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

Abstract

The diagonal metric tensor whose components are functions of one spatial coordinate is considered. Einstein's field equations for a perfect-fluid source are reduced to quadratures once a generating function, equal to the product of two of the metric components, is chosen. The solutions are either static fluid cylinders or walls depending on whether or not one of the spatial coordinates is periodic. Cylinder and wall sources are generated and matched to the vacuum (LeviCivita) space-time. A match to a cylinder source is achieved for − 1 2 < σ < 1 2 , where σ is the mass per unit length in the Newtonian limit σ → 0, and a match to a wall source is possible for |σ| > 1 2 , this case being without a Newtonian limit; the positive (negative) values of σ correspond to a positive (negative) fluid density. The range of σ for which a source has previously been matched to the Levi-Civita metric is 0 ≤ σ < 1 2 for a cylinder source.

Static Cylindrical Symmetry and Conformal Flatness

“…Thus for example one could assume µ = constant, P z = P φ = P r , then we can integrate (11) to obtain…”

Static fluid cylinders and their fields: global solutions

“…which is not bounded from above-solutions with unbounded m T have also been found analytically [18]. We can use mass per unit coordinate length, M 1 (6.12), with no upper bound (see figure 4).…”

The Static Cylinder, Gyroscopes and the C-Metric

“…On the other hand, it is clear that in the case of a plane source we should not expect φ to behave like an angle coordinate (see also [12] on this point). Therefore , on the basis of all comments above, we are inclined to think (as in [18]) that the σ = 1/2 case corresponds to an infinite plane.…”

“…However, as it has been recently emphasized [12], [18], Kretschmann scalar may not be a good measure of the strength of the gravitational field. Instead, those authors suggest that the acceleration of the test particle represents more suitably the intensity of the field.…”