## A6 Conformal rescalings

In accordance with the conformal rescaling (with H > 0 smoothly varying) we adopt the abstract-index relations gab H-2 gab, eab H ab, sab H-1 ab a-b- H EA'B', Sa'b' H-1 ab'. so that the action of va on a general quantity written with spinor indices is generated by Vaa-0 Vaa-0, Vaa- b Vaa' b - Yba- a, Vaa- b Vaa' b'- Yab- a-, where the treatment of a quantity with many lower indices being built up from these rules, one term for each index. (Upper indices have a corresponding treatment, but...

## A5 Bianchi identities

The general Bianchi identity V oRbc de 0, in spinor-indexed form, becomes (P& R 4.10.7, 4.10.8) Vg Wabcd VfB > CD)A-B- and Vca'Ocda-b-+ Vdb-R 0. When R is a constant a situation that arises with Einstein's equations when the sources are massless we have VCA Qcda-b- 0, whence VgWabcd V 'Ocda-b-, the symmetry in BCD on the right being implied. Incorporating the Einstein equation, with massless sources, we get (see P& R 4.10.12). Note that when Tabc-d- 0, we obtain the equation (P& R...

## Of Time

An Extraordinary New View of the Universe Roger Penrose has asserted his right under the Copyright, Designs and Patents Act 1988 to be identified as the author of this work This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition, including this condition, being...

## B3 The role of the phantom field

I shall refer to H, regarded as a particular instance of the massless self-coupled conformally invariant field to, as the phantom field. B4 It does not provide us with physically independent degrees of freedom its presence (in the gab-metric) simply allows us the scaling freedom that we need, so that we can rescale the physical metric to obtain a smooth metric gab, conformal to Einstein's physical metric, which smoothly covers each of the joins from one aeon to the next. With the aid of such...

## Appendix B Equations at crossover

As in Appendix A, conventions including the use of abstract indices are as in Penrose and Rindler (1984, 1986), but with the cosmological constant here denoted by A, rather than by the 'A' of that work, the scalar curvature quantity there referred to as 'A' being 24R. There are some aspects of the detailed analysis presented in what follows that are incomplete and provisional, and it is likely that refinements of these proposals will be needed for a more complete treatment. Nevertheless, we do...

## Preface

One of the deepest mysteries of our universe is the puzzle of whence it came. When I entered Cambridge University as a mathematics graduate student, in the early 1950s, a fascinating cosmological theory was in the ascendant, known as the steady-state model. According to this scheme, the universe had no beginning, and it remained more-or-less the same, overall, for all time. The steady-state universe was able to achieve this, despite its expansion, because the continual depletion of material...

## B11 The matter content of Vv

To see what our equations look like physically in the post-big-bang region Vv, we must rewrite things in terms of the 'reverse-hatted' quantities, with the metric gab to2gab, with Q to-1. As mentioned earlier, I shall write the total post-big-bang energy tensor as Uab, in order to avoid confusion with the conformally rescaled energy tensor of the (massless) matter entering Vv from VA Since Tab is traceless and divergence-free, this must hold for fab also (the scalings being in accordance with...

## Understanding the way the Big Bang was special

Let us return to the basic question that we have been trying to address in this part, namely the issue of how our universe happened to come about with a Big Bang that was so extraordinarily special yet special in what appears to have been a very peculiar way where, with regard to gravity, its entropy was enormously low in comparison with what it might have been, but the entropy was close to maximum in every other respect. This issue tends to be muddied, in most modern cosmological...

## Epilogue

Tom looked incredulously at Aunt Priscilla, then he said And that's the craziest idea I ever heard of Tom strode off to make his way to his aunt's car which would drive him home, and his aunt followed a short way behind. But presently he paused, to examine the raindrops falling on a large pond, to one side of the mill. The rain had by now tailed off considerably, to form a faint drizzle, and the impacts of individual raindrops were now clearly seen. Tom watched them for a while and he couldn't...

## B1 The metrics gab Qab and gab

We examine the geometry in the neighbourhood of a crossover 3-surface k, in accordance with the ideas of Part 3, where it is assumed that there is a collar V, of smooth conformal space-time containing k, which extends both to the past and to the future of k, within which only massless fields are present within V prior to the crossover U We choose a smooth metric tensor gab in this collar, consistent with the given conformal structure locally at least and in an initially somewhat arbitrary way....

## The structure of CCC

There are various aspects of this proposal that require a good deal more detailed attention than I have given above. One key issue concerns what the full contents of the universe might be likely to be in the very remote future. The discussion above concentrated mainly on the considerable background of photons that would be present, from starlight, from the CMB, and from black-hole Hawking evaporation. I have also considered that there would be a significant contribution to this background from...

## Appendix A Conformal rescaling 2spinors Maxwell and Einstein theory

Most of the detailed equations that I give here take advantage of the 2-spinor formalism. This is not a matter of necessity, since the more familiar 4-tensor description could have been provided pretty well throughout, as an alternative. However, not only is the 2-spinor formalism simpler when it comes to expressing conformal invariance properties (see A6), but it also provides a more systematic overview when it comes to understanding the propagation of massless fields and the corresponding...

## B7 Dynamics across k

How do we expect that our dynamical equations will allow us to propagate across kin an unambiguous way I am supposing that in the remote future of the earlier aeon, Einstein's equations hold, all sources being massless and propagating according to well-defined deterministic conformally invariant classical equations. We may suppose that these are Maxwell's equations, the Yang-Mills equations without mass, and things like the Dirac-Weyl equation Vaa' > a 0 (the Dirac equation in the zero-mass...

## Prologue

With his eyelids half closed, as the rain pelted down on him and the spray from the river stung his eyes, Tom peered into the swirling torrents as the water rushed down the mountainside. 'Wow', he said to his Aunt Priscilla, an astrophysics professor from the University of Cambridge, who had taken him to this wonderful old mill, preserved in excellent working order, 'is it always like this No wonder all that old machinery can be kept buzzing around at such great speed.' 'I don't think it's...

## A4 Massless gravitational sources

In Appendix B, we shall be particularly concerned with the Einstein field equations when the (symmetric) source tensor Tab is trace-free since this is appropriate for massless (i.e. zero rest-mass) sources, telling us that the spinor-indexed quantity Taba-b- Ta-b-ab Tab has the symmetry The divergence equation VaTab 0, i.e. Vaa'Taba-b- 0, can be re-expressed Vg Tcda-b- V B Tcd)a-b'. The Einstein equations above are now (P& R 4.6.32) Oaba-b- 4nGTab, R 4A. When rest-mass is present, so that...

## VK o

This has some curious consequences, which are of considerable importance for CCC. As we approach + from its past, we need to use a conformal factor n which tends to zero smoothly, 320 but with a nonzero normal derivative. The geometrical meaning of this is illustrated in Fig. 3.5. The conformal invariance of the wave-propagation equation for K implies that it attains finite (and usually non-zero) values on +, these values determining the strength (and polarization) of the gravitational...

## A8 Scaling of zero restmass energy tensors

It should be noted that for an energy tensor Tab that is trace-free (Taa 0), we find that the scaling (P& R 5.9.2) preserves the conservation equation VaTab 0, since we find In Maxwell theory, we have an expression for the energy tensor in terms of Fab that translates into spinor form as (P& R 5.2.4) Tab 2 VAB ty A B '. In the case of Yang-Mills theory, we simply have extra indices For a massless scalar field, subject to the equation ( + f)0 0 considered earlier (P& R 6.8.30), we have...

## B10 To eliminate spurious gmetric freedom

An issue that presents itself at this stage is that, according to the requirements of CCC, we want a unique propagation into Vv. This would not be problematic were it not for the unwanted additional freedom arising from an arbitrariness in the conformal factor. As things stand, this freedom provides us with some spurious degrees of freedom, which would inappropriately influence the non-conformally-invariant gravitational dynamics of Vv. These spurious degrees of freedom need to be eliminated in...

## Acknowledgements

I am very grateful to many friends and colleagues for their important inputs, and for sharing their thoughts with me relating to the cosmo-logical scheme that I am putting forward here. Most importantly, detailed discussions with Paul Tod, concerning the formulation of his proposal for a conformal-extension version of the Weyl curvature hypothesis have been crucially influential, and many aspects of his analysis have proved vital to the detailed development of the equations of conformal cyclic...

## A3 Spacetime curvature quantities

The (Riemann-Christoffel) curvature tensor Rabcd has the symmetries Rabcd R ab cd Rcdab, R abc d 0, and relates to commutators of derivatives via (P& R 4.2.31) This fixes the choice of sign convention for Rabcd. We here define the Ricci and Einstein tensors and the Ricci scalar, respectively, by Rac Rabcb, Eab Rgab - Rab, where R Raa, and the Weyl conformal tensor Cabcd is defined by (P& R 4.8.2) Cabcd Rabcd -2 R alcgbf + fRg acgb d, this having the same symmetries as Rabcd but, in...

## B8 Conformally invariant Dab operator

To help us to understand the physical implications for Vv, and to see how the Einstein equations for that region will operate, let us first examine Tab Q explicitly Tab Q Q2 QVa(A'Vb')bQ-1+ OaBA'B' which, with to - Q-1, we can rewrite as Va(a-Vb-)b + OABAB-jto 4nGto3Tab Q . This is an interesting equation in that the 2nd-order operator on the left, when acting on a scalar quantity of conformal weight 1 (where the extra symmetry over AB plays no role when the operator acts, as here, on a...

## N dft

The two most important things about n are, first, that it remains smooth over the crossover 3-surface and, second, that it is unchanged under the replacement ft ft-1. In CCC, we try to demand that n indeed be a smoothly varying quantity over the crossover, so that if we take n to define the required scaling information, rather than ft, then we can imagine that the transition fti ft-1 at crossover can be achieved while n remains smooth across it. This requires certain mathematical conditions to...

## A2 Massless freefield Schrodinger equation

This last equation is the case n 2 of the massless free-field equation (P& R 4.12.42), or 'Schrodinger equation' A2 for a massless particle of spin n ( > 0) Vaa'(abc e 0, where > abc e has n indices and is totally symmetric For the case n 0, the field equation is usually taken to be 0, where the D'Alembertian operator is defined by but in curved space-time, we need the operator Va to refer to covariant differentiation, and the form of equation (P& R 6.8.30) will be preferred here, as...

## VF o

This (conformally invariant) set of equations governs the propagation of electromagnetic waves (light) and it can also be regarded as the quantum-mechanical Schrodinger equation satisfied by individual free photons (see 3.4 and Appendix A2, A6). In the case of gravity, the source 0 -tensor E (Einstein tensor, taking the place of J see 2.6) does not have a scaling behaviour which provides conformal invariance for the equations, but there is a conformally invariant analogue of VF 0, which governs...

## Earlier preBig Bang proposals

The scheme of CCC may be contrasted with a number of other proposals for pre-Big-Bang activity, which had been put forward previously. Even among the earliest cosmological models consistent with Einstein's general relativity, namely those of Friedmann put forward in 1922, there was one that became referred to as the 'oscillating universe'. This terminology seems to have arisen from the fact that for the closed Friedmann model without cosmological constant (K> 0, A 0 see Fig. 2.2(a)), the...

## B2 Equations for VA

In what follows, I first consider equations relating to the region VA and deal with Vv afterwards (see B11). We can express the transformation law of the Einstein (and Ricci) tensor as (P& R 6.8.24) i> ABA-B-- Oaba-b- HVa(a'Vb')bH-1 - H-1V a(a-Vb-)bH This last equation has considerable pure-mathematical interest, being an instance of what is referred to as the Calabi equation.iB3 But it also has physical interest, being the equation for a conformally invariant self-coupled scalar field w...

## B4 The normal N to k

We observe that Q , as J* ( k) is approached from below, since the role of Q is to scale up the finite g-metric at J* by an infinite amount, to become the remote future of the earlier aeon. However, we find that the quantity approaches zero from below, at J+, in a smooth way (the minus sign being needed for what follows), and it does this so that the quantity is non-zero on the cross-over 3-surface k( J+), and so provides us, at points of k with a future-pointing timelike 4-vector N normal to k...

## A7 Yang Mills fields

It is important to observe that the Yang-Mills equations, that form the basis of our current understanding of both the strong and the weak forces of particle interactions, are also conformally invariant, so long as we can ignore the introduction of mass which may be taken to be through the subsequent agency of the Higgs field. The Yang-Mills field strengths can be described by a tensor quantity (a 'bundle curvature') where the (abstract) indices 0, r, refer to the internal symmetry group (U(2),...

## The robustness of the entropy concept

Matters concerning the entropy of the entire cosmos can be left aside for the time being. For the moment, we can just appreciate the value of Boltzmann's formula, for it provides us with an excellent notion of what the entropy of a physical system should be actually defined to be. Boltzmann put forward this definition in 1875, and it represented an enormous advance on what had gone before, 15 so that it now becomes possible to apply the entropy concept in completely general situations, where no...

## Why is the past different

Why has our reasoning gone so sadly astray this being apparently just the same reasoning that seemed convincingly to lead us to expect that the Second Law, with overwhelming probability, must hold for the future evolution of an ordinary physical system The trouble with the reasoning, as I have provided it, lies in the assumption that the evolution can be regarded as effectively 'random' in relation to the coarse-graining regions. Of course it is not really random, as noted above, since it is...

## B12 Gravitational radiation at k

One feature of the infinite conformai rescaling of the metric, as we pass from VA (with metric gab) to Vv (with metric gab) via k(with metric gab) is the way in which gravitational degrees of freedom, initially present and described in the g-metric by tyabcd (usually non-zero at k), become transferred to other quantities in the g-metric. Whereas we have (A9, P& R 6.8.4) ABCD ty ABCD - W tyABCD - W21pABCD so that gravitational radiation is very greatly suppressed in the big bang. However, the...

## Connecting with infinity

Physically, what would the material universe have been actually like, far far back in time, very soon after the Big Bang One thing in particular it would have to have been hot extremely hot. The kinetic energy in the motions of particles around at that time would have been so enormous as to have completely overwhelmed the particles' comparatively tiny rest energies (E mc2, for a particle of rest-mass m). Thus, the rest-mass of the particles would have been effectively irrelevant as good as zero...

## Notes

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...

## Observational implications

The question I now wish to address is whether we can find any specific evidence either for or against the actual validity of CCC. It might have been thought that any evidence concerning a putative 'aeon' existing prior to our Big Bang must be well beyond any observational access, owing to the absolutely enormous temperatures arising at the Big Bang that would seem to obliterate all information, thereby separating us from all that supposed previous activity. We should bear in mind, however, that...

## A1 The 2spinor notation Maxwell equations

The 2-spinor formalism employs quantities with abstract spinor indices (for the complex 2-dimensional spin-space) for which I use italic capital Latin letters, either unprimed (A, B, C, ) or primed (A', B', C', ), these being interchanged under complex conjugation. The (complexified) tangent space at each space-time point is the tensor product of the unprimed with the primed spin-space. This enables us to adopt the abstract-index identification where the italic lower-case Latin index letters a,...

## B9 Keeping the gravitational constant positive

We can gain more insights into the interpretation of the physics that is implied for us by CCC, if we examine the interaction between the mass-less gravitational source fields, as described by Tab, and the gravitational field (or 'graviton field') tyabcd, as implied by the equation (P& R 4.10.12) of A5, taken in 'hatted form', and rewritten in terms of w - H-1. We have V (- w abcd) 4nGVg'((- wfTcDA-B-), from which we derive the equivalent equation, in terms of the 'unhatted' quantities,...

## The inexorable increase of entropy into the future

Let us try to get some understanding of why it is to be expected that the entropy should increase when a system evolves into the future, as the Second Law demands. Suppose we imagine that our system starts off in a state of reasonably low entropy so that the point p, which is to move through phase space T thereby describing the time-evolution of the system, starts off at a point p0 in a fairly small coarse-graining region 0 (see Fig.1.15). We must bear in mind that, as noted above, the various...

## Tt

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. 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...

## B6 The reciprocal proposal

There is of course the awkwardness, in our particular situation here, that in describing the transition fromVA to Vv we do not have a smoothly varying quantity in either H or w which describes the scaling back to both Einstein metrics gab and gab in a uniform way. But an appropriate proposal for addressing this issue does indeed appear to be to adopt the reciprocal proposal w - H-1, referred to above, and it is then convenient to consider the 1-form n, defined by since this 1-form is then...

## Squaring the Second

Let us therefore return to the question which started out this whole enterprise, namely the origin of the Second Law. The first point to be made is that there is a conundrum to be faced. It is a conundrum that appears to confront us irrespective of CCC. The issue has to do with the evident fact that the entropy of our universe or the current aeon, if we are considering CCC seems to be vastly increasing, despite the fact that the very early universe and the very remote future appear to be...

## The ubiquitous microwave background

In the 1950s, a popular theory of the universe was one referred to as the steady state model, a proposal first put forward by Thomas Gold and Hermann Bondi in 1948, and soon taken up in more detail by Fred Hoyle, 213 who were all at Cambridge University at the time. The theory required there to be a continual creation of material throughout space, at an extremely low rate. This material would have to be in the form of hydrogen molecules each being a pair consisting of one proton and one...

## Phase space and Boltzmanns definition of entropy

We are still not finished with the definition of entropy, however, for what has been said up to this point only half addresses the issue. We can see an inadequacy in our description so far by considering a slightly different example. Rather than having a can of red and blue paint, we might consider a bottle which is half filled with water and half with olive oil. We can stir it as much as we like, and also shake the bottle vigorously. But in a few moments, the olive oil and the water will...

## Our expanding universe

The Big Bang what do we believe actually happened Is there clear observational evidence that a primordial explosion actually took place from which the entire universe that we know appears to have originated And, central to the issues raised in Part 1 is the question how can such a wildly hot violent event represent a state of extraordinarily tiny entropy Initially, the main reason for believing in an explosive origin for the universe came from persuasive observations by the American astronomer...

## Black holes and spacetime singularities

In most physical situations, where the effects of gravity are comparatively small, the null cones deviate only slightly from their locations in Minkowski space M. However, for a black hole, we find a very different situation, as I have tried to indicate in Fig. 2.24. This space-time picture represents the collapse of an over-massive star (perhaps ten, or more, times the mass of our Sun) which, having exhausted its resources of internal (nuclear) energy, collapses unstoppably inwards. At a...

## Spacetime null cones metrics conformal geometry

When, in 1908, the distinguished mathematician Hermann Minkowski who had coincidentally been one of Einstein's teachers at the Zurich Polytechnic demonstrated that he could encapsulate the basics of special relativity in terms of an unusual type of 4-dimensional geometry, Einstein was less than enthusiastic about the idea. But later he realized the crucial importance of Minkowski's geometric notion of space-time. Indeed, it formed an essential ingredient of his own generalization of Minkowski's...

## A9 Weyl tensor conformal scalings

The conformal spinor Wabcd encodes the information of the conformal curvature of space-time, and it is conformally invariant (P& R 6.8.4) We note the curious (but important) discrepancy between this conformal invariance and that needed to preserve satisfaction of the massless free field equations, where there would be a factor H-1 on the right. To accommodate this discrepancy, we can define a quantity tyabcd which is everywhere proportional to Wabcd, but which scales according to and we find...

## The relentless march of randomness

The Second Law of thermodynamics what law is this What is its central role in physical behaviour And in what way does it present us with a genuinely deep mystery In the later sections of this book, we shall try to understand the puzzling nature of this mystery and why we may be driven to extraordinary lengths in order to resolve it. This will lead us into unexplored areas of cosmology, and to issues which I believe may be resolved only by a very radical new perspective on the history of our...

## CCC and quantum gravity

The scheme of CCC provides us with a different outlook on various intriguing issues in addition to the Second Law that have confronted cosmology for many years. In particular, there is the question of how we are to view the singularities that arise in the classical theory of general relativity, and of how quantum mechanics enters into this picture. We find that CCC has something particular to say, not only about the nature of the Big-Bang singularity, but also about what happens when we try to...

## Entropy as state counting

But how does the physicist's notion of 'entropy', as it appears in the Second Law, actually quantify this 'randomness', so that the self-assembling egg can indeed be seen to be overwhelmingly improbable, and thereby rejected as a serious possibility In order to be a bit more explicit about what the entropy concept actually is, so that we can make a better description of what the Second Law actually asserts, let us consider a physically rather simpler example than the breaking egg. The Second...