Cycles of Time An Extraordinary New View of the Universe ROGERPENROSE Copyright © Roger Penrose 2010. All rights reserved. Contents Click entry name to see diagram Figs. 1.1 and 1.2 4 Figs. 1.3 and 1.4 5 Figs. 1.5 and 1.6 6 Figs. 1.7 and 1.8 7 Figs. 1.9 and 1.10 8 Figs. 1.11, 1.12, and 1.13 9 Figs. 1.14 and 1.15 10 Figs. 1.16 and 2.2 11 Fig. 2.3 12 Figs. 2.4 and 2.5 13 Figs. 2.7 and 2.8 14 Figs. 2.9 and 2.10 15 Figs. 2.11 and 2.12 16 Figs. 2.13 and 2.14 17 Figs. 2.15 and 2.16 18 Figs. 2.17 and 2.18 19 Figs. 2.19 and 2.20 20 Figs. 2.21 and 2.22 21 Figs. 2.23 and 2.24 22 Figs. 2.25 and 2.26 23 Figs. 2.27 and 2.28 24 Figs. 2.29 and 2.30 25 Figs. 2.31 and 2.32 26 Figs. 2.33 and 2.34 27 Figs. 2.35 and 2.36 28 Figs. 2.37 and 2.38 29 Figs. 2.39 and 2.40 30 Figs. 2.41 and 2.42 31 Figs. 2.43 and 2.44 32 Fig. 2.45 33 Section 2.6 34 Figs. 2.46 and 2.47 35 Figs. 2.49 and 3.1 36 Fig. 3.2 37 Figs. 3.3 and 3.4 38 Section 3.2 39 - 40 Figs. 3.5 and 3.6 41 Figs. 3.7, 3.8, and 3.9 42 Figs. 3.10 and 3.11 43 Figs. 3.12 and 3.13 44 Figs. 3.14 and 3.15 45 Figs. 3.16 and 3.17 46 Figs. 3.18 and 3.19 47 Figs. 3.20 and 3.21 48 Figs. 3.22 and 3.23 49 Figs. 3.24 and 3.25 50 Figs. 3.26 and 3.27 51 Figs. 3.28 and 3.29 52 Fig. 1.1 An egg rolling off a table, falling to the ground and smashing according to time-reversible dynamical laws. N N N Fig. 1.2 N×N×N cubical crate, each compartment containing a red or blue ball. 4 N=kn N=kn n n N=kn n intermediate box crate compartment with red or blue ball Fig. 1.3 The compartments are grouped together into k3 boxes, each of size n×n×n. z1 p1 y1 P=(p1, p2,..., pn) x1 z2 z1 zq p2 z2 yq y2 y1 y2 x2 xq x2 % x1 configuration space of 1 2 zq q point particles p , p ,..., p yq pq is a 3 -dimensional space xq % Fig. 1.4 Configuration space of q point particles p1,p2,… ,pq is a 3q-dimensional space. 5 % differing detailed configurations that appear to be microscopically identical are represented in the same coarse-graining region Fig. 1.5 A coarse-graining of % %. point particles 6 dimensions momentum: 3 numbers position: 3 numbers ! Fig. 1.6 The phase space has twice as many dimensions as . + +% 6 pt t evolution curve - equal volumes o phase 0 space - + Fig. 1.7 Point p moves along an evolution curve in the phase space . + Fig. 1.8 Impression of coarse-graining in higher dimensions. 7 phase space phase space × / + = × Fig. 1.9 The phase space( considered+ / by the experimenter is but a small factor of that which involves all the external degrees of freedom in the galaxy. y z y z = × + Fig. 1.10 Product space where is a plane and is a line. ( + / + / 8 coarse-graining coarse-graining region × in region in coarse-graining region in - . ( - + . / . = × + Fig. 1.11 Coarse-graining/ region in the product space as a product of coarse- graining regions in the factors. ( + / two very close points of in different coarse-graining regions + coarse-graining boundary fuzzy boundary (ignore points in this region) Fig. 1.12 ‘Fuzziness’ at the boundaries separating one coarse-graining region from the next. line of red dye Fig. 1.13 Two snug-fitting glass tubes and viscous fluid between, with line of red dye. 9 Fig. 1.14 The handle is turned several times spreading out the line of dye. The handle is then turned back the same number of times and the line reappears in apparent violation of the Second Law. –3 –2 ´3 –1 0 ´1 p0 ´2 p1 ℛ 1 ℛ ℛ ℛ ℛ 2 ℛ p2 ℛ ℛ p3 ℛ thermal 3 equilibrium max ℛ ℛ Fig. 1.15 The system starts off at a point of p0 in a fairly small coarse-graining region . 10 0 ℛ (a) (b) Fig. 1.16 As the dimension n increases, the typical number of neighbouring coarse- graining regions increases rapidly. (a) n=2 with typically 6 neighbours. (b) n=3 with typically 14 neighbours. K<0 K>0 K=0 Big Crunch time Big Bang Big Bang Fig. 2.2 Space-time histories for Friedmann’s cosmological models in which the spatial curvature of the universe is positive, zero, and negative (left to right). 11 (a) (b) (c) Fig. 2.3 The three basic kinds of uniform plane geometry as illustrated by Maurits C. Escher: (a) elliptic (positive, K>0); (b) Euclidean (flat, K=0); (c) hyperbolic (negative, K<0). Copyright M. C. Escher Company (2004). 12 R t >0 Fig. 2.4 Expansion rate of the universe for positive , with eventual exponential growth. " Λ space sections may be finite or infinite WE are somewhere around here Inflation? time Not visible on this Big Bang scale Fig. 2.5 Space-time expansion of the universe. Picture with positive (suggest- ively drawn so as not be be biased as to the value of K). Λ 13 phase space + (increasingBig Bang universe evolving size) Fig. 2.7 Adiabatic expansion of the universe depicted as a succession of maximal coarse-graining regions of equal volume. Gas in a box (a) time entropy Gravitating bodies (b) black hole Fig. 2.8 (a) Gas is initially constrained within a small compartment in the corner of a box before being released and distributing itself uniformly throughout the box. (b) In a galactic-sized box, stars are initially uniformly distributed but collect together in clumps over time: a uniform distribution in this case is not the one with highest entropy. 14 Fig. 2.9 Photons arriving at the Earth’s surface from the Sun have higher energy (shorter wavelength) than those returned to space by the Earth. Given an overall energy balance (the Earth does not get hotter over time), there must be more photons leaving than arriving; that is, the energy arriving has lower entropy than that departing. particles in uniform motion space, at noon the day after tomorrow space, at noon tomorrow time space, at noon today space, at noon yesterday space, at noon the day before yesterday Fig. 2.10 Space-time before Minkowski. 15 walker A walker B events simultaneous time with X, says A several weeks X Andromeda Galaxy events simultaneous walkers pass with X, says B at event X Fig. 2.11 Two walkers amble past one another, but the event X of their passing is judged by each to be simultaneous with events on Andromeda differing by several weeks. 3-space description + p − (a) (b) 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. 16 tangent vector + − accelerating massive massive photon particle particles 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. 0 massive particles photon Fig. 2.14 Non-uniform null cones in general relativity. 17 length of curve measured by g angle between % q curves determined p by g l 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 s b q r p a Fig. 2.16 ‘Straight lines’ (geodesics) in conformal representation of hyperbolic geometry are circular arcs meeting the boundary circle at right angles. 18 2 according to g according to g lengths differ, but angles agree Fig. 2.17 Conformal structure does not fix length measure,Ω but it does fix angles via the ratio of length measures in different directions at any point. Length measure can be rescaled up or down at different points without affecting the conformal structure. Lorentzian- null orthogonal timelike null spacelike 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. 19 ruler history (or train) time Minkowski pq space not simultaneous in 0 ruler’s rest-frame so separation qp is ruler not ruler’s length Fig. 2.19 A spacelike separation between points p and q in is not directly meas- ured by a ruler that is a 2-dimensional strip. 0 s time p equal 0 intervals q r back front Fig. 2.20 The ruler (or train) measures the separation pq only when they are simul- taneous, so light signals and clocks are needed instead. 20 particle of mass m Planck: E=hv Einstein: E=mc2 ∴ =m× , 2 ν frequency ν 4 5 ℎ Fig. 2.21 Any stable massive particle behaves as a very precise quantum clock.