Gravitomagnetic Jets

The sense of helical motion about the direction of the magnetic field is positive (negative) for a test particle of negative (positive) electric charge. If the initial velocity of the particle is normal to the direction of the magnetic field, then the particle simply moves along a circle in the plane perpendicular to the field. The purpose of this paper is to study the analogous situation for the motion of free test particles in a gravitomagnetic field. The nonlinearity of this field implies that only a rough similarity may be anticipated. We investigate geodesic motion in a rotating dynamic spacetime region that is a Ricci-flat solution of Einstein’s equations with cylindrical symmetry [1].

Though this time-dependent gravitational case is considerably more complicated than the magnetic case, we find qualitatively similar phenomena. In fact, the helical motions up and down parallel to the axis of symmetry are reminiscent of the double-jet structure of certain high-energy astrophysical sources.

Quasars and active galactic nuclei generally exhibit distinct relativistic outflows. These jets are conjectured to originate from massive rotating black holes surrounded by accretion disks; the outflows are focused beams of relativistic particles that proceed up and down along the rotation axis of the black hole (see, for instance, [2]). Similar phenomena have been observed in other high-energy astrophysical sources such as the Galactic X-ray binary systems. While our results suggest a mechanism for astrophysical jet formation, the more complicated physical process involves general relativistic MHD [3,4]. The present worktogether with previous efforts [5][6][7][8][9][10]-contributes to the purely gravitational aspects of this fundamental problem in astrophysics.

For a subclass of the Ricci-flat solutions under consideration, we show that the geodesic equations have families of special exact solutions that we call gravitomagnetic jets. More precisely, a gravitomagnetic jet is a set of special geodesics. These generally exhibit helical motions about the axis of cylindrical symmetry and their union is a non-compact connected invariant manifold that attracts all nearby geodesics. While we highlight features of these jets in the source-free cylindrical spacetime region of interest and provide strong (numerical) evidence that these families are attractors, the interesting question of the nature of the external matter currents that could generate such a gravitational field remains beyond the scope of our present investigation.

According to general relativity, a rotating mass generates a relativistic, and hence non-Newtonian, gravitomagnetic field that is due to mass current. The exterior gravitomagnetic field of the Earth has recently been directly measured via Gravity Probe B (GP-B) [11].

On the theoretical side, solutions of Einstein’s equations in the case of axial symmetry have received attention for a long time (see Ch. VIII of [12] and references cited therein). In particular, rotating solutions with cylindrical symmetry have been investigated by a number of authors (see [13][14][15] and references therein).

Previous interesting work on exact cylindrically symmetric gravitational fields in connection with the origin and structure of astrophysical jets has mainly involved the study of geodesics in the interior of time-independent rigidly rotating dust cylinders [16]. The behavior of free test particles in this case is directly influenced by the gravitational attraction of the rotating dust particles; to avoid this circumstance, we concentrate here on certain sourcefree gravitational fields that happen to depend exponentially upon time. These Ricci-flat fields are perhaps more representative of the strongly time-dependent near-zone exteriors of accreting and growing gravitationally collapsed configurations where astrophysical jets are expected to originate. The temporal variation of the gravitational fields considered in our work leads to a significant and surprising feature of the special exact solutions of the geodesic equations. The speeds of test particles in gravitomagnetic jets start from values that are always above a certain minimum speed, rapidly increase along their paths and asymptotically approach the speed of light. The minimum speed represents the speed of circular motion perpendicular to the axis of rotation. The range of this minimum speed turns out to be from zero up to about 0.63 c.

Is the main general relativistic problem of high-energy astrophysical

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