1.1 Hamiltonian theory is a framework that encompasses all of standard classical physics and which provides the essential link to quantum mechanics. See R. Penrose (2004), The Road to Reality, Random House, Ch.20.
1.2 Planck's formula: E=hv . For an explanation of the symbols, see Note 2.18.
1.3 Erwin Schrodinger (1950), Statistical thermodynamics, Second edition, Cambridge University Press.
1.4 The term 'product' is consistent with the multiplication of ordinary integers in that the product space of an m-point space with an n-point space is an mn-point space.
1.5 In 1803 the mathematician Lazare Carnot published Fundamental principles of equilibrium and movement where he noted the losses of 'moment of activity', i.e. the useful work done. This was the first-ever statement of the concept of transformation of energy or entropy. Sadi Carnot went on to postulate that 'some caloric is always lost' in mechanical work. In 1854 Clausius developed the idea of 'interior work', i.e. that 'which the atoms of the body exert on each other' and 'exterior work', i.e. that 'which arises from foreign influences [to] which the body may be exposed'.
1.6 Claude E. Shannon, Warren Weaver (1949), The mathematical theory of communication, University of Illinois Press.
1.7 In mathematical terms, the problem comes about because macroscopic indistinguishability is not what is called transitive, i.e. states A and B might be indistinguishable and states B and C indistinguishable, yet with A and C distinguishable.
1.8 The 'spin' of an atomic nucleus is something which requires considerations of quantum mechanics for a proper understanding, but for a reasonable physical picture, one may indeed imagine that the nucleus is 'spinning' about some axis, as might a cricket ball or baseball. The total value of this 'spin' comes about partly from the individual spins of the constituent protons and neutrons and partly through their orbital motions about one another.
1.9 E. L. Hahn (1950), 'Spin echoes'. Physical Review 80, 580-94.
1.10 J.P. Heller (1960), 'An unmixing demonstration'. Am J Phys 28 348-53.
1.11 It may be, however, that in the context of black holes the entropy concept does acquire some measure of genuine objectivity. We shall be examining this issue in §§2.6 and 3.4.
2.1 Various other possible interpretations of the red shift have been put forward from time to time, one of the most popular being some version of a 'tired light' proposal, according to which the photons simply 'lose energy' as they travel towards us. Another version proposes that time progressed more slowly in the past. Such schemes turn out to be either inconsistent with other well-established observations or principles, or 'unhelpful', in the sense that they can be re-phrased as being equivalent to the standard expanding-universe picture, but with unusual definitions of the measures of space and time.
2.2 A. Blanchard, M. Douspis, M. Rowan-Robinson, and S. Sarkar (2003), 'An alternative to the cosmological "concordance model"'. Astronomy & Astrophysics 412, 35-44. arXiv:astro-ph/0304237v2 7 Jul 2003.
2.3 This term was introduced in a BBC radio broadcast on 28 March 1949, as a somewhat derogatory description, by Fred Hoyle who had been a strong supporter of the rival 'steady state theory', see
§2.2. In this book, when referring to that particular event that apparently occurred some 1.37 x 1010 years ago, I shall adopt the capitalized form of this term 'Big Bang', but when referring to other similar occurrences which may occur either in reality or in theoretical models, I shall tend to use 'big bang' without specific capitalization.
2.4 Dark matter is not 'dark' (like the large, visibly dark dust regions, clearly seen from their obscuring effects), but is, more appropriately, invisible matter. Moreover, what is referred to as 'dark energy' is quite unlike the energy possessed by ordinary matter which, in accordance with Einstein's E=mc2, has an attractive influence on other matter. Instead it is repulsive, and its effects appear, so far, to be fully in accord with the presence of something quite unlike ordinary energy, namely the cosmological constant introduced by Einstein in 1917, and taken into consideration by virtually all standard cosmology texts since then. This constant is indeed necessarily constant, and so, quite unlike energy, it has no independent degrees of freedom.
2.5 Halton Arp and 33 others, 'An open letter to the scientific community'. New Scientist, May 22, 2004.
2.6 A pulsar is a neutron star—an extraordinarily dense object, around 10 kilometres across, with a mass somewhat more than that of the Sun—which has an enormously strong magnetic field and rapidly rotates, sending precisely repeated bursts of electromagnetic radiation detectable here on Earth.
2.7 Curiously, Friedmann himself did not actually explicitly address the easiest case where the spatial curvature is zero: Zeitschrift fur Physik 21 326-32.
2.8 That is, apart from possible topological identifications, which do not concern us here.
2.9 In both the cases K = 0 and K<0 there are topologically closed-up versions (obtained by identifying certain distant points in the spatial geometry with each other) in which the spatial geometry is finite. However, in all these situations, global spatial isotropy is lost.
2.10 A supernova is an extraordinarily violent explosion of a dying star (of mass somewhat greater than our own Sun), allowing it to achieve a brightness that, for a few days, exceeds the output of the entire galaxy within which it resides. See §2.4.
2.11 S. Perlmutter et al. (1999), Astrophysical J 517 565. A. Reiss et al. (1998), Astronomical J 116 1009.
2.12 Eugenio Beltrami (1868), 'Saggio di interpretazione della geometria non-euclidea', Giornale di Mathematiche VI285-315. Eugenio Beltrami (1868), 'Teoria fondamentale degli spazii di curvatura costante', Annali Di Mat, ser. II 2 232-55.
2.13 H. Bondi, T. Gold (1948), 'The steady-state theory of the expanding universe', Monthly Notices of the Royal Astronomical Society 108 252-70. Fred Hoyle (1948), 'A new model for the expanding universe', Monthly Notices of the Royal Astronomical Society 108 372-82.
2.14 I learnt a great deal of physics and its excitement from my close friend Dennis Sciama, a strong adherent of the steady-state model at that time, in addition to attending inspirational lectures by Bondi and Dirac.
2.15 J.R. Shakeshaft, M. Ryle, J.E. Baldwin, B. Elsmore, J.H. Thomson (1955), Mem RAS 67 106-54.
2.16 Temperature measures in fundamental physics tend to be given in units of 'Kelvin' (denoted simply by the letter 'K', following the temperature measure, which refers to the number of centigrade (or Celsius) units above absolute zero.
2.17 Abbreviations CMBR, CBR, and MBR are also sometimes used.
2.18 For a given temperature T, Planck's formula for the black-body intensity, for frequency v, is 2hv3/(ehv/kT-1), where h and k are Planck's and Boltzmann's constants, respectively.
2.19 R.C. Tolman (1934), Relativity, thermodynamics, and cosmology, Clarendon Press.
2.20 The local group of galaxies (the galactic cluster that includes the solar system's Milky Way galaxy) appears to be moving at about 630 km s-1 relative to the reference frame of the CMB. A. Kogut et al. (1993), Astrophysical J 419 1.
2.21 H. Bondi (1952), Cosmology, Cambridge University Press.
2.22 A curious exception appears to be provided by volcanic vents at odd places on the ocean floor upon which colonies of strange life forms depend. Volcanic activity results from heating due to radioactive material, this having been originated in some other stars which, at some distant time in the past had spewed out such material in supernova explosions. The low-entropy role of the Sun is then taken over by such stars, but the general point made in the text remains unchanged.
2.23 Slight corrections to this equation come from, on the one hand, the small amount of heating due to radioactive material referred to in Note 2.22, and, on the other, effects coming from the burning of fossil fuels and global warming.
2.24 This general point seems to have been first made by Erwin Schrodinger in his remarkable 1944 book What is life?
2.25 R. Penrose (1989), The emperor's new mind: concerning computers, minds, and the laws of physics, Oxford University Press.
2.26 It is quite a common terminology to refer to this null cone as a 'light cone', but I prefer to reserve that terminology for the locus in the whole space-time that is swept out by the light rays through some event p. The null cone, on the other hand (in the sense being used here), is a structure defined just in the tangent space at the point p (i.e. infinitesimally at p).
2.27 To be explicit about Minkowski's geometry, we can choose some arbitrary observer rest-frame and ordinary cartesian coordinates (x,y,z) to specify the spatial location of an event, with a time coordinate t for that observer's time coordinate. Taking space and time scales so that c = 1, we find that the null cones are given by dt2 — dx2 — dy2 — dz2 = 0. The light cone (see Note 2.26) of the origin is then t2—x2 —y2—z2 = 0.
2.28 The concept of mass being referred to here ('massive', 'massless') is that of rest-mass I shall return to this matter in §3.1.
2.29 As we recall from §1.3, the ordinary equations of dynamics are reversible in time, so that, as far as dynamical behaviour—as governed by the submicroscopic ingredients of a physical system— is concerned, we might equally say that causation can propagate from future to past. The notion of 'causation' used in the text is, however, in accordance with standard terminology.
2.30 Length=/yjgijdxidx' See R. Penrose (2004), The Road to Reality, Random House, Fig. 14.20, p. 318.
2.31 J.L. Synge (1956) Relativity: the general theory. North Holland Publishing.
2.32 It is the existence of this natural metric that actually undermines completely the seemingly penetrating analysis made by Poincare, when he argued that the geometry of space is basically a conventional matter, and that Euclidean geometry, being the simplest, would therefore always be the best geometry to use for physics! See Poincare Science and Method (trans Francis Maitland (1914)) Thomas Nelson.
2.33 The rest energy of a particle is its energy in the rest frame of the particle, so there is no contribution to this energy (kinetic energy) from the motion of the particle.
2.34 The 'escape velocity' is the speed, at the surface of a gravitating body, that an object needs to acquire in order that it can escape completely from that body and not fall back to its surface.
2.36 See appendix of R. Penrose (1965), 'Zero rest-mass fields including gravitation: asymptotic behaviour', Proc.Roy. Soc. A284 159-203. The argument is slightly incomplete.
2.37 The somewhat bizarre circumstance underlying this is related in my book (1989), The emperor's new mind, Oxford University Press.
2.38 The existence of a trapped surface is an example of what we now tend to refer to as a 'quasi-local' condition. In this case, we assert the presence of a closed spacelike topological 2-surface (normally a topological 2-sphere) whose future-pointing null normals are, at the surface, all converging into the future. In any space-time, there will be local patches of spacelike 2-surface whose normals have this property, so the condition is not a local one; a trapped surface occurs, however, only when such patches can join up to form a closed (i.e. of compact topology) surface.
2.39 R. Penrose (1965), 'Gravitational collapse and space-time singularities', Phys. Rev. Lett. 14 57-9. R. Penrose (1968), 'Structure of space-time', in Batelle Rencontres (ed. C.M. deWitt, J.A. Wheeler), Benjamin, New York.
2.40 The only requirement that a non-singular space-time needs to have in this context—and which the 'singularity' would turn out to prevent—is what is called 'future null completeness'. This requirement is that every null geodesic can be extended into the future to an indefinitely large value of its 'affine parameter'. See S.W. Hawking, R. Penrose (1996), The nature of space and time, Princeton University Press.
2.41 R. Penrose (1994), 'The question of cosmic censorship', in Black holes and relativistic stars (ed. R.M. Wald), University of Chicago Press.
2.42 R. Narayan, J.S. Heyl (2002), 'On the lack of type I X-ray bursts in black hole X-ray binaries: evidence for the event horizon?', Astrophysical J 574 139-42.
2.43 The idea of a strict conformal diagram was first formalized by Brandon Carter (1966) following the more relaxed descriptions of schematic conformal diagrams that I had been systematically using from around 1962 (see Penrose 1962, 1964, 1965). B. Carter (1966), 'Complete analytic extension of the symmetry axis of Kerr's solution of Einstein's equations', Phys. Rev. 1411242-7. R. Penrose (1962), 'The light cone at infinity', in Proceedings of the 1962 conference on relativistic theories of gravitation, Warsaw, Polish Academy of Sciences. R. Penrose (1964), 'Conformal approach to infinity', in Relativity, groups and topology. The 1963 Les Houches Lectures (ed. B.S. DeWitt, C.M. DeWitt), Gordon and Breach, New York. R. Penrose (1965), 'Gravitational collapse and space-time singularities', Phys. Rev. Lett. 14 57-9.
2.44 Coincidentally, the Polish word 'skraj' is pronounced in the same way as 'scri' and means a boundary (albeit usually of a forest).
2.45 In a time-reversed steady-state model, an astronaut, in free motion, following such an orbit, would encounter the inward motion of ambient material passing at greater and greater velocity until it reaches light speed, with infinite momentum impacts, in a finite experienced time.
2.46 J.L. Synge (1950), Proc. Roy. Irish Acad. 53A 83. M.D. Kruskal (1960), 'Maximal extension of Schwarzschild metric', Phys. Rev. 119 1743-5. G. Szekeres (1960), 'On the singularities of a Riemannian manifold', Publ. Mat. Debrecen 7 285-301. C. Fronsdal (1959), 'Completion and embedding of the Schwarzchild solution', Phys Rev. 116 778-81.
2.47 S.W. Hawking (1974), 'Black hole explosions?', Nature 248 30.
2.48 The notions of a cosmological event horizon and particle horizon were first formulated by Wolfgang Rindler (1956), 'Visual horizons in world-models', Monthly Notices of the Roy. Astronom. Soc. 116 662. The relation of these notions to (schematic) conformal diagrams were pointed out in R. Penrose (1967), 'Cosmological boundary conditions for zero rest-mass fields', in The nature of time (pp. 42-54) (ed. T. Gold), Cornell University Press.
2.49 This is meant in the sense that C-(p) is the (future) boundary of the set of points that can be connected to an event p by a future-directed causal curve.
2.50 Following my own work showing the inevitability of singularities arising in a local gravitational collapse (see Note 2.36 for 1965 reference), referred to in §2.4, Stephen Hawking produced a series of papers showing how such results could also be obtained which apply more globally in a cosmological context (in several papers in the Proceedings of the Royal Society (see S.W. Hawking, G.F.R. Ellis (1973), The large-scale structure of space-time, Cambridge University Press). In 1970, we combined forces to provide a very comprehensive theorem covering all these types of situation: S.W. Hawking, R. Penrose (1970), 'The singularities of gravitational collapse and cosmology', Proc. Roy. Soc. Lond. A314 529-48.
2.51 I first presented this kind of argument in R. Penrose (1990), 'Difficulties with inflationary cosmology', in Proceedings of the
14th Texas symposium on relativistic astrophysics (ed. E. Fenves), New York Academy of Science. I have never seen a response from supporters of inflation.
2.52 D. Eardley ((1974), 'Death of white holes in the early universe', Phys. Rev. Lett. 33 442-4) has argued that white holes in the early universe would be highly unstable. But that is not a reason for their not being part of the initial state, and it is perfectly consistent with what I am saying here. The white holes could well disappear, at various rates, just as in the opposite time direction, black holes can form, at various rates.
2.53 Compare A. Strominger, C. Vafa (1996), 'Microscopic origin of the Bekenstein-Hawking entropy', Phys. Lett. B379 99-104. A. Ashtekar, M. Bojowald, J. Lewandowski (2003), 'Mathematical structure of loop quantum cosmology', Adv. Theor. Math. Phys. 7 233-68. K. Thorne (1986), Black holes: the mebrane paradigm, Yale University Press.
2.54 Elsewhere I have given this figure with the second exponent as '123' rather than '124', but I am now pushing the value up so as to include a contribution from the dark matter.
2.55 Dividing 1010124 by 101089, we get lO10124"1089 = 101012\ as near as makes no difference.
2.56 R. Penrose (1998), 'The question of cosmic censorship', in Black holes and relativistic stars (ed. R.M. Wald), University of Chicago Press. (Reprinted J. Astrophys. 20 233-48 1999)
2.57 See Appendix A3 re Ricci tensor.
2.59 There will, however, be non-linear effects concerning how the different effects of lenses along a line of sight 'add up'. I am ignoring these here.
2.60 A. O. Petters, H. Levine, J. Wambsganns (2001), Singularity theory and gravitational lensing, Birkhauser.
2.61 R. Penrose (1979), 'Singularities and time-asymmetry', in S. W. Hawking, W. Israel, General relativity: an Einstein centenary survey, Cambridge University Press, pp. 581-638. S. W. Goode, J. Wainwright (1985), 'Isotropic singularities in cosmological models',
Class. Quantum Grav. 2 99-115. R. P. A. C. Newman (1993), 'On the structure of conformal singularities in classical general relativity', Proc. R. Soc. Lond. A443 473-49. K. Anguige and K. P. Tod (1999), 'Isotropic cosmological singularities I. Polytropic perfect fluid spacetimes', Ann. Phys. N.Y. 276 257-93.
3.1 A. Zee (2003), Quantum field theory in a nutshell, Princeton University Press.
3.2 There are good theoretical reasons (to do with electric charge conservation) for believing that photons are actually strictly mass-less. But as far as observations are conserved, there is an upper limit of m < 3 x 10-27 eV on the mass of the photon. G.V. Chibisov (1976), 'Astrofizicheskie verkhnie predely na massu pokoya fotona', Uspekhi fizicheskikh nauk 119 no. 3. 19 624.
3.3 There is a common use of the term 'conformal invariance' among some particle physicists, which is much weaker than the one being used here, namely that the invariance is a mere 'scale invariance', demanded only for the far more restricted transformations gi^Q2g for which Q is a constant.
3.4 There can, however, be an issue with regard to what is referred to as a conformal anomaly, according to which a symmetry of the classical fields (here the strict conformal invariance) may not hold exactly true in the quantum context. This will not be of relevance at the extremely high energies that we are concerned with here, though it could perhaps be playing a role in the way that conformal invariance 'dies off' as rest-mass begins to be introduced.
3.5 D.J. Gross (1992), 'Gauge theory - Past, present, and future?', Chinese J Phys. 30 no. 7.
3.6 The Large Hadron Collider is intended to collide opposing particle beams at an energy of 7x1012 electronvolts (1.12 J per particle, or lead nuclei at an energy of 574 TeV (92.0 J per nucleus.
3.7 The issue of inflation is discussed in §§3.4 and 3.6
3.8 S. E. Rugh and H. Zinkernagel (2009), 'On the physical basis of cosmic time', Studies in History and Philosophy of Modern Physics 40 1-19.
3.9 H. Friedrich (1983), 'Cauchy problems for the conformal vacuum field equations in general relativity', Comm. Math. Phys. 91 no. 4, 445-72. H. Friedrich (2002), 'Conformai Einstein evolution', in The conformai structure of spacetime: geometry, analysis, numerics (ed. J. Frauendiener, H. Friedrich) Lecture Notes in Physics, Springer. H. Friedrich (1998), 'Einstein's equation and conformai structure', in The geometric universe: science, geometry, and the work of Roger Penrose (eds. S.A. Huggett, L.J. Mason, K.P. Tod, S.T. Tsou, and N.M.J. Woodhouse), Oxford University Press.
3.10 An example of such an inconsistency problem is the so-called grandfather paradox in which a man travels back in time and kills his biological grandfather before the latter met the traveller's grandmother. As a result, one of the traveller's parents (and by extension the traveller himself) would never have been conceived. This would imply that he could not have travelled back in time after all, which means the grandfather would still be alive, and the traveller would have been conceived allowing him to travel back in time and kill his grandfather. Thus each possibility seems to imply its own negation, a type of logical paradox. René Barjavel (1943), Le voyageur imprudent (The imprudent traveller). [Actually, the book refers to an ancestor of the time traveller not his grandfather.]
3.11 This measure on T is a power of 'dpAdx', where dp refers to the momentum variable corresponding to the position variable x; see for example R. Penrose (2004), The road to reality, §20.2. If dx scales by a factor Q, then dp scales by Q-1. This scale invariance on T holds independently of any conformal invariance of the physics being described.
3.12 R. Penrose (2008), 'Causality, quantum theory and cosmology', in On space and time (ed. Shahn Majid), Cambridge University Press. R. Penrose (2009), 'The basic ideas of Conformal Cyclic Cosmology', in Death and anti-death, Volume 6: Thirty years after Kurt Godel (1906-1978) (ed. Charles Tandy), Ria University Press, Stanford, Palo Alto, CA.
3.13 Recent experiments at the Super-Kamiokande water Cherenkov radiation detector in Japan give a lower limit of the proton halflife of 6.6 x 1033 years.
3.14 Primarily pair annihilation; I am grateful to J.D. Bjorken for making this issue clear to me. J.D. Bjorken, S.D. Drell (1965), Relatavistic quantum mechanics, McGraw-Hill.
3.15 The observational situation, concerning neutrinos, at the moment, is that the differences between their masses cannot be zero, but the possibility of one of the three types of neutrino being mass-less seems still to be a technical possibility. Y. Fukuda et al. (1998), 'Measurements of the solar neutrino flux from Super-Kamiokande's first 300 days', Phys. Rev. Lett. 81 (6) 1158-62.
3.16 These operators are the quantities constructible from the generators of the group which commute with all the group elements.
3.17 H.-M. Chan and S. T. Tsou (2007), 'A model behind the standard model', European Physical Journal C52, 635-663.
3.18 Differential operators measure how the quantities that they act on vary in space-time; see the Appendices to see the explicit meanings of the 'V' operators used here.
3.19 R. Penrose (1965), 'Zero rest-mass fields including gravitation: asymptotic behaviour', Proc. R. Soc. Lond. A284 159-203.
3.20 In fact, in Appendix B1, my conventions as to whether g or g is Einstein's physical metric will be opposite to this, so that it will be 'Q-1' that tends to zero.
3.21 This depends upon the nature of the matter at U- being that of radiation, as in Tolman's radiation model described in §3.3, rather than the dust of Friedmann's model.
3.22 The 'differential' dQ/(1-Q2) is interpreted, according to Cartan's calculus of differential forms as a 1-form, or covector, but its invariance under Q^Q-1 is easily checked using standard rules of calculus: see, e.g., R. Penrose (2004), The road to reality, Random House.
3.23 I personally find the modern tendency to refer to 'dark energy' as contributing to the universe's matter density rather inappropriate.
3.24 Even obtaining a value that is too large by 120 orders of magnitude requires some act of faith in a 'renormalization procedure', without which the value would be obtained instead (see §3.5).
3.25 Determinations based on celestial mechanics provide constraints on the variation of G of (dG/dt)/G0 < 10-12/year.
3.26 R.H. Dicke (1961), 'Dirac's cosmology and Mach's principle', Nature 192 440-441. B. Carter (1974), 'Large number coincidences and the anthropic principle in cosmology', in IAU Symposium 63: Confrontation of Cosmological Theories with Observational Data, Reidel, pp. 291-98.
3.27 A. Pais (1982), Subtle is the Lord: the science and life of Albert Einstein, Oxford University Press.
3.28 R.C. Tolman (1934), Relativity, thermodynamics, and cosmology, Clarendon Press. W. Rindler (2001) Relativity: special, general, and cosmological. Oxford University Press.
3.29 The strict notion of analytic continuation is described in R. Penrose (2004), The Road to Reality, Random House.
3.30 A so-called 'imaginary number' is a quantity a which squares to a negative number, such as the quantity i, which satisfies i2 = -1. See R. Penrose (2004), The road to reality, Random House, §4.1.
3.31 B. Carter (1974), 'Large number coincidences and the anthropic principle in cosmology', in IAU Symposium 63: Confrontation of Cosmological Theories with Observational Data, Reidel, pp. 291-8. John D. Barrow, Frank J. Tipler (1988), The anthropic cosmological principle, Oxford University Press.
3.32 L. Susskind, 'The anthropic landscape of string theory arxiv:hep-th/0302219'. A. Linde (1986), 'Eternal chaotic inflation', Mod. Phys. Lett. A1 81.
3.33 Lee Smolin (1999), The life of the cosmos, Oxford University Press.
3.34 Gabriele Veneziano (2004), 'The myth of the beginning of time', Scientific American, May.
3.35 Paul J. Steinhardt, Neil Turok (2007), Endless universe: beyond the big bang, Random House, London.
3.36 See, for example, C.J. Isham (1975), Quantum gravity: an Oxford symposium, Oxford University Press.
3.37 Abhay Ashtekar, Martin Bojowald, 'Quantum geometry and the Schwarzschild singularity'. http://www.arxiv.org/gr-qc/0509075
3.38 See, for example, A. Einstein (1931), Berl. Ber. 235 and A. Einstein, N. Rosen (1935), Phys. Rev. Ser. 2 48 73.
3.41 There is good evidence for some far larger black holes in some other galaxies, the present record being an absolutely enormous black hole of mass ~1.8 x 1010 M©, about the same mass as an entire small galaxy, but there may also be a good many galaxies whose black holes are much smaller than our ~4 x 106 M© hole. The exact figure suggested in the text is far from being crucially important for the argument. My guess would be that it is actually somewhat on the low side.
3.42 J. D. Bekenstein (1972), 'Black holes and the second law', Nuovo Cimento Letters 4 737-740. J. Bekenstein (1973), 'Black holes and entropy', Phys. Rev. D7, 2333-46.
3.43 J. M. Bardeen, B. Carter, S.W. Hawking (1973), 'The four laws of black hole mechanics', Communications in Mathematical Physics 31 (2) 161-70.
3.44 In fact, a stationary black hole (in vacuum) needs only 10 numbers to characterize it completely: its location (3), its velocity (3), its mass (1), and its angular momentum (3), despite the vast numbers of parameters that would be needed to describe the way it was formed. Thus, these 10 macroscopic parameters would seem to label an absolutely enormous region of phase space, giving a huge entropy value by Boltzmann's formula.
3.46 L. Susskind (2008), The black hole war: my battle with Stephen Hawking to make the world safe for quantum mechanics, Little, Brown.
3.47 D. Gottesman, J. Preskill (2003), 'Comment on "The black hole final state"', hep-th/0311269. G.T. Horowitz, J. Malcadena (2003), 'The black hole final state', hep-th/0310281. L. Susskind (2003), 'Twenty years of debate with Stephen', in The future of theoretical physics and cosmology (ed. G.W. Gibbons et al.), Cambridge University Press.
3.48 It was early pointed out by Hawking that the pop itself would, technically, represent a momentary 'naked singularity' in violation of the cosmic-censorship conjecture. This is basically the reason why the hypothesis of cosmic censorship is restricted to classical general relativity theory. R. Penrose (1994), 'The question of cosmic censorship', in Black holes and relativistic stars (ed. R.M. Wald), University of Chicago Press.
3.49 James B. Hartle (1998), 'Generalized quantum theory in evaporating black hole spacetimes', in Black Holes and Relativistic Stars (ed. R.M. Wald), University of Chicago Press.
3.50 It is a well-known result of quantum theory referred to as the 'no-cloning theorem' which forbids the copying of an unknown quantum state. I see no reason why this should not apply here. W.K. Wootters, W.H. Zurek (1982), 'A single quantum cannot be cloned', Nature 299 802-3.
3.51 S.W. Hawking (1974), 'Black hole explosions', Nature 248 30. S.W. Hawking (1975), 'Particle creation by black holes', Commun. Math. Phys. 43.
3.52 For Hawking's newer argument, see 'Hawking changes his mind about black holes', published online by Nature (doi:10.1038/news040712-12). It is based on conjectural ideas that have relations to string theory. S.W. Hawking (2005), 'Information loss in black holes', Phys. Rev. D72 084013.
3.53 Schrodinger's equation is a complex first-order equation, and when time is reversed, the 'imaginary' number i must be replaced by —i (i = V —1); see note 3.30.
3.54 For further information, see R. Penrose (2004), The road to reality, Random House, Chs 21-3.
3.55 W. Heisenberg (1971), Physics and beyond, Harper and Row, pp. 73-6. See also A. Pais (1991), Niels Bohr's times, Clarendon Press, p. 299.
3.56 Dirac appears to have taken no interest in the issue of 'interpreting' quantum mechanics as it stands, in order to resolve the measurement issue, taking the view that current quantum field theory is, in any case, indeed just a 'provisional theory'.
3.57 P.A.M. Dirac (1982), The principles of quantum mechanics. 4th edn. Clarendon Press [1st edn 1930].
3.58 L. Diosi (1984), 'Gravitation and quantum mechanical localization of macro-objects', Phys. Lett. 105A 199-202. L. Diosi (1989), 'Models for universal reduction of macroscopic quantum fluctuations', Phys. Rev. A40 1165-74. R. Penrose (1986), 'Gravity and state-vector reduction', in Quantum concepts in space and time (eds. R. Penrose and C.J. Isham), Oxford University Press, pp. 129-46. R. Penrose (2000), 'Wavefunction collapse as a real gravitational effect', in Mathematical physics 2000 (eds. A. Fokas, T.W.B. Kibble, A.Grigouriou, and B.Zegarlinski), Imperial College Press, pp. 266-282. R. Penrose (2009), 'Black holes, quantum theory and cosmology' (Fourth International Workshop DICE 2008), J. Physics Conf. Ser. 174 012001. doi: 10.1088/1742-6596/174/1/012001
3.59 There is always a problem, when dealing with a universe which might be spatially infinite, that total values of quantities like entropy would come out as infinite. This is not too important a point, however, as with the assumption of a general overall spatial homogeneity, one can work, instead, with a large 'co-moving volume' (whose boundaries follow the general flow of matter).
3.60 S.W. Hawking (1976), 'Black holes and thermodynamics', Phys. Rev. D13(2) 191. G.W. Gibbons, M.J. Perry (1978), 'Black holes and thermal Green's function', ProcRoy. Soc. Lond. A358 467-94. N.D. Birrel, P.C.W. Davies (1984), Quantum fields in curved space, Cambridge University Press.
3.61 Personal communication by Paul Tod.
3.63 I think that my own viewpoint with regard to the 'information loss' that gives rise to black-hole entropy differs from that frequently expressed, in that I do not regard the horizon as the crucial location for this (since horizons are not locally discernable, in any case), but I take the view that it is really the singularity that is responsible for information destruction.
3.65 W.G. Unruh (1976), 'Notes on black hole evaporation', Phys. Rev. D14 870.
3.66 G.W. Gibbons, M.J. Perry (1978), 'Black holes and thermal Green's function', Proc Roy. Soc. Lond, A358 467-94. N.D. Birrel, P.C.W. Davies (1984), Quantum fields in curved space, Cambridge University Press.
3.67 Wolfgang Rindler (2001), Relativity: special, general and cosmo-logical, Oxford University Press.
3.68 H.-Y. Guo,C.-G. Huang, B. Zhou(2005), Europhys.Lett.72 1045-51.
3.69 It might be objected that the region covered by the Rindler observers is not the whole of M, but this objection applies also to D.
3.70 J. A. Wheeler, K. Ford (1995), Geons, black holes, and quantum foam, Norton.
3.71 A. Ashtekar, J. Lewandowski (2004), 'Background independent quantum gravity: a status report', Class. Quant. Grav. 21 R53-R152. doi:10.1088/0264-9381/21/15/R01, arXiv:gr-qc/0404018.
3.72 J.W. Barrett, L. Crane (1998), 'Relativistic spin networks and quantum gravity', J. Math. Phys. 39 3296-302. J.C. Baez (2000), An introduction to spin foam models of quantum gravity and BF theory. Lect. Notes Phys. 543 25-94. F. Markopoulou, L. Smolin (1997), 'Causal evolution of spin networks', Nucl. Phys. B508 409-30.
3.73 H.S. Snyder (1947), Phys. Rev. 71(1) 38-41. H.S. Snyder (1947), Phys. Rev. 72(1) 68-71. A. Schild (1949), Phys. Rev. 73, 414-15.
3.74 F. Dowker (2006), 'Causal sets as discrete spacetime', Contemporary Physics 47 1-9. R.D. Sorkin (2003), 'Causal sets: discrete gravity', (Notes for the Valdivia Summer School), in Proceedings of the Valdivia Summer School (ed. A. Gomberoff and D. Marolf), arXiv:gr-qc/0309009.
3.75 R. Geroch, J.B. Hartle (1986), 'Computability and physical theories', Foundations of Physics 16 533-50. R.W. Williams, T. Regge (2000), 'Discrete structures in physics', J.Math.Phys. 41 3964-84.
3.76 Y. Ahmavaara (1965), J. Math. Phys. 6 87. D. Finkelstein (1996), Quantum relativity: a synthesis of the ideas of Einstein and Heisenberg, Springer-Verlag.
3.77 A. Connes (1994), Non-commutative geometry, Academic Press.
S. Majid (2000), 'Quantum groups and noncommutative geometry', J. Math. Phys. 41 (2000) 3892-942.
3.78 B. Greene (1999), The elegant universe, Norton. J. Polchinski (1998), String theory, Cambridge University Press.
3.79 J. Barbour (2000), The end of time: the next revolution in our understanding of the universe, Phoenix. R. Penrose (1971), 'Angular momentum: an approach to combinatorial space-time', in Quantum theory and beyond (ed. T. Bastin), Cambridge University Press.
3.80 For an explanation of twistor theory, see R. Penrose (2004), The road to reality, Random House, ch. 33.
3.81 G. Veneziano (2004), 'The myth of the beginning of time', Scientific American (May). See also Note 3.34.
3.82 R. Penrose (2004), The road to reality, Random House, §28.4.
3.83 'Realizing' a quantum fluctuation as an actual irregularity in a classical matter distribution actually requires a manifestation of the R-process referred to towards the end of §3.4, which is not part of a unitary evolution U.
3.84 D.B. Guenther, L.M. Krauss, P. Demarque (1998), 'Testing the constancy of the gravitational constant using helioseismology', Astrophys. J. 498 871-6.
3.85 In fact, there are standard procedures for taking into account the evolution from U- to WC This was not applied in Hajian's preliminary analysis of the CMB data (to be described shortly in the text), however.
3.86 Such distortions of circular shape could also occur in the previous aeon, though my guess is that this would be a smaller effect. In any case, if these occur, their effects would be much harder to deal with, and would be a great nuisance for the analysis, for a multitude of reasons.
3.87 V.G. Gurzadyan, C.L. Bianco, A.L. Kashin, H. Kuloghlian, G. Yegorian (2006), 'Ellipticity in cosmic microwave background as a tracer of large-scale universe', Phys. Lett. A 363 121-4. V.G. Gurzadyan, A.A. Kocharyan (2009), 'Porosity criterion for hyperbolic voids and the cosmic microwave background',
Astronomy and Astrophysics 493 L61-L63 [DOI: 10.1051/0006361:200811317]
A.1 R. Penrose, W. Rindler (1984), Spinors and space-time, Vol. I: Two-spinor calculus and relativistic fields, Cambridge University Press. R. Penrose, W. Rindler (1986), Spinors and space-time, Vol. II: Spinor and twistor methods in space-time geometry, Cambridge University Press.
A.2 P.A.M. Dirac (1982), The principles of quantum mechanics, 4th edn. Clarendon Press [1st edn 1930]. E.M. Corson (1953) Introduction to tensors, spinors, and relatavistic wave equations. Blackie and Sons Ltd.
A.3 C.G. Callan, S. Coleman, R. Jackiw (1970), Ann. Phys. (NY) 59 42. E.T. Newman, R. Penrose (1968), Proc. Roy. Soc., Ser. A 305 174.
A.4 This is the spin-2 Dirac-Fierz equation, in the linearized limit of general relativity. Dirac, P.A.M. (1936), Relativistic wave equations. Proc. Roy. Soc. Lond. A155, 447-59. M. Fierz, W. Pauli (1939), 'On relativistic wave equations for particles of arbitrary spin in an electromagnetic field', Proc. Roy. Soc. Lond. A173 211-32.
B.1 It may well be that the present formalism should be modified so that a decaying rest-mass in VA, in accordance with §3.2, is also incorporated. However, this would be likely to complicate matters considerably, so for the moment I am restricting attention to situations that can be well treated with the assumption that our 'collar' contains no rest-mass in VA.
B.2 I do not believe that A=A is, in itself, is a big assumption; it is just an issue of convenience. As things stand, it is merely a matter of arranging that any changes in physical constants that might occur from one aeon to the next are taken up by other quantities. As a further comment, it may be remarked that an alternative to the standard 'Planck units' introduced in §3.2, one might consider replacing the condition G = 1 by A = 3, as this fits in well with the formalism of CCC as presented here.
B.3 E. Calabi (1954), 'The space of Kahler metrics', Proc. Internat. Congress Math. Amsterdam, pp. 206-7.
B.4 Phantom field: this term has also been used, in the literature, in various other somewhat different senses.
B.7 The full freedom is given by the replacement Qi^(AQ+B)/(BQ+A), with A and B constant, whereby n^n. But this ambiguity is dealt with by the demand that Q have a pole (and rn a zero) at X.
B.8 K.P. Tod (2003), 'Isotropic cosmological singularities: other matter models', Class. Quant. Grav. 20 521-34. [DOI: 10.1088/02649381/20/3/309]
B.10 This operator was apparently introduced, in effect, by C.R. LeBrun ((1985), 'Ambi-twistors and Einstein's equations', Classical Quantum Gravity 2 555-63) in his definition of 'Einstein bundle' in twistor theory. It forms part of a much more general family of operators introduced by Eastwood and Rice (M.G. Eastwood and J.W. Rice (1987), 'Conformally invariant differential operators on Minkowski space and their curved analogues', Commun. Math. Phys. 109 207-28, Erratum, Commun. Math. Phys. 144 (1992) 213). It has relevance also in other contexts (M.G. Eastwood (2001), 'The Einstein bundle of a nonlinear graviton', in Further advances in twistor theory vol III, Chapman & Hall/CRC, pp. 36-9. T.N. Bailey, M.G. Eastwood, A.R. Gover (1994), 'Thomas's structure bundle for conformal, projective, and related structures', Rocky Mtn. Jour. Math. 24 1191-217.) It has come to be known as the 'conformal to Einstein' operator. see also footnote on p.124 of R. Penrose, W. Rindler (1986), Spinors and space-time, Vol. II: Spinor and twistor methods in space-time geometry, Cambridge University Press.
B.11 This interpretation was pointed out to me by K.P. Tod. In Penrose and Rindler (1986), the condition is referred to as the 'asymptotic Einstein condition'. R. Penrose, W. Rindler (1986), Spinors and space-time, Vol. II: Spinor and twistor methods in space-time geometry, Cambridge University Press.
B.12 There are other ways of seeing this effective sign change in the gravitational constant, one being in the comparison between the 'Grgin behaviour' of the radiation field and the 'anti-Grgin behaviour' of the gravitational sources as conformal infinity is crossed; see Penrose and Rindler (1986), §9.4, pp. 329-32. R. Penrose, W. Rindler (1986), Spinors and space-time, Vol. II: Spinor and twistor methods in space-time geometry, Cambridge University Press.
B.13 K.P. Tod, personal communication.
Was this article helpful?