Translation in cylindrically symmetric vacuum

Abstract

Starting from the stationary cylindrically symmetric solution, but with the coordinates z and φ interchanged, and supposing that it could describe the vacuum spacetime of a translating cylinder, we investigate its physical and geometrical properties. This hypothesis is not entirely new since it has already been considered in a previous paper describing a translating source. We show that this metric is geometrically related to the vacuum field produced by a stationary rotating cylindrical source, known as Lewis solution. However, we find new physical properties, different from those of the Lewis vacuum solution. Moreover, we show that no translating cylindrical dust solution can exist in General Relativity.

Stationary cylindrical anisotropic fluid and new purely magnetic GR solutions

“…Notice that in the nonrigid rotation case where Ω = 0, (103) implies µ ′ = 0 and thus a priori nonstatic spacetimes, at variance with what obtains for a rigidly rotating fluid, see Section IV F 1. We have therefore implicitly defined a class of purely electric interior spacetimes sourced by a stationary nonrigidly rotating anisotropic fluid whose metric functions are solutions of the seven differential equations ( 10)-( 14), ( 98) and ( 101) and of the timelike condition (6). Notice that four among these equations can be replaced by ( 15)-( 17) and (103) which are partially integrated equations, i. e., interesting simplifications for future analytic or numerical resolutions.…”

“…The vacuum solution outside a cylindrical source in translation along its symmetry axis is mathematically akin to the Lewis solution with exchanged z and φ coordinates. It has been shown that they are however physically different [6]. Nonvacuum cylindrically symmetric spacetime investigations date back to 1937 when van Stockum gave the metric solution for a rigidly rotating infinitely long dust cylinder [7].…”

Causal Geodesics in Cylindrically Symmetric Vacuum Spacetimes Using Hamilton-Jacobi Formalism

“…Back in 2014, Brito et al found the solution to the geodesic equations in Linet-Tian spacetime for few special cases [26]. Célérier et al explored radial, axial, and circular geodetic motions in a cylindrically symmetric translating spacetime, in 2019 [27]. In 2016, Hoseini et al came up with an analytic solution to the geodesic equation in a static cylindrically symmetric conformal spacetime [28].…”

Exact interior solutions for rigidly rotating stationary cylindrical fluids with azimuthal pressure

“…A vacuum spacetime gravitationally sourced by a cylinder of matter in translating motion along its symmetry axis is mathematically similar to a Lewis solution where the coordinates z and φ have been exchanged. Their physical properties are however different [5].…”