For those who are interested, the list of sponsoring organizations were: the University of Nice, the Mathematics faculties of Jussieu, Universite Diderot, Universite Pierre and Marie Curie, the ENS, the Ministry of Higher Education, the Astro-particle and Cosmology laboratory of Paris 7, and the Paris Observatory.
Not only were the speakers international celebrities in this subject, the titles of their lectures were equally succulent:
The windows were covered with thick velvet curtains, the lighting dank, the auditorium over-heated. I could not help reflecting that, if this is what France provides for the elite of its intelligent youth, what goes on in the rest of the school system? My impression is that, before 1968, if you were French, young, wanted an intellectual career but did not graduate from the ENS, you had to emigrate! It is said that things have changed since then.
I recognized some faces, certainly those of the speakers, but saw no-one that I knew from before. All the talks were in English, the generic language of science in today's world.
For me the opening talk by Roger Penrose was the most informative. Carlo Rovelli's talk was also of interest, though I was unfamiliar with some of the technical background. The talk by Dijkgraaf was frankly a disappointment: a survey talk about the facets and phases of theoretical quantum gravity, with overly slick power point pictures and little new content.
It was impossible to take notes for the talk by Alain Connes: the material was too dense, too rich, with several ideas being expressed simultaneously, in a good yet overly rapid English. At the cocktail party after the conference I expressed the view that he was the kind of person who had so many ideas in his mind at once that he couldn't convey them coherently. The student I was talking to replied: "Yes: it was indigestible!" Well: there is a flattering and not so flattering way of describing everything.
The impression that he gave was as a proselytizer for an analysis of cosmic structure based on the spectral properties of operators: All Is Eigenvalues, if I understood him correctly.
After an hour or so, just when we thought he was beginning to wind down, Alain Connes began a whole new presentation, a list of arguments in defense of his reasons for introducing p-adic numbers into physics! This meant explaining to most of the audience what p-adic numbers are; but soon after he'd begun doing this he got side-tracked into the subject of operator algebras of characteristic 1, the connection being that they also violate the Archimedean Axiom of Hilbert's reform of Euclidean Geometry.
I didn't stay for the talk by Maxim Kontsevich talk. It was not only that the talk by Alain Connes had so thoroughly bewildered me, but also because of a personal ordeal, which I will describe in its place, had consumed the two-hour lunch break. I therefore left after the talk by Connes, rested up and came back for the cocktail party. This resulted in several agreeable conversations, nothing noteworthy.
Owing to the factors described above, this report will confine itself to the talks by Roger Penrose and Carlo Rovelli.
And, as much as any fabled Old Testament prophet, he came well-supplied with revelations! That we were going to be the lucky ones to receive his new ideas became apparent from the first transparency that he manipulated onto the platen of the over-head projector. On it were drawn words in huge letters:
He asked us to forget Einstein's metric General Relativity (GR), which is governed by a group of 10 generators (The Poincare group), and confine our attention to Conformal Geometry, which has 9 generators and lacks a metric. In ordinary GR, there is a light cone at every point in space-time. On the horizon of a Black Hole these cones point inward. Conformal Geometry, he explained, is the geometry of the light cones, ignoring both the individual points and the metric. (Julian Barbour has interesting things to say about this approach, which he calls "Shape Dynamics") Some additional structure is needed to account for apparent mono-directionality of time: the customary categories of past, present and future. His confomal geometry retains this structure. None of this is particularly new. Now, however, he asked us to think of mass as the mediator that determines the rate at which clocks tick. Rest mass is the basis for the metric structure that can measure the passage of time. In justification of this he wrote down these classical equations:
E =hν, where ν is frequency
E = Mc²
Eliminating E, one gets M = ν (c²/h), which can be interpreted as: rest-mass and frequency differ by a constant of proportionality.
Since it follows that Conformal Geometry has no metric structure, one concludes that it no massive particles and is the natural geometry for massless particles like photons. It also has no clocks, so one cannot speak of the "passage of time". As Roger Penrose expressed it:
"Mass breaks the (symmetry of the) metric geometry".
The next transparency that flashed onto the screen displayed the Standard Model, from the expansion of the universe after the initial Big Bang, to the future boundary of space-time. Normally Penrose, Hawking, Ellis and others talk about this boundary as being off at infinity. The new idea that Penrose brought to this conference was the idea that this "future boundary" is not infinitely far way, but occurs essentially at the moment when ALL of the Black Holes have disappeared, in the form of Hawking radiation!
In Stephen Hawking's theory of the second quantization of the fields of, Black Holes, they have a surface temperature inversely proportional to the mass of the Black Hole, which is a very natural extension of the fundamental magnitudes of heat, temperature and entropy.
This "temperature" is extremely tiny, far below even the temperature of the microwave background radiation, a few degrees above absolute zero. However, taking the long perspective, trillions of years down the road, the MBR will continue to fall until it falls below the mean temperature on the surfaces of the Black Holes. When this happens the Black Holes will all radiate out their frozen matter, no new ones will be formed, and, in an unbelievably distant future, every single one of f them vanish with a "pop "! The subsequent transparencies were spotted with little "pops!" of Black Holes going off.
When the last Black Hole "pops", Metric Geometry will disappear, only Conformal Geometry will reign, and our universe will have reached the future boundary of space-time that is described in the treatise The Large-Scale Structure of Space-Time of Hawking and Ellis.
Surprise: his smooth conformal boundary should not thought of as "The End of Everything", but as the seedbed of the next Big Bang!! Roger Penrose emphatically stated that he does not accept the doctrine of Cosmic Inflation, as enunciated by Alan Guth and others. This is because he doesn't think that the Big Bang was an explosion at a point, Instead, he posits an entire Cauchy surface, (a flat acausal "Now") at the beginning of time, from which our universe emerged, and from which future universes will emerge from the vanishing of matter and time in this one.
Penrose's objections to Inflation are not new. From previous talks and from "The Road to Reality" I've gathered that Roger Penrose believes that the 2nd Law of Thermodynamics is the most fundamental of all universal laws. Inflation violates the inexorable descent from a state of a high initial state of organization (low entropy) to the present level. Almost all of the entropy in the universe today, he argued, is in the Black Holes, and determines, among other things, the Hawking temperature.
Other objections to Inflation were presented: it does not explain, he claimed, even what it was designed to explain. Guth's Inflationary Hypothesis posits an instanton field that decayed and left us with a smooth universe. He doesn't see how this could possibly work, as it implies that the entropy of the universe at the beginning had to be far too low, without giving any reason for why it was so. What he suggests, and this can be found also in The Road to Reality , is that almost all of the entropy at the beginning was packaged inside "non-thermalized gravitational degrees of freedom".
To summarize: there was no mass in the flat conformal Cauchy surface of the Big Bang, but this developed very quickly. The way to get rid of mass is to claim that the Weyl Curvature on this conformal hypersurface was zero. Everything is dominated by the Ricci Tensor. Weyl Curvature somehow emerges and, through it, both mass and time.
Given a completely time symmetric universe, another one of articles of the Penrosean faith, mass and time will disappear at the far end, the Weyl Curvature will go to zero, a new Cauchy surface governed by a new Ricci Tensor will arise. In time (!), matter and time will be restored, and the whole process will begin again.
Roger Penrose has become a Hindu! He proposes an endless cycle of emergent and dying universes, each with its own Big Bang. The Bang itself is not an explosion but a full conformal Cauchy surface without time or matter, both of which emerge from the 2nd Law. In fact, as he told us "The Second Law drives the whole thing."
What is remarkable, he observed, is that the whole picture is classical; no quantum effects or quantum theory enter into his model.
As is usual in such conferences on "Geometry and Physics", or "Topology and Physics", the mathematicians elaborate an ingenious all-embracing theory, then try to find some physical experiment to justify the inclusion of the word "physics" in the title! Neither Penrose nor Rovelli departed from this pattern.
Penrose suggested that one might be able to examine the W-Map of the cosmos to find traces of the "popping" of the Black Holes in the universe before our present universe began! Even as the rings on a pond that develop when a stone is dropped into it will dissipate into a confused webs of peaks and troughs, so the "rippling gravity waves" formed by the vanishing of the last Black Holes in the previous universe may be detectible in the fine structure of the irregularities in Smoot's map of the background radiation!
He mentioned a team of astronomers that are investigating such a possible effect, and even displayed some frequency graphs that appear to support his hypothesis, but he readily admitted that they don't really show anything yet.
His construction is inspired by the work of Tullio Regge. Regge simplifies, for computational purposes, the 4-dimension space of General Relativity into a basic space of regular points and two 2-dimensional manifolds (topological defects), in which all the information about the curvature is located. The space of regular points is Lorentz-flat. He then develops the Regge calculus to compute metric information about actual space-time.
Okay, this is what I surmised from what he was saying, experts can no doubt pick up from here. Rovelli pointed that one can combine General Relativity and Quantum Field Theory quite successfully when one is speaking of very high or very low energies. There is however a wedge between these in which the model fails. It is here, he claims, that his model takes up the slack.
His spin manifolds with co-bordant boundaries form a category H1; a second category H2 is formed by the Hilbert spaces associated with the quantum operators on H1. The essence of his construction lies in the Functor that connects H1 with H2
Midway through Rovelli's talk the door down at the front of the auditorium opened. Into the auditorium marched a demonstration of students, kitchen workers and janitors! They were banging pots and crying out political slogans in a loud sing-song. They marched around the auditorium, then out the door at the back. A few minutes later they returned and went back to the door from which they'd entered. Before leaving, one of the students turned around and gave a speech to the effect that the audience of the conference needed to know that the salaries of the workers at the ENS was too low, that they worked in bad conditions, that the whole society was rotten, that the age of early retirement should be returned to 60, and full retirement to 65, that the workers and the students are united, and so on. Then they marched out.
Some members of the audience applauded. I also applauded because they'd wakened me up and I was able to follow the rest of Rovelli's talk. Rovelli commented that he would have been marching with them at their age. Then he returned to his lecture.
Rovelli then flashed a "master equation" onto the screen, a Feynman Action Integral that combined contributions from all features of his model, including a final chunk that he claimed represented the essence of General Relativity.
Obviously I know less about this subject than about the things that Roger Penrose was talking about. Like Penrose, Rovelli terminated the talk with his own “physical application"! One of the things that drops out of his model is the universe model based on the Robertson-Walker-Friedman solution to Einstein's Field Equations. His equation however, has a new constant of proportionality for proper time, and he suggested that there might be some physical means for measuring it. He had no idea of how one should go about doing so; it was his way of bringing the “physics" into a physics talk!
After Carlo Rovelli's lecture there was a 2-hour lunch break. I ate mine in the cafeteria of the Institute Poincaré. Around 1 o'clock began walking down the Boulevard St Michel looking for a bank that would change a 100€ note . None of the banks on this boulevard handle monetary transactions. I wasted half an hour looking for one and had to get back. My left foot has been causing me trouble for 2 months; despite this I had to walk as fast as I could for almost a mile to get back to the ENS in time for the lecture by Robbert Dijkgraaf. I raced through the door at 3 minutes after 2, plenty of time to reclaim my seat before he began speaking.
Following the conference and before the cocktail party, I visited the Post Office down the street. The woman there was very friendly. She explained that she wasn't allowed to change money, but that if I bought a stamp for a post-card (0.87€) to the US, she would break the bill for me! The card was mailed the next morning.
So, apart from having to nurse my foot for the rest of the day, the story has a happy ending. May we all be so lucky in the next universe!!