Cylindrical systems in general relativity

Abstract

With the arrival of the era of gravitational wave astronomy, the strong gravitational field regime will be explored soon in various aspects. In this article, we provide a general review over cylindrical systems in Einstein's theory of general relativity. In particular, we first review the general properties, both local and global, of several important solutions of Einstein's field equations, including the Levi-Civita and Lewis solutions and their extensions to include the cosmological constant and matter fields, and pay particular attention to properties that represent the generic features of the theory, such as the formation of the observed extragalactic jets and gravitational Faraday rotation. We also review studies of cylindrical wormholes, gravitational collapse and Hoop conjecture, and polarizations of gravitational waves. In addition, by rigorously defining cylindrically symmetric spacetimes, we clarify various (incorrect) claims existing in the literature, regarding to the generality of such spacetimes.

Landau levels in a gravitational field: the Levi-Civita and Kerr spacetimes case

“…Here, the constants a, b and K are all arbitrary -with a constraint between a and bwhile c is the speed of light. Note that the form (II.2) of the Levi-Civita metric we use here is not the one that one often encounters in the literature [28][29][30][31] (see also, the very nice recent review [32].) In fact, the first difference is that we have introduced here the fixed constant radius ρ * to allow us to keep inside the metric the radius ρ, describing the position of the particle away from the center of the cylinder, with the dimensions of a length.…”

“…This specific choice is merely made here for the sake of simplicity. In fact, had we chosen to use instead the more familiar form of the metric, in which both coordinates acquire the same metric component [31,32], the angular component of the metric would also acquire 2 a power function of ρ instead of having the above familiar factor ρ 2 . This would indeed only render our equations and analysis uselessly complicated.…”

“…Given that the Lewis spacetime reduces in the limit of zero radius of the cylinder to that of a rotating cosmic string (see, e.g., Ref. [32] and the references therein for more details about such a metric), which, in turn, has extensively been studied in Ref. [42], we are going to turn instead into the rotating spacetime represented by the Kerr metric.…”

Rotating Cylinders with Anisotropic Fluids in General Relativity

“…where the constant factors A, B, C, D depend on the values of w i as well as ρ 0 and ω 0 . If B = C = 0, we have two separate Liouville-type equations which are solved directly and completely, and it remains to substitute the solution to the first-order equation (23) to verify the solution and to obtain a relation between the integration constants. Apart from the case B = C = 0, in which the matrix of coefficients in Eqs.…”

“…Among motivations of such studies one can mention the relative simplicity of the gravitational field equations as compared to more realistic axial symmetry (to say nothing on the general nonsymmetric case) and the possible existence of linearly extended structures like cosmic strings in the Universe. In addition, cylindrical symmetry is the simplest one that admits rotation and is most suitable for studying rotational phenomena in GR, so not too surprising is the wealth of existing stationary (that is, assuming rotation) exact solutions to the Einstein equations with various sources of gravity: the cosmological constant Λ [3][4][5]; scalar fields with or without self-interaction potentials [6,7], including Λ as a constant potential [8]; dust in rigid or differential rotation [9][10][11], electrically charged dust [12], dust with a scalar field [13], perfect fluids with various equations of state, mostly p = wρ, w = const (in usual notations) [14][15][16][17][18]; some examples of anisotropic fluids [19][20][21] etc, see also references therein as well as reviews [22,23].…”