Behavior of Einstein-Rosen Waves at Null Infinity

TL;DR

The paper investigates the asymptotics of Einstein-Rosen waves at null infinity in four dimensions, focusing on directions relative to the translational symmetry axis. By performing a symmetry reduction to a three-dimensional description, the problem reduces to a decoupled wave equation for a scalar field $\psi$ on 3D Minkowski space, with the remaining metric function $\gamma$ determined by $\psi$, enabling analysis of both null and time-like infinity. In the time-symmetric case, the 4D null infinity exists in generic directions with Bondi-type fall-off and curvature peels, while in directions perpendicular to the axis the fall-off is too slow; for generic data, null infinity can acquire logarithmic behavior, but even then 4D asymptotics are better than naive expectations. The work shows that the 3D time-like infinity governs the 4D asymptotics and clarifies radiation concentration along the axis generators, providing a framework for Bondi-type expansions in axisymmetric spacetimes, including cases with logarithmic infinity. These results deepen understanding of how dimensional reduction informs the radiative structure and peeling properties of cylindrically symmetric gravitational waves.

Abstract

The asymptotic behavior of Einstein-Rosen waves at null infinity in 4 dimensions is investigated in {\it all} directions by exploiting the relation between the 4-dimensional space-time and the 3-dimensional symmetry reduction thereof. Somewhat surprisingly, the behavior in a generic direction is {\it better} than that in directions orthogonal to the symmetry axis. The geometric origin of this difference can be understood most clearly from the 3-dimensional perspective.