List of Figures

View Image Figure 1:
The embedding of Minkowski space into the Einstein cylinder ℰ.
View Image Figure 2:
The conformal diagram of Minkowski space.
View Image Figure 3:
Spacelike hypersurfaces in the conformal picture.
View Image Figure 4:
Waveforms in physical space-time.
View Image Figure 5:
Waveforms in conformal space-time.
View Image 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.
View Image 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.
View Image 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.
View Image Figure 9:
The geometry of the standard Cauchy approach. The green lines are surfaces of constant time. The blue line indicates the outer boundary.
View Image Figure 10:
The geometry of Cauchy-Characteristic matching. The blue line indicates the interface between the (inner) Cauchy part and the (outer) characteristic part.
View Image Figure 11:
The geometry of the conformal approach. The green lines indicate the foliation of the conformal manifold by hyperboloidal hypersurfaces.
View Image 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.
View Image 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.
View Image 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.
View Image Figure 15:
Decay of the radiation at null-infinity.
View Image Figure 16:
The radiation field ψ4 and the Bondi mass for a radiating A3-like space-time.
View Image Figure 17:
The numerically generated ‘Kruskal diagram’ for the Schwarzschild solution.
View Image Figure 18:
The rescaled Weyl tensor in a flat space-time obtained from flat initial data and random boundary data in the unphysical region.