Construction and application of variations on the cylindrical gravitational waves of Weber, Wheeler, and Bonnor

Abstract

To clarify certain nonlinear properties of strong gravitational field, we investigate cylindrically symmetric gravitational waves that are localized as regular wave packets in the space of radial and time coordinates. The waves are constructed by applying a certain kind of harmonic mapping method to the seed solutions with linear polarization, which are generalizations of the solution representing a cylindrical gravitational pulse wave discussed by Weber-Wheeler and Bonnor. The solutions obtained here, though their form is rather simple, show occurrence of strong mutual conversion between a linear mode and a cross mode apparently. The single localized wave shows the conversion in the vicinity of the symmetric axis where the self-interaction is strengthened, and the collision between multiple waves also causes the conversion. These phenomena can be thought to be the emergence of genuine nonlinearity that the Einstein gravity holds. Finally we discuss a simple, but interesting application of the solutions to the case of the Einstein-Maxwell system.

Cylindrical systems in general relativity

“…These studies recently have gained additional momenta, due to the close relations between asymptotically symmetric theorems of soft gravitons and GW memory [178,296]. Such studies in general have been carried out in spacetimes with plane symmetry, but to shed some new light on the nonlinear effects, recently cylindrically GWs have been also studied [240].…”

Cylindrical gravitational waves: C-energy, super-energy and associated dynamical effects

“…so that 5) and is called "C-energy," as it represents the total gravitational energy per unit length between the axis of symmetry and the cylindrical radius ρ at time t. It can be suggestively rewritten as…”

“…which can be solved perturbatively 5 . The C-energy density (5.4) turns out to be .11) and admits the following representation…”

“…Other solutions have been investigated since then for both vacuum and electro-vacuum spacetimes as well as in the presence of perfect fluid sources, either in the form of monochromatic (elementary) waves or as wave packets [2]. Numerical studies of general solutions also exist [3][4][5]. The asymptotic behavior of cylindrically symmetric spacetimes including Einstein-Rosen waves has been deeply discussed in [6][7][8] (see also references therein for a complete list of related works).…”

Observable effect of quantized cylindrical gravitational waves

“…The Einstein-Rosen wave is an exact solution of general relativity with two commuting Killing vectors and describes a cylindrical GW, so using it to study the aspects of nonlinearity originating from Einstein gravity is convenient and significant [12]. Historically, it played an important role in early attempts at defining the energy carried by gravitational waves [13,14], since the energy of GWs is difficult to be described locally due to the equivalence principle [15][16][17][18].…”