Cycles of Time: An Extraordinary New View of the Universe

by Roger Penrose

Fig. 1.12 ‘Fuzziness’ at the boundaries separating one coarse-graining region from the next.

Fig. 2.12 (a) Null cone at p in Minkowski’s 4-space; (b) 3-space description of the future cone as an expanding succession of concentric spheres originating at p.

Fig. 2.13 Null cones in , uniformly arranged. World lines of massive particles are directed within the cones and of massless ones along the cones.

Fig. 2.15 The metric g assigns lengths to curves and angles between them. The geodesic l provides the ‘shortest route between p and q′ in the metric g.

Fig. 2.16 ‘Straight lines’ (geodesics) in conformal representation of hyperbolic geometry are circular arcs meeting the boundary circle at right angles.

While the conformal structure does not fix the length measure, it does fix the ratios of the length measures in different directions at any point—

Minkowskian if it is flat and Lorentzian if it is curved.

Fig. 2.18 ‘Orthogonality’ of spacelike and timelike directions in Lorentzian space-time, represented in a Euclidean picture for which the null cone is right-angled.

Fig. 2.19 A spacelike separation between points p and q in is not directly measured by a ruler that is a 2-dimensional strip.

Fig. 2.20 The ruler (or train) measures the separation pq only when they are simultaneous, so light signals and clocks are needed instead.

(h being Planck’s constant), telling us that this particle’s rest energy defines for it a particular frequency v of quantum oscillation (see Fig. 2.21).

Fig. 2.21 Any stable massive particle behaves as a very precise quantum clock.

In fact the quantum frequency of a single particle is extremely high, and it cannot be directly harnessed to make a usable clock. For a clock that can be used in practice, we need a system containing many particles, combined together and acting appropriately in concert. But the key point is still that to build a clock we do need mass

Fig. 2.22 Bowl-shaped 3-surfaces mark off the successive ‘ticks’ of identical clocks.

Fig. 2.23 A timelike geodesic line l is characterized by the fact that for any two points p and q on l, not too far apart, the longest local curve from p to q is in fact a portion of l.

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

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

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

Fig. 2.37 Key for strict conformal diagrams.

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

There is an algebraic procedure known as contraction (or transvection) which allows us to connect a lower index to an upper one (rather in the manner of chemical bonding), thereby removing these two indices from the final expression—

Fig. 2.47 The presence of Weyl curvature surrounding a gravitating body (here the Sun) can be seen in the distorting (non-conformal) effect that it has on the background field.

Fig. 2.49 Schematic conformal diagram of Paul Tod’s proposal for a form of ‘Weyl curvature hypothesis’; asserting that the Big Bang provides a smooth boundary to the space-time .

Fig. 3.5 The conformal scale factor goes cleanly from positive to negative at crossover, the curve having a slope that is neither horizontal nor vertical. Here ‘conformal time’ just refers to ‘height’ in a suitable conformal diagram.

Fig. 3.6 The gravitational field is measured by the tensor K, propagates according to a conformally invariant equation, and so generally attains finite non-zero values at

Fig. 3.11 Comparison between the behaviours of the conformal factors ω for (a) Friedmann’s dust and (b) Tolman’s radiation. Only the latter (b) is consistent with CCC. (See Fig. 3.5 and Appendix B for the terminology and notation.)

Fig. 3.13 Conformal diagrams drawn irregularly (to suggest a lack of symmetry) to indicate (a) gravitational collapse to a black hole; (b) collapse followed by Hawking evaporation. The singularity remains spacelike according to strong cosmic censorship.

Fig. 3.15 A Hawking-evaporating black hole: (a) conventional space-time picture; (b) strict conformal diagram. Loss of internal degrees of freedom may be considered to result only as the ‘pop’ occurs, this being the picture suggested according to the time-slices given by unbroken lines. Alternatively, according to the time-slices given by the broken lines, the loss occurs gradually over the whole history of the black hole.

Fig. 3.16 The information loss in black holes does not affect local phase space (compare Fig. 1.9), though it contributes to the total, prior to loss.

Fig. 3.19 For a cosmological event horizon there is no information loss (unlike the case of a black hole), as is evident from the all-encompassing nature of a family of global time-slices.

Fig. 3.22 A ‘radius of curvature’ is a reciprocal measure of curvature, which is small when curvature is large and large when curvature is small. Quantum gravity is commonly argued to become dominant when space-time curvature radii approach the Planck length, but CCC maintains this applies only to Weyl curvature.

Fig. 3.27 When the gravitational wave burst encounters the crossover 3-surface, it gives the initial material of the succeeding aeon a ‘kick’ in the direction of the wave.

Fig. 3.28 We appear to be about ⅔ of the way up our aeon, in a conformal diagram. If this applies also to the earliest black-hole encounters in the previous aeon, then a cut-off in angular correlations at 60° is to be expected.

Fig. 3.29 Twisting the CMB sky (using the formula θ′ = θ, φ′ = φ + 3aπθ2 – 2aθ3) in spherical polar coordinates. This sends circles into more elliptical shapes.