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.

In this paper we systematically study a recently proposed model of spherically symmetric polymer black/white holes by Bodendorfer, Mele, and Münch (BMM), which generically possesses five free parameters. However, we find that, out of these five parameters, only three independent combinations of them are physical and uniquely determine the local and global properties of the spacetimes. After exploring the whole 3-dimensional (3D) parameter space, we show that the model has very rich physics, and depending on the choice of these parameters, various possibilities exist, including: (i) spacetimes that have the standard black/white hole structures, that is, spacetimes that are free of spacetime curvature singularities and possess two asymptotically flat regions, which are connected by a transition surface (throat) with a finite and nonzero geometric radius. The black/white hole masses measured by observers in the two asymptotically flat regions are all positive, and the surface gravity of the black (white) hole is positive (negative). In this case, there also exist possibilities in which the two horizons coincide, and the corresponding surface gravity vanishes identically. (ii) Spacetimes that have wormholelike structures, in which the two masses measured in the two asymptotically flat regions are all positive, but no horizons exist, neither a trapped (black hole) horizon nor an anti-trapped (white hole) horizon. (iii) Spacetimes that still possess curvature singularities, which can be either hidden inside trapped regions or naked. However, such spacetimes correspond to only some limit cases. In particular, the necessary (but not sufficient) condition is that at least one of the two “polymerization” parameters vanishes. These results are not in conflict to the Hawking-Penrose singularity theorems, as the effective energy-momentum tensor, purely geometric and resulted from the “polymerization” quantization, satisfies none of the three (weak, strong or dominant) energy conditions in any of the two asymptotically flat regions for any choice of the three independent free parameters, although they can hold at the throat and/or at the two horizons for some particular choices of them. In addition, it is true that quantum gravitational effects are mainly concentrated in the region near the throat, however, in this model even for solar mass black/white holes, such effects can be still very large at the black/white hole horizons, again depending on the choice of the parameters. Moreover, in principle the ratio of the two masses (for both of the black/white hole and wormhole spacetimes) can be arbitrarily large.

The properties of interior spacetimes sourced by stationary cylindrical anisotropic fluids are here analytically studied for both nonrigid and rigid rotation. As regards nonrigid rotation, this is, to our knowledge, the first work dedicated to such a study. We give here a complete equation set describing these spacetime properties. In particular, we focus our attention on both nonrigid and rigid rotation gravito-electromagnetic features and are thus led to display strong hints in favor of conjecturing purely electric Weyl tensor existence in this framework. We have also been able to characterize new purely magnetic physically consistent spacetimes and have found new rigidly rotating exact solution classes to the five Einstein’s field equations pertaining to the issue and the two purely magnetic constraints we have derived for this purpose. This should be considered as a prominent result, since extremely few purely magnetic exact solutions are available in the literature.

The Lewis solutions describe the exterior gravitational field produced by infinitely long rotating cylinders, and are useful models for global gravitational effects. When the metric parameters are real (Weyl class), the exterior metrics of rotating and static cylinders are locally indistinguishable, but known to globally differ. The significance of this difference, both in terms of physical effects (gravitomagnetism) and of the mathematical invariants that detect the rotation, remain open problems in the literature. In this work we show that, by a rigid coordinate rotation, the Weyl class metric can be put into a ‘canonical’ form where the Killing vector field ∂ _t is time-like everywhere, and which depends explicitly only on three parameters with a clear physical significance: the Komar mass and angular momentum per unit length, plus the angle deficit. This new form of the metric reveals that the two settings differ only at the level of the gravitomagnetic vector potential which, for a rotating cylinder, cannot be eliminated by any global coordinate transformation. It manifests itself in the Sagnac and gravitomagnetic clock effects. The situation is seen to mirror the electromagnetic field of a rotating charged cylinder, which likewise differs from the static case only in the vector potential, responsible for the Aharonov–Bohm effect, formally analogous to the Sagnac effect. The geometrical distinction between the two solutions is also discussed, and the notions of local and global staticity revisited. The matching in canonical form to the van Stockum interior cylinder is also addressed.