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The study of the dynamics of magnetic fields in galaxies is one of important problems in formation and evolution of galaxies. In this paper, we present the exact relativistic treatment of a rotating disk surrounded by a magnetized material halo. 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.

In the observational context, many ambiguities still exist about the main constituents, geometry, and dynamics (thermodynamics) of the galactic disk-haloes. However, there are several different observations which probe the galactic and surrounding galactic magnetic field. For instance, it can be measured by nonthermal radio emission from energy equipartition that results from the interaction of magnetic energy with relativistic particles which can play a role in the formation of arms in spiral galaxies (see Krause’s “Magnetic Fields and Halos in Spiral Galaxies” [

It is important to stress that magnetic fields are found mainly in interstellar medium and can be found in every type of galaxies but remarkably noticed in spiral galaxies (see [

The superposition of a static or stationary thin disk with a black hole has been considered in [

The thin disks with magnetic fields studied in [

Moreover, interestingly, magnetic fields seem to play an important role in the formation of jets (resulting from collimated bipolar outflows of relativistic particles) and the accretion disk near supermassive black holes [

The presence of the electric field on the dark matter halo models has been considered in [

In this paper, we do not focus on the whys and wherefores of the detailed discussion on the probes and relevance of magnetic fields on disks but we apply a standard galaxy modeling 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 [

Accordingly, we show that the rotating disk-haloes with isotropic pressure, stress tensor, and heat flow generalize the static disk-haloes obtained in [

The paper is organized as follows. In Section

In this section we consider the conventional treatment of rotating galaxies modelled as a stationary thin disk and, correspondingly, we associate the magnetized halo with the region surrounding the disk. This is motivated from the fact that the magnetic field is remarkably noticed on spiral galaxies and can play a fundamental role in formation of arms. To do so, we formulate the distributional Einstein-Maxwell field equations assuming axial symmetry [

To solve the Einstein-Maxwell equations (

The discontinuity in the

In order to reduce the complexity of the last field equation systems we assume that the halo’s electric current density vanishes (i.e.,

So far, by using the inverse method and the distributional formulation of the Einstein-Maxwell equations, we have obtained the separate energy-momentum tensor of the disk and halo. In addition, we have discussed a method to determine its components in terms of an arbitrary harmonic function. 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-haloes models in such a way that it can describe reasonably physical sources. We will now study the possible features of the disk by assuming that it is possible to express its energy-momentum tensor in the canonical form:

By using the results obtained in the precedent section, we can write the surface energy density of the disk and the energy density of the halo can written as

It is important to remark that

As an example of application of the formalism described in the precedent sections, we now consider the magnetized haloes surrounding the rotating disks generated by a generalization of the Kuzmin-disk potential in the form [

It is worth noticing that the mass surface density as well as the isotropic pressure of the disk decays very rapidly (as

To illustrate the results corresponding to the principal quantities describing the halo in Figure

Surface plots of the energy densities (a)

In Figure

Surface plots of the pressures (a)

To proceed further, we evaluate the constants of motion. Therefore, from (

The motion of a test particle of charge

The equations of motion of the test particle can be derived from (

Surface plot of the velocity (a)

Surface plot of the velocity (a)

We used the formalism presented in [

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.,

The expressions obtained here are the generalization of the expressions obtained for the conformastatic disk-haloes without isotropic pressure, stress tensor, or heat flow presented in [

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 haloes 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 a well-behaved exact general relativistic rotating disk surrounded by a well-behaved magnetized “material” halo. 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-haloes systems and applications, for example, relativistic generalization of alpha-effect which will be discussed somewhere.

We write metric (

The circular velocity of the system disk-halo can be modelled by a fluid spacetime whose circular velocity

The authors declare that there is no conflict of interests regarding the publication of this paper.