Abstract
We present the Riemann and Ricci tensors for a fully general nontwisting and shear-free geometry in arbitrary dimension D. This includes both the nonexpanding 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., a 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.