Figure 1

Schematic conformal diagrams of the Robinson-Trautman exact spacetimes which exist for any smooth initial data prescribed on $u_0$. Pure radiation field is present in the shaded region $u<0$. Near $u=0$ the solutions approach the Vaidya metric, and can be extended to flat Minkowski region $u>0$. Thick line indicates the curvature singularity at $r=0$ whereas double line represents future conformal infinity ${\mathcal{I}}^{+}$ at $r=\infty$ ($\Lambda =0$ is assumed). The global structure depends on the value of the parameter $\mu$ of the linear mass function (8): left diagram corresponds to $\mu >1/16$, the right one applies when $\mu \le 1/16$.Reuse & Permissions

Figure 2

Possible extensions of the Robinson-Trautman radiative spacetimes into the region $u<u_0$. Pure radiation is present only in the shaded region, everywhere else it is a vacuum solution. For $u\in (u_1,u_0)$ the mass function is constant, $m(u_0)=-\mu u_0$, but the spacetime is not spherically symmetric—it is not the Schwarzschild solution ($\mu >1/16$ on the left, $\mu \le 1/16$ on the right).Reuse & Permissions

Figure 3

Time-reversed version of Fig. 2 represents the advanced form of the Robinson-Trautman spacetimes (30) which describes an ingoing flow of radiation.Reuse & Permissions