Peace in our time

It seems that enlightened self-interest has worked its magic, and the Formula 1 teams' association (FOTA) have reached a compromise settlement with the governing body, the FIA, in F1's 'budget-cap' row. Let us recall that FIA President Max Mosley was implacably demanding the imposition of a £40 million budget-cap in 2010. That will now not happen. Rather, a budget cap in the region of that figure will be introduced for 2011, but numerous pieces of expenditure, such as engine, driver, hospitality and marketing costs, will be excluded from the cap. This, then, must constitute a victory of sorts for the teams. Max, however, would no doubt suggest that his initial demands were merely an opening bargaining position, and that he has obtained what he really wanted all along.

So, after three months of remarkable political turbulence in Formula 1, the diffuser issue is sorted, the McLaren issue is sorted for the moment, and the budget-cap issue is also apparently sorted. Which leaves us with a Formula 1 season which is being dominated by a single driver and team. Not good for business, that. As The Times's Edward Gorman comments:

In the past a runaway leader has attracted the attention of the FIA, which has stepped in to find something illegal on a hot car.

It's the type of comment which many people think is true, but which few are willing to state in print. If true, it entails that the Formula 1 World Championship is manipulated by the governing body for financial ends; if false, then it is libellous. Either way, it's the type of comment which could attract the attention of the FIA's lawyers, given the difficulty of proving that the championship is manipulated.

On this occasion, it seems unlikely that the FIA will intervene, given the ongoing need to curtail the power of the manufacturer-teams in FOTA. Moreover, Ferrari are beginning to close on Team Brawn, and whilst the championship may be out of their reach, race victories appear to be imminent.

Is the discussion of free will an illusion?

Biologist Martin Heisenberg writes an article for Nature which purports to address the issue of free will, but ultimately does nothing of the kind.

Heisenberg describes the actual research around which the article is constructed, as follows:

My lab has demonstrated that fruit flies, in situations they have never encountered, can modify their expectations about the consequences of their actions. They can solve problems that no individual fly in the evolutionary history of the species has solved before. Our experiments show that they actively initiate behaviour. Like humans who can paint with their toes, we have found that flies can be made to use several different motor outputs to escape a life-threatening danger or to visually stabilize their orientation in space.

The 'expectations' of fruit flies?

Let us be generous, and accept that this term is used metaphorically. The problem with Heisenberg's article owes far more to the general thrust of the argument, which is merely to claim that animals are capable of adapting their behaviour, that "behavioural output can be independent of sensory input." Yet, as Heisenberg admits himself, "the idea that animals act only in response to external stimuli has long been abandoned, and it is well established that they initiate behaviour on the basis of their internal states, as we do." But given that this fact is well-established, it is difficult to see what Heisenberg thinks has been newly discovered in his lab.

Let us accept that Heisenberg's lab have correctly interpreted their empirical data, and that fruit lies are indeed capable of adapting to their environment. This would constitute a type of learning, but it is difficult to see how this bears upon the issue of free will. Neural networks, for example, are capable of learning, and there is a body of literature which demonstrates that recurrent neural networks can be trained to behave like deterministic finite-state automata (DFA). Fruit-fly learning and subsequent behaviour could be represented by such a neural network, but a neural network that can be trained to behave like a DFA is hardly considered to be the epitome of freely-willed behaviour. Neural networks themselves can be either deterministic or stochastic (i.e., random), but both types of causation are distinct from Heisenberg's notion of freely-willed behaviour as "self-generated," (i.e., neither determined, nor random).

If fruit flies are indeed capable of adapting to their environment, then this would be inconsistent with a behaviouristic interpretation of fluit fly behaviour (i.e., an interpretation which denies that fruit flies possess internal states), but it is perfectly consistent with a deterministic interpretation of their behaviour (as well as being quite irrelevant to the issue of free will). Without internal states, there can be no variation in the output response to input stimuli, but with internal states, the response to a stimulus can vary depending upon the internal state, and the internal state can be the result of prior learning.

So Heisenberg's lab have perhaps found evidence for the existence of internal states in fruit flies, but such a finding is of no relevance to the issue of free will.

Parallel universes and wife-swapping

Today, between fields of luminous rapeseed, I drove up to Oxford to see Peter Byrne deliver a talk on the life of Hugh Everett III, the inventor of the many-worlds interpretation of quantum mechanics. Peter is close to completing a biography of Everett, and has been able to draw upon numerous original manuscripts and notes, retrieved with the assistance of Everett's son, Mark (singer and songwriter of Eels), from the family basement.

One thing which struck most of the audience, I think, was the lengths to which Everett's thesis supervisor, the legendary John Archibald Wheeler, went to re-cast Everett's work, not as a radical alternative to Bohr's prevailing Copenhagen interpretation, but as a generalisation of the existing measurement theory. Wheeler appeared to regard Bohr with great awe, and was most anxious not to upset him in any way.

Everett had been fascinated by game theory and operational research prior to starting his work in quantum theory, and after his PhD he immediately left academia to work for the Pentagon's Weapons Systems Evaluation Group (WSEG). Here, he was involved in tasks such as calculating the number of worldwide casualities from a full-scale nuclear war. The audience were amused to see a certificate that Everett received upon completion of his first course at WSEG, in which the congratulatory text was inscribed upon a schematic drawing of a mushroom cloud.

Everett, however, does not appear to have been a warm, affectionate individual, and I must admit that I recognise in a number of my own colleagues the same semi-autistic symptoms which Everett exhibited. After a time at WSEG, Everett and his wife wrote a letter to their friends, all of whom were fellow WSEG employees, declaring that their marriage was now an open one. Everett then proceeded to sleep with most of his colleagues' wives.

Everett became increasingly unhappy, drinking and smoking copiously until he died of a heart attack in 1982. His daughter committed suicide in 1996, writing in her suicide note that she wished to be placed with the garbage, so that she might inhabit the parallel universe closest to her father's.

Angelons and de-mons

Midnight in a subterranean laboratory straddling the Swiss-French border. Father Vaux Vectra is alone at his desk, transfixed by the image on his display screen. He gazes with growing fear and wonder at the lambent pattern of particle tracks, and struggles to absorb the full implications.

Father Vectra had known for years that his colleagues at CERN were not looking in the right direction, were not searching for the right things in the morass of data generated by the Large Hadron Collider. They might have found the Higgs boson, but Father Vectra knew that there was much more at stake here, that there was something of truly metaphysical significance hidden in the data, waiting to be found.

A knock at the door!

Father Vectra's heart spasms with alarm against his ribcage. No-one else should be here! Vectra worked alone, and had studied the work roster with great care to ensure that he would be totally free from interruption.

"Who's there?"

Silence.

Vectra scowls, levers himself out of his chair, and walks with a mixture of irritation and trepidation to the door. "I say, who is that? Are you security? Work control have cleared me to work the late shift, you know..."

More silence.

Vectra pulls his keys out of his pocket, unlocks the door, and turns the handle. It is the last action that he performs alive.

***********************************************************************************

Robert Wheldon's sense of unease was suddenly transformed into outright alarm. "Surely the Dawkinati cannot still exist?" he muttered aloud. "This is worse than I thought!"

"What does it all mean?" asked Danica.

"It looks like Vectra was murdered by an acolyte of a secret cadre, thought to have dissolved decades ago, devoted to the destruction of the Roman Catholic Church."

"That's terrible!" exclaimed Danica. "But why would they kill a physicist like Vectra?"

"Well, according to Vectra's theories and calculations," said Weldon, holding a sheaf of Vectra's hand-written notes, "the material universe we see around us, with its 4 space-time dimensions, its photons and electrons, its carbon and silicon atoms, its nucleic acids and proteins, its oceans, forests and mountains, its red giants and neutron stars, its galaxies, stellar nurseries, and supermassive black holes: all of that is merely appearance and illusion, a transient, local phenomenon. Beneath it all is a 10-dimensional world, described by a unification of string theory and Manichaeic cosmology, in which the forces of good and evil fight an eternal war. Vectra believed that God has compactified 6 of the dimensions into a Calabi-Yau manifold, forcing the exponential inflationary expansion of the other 4, and the creation of the world we see around us.

"Vectra went looking for evidence of God's signature in the moduli fields which keep the 6 dungeon dimensions compactified, and it looks like he found it just before he was killed. With knowledge of these moduli fields, and their quanta of excitation, the angelons and de-mons, it will be possible for the Dawkinati to open up the 6 hidden dimensions temporarily, allowing the Christ to collide with the AntiChrist!"

"We've got to get to the Vatican before it happens," cried Danica. "I've got a helicopter..."

Jenson's secret

You're Jenson Button, and you have a secret. You have a type of knowledge which the Greeks referred to as a techne, a special skill, which in your case is so finely developed, so sensitive, that you can see and feel things which no other driver in Formula 1 can detect. You've known this all along, and it's provided you with a kernel of inner confidence that has sustained you through the wilderness years. Now, finally, it's time to draw upon that well of secret knowledge.

To understand this, it's necessary to appreciate that each corner is not merely a strip of asphalt with a specific radius of curvature; that might be all that the casual observers are capable of seeing, but in reality each corner, (in combination with the characteristics of the car), provides the conditions which create a sheaf of potential dynamical paths. Each path is defined not just by a geometrical trajectory, but by the speed profile over that trajectory. The telemetry traces retrospectively plot these profiles like paths across a valley, the elevation at each point representing the speed, negative gradients corresponding to deceleration, positive gradients to acceleration. Each path has a different combination of initial braking point, turn-in point, deceleration profile, apex, and acceleration profile, and there's an infinite number of these possible paths through a corner. The points of minimum speed form a basin in the sheaf of dynamical paths, and when the paths are projected down into trajectories through the corner, the set of apices form an extended patch on the road surface. Within the sheaf however, is a single, mathematically optimum path, the one that minimises the time spent in the corner. Your ability as a Grand Prix driver is determined by how closely and how consistently you can approach this optimum path through each corner.

Your mind's eye interacting with your visual field, you can almost see this bundle of possible paths, the optimal one running like a golden thread through a tapestry. You know that you cannot keep your car adhered to the golden thread anything other than briefly, but you can keep it within a small neighbourhood of that path, and if the car is as you like it, you can approach it perhaps closer than anyone else.

With the steering wheel cradled in your hands like you're holding the eggshell of a rare Tibetan albatross, the last of its kind, and your inputs smooth as silk, you avoid the corrections which will send the path of your car zig-zagging across the contours of the dynamical landscape. If you could only see the filigree, gossamer detail of the contours at high resolution within this landscape, you would appreciate the importance of being smooth, of minimising needless deviations. It would almost be vulgar to do otherwise, for this is as much an aesthetic sensibility as a utilitarian method.

It's time to follow that golden thread.

Primeval music

The European Space Agency's Planck satellite is due to launch from French Guiana on May 14th. Pending a successful deployment, Planck will measure the temperature of the cosmic microwave background radiation (CMBR) across the entire celestial sphere, with greater sensitivity and spatial resolution than achieved by its predecessor, NASA's WMAP satellite. The variations in the temperature of the CMBR reflect variations in the density of matter when the universe was 380,000 years old, at the time of so-called 'recombination' when atomic nuclei captured previously free electrons.

New Scientist duly have an article to herald the launch, which claims that "these so-called anisotropies are believed to be due to inflation...During inflation, quantum fluctuations in space-time were extended to cosmological scales: by the time the CMB was released, these fluctuations had led to variations in the distribution of matter across the universe. Denser regions of the universe produced CMB photons slightly colder than average, and vice versa."

In fact, whilst it is claimed by cosmologists that temperature fluctuations more than a few degrees across are the imprint of fluctuations present at the end of the inflationary period, fluctuations smaller than a degree are believed to be the result of acoustic oscillations in the plasma of baryons, electrons and photons present between the end of inflation and the time of recombination. These small-scale fluctuations are therefore the visible remnant of the earliest sound waves in the universe.

For the large angular-scale fluctuations, the denser regions redshifted the light climbing out of those regions, and therefore produce cooler spots in the CMBR; in contrast, for the small angular-scale fluctuations, denser regions were regions where the plasma was hotter, hence these denser regions produce hotter spots in the CMBR.

Marcus du Sautoy and Formula 1

Marcus du Sautoy, the Charles Simonyi Professor for the Public Understanding of Science at Oxford University, makes an attempt in The Times to find a mathematical perspective on Formula 1. He begins:

This year Formula One teams that agree to race within a strict budget rather than relying on unlimited funds to develop their cars are being rewarded with a number of perks under a scheme called the Kinetic Energy Recovery System [KERS].

Marcus seems to have got his wires slightly crossed here: the opportunity to exploit greater technical freedom in exchange for a voluntary budget cap, does not arrive until 2010, whilst KERS is already available to anyone this year, irrespective of budget. Not an auspicious start then.

However, Marcus then redeems himself by posing an interesting question which does, by analogy, help to explain why the teams with double-decker diffusers have done so much better this year than the teams with KERS alone:

You arrive at an airport for a flight connection but the timing is tight. You discover that your next plane is leaving from the far end of the airport. There is a moving walkway for part of the journey. You’ve got enough energy to do a short burst of running, but otherwise you’ll walk at a constant speed...You want to get to the gate as quickly as possible, so the question is: when should you use your burst of energy? Should you run on or off the moving walkway?

The answer is that the button should be hit off the walkway:

Our twins are walking together towards the walkway when one twin decides to press the boost button to get him to the walkway before his brother. He reaches it D metres ahead of his brother. But as soon as he steps on to the walkway the distance between them begins to increase even more. When the second twin steps on to the walkway he presses his booster. Since both are on the walkway, this will allow him to catch up only D metres on his brother. But his brother is more than D metres away by now.

Marcus thinks that an analogy can be made with the question of whether the KERS boost button should be used on the slow part or the fast part of a circuit, but in practice, it can only be used to benefit on the straights; if used in the corners it would simply result in wheelspin.

The airport scenario does, however, still provide a good F1 analogy in the following sense: more time is spent, per unit distance covered, off the walkway, hence the greatest benefit is gained by hitting the boost button there. Similarly, in Formula 1, more time is spent, per unit distance covered, in the corners than on the straights, hence in terms of lap-time alone, the greatest benefits accrue from improving the cornering characteristics of the cars than by improving straightline speed. As a fresh demonstration of this, the teams which have developed a cornering advantage this year (the double-decker diffuser teams), are significantly faster than those which initially developed only a straightline speed advantage (the KERS teams).

Penrose's conformally cyclic cosmology

Of all the cosmological models proposed in recent years, perhaps the most ingenious is Roger Penrose's conformally cyclic cosmology. Penrose elaborated this idea at length in an outstanding contribution to the 2008 collection, On Space and Time.

Penrose's cyclic cosmological model is a particular application of his conformal compactification construction, which takes any space-time, whether or not it contains singularities, and whether or not it is infinite in time or space, and constructs a finite (compactified) space-time with boundary, whose metric is related to the original metric of space-time by a locally variable scale factor Ω, called the conformal factor. This transformation preserves the causal structure of the original space-time, even if it doesn't preserve lengths and times. The boundary of the conformal compactification contains components which correspond to singularities, and components which correspond to spacelike infinity, timelike infinity, (and null infinity).

Now, in an open or flat Friedmann-Robertson-Walker cosmological model with no cosmological constant, whilst the past 'Big-Bang' singularity corresponds to a spacelike hypersurface in the boundary of the conformal compactification, the future timelike infinity corresponds to a single point. Current astronomical evidence, however, suggests that the expansion of our universe is accelerating due to the presence of a hypothetical dark-energy, which at least mimics the effect of a cosmological constant. The far future of our universe is therefore representable by de Sitter space-time, and in this type of space-time the future timelike boundary is a spacelike hypersurface. Penrose made the simple observation that the future conformal boundary of such a space-time can therefore be joined to the spacelike initial conformal boundary of a Friedmann-Robertson-Walker model to make a cyclic universe.

The fact that the universe is very small and hot at the beginning, and very large and cold in the far future, is not a problem, argues Penrose, because both the early universe and far future universe contain only conformally invariant, massless particles. Without massive particles, there is no way of defining lengths or times, hence the only physically meaningful structure is the conformal structure, i.e., the causal structure. By compressing the conformal factor towards the far future, and expanding it towards the beginning, the geometry of the future conformal boundary can be joined seamlessly to the initial conformal boundary. In other words, the conformal factor Ω must tend to zero as time t tends to ∞, to compress the infinite future into a finite conformal time, and Ω must tend to ∞ as t tends to 0, to stretch the metric as it tends towards the Big-Bang. The conformal metric then matches on the two boundary components, and the components can be identified. The Weyl curvature is zero on both the future boundary and past boundary, hence the Big Bang is still well-defined in the cyclic model as the unique hypersurface on which the Weyl curvature vanishes.

Penrose suggests that cosmic inflation doesn't occur after the Big Bang, but before it! The accelerating expansion of the universe that we currently observe, is identified as the onset of inflation. It is this inflation, proposes Penrose, that generates the scale-invariant spectrum of density perturbations in the post-Big Bang universe.

Penrose proposes that the far future of our universe contains only electromagnetic radiation and gravitational radiation. The electromagnetic radiation comes from the cosmic background radiation of the Big Bang, from stars, and from the eventual evaporation of black holes. The gravitational radiation, meanwhile, comes mostly from the coalescence of black holes. Penrose proposes that massive particles such as electrons, either annihilate with massive particles of opposite charge (positrons), or decay by some as-yet undiscovered mechanism.

The cycle of Penrose's model is one in which the universe 'begins' in a conformally invariant state, with zero Weyl curvature, but in which there is a normal derivative to the Weyl curvature. This seems to trigger the formation of massive particles. The matter then clumps together into stars and galaxies, such clumping increasing the Weyl curvature and decreasing the Ricci curvature. Eventually, much of the matter is swept into black holes, where the Weyl curvature diverges, but the Ricci curvature is zero. The matter which isn't swept into black holes decays or annihilates into radiation, and the black holes eventually evaporate themselves into radiation. The Weyl curvature thereby returns to zero, all particles are massless again, and conformal invariance resumes. Gravitational radiation, however, never thermalizes, and this appears to be responsible for the normal derivative to the Weyl curvature, which triggers the formation of massive particles in the next cycle.

Two potential problems spring to mind. Firstly, following an argument by Gibbons and Hawking, de Sitter space-time is widely believed to possess a minimum temperature due to its cosmological constant. With the value of the cosmological constant we observe, this temperature is about 10-30 Kelvin. A black hole will only evaporate if the temperature of its horizon is greater than the temperature of surrounding space. The temperature of a black hole is inversely proportional to its mass, and a black hole which grows large enough that its temperature drops below 10-30 Kelvin would never evaporate. However, such a black hole would have a mass approximately equal to the current observable universe, so the formation of such a black hole may well be impossible in a universe whose contents are diluted by the accelerating expansion of dark energy.

The second problem is that if the quantum fields in the far future of our universe can be treated as quantum fields in thermal equilibrium in de Sitter space-time, then because such a universe is eternal, quantum fluctuations ensure the spontaneous generation, at a constant rate, of anything you care to name, including massive particles and black holes. This would prevent our universe from ever reaching an exact state of conformal invariance in the far future. However, because gravitational radiation never reaches thermal equilibrium, one could perhaps argue that the quantum fields in the far future of our universe cannot be treated as quantum fields in thermal equilibrium in de Sitter space-time.

Penrose's proposal remains fascinating and elusive.

Adrian Newey and Rayleigh-Taylor instability

After dominating the recent Chinese Grand Prix, a race run in treacherously wet conditions, Red Bull designer Adrian Newey was asked by Autosport's Edd Straw to explain why his car is so quick in the wet:

"That's a good question!" he replied, "The fundamental characteristics of the car - it seems to be reasonably well balanced with a decent level of downforce - are a good start for the wet. But it gets into the fine details for the last couple of tenths and it's a bit harder to nail down why."

Whilst the Red Bull is undeniably competitive this year, for the moment it is bereft of a double-decker diffuser, and the evidence from the other races this year suggests that the Brawn is intrinsically a quicker car; the Brawn is also 'reasonably well balanced with a decent level of downforce'. Hence, the advantage of the Red Bull in wet conditions requires something of an explanation.

One could take Adrian's response at face value, and accept that Red Bull don't really understand it themselves. Or, more intriguingly one might recall that Adrian has spent some time looking at the fluid mechanics of yachts, and has often flirted with the idea of working on an entry for the America's Cup. What fascinates Adrian about yachts, in particular, is the fact that they cleave through two different fluids: air and water. In this sense, the hydrodynamical challenage of yachting design is quite different to that faced by an automobile aerodynamicst, who must study only the effect of one fluid, the air. Except, that is, when it rains, and a film of water lies upon the surface of the road...

One might hypothesise, therefore, that Adrian has learnt something from his yachting research, which can be applied to good effect in setting-up a Formula 1 car for wet-weather conditions. When it rains in motorsport, the generic response is simply to increase ride heights and wing angles. It might, however, be possible to exploit the complex processes that occur at the contact boundary between the air and water when accelerated by the aerodynamic surfaces of a Formula 1 car. Liquid water is denser than air, and when fluids of different densities are subjected to acceleration, Rayleigh-Taylor instabilities result in turbulent mixing of the fluids. Could one, for example, use the mixing of the air and water on the underside of the wings and beneath the floor of the car, to create a turbulent boundary layer that keeps the flow attached for slightly longer, thereby increasing downforce and reducing drag? The effect would be the same as that created by the dimpling on the surface of a golf ball.

Perhaps the Red Bull's wet-weather advantage over the Brawn is simply due to the fact that it is harder on its tyres in all conditions, and therefore capable of generating the necessary heat to 'switch the tyres on' in wet conditions. And it should also be noted that once the layer of water on the road surface has been transformed into a spray, the water droplets in the spray do not follow aerodynamic streamlines. Nevertheless, one feels that the hydrodynamics of Formula 1 cars displacing both air and a surface layer of water, is a science which has yet to be fully explored or exploited.