In this note we observe that the exact Maxwell-Einstein equations in the background metric of a spinning string can be solved analytically. This allows us to construct an analytical model for the magnetosphere which is approximately force free near to the spinning string. As in the case of a Kerr black hole in the presence of an external magnetic field the spinning string will acquire an electric charge which depends on the vorticity carried by the spinning string. The self-generated magnetic field and currents strongly resemble the current and magnetic field structure of the jets associated with active galaxies as they emerge from the galactic center.
In this note we observe that the exact Maxwell--Einstein equations in the background metric of a spinning string can be solved analytically. This allows us to construct an analytical model for the magnetosphere which is approximately force free near to the spinning string. As in the case of a Kerr black hole in the presence of an external magnetic field the spinning string will acquire an electric charge which depends on the vorticity carried by the spinning string. The self--generated magnetic field and currents strongly resemble the current and magnetic field structure of the jets associated with active galaxies as they emerge from the galactic center. The physical origin of the well--collimated jets seen emerging from the massive compact objects found at the centers of many galaxies has long been a mystery. Although the general idea [1] that the energy to power these jets comes from the electromagnetic torque exerted on the compact object by an accretion disk has long been accepted, it has never been demonstrated in detail how this might work. One problem is that in the case of a Kerr black hole model for the compact object the relevant equations are sufficiently nonlinear and intractable that one must rely on numerical methods. This complexity has so far prevented computational astrophysicists from either proving or disproving that externally generated magnetic fields interacting with a Kerr black hole can produce the observed jet structures. In this paper we suggest an alternative approach. We observe that the exact Maxwell--Einstein equations for the case of a spinning string surrounded by a static cylindrically symmetric configuration of charge and current densities can be solved analytically. This result allows us to find an analytical model for the magnetosphere that is very nearly force--free near to the spinning string, and confining farther away. Remarkably the structure of the self--generated magnetic field and currents in this model strongly resemble that inferred from mm--wave VLBLI observations of the astrophysical jets associated with the massive compact objects at the centers of active galaxies [2]. The success of this simple model suggests that the powerful jets seen emerging from the nuclei of active galaxies actually originate inside the compact object itself.
In 1987 P. Mazur discovered a remarkable solution to the Einstein equations in which space–time is flat everywhere except for a torsion–like singularity along a straight line where general relativity breaks down [3]. If we identify this straight line with the z–axis, the metric for the external space–time in cylindrical coordinates is
where A is a constant defining the vorticity of the spinning string (A plays a role for the spinning string space–time similar to the angular momentum per unit mass of a black hole). Electromagnetic fields in this space–time background satisfy the covariant Maxwell equations:
where J α = ρdx α /dt, g 00 = (1–A 2 /r 2 )/c 2 , g 0φ = A/cr 2 , g φφ = –1/r 2 , g rr = g zz = –1, and -g= c 2 r 2 . If the charge and current densities depend only on r, the equations for the physical currents j α = √|gαα|J α become j r = 0
( An exact analytic solution for these equations can be found by noting that combining the equations for jφ and ρ yields first order linear differential equations for Er and Bz that can be solved by elementary methods. In particular, Eq’s. (3) imply that for a stationary configuration of charge and current densities surrounding the spinning string the exterior electric and magnetic fields have the form
) B ! = µ 0 2" r I z (r) where q(r), Iφ(r) and Iz(r) are the net charge per unit length, azimuthal current per unit length, and total current flowing in the z–direction, all contained within a cylindrical shell with outer radius r and inner radius r0 . Iφ is the total azimuthal current per unit length, Iz is the total current flowing in the z–direction, and B0 is the external magnetic field, which for the purposes of this paper we assume is parallel to the z–axis. One immediate implication of eq’s 4 is that if A and Iφ are nonzero, then the charge per unit length of the spinning string must be nonzero. This is reminiscent of the result [4] that a rotating black hole in the presence of an external magnetic field must acquire a charge. Eq.s (4) describe the electromagnetic fields surrounding a spinning string with fixed external charges and currents. However, in a vacuum where the charges are free to move a fixed configuration of charges and currents will in general not be stable because of magneto–hydrodynamic forces. Substituting the solution (4) into the force–free condition j x B = ρE we find that in order for the configuration of fields and charges to be stable the current and charge densities must satisfy the integral equation:
We expect that in general this equation must be solved numerically. However, since some of the terms in Eq. ( 5) are also singular at r=√2A, it
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