the behavior of energy densities µ ± on the halo as a function of r and z. In each case, we plot µ ± 0 (r, z) [ Fig. 3(a)] and µ ± 1 (r, z) [ Fig. 3(b)] for the indicate values of the parameters. It can be seen that the energy density is everywhere positive and vanishes sufficiently fast as r increases.
In Fig. 4, we show the behavior of pressure P ± on the halo as a function of r and z. In each case, we plot P ± 0 (r, z) [ Fig. 4(a)] and P ± 1 (r, z) [ Fig. 4(b)] for the indicate values of the parameters. We can see that pressure is always positive and behaves as the energy density of the halo. Thus, we can see that the behavior of these quantities are in agreement with the results published in [5]. Moreover, we also computed these functions for other values of the parameters within the allowed range and in all cases we have found a similar behavior.

A. The constants of motion
To proceed further, we evaluate the constants of motion. As we know, the line element (8) must reduce to the Minkowski metric at spatial infinity. This means that the gravitational and magnetic fields vanish at large distances from the gravitational source, i.e., it is asymptotically flat. This requires that the constants k 3 k 4 = −k 5 = −1 and k 8 = 0. Therefore, from (26) we have Then, for the solution (55a) we may write whereR 2 ≡r 2 +z 2 . This follows that the metric potentials g tt and g tϕ for R → ∞ in the disk (z = 0 ) become This implies, as is well known (See [13]), that the total mass-energy of space-time associated with the disk is On the other hand, in (x, y, z) coordinates we find that As an application, we use the same procedure as in [14] and see that the angular momentum L M 0 is in the z-direction and is given by According to (56a) the magnetic field is where e α are unit basis vectors in cylindrical coordinates. Accordingly, by expressing the components of the magnetic field in Cartesian coordinates and taking the limit as R → ∞ of B 0 (x, y, z) and by using the formula (44.4) of Landau and Lifshitz [15] we may conclude that the magnetic momentum may be written as We thus see that constants k and k ω defines the gyromagnetic ratio L M 0 /L B0 = (kk ω )/2.

V. MOTION OF A CHARGED TEST PARTICLE IN THE HALO
It is interesting to describe the motion of a particle "falling" in the halo, this kind of motion is called electrogeodesic. In the same fashion as in [15], the equation of motion of a charged particle in a gravitational and electromagnetic fields (electrogeodesic equation) is obtained by where e and m are the charge and the mass of the particle, respectively. The velocity of the particle as measured by the local observers is given by Here, the 3-velocity v and the angular velocity Ω of the particle as measured by the local observers are given by and respectively. All the quantities depend on r and z. In Fig. 5(a) and Fig. 5(a) we show the behavior of the velocity v 2 0 and v 2 1 of a charged particle following an electrogeodesic motion on the halo for the values of indicated parameters, respectively. Additionally, in Fig. 6(b) and Fig. 6(b), we plot the z-slices of the surface plot of the velocity and v 2 0 and v 2 1 for the indicated values of the parameters, respectively. These curves are obtained via vertical slices of the surface v 2 = v 2 (r, z) (a vertical slice is a curve formed by the intersection of the surface v 2 = v 2 (r, z) with the vertical planes). For each curve, we can see that the velocity is always less than 1, its maximum occurs around r = 0, and it vanishes sufficiently fast as r increases. It can also be observed that the maximum of the velocity decreases as the values of z increases. We also computed these functions for other values of the parameters within the allowed range and in all cases we found a similar behavior. Naturally, the description of the motion of charged particles on disk here deduced is in agreement with the results of analysis of the electrogeodesic motion of the particle in the magnetized disks discussed in [10].

VI. CONCLUDING REMARKS
We used the formalism presented in [6] to model an exact relativistic rotating disk surrounded by a magnetized halo. The model was obtained by solving the Einstein-Maxwell distributional field equations. In doing so, we introduced an auxiliary harmonic function that determines the functional dependence of the metric components and the electromagnetic potential. Accordingly, we separated the total energy-momentum tensor of the system disk-halo. Additionally, we expressed the energy momentum tensor of the halo as the sum of two distributional contributions, one due to the electromagnetic part and the other due to a material part. As we can see, due to that the spacetime here considered is non-static (conformastationary), the distributional approach of the Einstein-Maxwell equations allows us to work with a strongly non-lineal partial equations system. We considered, for simplicity, the astrophysical consistent case in that there is no electric charge on the halo. We obtained that the charge density on the disk is zero.
In order to analyze the physical content of the energy-momentum tensor of the halo and disk, we projected each tensor, in the canonical form, in the comoving frame defined by the local observers tetrad. This analysis allowed us to give a complete dynamical description of the system in terms of two parameters (i.e β and k ω ) which determine the matter content of the sources. Indeed, the parameter β in the metric vanishes when it is equal to the isotropic pressure and the anisotropic tensor on the material constituting the disk. Similarly, when the parameter k ω is equal to zero the heat flux on the disk and halo vanishes, a feature of the static systems. So, in this paper we presented, for first time, the complete analysis of the most general energy-momentum tensor of a disk-halos system obtained from an exact conformastationary axially symmetric solutions of the Einstein-Maxwell equations.
The expressions obtained here are the generalization of the obtained for the conformastatic disk-halos without isotropic pressure, stress tensor or heat flow presented in [6]. Moreover, when we take simultaneously k ω = 0 and β = 1, we obtain its corresponding electrized disk-halos version. Furthermore, our results are compatibles with the description of the relativistic models of perfect fluid disks in a magnetic field presented in [10] and the halo presented in [5]. Furthermore, we have shown that the description of the motion of charged particles on the disk and is in agreement with the results of analysis of particles motion in the magnetized disks discussed in [10] We have considered specific solutions in which the gravitational and magnetic potential are completely determined by a "generalization" of the Kuzmin-disk potential. Accordingly, we have generated relativistic exact solutions for magnetized halos surrounding rotating disks from a Newtonian gravitational potential of a static axisymmetric distribution of matter. The solution obtained is asymptotically Minkowskian in general and turns out to be free of singularities.
In short, we concluded that we have presented an exact general relativistic well-behaviored rotating disk surrounded by a well-behaviored magnetized halo "material". In our description we do not impose restriction on the kind of "material" constituting the system disk-halo. Consequently, we can speculate that the halo could be made of magnetized dark matter. This work provides a solid footing to refine future studies of relativistic disk-halos systems.
The circular velocity of the system disk-halo can be modelled by a fluid space-time whose circular velocity V α can be written in terms of two Killing vectors t α and ϕ α , where is the angular velocity of the fluid as seen by an observer at rest at infinity. The velocity satisfy the normalization V α V α = −1. Accordingly, for the metric (B1 ) we have with consequently we write the velocity as where is the velocity as measured by the local observers.