Figure 1:
The embedding of Minkowski space into the Einstein cylinder . 

Figure 2:
The conformal diagram of Minkowski space. 

Figure 3:
Spacelike hypersurfaces in the conformal picture. 

Figure 4:
Waveforms in physical spacetime. 

Figure 5:
Waveforms in conformal spacetime. 

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 which is in its future. Note that the two cones and are separated to indicate the nontrivial 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 that is in its future. Note that the ingoing surface may develop selfintersections and caustics. 

Figure 8:
The geometry near spacelike infinity: The “point” has been blown up to a cylinder that is attached to and . The physical spacetime is the exterior part of this “stovepipe”. The “spheres” and 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 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 CauchyCharacteristic 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 axisymmetric solution of the Yamabe equation with on the boundary (top) and its third differences in the direction (bottom). The location of 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 are shown. 

Figure 14:
Upper corner of a spacetime with singularity (thick line). The dashed line is , 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 nullinfinity. 

Figure 16:
The radiation field and the Bondi mass for a radiating A3like spacetime. 

Figure 17:
The numerically generated ‘Kruskal diagram’ for the Schwarzschild solution. 

Figure 18:
The rescaled Weyl tensor in a flat spacetime obtained from flat initial data and random boundary data in the unphysical region. 
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