Although it’s well written and the development of his line
of reasoning is clear it’s not at all easy to comprehend. I did mathematics as a secondary subject to
an engineering degree and the concepts that he uses go well beyond that
level. If you really want to understand
his arguments properly you need to have studied, to graduate level, Einstein’s Theory
of General Relativity and his use of tensors, together with statistical thermodynamics
and quantum theory, as well as have a familiarity with current and historical
cosmological models and the physics and mathematics of black holes. People have written entire books and devoted
their lives to researching these subjects and he assumes that you are up to
date with their findings. I found that
it was just about possible to follow the outline of his ideas and the very numerous
diagrams helped a lot.

Whilst he’s prepared to speculate in certain areas he sticks
rigidly to the correctness of the second law of thermodynamics. He emphasizes throughout the definition of entropy derived from statistical mechanics. Current experimental evidence indicates that the universe is expanding at an increasing rate. It’s widely accepted that at the Big Bang the entropy of the universe was low and has increased continually thereafter. During the evolution of the universe many black holes form, some of them at the centre of galaxies are extremely massive. At one stage Stephen Hawking thought that matter entering a black hole entailed a destruction of entropy, which is against the second law of thermodynamics, but he withdrew that opinion in 2004. According to Stephen Hawking’s widely accepted theoretical research, a black hole very very gradually evaporates by radiating at a very low temperature until it eventually disappears altogether with a small pop”. Thus the entropy swallowed by the black hole is eventually released in the form of radiation.

But not all matter will end up in black holes, whatever the age of the universe. Some matter will just cool down and stay there. Here Penrose is forced to accept a speculation that allows him to dispose of this matter over a very long period of time. He postulates that matter in itself has a propensity to decay and the decay products to annihilate each other and disappear as radiation, to return, if you like, to its equivalence in energy. The reason that he is forced down this route of allowing matter to evaporate, as well as the black holes, is that his mathematical construct of Cyclic Conformal Cosmology requires that time becomes impossible to measure at the limiting condition. If there's no matter and just radiation, then there’s no way to construct even an atomic clock, so time itself becomes irrelevant in the expression of space time and by using conformal mathematical concepts he postulates that the cosmological cycle restarts from this state.

I’m afraid that this is where my understanding fails me. In the way that he presents it diagrammatically it seems possible. Mathematically you are equating the low entropy state of the origins of the Big Bang with the low entropy state that would exist once all of the black holes and matter had turned into low temperature radiation but how the latter, in the form of dispersed radiation, turns into the former, a singularity containing the entire energy of the cosmos, leaves me perplexed. It can be made to work mathematically, but as in many things mathematical, it does not seem to relate to any normal concept of reality. It’s an interesting idea nonetheless.

In the very last part of the book he speculates that
it might be possible to detect the results of gravitational waves resulting
from close encounters between massive black holes which took place

*BEFORE*the big bang. These would show as ripples appearing in the Cosmic Microwave Background Radiation. He persuaded David Spergel and his research assistant Amir Hajian at Princeton University to look for these in the CMB data and, when spurious spikes from regions around the galactic plane were suppressed, the results were not encouraging. Then they tried using elliptical shapes instead of circles to look for correlations and they had more success but this might be due to lensing by the gravitational curvature of space time introduced by massive objects in the current aeon as the CMB radiation traverses our universe. Roger Penrose hopes that by correcting for these anomalies he will be able to confirm his ideas about Conformal Cyclic Cosmology.
I wish him good luck.

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