List of Figures

	Figure 1: The embedding of Minkowski space into the Einstein cylinder $ℰ$ .
	Figure 2: The conformal diagram of Minkowski space.
	Figure 3: Spacelike hypersurfaces in the conformal picture.
	Figure 4: Waveforms in physical space-time.
	Figure 5: Waveforms in conformal space-time.
	Figure 6: The physical scenario: The figure describes the geometry of an isolated system. Initial data are prescribed on the blue parts, i.e. on a hyperboloidal hypersurface and the part of $− ℐ$ which is in its future. Note that the two cones $ℐ +$ and $ℐ −$ are separated to indicate the non-trivial transition between them.
	Figure 7: The geometry of the asymptotic characteristic initial value problem: Characteristic data are given on the blue parts, i.e. an ingoing null surface and the part of $− ℐ$ that is in its future. Note that the ingoing surface may develop self-intersections and caustics.
	Figure 8: The geometry near space-like infinity: The “point” $i0$ has been blown up to a cylinder that is attached to $− ℐ$ and $+ ℐ$ . The physical space-time is the exterior part of this “stovepipe”. The “spheres” $± I$ and $0 I$ are shown in blue and light green, respectively. The brown struts symbolize the conformal geodesics used to set up the construction. Note that they intersect $ℐ$ and continue into the unphysical part.
	Figure 9: The geometry of the standard Cauchy approach. The green lines are surfaces of constant time. The blue line indicates the outer boundary.
	Figure 10: The geometry of Cauchy-Characteristic matching. The blue line indicates the interface between the (inner) Cauchy part and the (outer) characteristic part.
	Figure 11: The geometry of the conformal approach. The green lines indicate the foliation of the conformal manifold by hyperboloidal hypersurfaces.
	Figure 12: An axi-symmetric solution of the Yamabe equation with $u = 1$ on the boundary (top) and its third differences in the $z$ -direction (bottom). The location of $ℐ$ is clearly visible.
	Figure 13: The difference between two solutions of the Yamabe equation obtained with different boundary conditions outside. Only the values less than $0.005$ are shown.
	Figure 14: Upper corner of a space-time with singularity (thick line). The dashed line is $ℐ$ , while the thin line is the locus of vanishing divergence of outgoing light rays, i.e. an apparent horizon.
	Figure 15: Decay of the radiation at null-infinity.
	Figure 16: The radiation field $ψ4$ and the Bondi mass for a radiating A3-like space-time.
	Figure 17: The numerically generated ‘Kruskal diagram’ for the Schwarzschild solution.
	Figure 18: The rescaled Weyl tensor in a flat space-time obtained from flat initial data and random boundary data in the unphysical region.