Tt

Strict Sitter
Fig. 2.36 De Sitter space-time: (a) represented (with 2 spatial dimensions suppressed) in Minkowski 3-space; (b) its strict conformal diagram; (c) cut in half, we get a strict conformal diagram for the steady-state model.

Whatever view one might take on the physics of the matter, I indicate this kind of incompleteness by a slightly jagged line in my strict conformal diagrams. The one remaining type of line that I am using in these diagrams is an internal dotted line, to denote a black hole's event horizon. I am using all these five kinds of line (broken for symmetry axis, bold for infinity, wiggly for a singularity, slightly jagged for incompleteness, and dotted for a black hole's horizon) and two kinds of spot (black representing a single point in the 4-space, white tracing out an S2) consistently in my strict conformal diagrams, as given in the key in Fig. 2.37.

A strict conformal diagram for the Oppenheimer-Snyder collapse to a black hole is given in Fig. 2.38(a). This arises from 'gluing together' a portion of a collapsing Friedmann model and a portion of the Eddington-Finkelstein extension of the original Schwarzschild solution, as shown in the strict conformal diagrams Fig. 2.38(b),(c); see also Fig. 2.39. Schwarzschild found his solution of Einstein's equations in 1916, shortly after Einstein published the equations for his general theory of relativity. This solution describes the external gravitational field of a static spherically-symmetrical body (such as a star), and it can be extended inwards, as a static space-time, down to its Schwarzschild radius

where M is the mass of the body and G is Newton's gravitational constant. For the Earth this radius would be about 9 mm, for the Sun, about 3 km—but in these cases the radius would be well within the body and would be theoretical distances of no immediate relevance to the spacetime geometry, as this Schwarzschild metric holds only for the external region. See the strict conformal diagram Fig. 2.39(a).

symmetry axis infinity —/»www^ singularity incompleteness

/' black-hole horizon point of boundary ° sphere (S2) on boundary

Fig. 2.37 Key for strict conformal diagrams.

Fig. 2.38 The Oppenheimer-Snyder model of collapse to a black hole: (a) strict conformal diagram constructed from gluing together; (b) left part of time-reverse of Friedmann model (Fig. 2.34(b)) and (c) right part of Eddington-Finkelstein model (Fig. 2.39(b)). (In local models, such as these, A is ignored, so J is treated as null.)

Fig. 2.38 The Oppenheimer-Snyder model of collapse to a black hole: (a) strict conformal diagram constructed from gluing together; (b) left part of time-reverse of Friedmann model (Fig. 2.34(b)) and (c) right part of Eddington-Finkelstein model (Fig. 2.39(b)). (In local models, such as these, A is ignored, so J is treated as null.)

Fig. 2.39 Strict conformal diagrams of a spherically symmetrical (A = 0) vacuum: (a) original Schwarzschild solution, external to the Schwarzschild radius; (b) extension to Eddington-Finkelstein collapse metric; (c) full extension to Kruskal/ Synge/Szekeres/Fronsdal form.

For a black hole, however, the Schwarzschild radius would be at the horizon. At this radius, the Schwarzschild form of the metric goes singular, and the Schwarzschild radius was originally thought of as an actual singularity in space-time. However it was found, initially by Georges Lemaitre in 1927, that if we abandon the requirement that the spacetime remain static, it is possible to extend it in a completely smooth way. A simpler description of this extension was found by Arthur Eddington in 1930 (although he omitted to point out what it had achieved in this respect), this description being rediscovered, and its implications clearly enunciated, by David Finkelstein in 1958; see the strict conformal diagram of this in Fig. 2.39(b). What is referred to as the 'maximal extension of the Schwarzschild solution' (often called the Kruskal-Szekeres extension, although an equivalent, though more complicated description was found much earlier by J.L. Synge[246]) is given in the strict conformal diagram of Fig. 2.39(c).

In §3.4, we shall come to another feature of black holes which, though an extremely tiny effect at the present time, will ultimately have a crucial significance for us. Whereas according to the classical physics of Einstein's general relativity a black hole ought to be completely black, an analysis carried out by Stephen Hawking in 1974 showed that when effects of quantum field theory in curved space-time backgrounds are brought into the picture, a black hole ought to have a very tiny temperature T, which is inversely proportional to the hole's mass. For a black hole of 10 Mo, for example, this temperature would be the extraordinarily tiny, around 6x 10-9K which may be compared with the record low temperature, as of 2006, produced in the laboratory of ~ 10-9K achieved at MIT. This is about as warm as the black holes around today are likely to be. Larger black holes would be even colder, and the temperature of the ~4 000 000Mo black hole at the centre of our galaxy would be only about 1.5 x 10-14 K. Taking the ambient temperature of our universe, at the present time to be that of the CMB, we find that it has the immensely hotter value of ~ 2.7 K.

Yet, if we take the very very long view, and bear in mind that the exponential expansion of our universe will, if it continues indefinitely, lead to a vast cooling in the CMB, we would expect it to get down to the temperature of even the largest black holes that are likely ever to arise. After that, the black hole will start to radiate away its energy into the surrounding space, and in losing energy it must also lose mass (by Einstein's E=mc2). As it loses mass, it will get hotter, and gradually, after an incredible length of time (perhaps up to around 10100—i.e. a 'googol'—years, for the largest black holes around today) it shrinks away completely, finally disappearing with a 'pop'—this final explosion being hardly worthy of the name 'bang', as it would be likely to be only around the energy of an artillery shell which is something of an anticlimax after all that wait!

Of course, this is extrapolating our present physical knowledge and understanding to an enormous degree. But Hawking's analysis is well in accordance with accepted general principles, and these principles do not seem to allow us to escape this overall conclusion. Accordingly, I am accepting it as a plausible account of a black hole's eventual fate. Indeed, this expectation will form an important ingredient to the scheme that I shall be presenting in Part 3 of this book. In any case, it is of relevance to present a sketch of this process in Fig. 2.40, together with its strict conformal diagram, in Fig. 2.41.

\W

Hawking radiation radiation

x\ I Y

/

time

/ 1

(very long!) ■ 1 ,

singularity

black

///

collapsing material

Fig. 2.40 Hawking-evaporating black hole.

Fig. 2.40 Hawking-evaporating black hole.

Fig. 2.41 Strict conformai diagram of Hawking-evaporating black hole.

Of course, most space-times do not possess spherical symmetry, and a description in terms of a strict conformal diagram may not even supply a reasonable approximation. Nevertheless, the notion of a schematic conformal diagram can frequently be of considerable value for clarifying ideas. Schematic conformal diagrams do not have the clear-cut rules that govern the strict ones, and sometimes one needs to imagine that the diagram is presented in 3 (or even 4) dimensions for its implications to be fully appreciated. The basic point is to make use of two of the ingredients of conformal representations of space-times which make infinite quantities finite. These are, on the one hand, the bringing into our finite comprehension the infinite regions of space and time that we have seen in our strict conformal diagrams, that have been depicted by bold-line boundaries and, on the other, the folding out of those regions that are infinite in a different sense, namely the space-time singularities that in our strict diagrams have been denoted by wiggly-line boundaries. The first has been achieved by a conformal factor (the 'H' of g^H2g in §2.3) which has been allowed to tend smoothly to zero, so that the infinite regions are 'squashed' down to something finite. The second has been achieved, by a conformal factor that has been allowed to become infinite, so that the singular regions have been rendered finite and smooth by 'stretching out'. Of course, we may not be guaranteed that such pro -cedures will actually work, in any particular case. Nevertheless, we shall find that both these procedures have important roles to play in the ideas that we shall be coming to, and the combination of the two will be central to what I am proposing in Part 3.

To end this section, it will be useful to present one context in which both these procedures can be particularly illuminating, namely with regard to the issue of cosmological horizons. In fact, there are two distinct notions that, in the cosmological context, are referred to as 'horizons'.[248] One of these is what is known as an event horizon; the other a particle horizon.

Let us first consider the notion of a cosmological event horizon. It is closely related to that of a black hole's event horizon, though the latter has a more 'absolute' character in the sense that it is less dependent upon some observer's perspective. Cosmological event horizons occur when the model possess a which is spacelike as with all those Friedmann A > 0 models exhibited in the strict conformal diagrams of Fig. 2.35 and in the de Sitter model D of Fig. 2.36(b), but the idea applies also in situations of a spacelike where no symmetry is assumed (this being a general feature of A> 0). In the schematic conformal diagrams of Fig. 2.42(a),(b), I have indicated (for 2 or 3 space-time dimensions, respectively) the region of space-time that is in principle observable to an observer O (considered to be immortal!) with world-line l terminating at a point o+ on This observer's event horizon C~(o+) is the past light cone of o+.[249] Any event that occurs outside C-(o+) will forever remain unobservable to O. See Fig. 2.43. We notice, however, that the exact location of the event horizon is very much dependent on the particular terminal point o+.

Fig. 2.42 Schematic conformal diagrams of cosmological event horizons occurring when A > 1: (a) 2-dimensional; (b) 3-dimensional.

Fig. 2.43 The event horizon of the immortal observer O represents an absolute boundary to those events that are ever observable to O, this horizon itself dependent on O's choice of history. A change of mind at X can result in a different event horizon.

Particle horizons, on the other hand, arise when the past boundary— normally taken to be a singularity rather than infinity—is spacelike. In fact, as may be gleaned from those strict conformal diagrams depicted here in which singularities appear, a spacelike character is the norm for space-time singularities. This is closely related to the issue of 'strong cosmic censorship', which I shall touch upon in the next section. Let us call this initial singular boundary U-. If the event o is the space-time location of some observer O, then we may consider the past light cone V- (o) of o, and see where it meets U-. Any particles that originate on U- outside this intersection will never enter the region visible to the observer at the event o, although if O's world-line is allowed to be extended into the future, then more and more particles will come into view. It is usual to consider the actual particle horizon of the event o to be the locus traced out by idealized galaxy world-lines, originating at the intersection of V -(o) with U-. See Fig. 2.44.

Fig. 2.44 Schematic conformal diagrams of particle horizons in (a) 2 dimensions, (b) 3 dimensions.
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