Past horizons in Robinson-Trautman spacetimes with a cosmological constant

We study past horizons in the class of type II Robinson-Trautman vacuum spacetimes with a cosmological constant. These exact radiative solutions of Einstein’s equations exist in the future of any sufficiently smooth initial data, and they approach the corresponding spherically symmetric Schwarzschild–(anti-)de Sitter metric. By analytic methods we investigate the existence, uniqueness, location, and character of the past horizons in these spacetimes. In particular, we generalize the Penrose-Tod equation for marginally trapped surfaces, which form such white-hole horizons, to the case of a nonvanishing cosmological constant, and we analyze the behavior of its solutions and visualize their evolutions. We also prove that these horizons are explicit examples of an outer trapping horizon and a dynamical horizon, so that they are spacelike past outer horizons.

Figure 1

Schematic Penrose conformal diagram (for a fixed $\zeta$) of Robinson-Trautman exact spacetimes with $\Lambda >0$ (the shaded region) which exist for any smooth initial data prescribed on $u_i$. For $u\to +\infty$ the solutions approach the spherically symmetric Schwarzschild–de Sitter metric, and can be extended through the future event horizon ${\mathcal{H}}^{+}$ to the interior of the corresponding black hole (with the horizon at $r_h$). Zigzag lines indicate the curvature singularities at $r=0$, thick lines at $r=\infty$ represent future and past de Sitter-like conformal infinities ${\mathcal{I}}^{+}$ and ${\mathcal{I}}^{-}$, and $r_c$ locates the cosmological horizon. The null past event horizon is not well defined in the Robinson-Trautman region because the metrics diverge as $u\to -\infty$ (dotted lines). Instead, the white hole is localized by ${\mathcal{H}}^{-}$, which is the past horizon defined by marginally past-trapped surfaces ${\mathcal{T}}_u$ (see also Fig. 5). The boundary ${\mathcal{H}}_c$ indicates another expected horizon, namely, the de Sitter-like cosmological horizon in the dynamical Robinson-Trautman region which contains gravitational radiation.

Figure 2

Typical axially symmetric initial data at $u_i=0$ for the Robinson-Trautman equation (left), and the corresponding marginally past-trapped surface (right). The angles $\theta$, $\varphi$ are taken as standard spherical polar coordinates, while the radial distance is determined by $1+g$ and $\mathcal{R}$, respectively.

Figure 3

Visualization of the asymptotic behavior of typical solutions to the Robinson-Trautman equation (top) and the Penrose-Tod equation for the past horizon ${\mathcal{H}}^{-}$ (bottom). The pictures show their evolution given by the $u$ dependence. The right parts of each of the four pictures represent the section $\varphi =0$, while the left parts are the sections $\varphi =\pi$. For large $u$, the shapes shown become circular and thus, with $\varphi$ reintroduced, the corresponding 2-surfaces become spherically symmetric.

Figure 4

Past horizon ${\mathcal{H}}^{-}$ in a nonaxially symmetric Robinson-Trautman spacetime with a positive cosmological constant. The pictures visualize four marginally past-trapped 2-surfaces ${\mathcal{T}}_u$ for sections $u=0$, 0.002, 0.005, and 0.05, respectively. Their radius is $\mathcal{R}=r_h(1+h(\theta ,\varphi ,u))$, where $\theta$, $\varphi$ are taken as spherical coordinates. For large $u$, the marginally past-trapped surfaces approach a round sphere of radius $r_h$.

Figure 5

Schematic representation of the global structure of a Robinson-Trautman spacetime with $\Lambda >0$ for section $\theta =\mathrm{const}$ (the equatorial plane $\theta =\frac{\pi }{2}$, say), where $\varphi =\arg \zeta$. Here $u=u_i$ is the initial null boundary, and the horizontal circular ring ${\mathcal{I}}^{+}$ denotes future de Sitter-like conformal infinity at $r=\infty$. The black-hole singularity at $r=0$ is hidden behind the future event horizon ${\mathcal{H}}^{+}$ given by the null cone $u=+\infty$, while the white-hole singularity at $r=0$ is localized by the past horizon ${\mathcal{H}}^{-}$, which is formed by marginally past-trapped surfaces ${\mathcal{T}}_u$. For more details see the text.