Orbits in a Magnetic Universe

Abstract

A cylindrically symmetric parallel bundle of magnetic lines of force, in equilibrium under their mutual gravitational attraction (``magnetic universe''), has recently received attention. While a Newtonian analysis suggests that the equilibrium is unstable, the complete general relativity analysis shows that the equilibrium is stable. This discrepancy may have to do with the unusually slow falloff of the gravitational field at large distances in this geometry. In order to understand the gravitational field of the static magnetic universe somewhat better, we have studied its timelike and lightlike geodesics (i.e., the orbits in it of electromagnetically neutral test particles with unit or zero rest mass). Since the density of magnetic flux-and energy and stress and, therefore, ``gravitating mass''-is approximately uniform in the vicinity of the axis, the motion of test particles there is like that in a Newtonian simple harmonic oscillator field. ``Vicinity'' here means within a small fraction ρ of the range radius ā = (6.96/B0) × 1024 cm (B0 is the magnetic field on the axis measured in gauss). As is to be expected from the universality of the angular frequency ω0 in the harmonic oscillator field and the relation: orbital velocity ≅ ω0ρ, no motion can get too far from the axis. Otherwise the physical orbital velocity would exceed the speed of light. It is in this way that the strength of the attractive field, though it does not remain strictly of the harmonic oscillator type as one proceeds outward, implies that there is a critical straddling radius ρ = 1/√3. Circular or circular helical light tracks occur only at the critical radius, and with B0 = 105 G, the time required for light to circumnavigate the critical circle is about 200 years. The cylinder marked out by this radius plays a unique limiting role: All particles, whether of zero or nonzero mass, and no matter what their initial positions and velocities (except in the one singular subcase of light tracks parallel to the cylindrical axis), must have their orbits lying wholly or partially within the cylindrical region ρ < 1/√3; hence the use of the adjective ``straddling.'' Constants of motion which correspond closely to ζ-component linear momentum, angular momentum, and energy in Newtonian mechanics are defined. Bounds are placed on these dynamical constants and on the apsidal radii by the requirement that the range of motion be real. Finally, the magnetic universe is complete in the sense that ``no news can enter or leave''-all orbits are of infinite duration.