Robinson-Trautman spacetimes in higher dimensions

TL;DR

This paper extends Robinson-Trautman solutions to higher dimensions, fully integrating Einstein's equations with a cosmological constant and radiation, revealing unique geometric and algebraic properties distinct from four-dimensional cases.

Contribution

It provides the complete explicit family of higher-dimensional Robinson-Trautman spacetimes, highlighting their geometric structure and limitations compared to four-dimensional solutions.

Findings

Transverse space must be a Riemannian Einstein space.

Weyl tensor is necessarily of algebraic type D.

Includes higher-dimensional Schwarzschild and Vaidya solutions, but no C-metric analog.

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.