An interpretation of Robinson-Trautman type N solutions

TL;DR

This work clarifies the physical interpretation of Robinson–Trautman type N vacuum solutions by examining their weak-field limits to Minkowski, de Sitter, and anti-de Sitter backgrounds and by constructing explicit sandwich-wave families that interact with cosmic-string-like defects. Introducing the RTN(Λ,ε) class with an arbitrary holomorphic F(ζ,u), it analyzes wavefront geometry, singularities, and impulsive limits, showing how snapping or decaying strings can generate expanding gravitational waves in curved backgrounds. The study reveals how string-induced deficit angles propagate with the waves and characterizes the global structure and singularities, including how Λ shifts the geometry of the background and wave regions. The results provide a tractable, explicit framework for modeling gravitational radiation coupled to topological defects and suggest natural generalizations to more complex wave profiles and multi-wave scenarios.

Abstract

The Robinson-Trautman type N solutions, which describe expanding gravitational waves, are investigated for all possible values of the cosmological constant Lambda and the curvature parameter epsilon. The wave surfaces are always (hemi-)spherical, with successive surfaces displaced in a way which depends on epsilon. Explicit sandwich waves of this class are studied in Minkowski, de Sitter or anti-de Sitter backgrounds. A particular family of such solutions which can be used to represent snapping or decaying cosmic strings is considered in detail, and its singularity and global structure is presented.

Figures (7)

• Figure 1: Families of null cones given by $u=u_0$ foliate Minkowski space-time in different ways in the three cases for which $\epsilon=1,0,-1$. (One spatial dimension ($y$) is suppressed).
• Figure 2: Three typical members of the family of null cones $u=$ const. which foliate part of Minkowski space in the case when $\epsilon=-1$. (One spatial dimension ($y$) has been suppressed.)
• Figure 3: Families of null cones in de Sitter space given by $u=u_0$ are sections of a hyperboloid in a 5-dimensional Minkowski space. With two dimensions ($Z_2$ and $Z_3$) suppressed, they appear as straight (null) lines. These foliate parts of the hyperboloid in different ways for the three cases in which $\epsilon=1,0,-1$. The vertices are located respectively along timelike, null and spacelike lines.
• Figure 4: Families of null cones $u=u_0$ in anti-de Sitter space are sections of a hyperboloid in a 5-dimensional Minkowski space. With the spatial dimensions $Z_2$ and $Z_3$ suppressed, they are straight (null) lines. These foliate parts of the hyperboloid in different ways for the three cases in which $\epsilon=1,0,-1$, with vertices located respectively along timelike, null and spacelike lines.
• Figure 5: The shaded regions represent Robinson--Trautman sandwich waves at some fixed time for different values of $\epsilon$. The expanding spherical (hemispherical for $\epsilon=-1$) wave surfaces are given by $u =$ const. The region behind the wave is Minkowski or (anti-)de Sitter, while the region ahead of the wave is Minkowski or (anti-)de Sitter with a deficit angle representing a cosmic string. The dashed lines denote the boundaries of the coordinate system adopted.