Conformal diagrams and conformal boundaries

There is a convenient way of representing space-time models in their entirety, especially in the case of models possessing spherical symmetry, as in the case of the Oppenheimer-Snyder and Friedmann space-times. This is by the use of conformal diagrams. I shall distinguish two types of conformal diagram here, the strict and the schematic conformal diagrams.[2 43] We shall be seeing something of the utility of each.

Let us start with the strict conformal diagrams, which can be used to represent space-times (here denoted by M) with exact spherical symmetry. The diagram would be a region © of the plane, and each point in the interior of © would represent a whole sphere's worth (i.e. an SP's worth) of points of M. To get something of a picture of what is going on, we may lose one spatial dimension, and imagine rotating the region © about some vertical line off to the left (see Fig. 2.26)—this line being referred to as an axis of rotation. Then each point in © will trace out a circle (S1). This is good enough for our visual imaginations. But for the full 4-dimensional picture of our space-time M, we would need a 2-dimensional rotation, so each interior point of © has to trace out a sphere (S2) in M.

Often, in our strict conformal diagrams, we find that we have an axis of rotation which is part of the boundary to the region ©. Then those boundary points on the axis—represented in the diagram as a broken line—would each represent a single point (rather than an S2) in the 4-dimensional space-time, so that the entire broken line would

Fig. 2.26 Strict conformal diagram V used to represent space-times (here denoted by M) with exact spherical symmetry. The 2-dimensional region V is rotated (through a 2-dimensional sphere S2) to make the 4-space M.

also represent a single line in M. Fig. 2.27 gives us an impression of how the whole space-time M is constituted as a family of 2-dimensional spaces identical to V in rotation about the broken-line axis.

rotation axis rotation axis

Conformal Spacetime Diagram

We are going to think of M as a conformal space-time, and not worry too much about the particular scaling that gives M its full metric g.

Thus, in accordance with the final sentence of §2.3, M is provided with a full family of (time-oriented) null cones. In accordance with this, D itself, being a 2-dimensional subspace of M, inherits from it a 2-dimensional conformal space-time structure, and has its own 'time-oriented null cones'. These simply consist of a pair of distinct 'null' directions at each point of D that are deemed to be oriented towards the future. (They are just the intersections of the planes defining the copies of D with the future null cones of M; see Fig. 2.28.)

Fig. 2.28 The 'null cones' in D, angled at 45° to the vertical, are the intersections of those in M with an embedded D.

In a strict conformal diagram, we endeavour to arrange all these future null directions in D to be oriented at 45° to the upward vertical. To illuminate the situation, I have drawn, in Fig. 2.29, a conformal diagram for the entire Minkowski space-time M, the radial null lines being drawn at 45° to the upward vertical. In Fig. 2.30, I have tried to indicate how this mapping is achieved. We see that Fig. 2.29 exhibits an important feature of conformal diagrams: the picture is of merely a finite (right-angled) triangle, despite the entire infinite space-time M being encompassed by the diagram. A characteristic feature of conformal diagrams is, indeed, that they enable the infinite regions of the space-time to be 'squashed down' so as to be encompassed by a finite picture. Infinity itself is also represented in the diagram. The two bold sloping boundary lines represent past null infinity J- and future null infinity J+, where every null rotation axis rotation axis

Fig. 2.28 The 'null cones' in D, angled at 45° to the vertical, are the intersections of those in M with an embedded D.

geodesic (null straight line) in M acquires a past end-point on J- and a future end-point on J+. (It is usual to pronounce the letter Jas 'scri'— meaning 'script I'.)[244] There are also three points, i-, i0, and i+ on the boundary, respectively representing past timelike infinity, spacelike infinity, and future timelike infinity, where every timelike geodesic in M acquires the past end-point i- and future end-point i+, and every spacelike geodesic closes into a loop via the point i0. (We shall be seeing, shortly, why i0 must indeed be considered as being just a single point.)

At this juncture, it may be helpful to recall the Escher print Fig. 2.3(c) providing a conformal picture of the entire hyperbolic plane. The bounding circle represents its infinity, in a conformally finite way, in an essentially similar manner to the way in which , J2— i-, i0, and i+ together represent infinity for M. In fact, just as we can extend the hyperbolic plane, as a smooth conformal manifold, beyond its conformal boundary to the Euclidean plane inside which it is represented (Fig. 2.31), we may also extend M, smoothly, beyond its boundary to a larger conformal manifold. In fact, M is conformally identical to a portion of the space-time model known as the Einstein universe £ (or the 'Einstein cylinder'). This is a cosmological model which is spatially a 3-sphere (S3) and completely static. Figure 2.32(a) gives an intuitive picture of this model (the one Einstein originally introduced his cosmological constant A in order to achieve, in 1917; see §2.1) and Fig. 2.32(b) provides a strict conformal diagram representing it. Note that in this diagram there are two separate 'axes of rotation', represented by the two vertical broken lines. This is completely consistent; we just think of the radius of the S2, which each point in the interior of the diagram represents, as shrinking down to zero as a broken line is approached. This also serves to explain the rather

Fig. 2.31 Extending the hyperbolic plane, as a smooth conformal manifold, beyond its conformal boundary to the Euclidean plane inside which it is represented.

Fig. 2.31 Extending the hyperbolic plane, as a smooth conformal manifold, beyond its conformal boundary to the Euclidean plane inside which it is represented.

curious-seeming fact that spatial infinity for M is conformally just the single point i0, for the radius of the S2 that it would seem to have represented has shrunk down to zero. The spatial S3 cross-sections of the space-time £ arise from this procedure. Figure 2.33(a) shows how M arises as a conformal subregion of £, and in fact how we can consider the entire manifold £ as made up, conformally, of an infinite succession of spaces M, where the \+ of each one is joined on to the of the next, and Fig. 2.33(b) shows how this is done in terms of strict conformal diagrams. It will be worth bearing this picture in mind when we come to consider the proposed model of Part 3.

r. i

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t time

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(a)

(b)

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Fig. 2.32 (a) Intuitive picture of the Einstein universe £ ('Einstein cylinder'); (b), (c) strict conformal diagrams of the same thing.

Fig. 2.32 (a) Intuitive picture of the Einstein universe £ ('Einstein cylinder'); (b), (c) strict conformal diagrams of the same thing.

Let us now consider the Friedmann cosmologies introduced in §2.1. The different cases K> 0, K=0, K<0, for A = 0, are illustrated in Fig. 2.34(a),(b),(c), respectively. Singularities are here represented as wiggly lines. Here I have introduced a notation whereby a white dot on the boundary represents an entire sphere S2, whereas the black dots '•' (which we already had in the case of M) represent single points. These white dots actually represent the boundary spheres of hyperbolic space, in the conformal representation that Escher used in the 2-dimensional case. The corresponding cases for positive cosmological constant (A > 0, where in

Fig. 2.33 To see why i0 is a single point. (a) M arises as a conformai subregion of £. The entire manifold £ can be considered to be made up, conformally, of an infinite succession of spaces M; (b) shows how this is done in terms of strict conformal diagrams.

Fig. 2.34 Strict conformal diagrams for the three different cases, K> 0, K = 0, K <0 for A = 0, of the Friedmann cosmologies.

the case K> 0 we assume that the spatial curvature is not large enough to overcome A and produce an ultimate re-collapse). These are illustrated in Fig. 2.35(a),(b),(c). An important feature of these diagrams may be pointed out here. The future infinity J2* of all these models is spacelike, as is indicated by the final bold boundary line being always more horizontal than

Fig. 2.34 Strict conformal diagrams for the three different cases, K> 0, K = 0, K <0 for A = 0, of the Friedmann cosmologies.

45°, in contrast with the future infinity that occurs when A=0 (in the cases illustrated in Fig. 2.34(b),(c) and Fig. 2.29), where the boundary is at 45°, so J+ is then a null hypersurface. This is a general feature of the relation between the geometrical nature of J+ and the value of the cosmological constant A, and it will have a key importance for us in Part 3.

Fig. 2.35 Strict conformal diagrams for Friedmann models with A> 0. (a) K>0; (b) K = 0; K<0.

These Friedmann models with A > 0 all have a behaviour in their remote future (i.e. near J+) which closely approaches de Sitter space-time D, a model universe that is completely empty of matter and is extremely symmetrical (being a Minkowskian analogue of a 4-dimensional sphere). In Fig. 2.36(a) I have sketched a 2-dimensional version of D, with only one spatial dimension represented (where the full de Sitter 4-space D would be a hypersurface in Minkowski 5-space), and I have given a strict conformal diagram for it in Fig. 2.36(b). The steady-state model, referred to in §2.2, is just one half of D, as shown in Fig. 2.36(c). Owing to the 'cut' through D that is required (jagged boundary), the steady-state model is actually what is called 'incomplete', in past directions. There are ordinary timelike geodesics—which could represent free motions of massive particles—whose time measure does not extend to earlier values than some finite value. This might well have been regarded as a worrying flaw in the model if it had applied to future directions, since it could apply to the future of some particle or space traveller,[245] but here we can simply say that such particle motions were never present.

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