The Levi–Civita space–time

Abstract

Author's internet addresses respectively: mfas@on.br, lherrera@conicit.ve, fmpaiva@on.br and nos@on.br. 1 AbstractWe consider two exact solutions of Einstein's field equations corresponding to a cylinder of dust with net zero angular momentum. In one of the cases, the dust distribution is homogeneous, whereas in the other, the angular velocity of dust particles is constant [1]. For both solutions we studied the junction conditions to the exterior static vacuum Levi-Civita spacetime. From this study we find an upper limit for the energy density per unit length σ of the source equal . We also find that the cluster of homogeneous dust has a superior limit for its radius, depending on the constant volumetric energy density ρ 0 .

The Static Cylinder, Gyroscopes and the C-Metric

“…Furthermore, this fact might support too the previous acceleration representation of the field intensity. It agrees with the result that the tangential speed W of a test particle [11] in a circular geodesics increases with σ, attaining W → ∞ for σ → 1/2. The source studied in [14] remains cylindrical for σ = 1/2, producing a cosmic string with finite radius.…”

“…This last argument in turn, was objected in the past on the basis that Kretschmann scalar decreases as σ increases from 1/4 to 1/2, suggesting thereby that the gravitational field becomes weaker [9], [11]. However, as it has been recently emphasized [12], [18], Kretschmann scalar may not be a good measure of the strength of the gravitational field.…”

“…The absence of circular test particle orbits for σ > 1/4, and the fact that most of the known material sources for LC, [4], [5], [6], [11] require σ ≤ 1/4, led to think that LC describes the field of a cylinder only if σ ranges within the (0,1/4) interval.…”

Perfect-fluid cylinders and walls - sources for the Levi-Civita spacetime

“…The explanation, suggested by the Newtonian case, that this is due to the gravitational attraction becoming strong enough to prevent any particle (including light) orbiting circularly [9] has been contrasted with the fact that a study of the curvature invariants suggests that the field gets weaker as σ increases from (where the metric is flat but accelerated) [2,7]. However, although it sounds strange, what matters here is not the field (i.e.…”

“…This well-defined Newtonian limit is the reason why the Levi-Civita space-time is usually said to represent the field outside an infinitely long static cylinder-thus the φ-coordinate is taken to be periodic-and cylinder sources have indeed been found for this space-time [4,5,6,7]. The sources found are of two types: static dust cylinders (composed of equal amounts of dust rotating in opposite senses around the axis to produce zero net angular momentum) which were shown to match to the Levi-Civita space-time for values of σ in the range 0 ≤ σ < 1 4 [5,7], and perfect-fluid cylinders, one example of which could be matched to the Levi-Civita exterior for 0 ≤ σ < .…”