Absence of gyratons in the Robinson-Trautman class

TL;DR

This paper proves that Robinson-Trautman spacetimes in any dimension cannot contain gyratonic sources, showing that off-diagonal metric components do not encode angular momentum but relate to gravitational wave amplitudes.

Contribution

It provides explicit expressions for curvature tensors in general non-twisting, shear-free geometries and demonstrates the absence of gyratons in Robinson-Trautman solutions across all dimensions.

Findings

Robinson-Trautman class does not admit gyratonic matter sources.

Off-diagonal metric components relate to gravitational wave amplitudes in D=4.

Explicit curvature tensors for general non-twisting, shear-free geometries.

Abstract

We present the Riemann and Ricci tensors for a fully general non-twisting and shear-free geometry in arbitrary dimension D. This includes both the non-expanding Kundt and expanding Robinson-Trautman family of spacetimes. As an interesting application of these explicit expressions we then integrate the Einstein equations and prove a surprising fact that in any D the Robinson-Trautman class does not admit solutions representing gyratonic sources, i.e., matter field in the form of a null fluid (or particles propagating with the speed of light) with an additional internal spin. Contrary to the closely related Kundt class and pp-waves, the corresponding off-diagonal metric components thus do not encode the angular momentum of some gyraton. Instead, we demonstrate that in standard D=4 general relativity they directly determine two independent amplitudes of the Robinson-Trautman exact gravitational waves.