Conformal Infinity

Abstract

The notion of conformal infinity has a long history within the research in Einstein's theory of gravity. Today, "conformal infinity" is related to almost all other branches of research in general relativity, from quantisation procedures to abstract mathematical issues to numerical applications. This review article attempts to show how this concept gradually and inevitably evolved from physical issues, namely the need to understand gravitational radiation and isolated systems within the theory of gravitation, and how it lends itself very naturally to the solution of radiation problems in numerical relativity. The fundamental concept of null-infinity is introduced. Friedrich's regular conformal field equations are presented and various initial value problems for them are discussed. Finally, it is shown that the conformal field equations provide a very powerful method within numerical relativity to study global problems such as gravitational wave propagation and detection.

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 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 which is in its future. Note that the two cones and 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 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 and . The physical space-time is the exterior part of this “stovepipe”. The “spheres” I^± and I⁰ 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 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 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 , 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.