Robinson-Trautman spacetimes in higher dimensions

TL;DR

The paper extends Robinson–Trautman spacetimes to D>4 by enforcing a geodesic, shear-free, expanding, hypersurface-orthogonal null congruence and solving Einstein’s equations with an arbitrary cosmological constant and aligned pure radiation. The authors derive a closed-form higher-dimensional RT class, showing the transverse space must be a Riemannian Einstein space and that the Weyl tensor is of algebraic type D (or O in special cases), with vacuum solutions corresponding to generalized Schwarzschild–Kottler–Tangherlini black holes. They obtain an explicit metric form ds^2 = (r^2/P^2) γ_{ij} dx^i dx^j − 2 du dr − 2H du^2 and a simplified Robinson–Trautman equation for P, highlighting that, unlike in D=4, C-metric-like radiative solutions are absent in D>4. The work reveals that higher-dimensional RT spacetimes admit far richer horizon geometries (arbitrary Einstein spaces) but simpler radiative content, and it suggests exploring relaxations of the shear-free condition or other matter couplings to access a broader radiative sector.

Abstract

As an extension of the Robinson-Trautman solutions of D=4 general relativity, we investigate higher dimensional spacetimes which admit a hypersurface orthogonal, non-shearing and expanding geodesic null congruence. Einstein's field equations with an arbitrary cosmological constant and possibly an aligned pure radiation are fully integrated, so that the complete family is presented in closed explicit form. As a distinctive feature of higher dimensions, the transverse spatial part of the general line element must be a Riemannian Einstein space, but it is otherwise arbitrary. On the other hand, the remaining part of the metric is - perhaps surprisingly - not so rich as in the standard D=4 case, and the corresponding Weyl tensor is necessarily of algebraic type D. While the general family contains (generalized) static Schwarzschild-Kottler-Tangherlini black holes and extensions of the Vaidya metric, there is no analogue of important solutions such as the C-metric.