Horizons in Robinson–Trautman spacetimes

Abstract

The past quasi-local horizons in vacuum Robinson-Trautman space-times are described. The case of a null (non-expanding) horizon is discussed. It is shown that the only Robinson-Trautman spacetime admitting such a horizon with sections diffeomorphic to S 2 is the Schwarzschild space-time. Weakening this condition leads to the horizons of the C-metric. Properties of the hypersurface r = 2m are examined.

Past horizons in Robinson-Trautman spacetimes with a cosmological constant

“…In the context of vacuum Robinson-Trautman spacetimes with Λ = 0, past (white-hole) horizons were already studied [11,14,15,16,17]. Following the approach outlined by Penrose [33,34], Tod in [11] explicitly derived the equation for an outer boundary of marginally past-trapped 2-surfaces at any constant retarded time u, and subsequently proved the existence and uniqueness of its smooth solutions.…”

“…In addition, they performed numerical simulations of its evolution (see also [16]). Recently, Natorf and Tafel [17] also investigated the past quasi-local horizons in vacuum Robinson-Trautman spacetimes. They showed that the marginally trapped 2-surfaces cross the surface r = 2m, and that the only spacetime which admits null nonexpanding horizon with sections diffeomorphic to S 2 is the Schwarzschild spacetime.…”

Horizons in Robinson-Trautman spacetimes

“…It is also shear-free due to the Raychaudhuri equation. Independly of topological assumptions on intersections of H with u=const we obtain the following result (see [13] for a proof)…”

“…Global structure, trapped surfaces and asymptotic behaviour of RT spacetimes were successfully studied by Penrose [2], Foster and Newman [3], Lukacs, Perjes, Porter and Sebestyen [4], Schmidt [5], Rendall [6], Tod [7], Singleton [8], Chruściel [9,10], Chruściel and Singleton [11], Chow and Lun [12] and others. In this communication we summarize our results [13] on trapped surfaces and quasi-local horizons in RT spacetimes. These geometrical objects play an important role in modern theory of black (or white) holes (see e.g.…”

“…In general, the answer is 'no' since solving condition H = 0 together with the RT equation shows that (see [13] for details)…”