## 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...