The Static Cylinder in General Relativity
For a certain range of a parameter m the Levi-Civita vacuum solution describes the exterior field of a massive static cylinder. When m = 0, the spacetime is flat, and when m is small and positive, the gravitational field can be reconciled with the corresponding Newtonian one. When m =1/4, the circular orbits of test particles become null, and for m > 1/4, all circular orbits are space like. Increasing m still further, one finds that when m = 1/2 spacetime again becomes flat. It was formerly thought that for m > 1/4 the Levi-Civita spacetime did not represent the field of a cylinder. Recently, however, it has been matched with interior cylindrical solutions throughout the range 0 ≤ m < 1/2. The strange behavior for 1/4 ≤ m ≤ 1/2 therefore requires explanation. In this paper I examine one of these interior solutions from this point of view. It seems that for 1/4 ≤ m < 1/2 the Levi-Civita solution does refer to a cylinder, but of increasing radius. As m approaches 1/2, the radius tends to infinity, suggesting that when m = 1/2, the bounding spatial surface is plane.
KeywordsGravitational Field Circular Orbit Test Particle Interior Solution Newtonian Theory
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