Aeroelasticity in F1

Formula One's latest technical controversy surrounds the aeroelastic front wings currently used by Red Bull and Ferrari. The ends of the front wings on these cars appear to be deforming at high aerodynamic loadings, thereby generating ground effect downforce. Courtesy of Darren Heath's photographs, estimates suggest that the front-wing endplates are deflecting by up to 24mm.

McLaren, in particular, are currently working hard to understand how the effect is achieved. A cursory literature search, however, suggests that the effect probably depends upon the orientation of the carbon-fibre plies in the front wing, and Red Bull and Ferrari may even be using a method of coupling the bending of the front wing to the twisting of the endplates.

City University's Aeronautics department point to research revealing "the effect of ply orientation on the dynamic and aeroelastic behaviour of composite wings." Bristol University's Aerospace Engineering department provides further details, explaining that "Laminated composite materials designed adequately can present elastic coupling properties that can be used to induce an adaptive change. For instance, a composite presenting in-plane elastic coupling that is loaded under normal loads experiences a shear deformation. The proposed morphing design consists of a wing made of laminated composite materials presenting elastic couplings so as to induce twist when the wing bends. This concept, if proven, could provide a passive actuation for the control of the wing twist." Elsewhere, there are claims that "most new helicopters have composite elements in the hub/root of the main blades that are used to replace functions usually done by hinges on older designs."

Fascinatingly, a group of researchers in the Netherlands have also just published a paper entitled 'Aeroelastic tailoring using lamination parameters - Drag reduction of a Formula One rear wing', which also proposes a bending-torsion coupling, in this case to reduce rear-wing induced drag.

If these front wing effects are genuinely dependent upon the precise orientation of the carbon-fibre plies, then it will be interesting to see if McLaren can react on a time-scale consistent with winning this year's World Championship...

Ferrari team orders

So what exactly was it that made Ferrari's decision to move Fernando Alonso past Felipe Massa in the closing stages of Sunday's German Grand Prix, so objectionable? Whilst team orders have nominally been banned since Ferrari's infamous prior offence with Schumacher and Barrichello in 2002, covert team orders have, of course, continued to be implemented. The teams have avoided issuing overt commands over the radio, but team orders have nevertheless been effected, either by issuing information which is interpreted as a coded instruction by the driver, or by timing pitstops in a manner which swaps the positions of the team-mates.

So, was it because victory in a Grand Prix was at stake? Well, it clearly wasn't this alone, because Massa himself had to sacrifice victory in the 2007 Brazilian Grand Prix in order that his then team-mate Kimi Raikkonen could win the World Championship. There was no media or public outcry on that occasion.

Was it because there are still many races to be run in the championship? Well, once again, clearly not, because in 2005 Juan-Pablo Montoya was forced to let McLaren team-mate Kimi Raikkonen past in the Hungarian and Belgian Grands Prix in order to assist Kimi's championship hopes, when the season had yet to reach its final stages. In the case of the Belgian Grand Prix, this cost Montoya a Grand Prix victory. McLaren even deliberately sabotaged Montoya's chances in the Canadian Grand Prix that year to permit Raikkonen first call at the pitstops induced by a safety-car.

Was it because the passing manoeuvre took place on-track, rather than during the pit-stops? Once again, it cannot be this alone, because at the 2008 German Grand Prix, Heikki Kovalainen let McLaren team-mate Lewis Hamilton past at exactly the same place, coming out of Turn 6, where Massa let Alonso past in this year's race. On that occasion, Kovalainen wasn't leading, but by letting Hamilton past he permitted Lewis to catch the leaders and win the race himself.

Whilst these factors exacerbate the offence, there is a crucial additional property which is common to both the 2002 Austrian Grand Prix scandal and Sunday's reprise: Massa, like Barrichello eight years ago, was labouring under the illusion that he would be permitted to beat his team-mate.

Ferrari could, if they so wished, have explained to Massa what they required of him before the race. Team Principal Stefano Domenicali could have told Massa beforehand that, 'if we're running 1-2 in the final stages of the race, with Fernando less than five seconds behind, then you must let him pass, for the sake of the championship.' The fact that Ferrari didn't do this, entails that they were hoping that the situation would never arise, that they could avoid de-motivating Massa by making their support for Alonso explicit. Thus, rather than acceding to a pre-agreed plan, we had a coded instruction, and a driver dis-illusioned whilst driving a car that was leading a Grand Prix.

Grand Prix racing will always be a confluence of sport, business and technology, but to maintain the revenue streams which depend upon the interest of millions of fans across the world, the teams need to understand that any coordination between team-mates must be seen to be done with the planned and grudging consent of both drivers.

Louvres, holes and algebraic topology

Autosport's technical triumvirate of Mark Hughes, Gary Anderson and Giorgio Piola, have spotted a beautiful touch on the engine cover McLaren were intending to introduce with their new exhaust-blown diffuser at last week's British Grand Prix. Whilst the teams have in recent years been forced, by regulation, to substitute single exit orifices in place of radiator exit louvres, McLaren have cleverly realised that if they cut the bodywork between each louvre in half, then the result is topologically identical to a single hole.

To recall, topology is the mathematical study of the connectedness and continuity of shapes and surfaces, irrespective of their geometry. Thus, the surface of a tea-cup is often said to be topologically identical to the surface of a doughnut, and the London Underground Map is said to preserve the topology of the capital city's subterranean transportation network, if not the actual length and shape of the tracks.

Now, the number of holes in a shape or surface M is typically characterised by an object from algebraic topology called the fundamental group π₁(M). Algebraic topology is essentially the use of groups to characterise the topological characteristics of shapes and surfaces. Recall that a group is a set of elements which is equipped with a binary product operation, a unary operation called the inverse, and a special element called the identity element.

To understand what the fundamental group is, we need another concept, called homotopy equivalence. Basically, two curves or loops are said to be homotopically equivalent if one can be continuously deformed into another. If a pair of curves cannot be deformed into each other, then they belong to different homotopy equivalence classes.

Now, if we consider the set of all loops beginning and ending at the same point p in a shape or surface, then we can tag one loop onto the end of another to form a new loop. This concatenation operation gives us a product operation between different homotopy equivalence classes of loops at a point. Furthermore, by simply running around a curve in the opposite direction, we have an inverse operation, and the degenerate loop consisting of the point p, serves as the identity element e of a group. The homotopy equivalence classes of loops at a point can thus be treated as a group, and this group is called the fundamental group π₁(M). (If the shape or surface is connected, it can be shown that the fundamental group at each point is isomorphic, hence the point chosen is arbitrary).

If a shape or surface has no holes in it, then all loops through an arbitrary point p can be continuously shrunk down to the point itself, hence the fundamental group consists of a single element π₁(M) = {e}. However, if the shape or surface has a single hole, then there will be at least two homotopy classes of loops through a point: those which can be deformed down to the point, and those which cannot, because they circle the hole, and cannot be shrunk any smaller than the hole. In this case, the fundamental group consists of at least two elements. If there are two holes, then the loops around both holes cannot be shrunk to a loop around one hole, and the loops around one hole cannot be shrunk to a point, hence the fundamental group will contain at least three elements.

In the case of a radiator exit with n louvres, the fundamental group of the surface will contain at least n+1 elements. By cutting through the bodywork between the louvres, however, it becomes impossible to form a loop around anything but the entire louvre collection. Thus, topologically speaking, there is only a single exit orifice. Ingenious.

Laser aerodynamics

One of the intentions of the new technical regulations introduced for the 2009 F1 season, was to exclude the existence of bargeboards between the trailing edge of the front tyres, and the leading edge of the sidepods. A crucial function of these devices was to guide the turbulent air from the front wing and front wheels, away from the vital airflow underneath the car, that ultimately feeds the diffuser.

Now, the teams responded to this with typical ingenuity, by shortening the sidepods, and installing mini-bargeboards in the newly created region of space. Nevertheless, the regulations were successful here in eliminating full-length bargeboards. When an item is banned in F1, however, the engineering tradition is to find some other means of achieving the same effect, so let us see if we can do just that. Note, however, that what follows is not intended to be a serious short-term practical recommendation, more an attempt to demonstrate the art of the possible.

Bargeboards were solid, 2-dimensional surfaces. When a solid is introduced into a viscous airflow, the surface of the solid provides a new boundary component, along which the airflow velocity must be zero. In the case of a bargeboard, this creates a stagnation point at the leading edge, and boundary layers down the inner and outer flanks of the boards. The stagnation point at the leading edge forces the streamlines of the airflow to go either side, hence the stagnation point functions as a branching point in the airflow. Thus, a solid, 2-dimensional surface is a hugely convenient device for enforcing the separation of airflow.

However, with bargeboards now banned, the question is whether there are other means of enforcing the separation of airflow in the region of space between the trailing edge of the front tyres and the leading edge of the sidepods. The rules prohibit solid substances in this region, so what else could we use?

Well, in principle we could use a plasma, but confining the plasma would be rather tricky, and would require the use of magnetic fields generated by superconducting magnets, which in turn would need to be cooled by liquid helium. That would be rather challenging to package.

So how about electromagnetic radiation? The regulations only prohibit the presence of solid substances in that sensitive region of space behind the front wheels. The space all around the car is already filled with natural and artificially generated radiation, so banning the presence of radiation would be very difficult. If electromagnetic radiation is permitted in the space between the front wheels and the sidepods, then it follows that coherent radiation, or laser light, is also permitted, and it may be that we can create virtual bargeboards out of laser light.

For aerodynamic purposes, the crucial property of radiation is that it is capable of applying and transferring pressure. Lasers provide the capability to inject pressure into an airflow at very precise locations, and could therefore be used, amongst other things, to create stagnation points in the airflow, and narrow high-pressure layers of air. By this means, lasers could, in principle, be used in F1 to replicate the function of bargeboards in achieving airflow separation.

Needless to say, this would not be the work of a moment. The radiation pressure would create high air temperatures as well as pressures, and the injection of heat energy into the airflow just in front of the sidepods is not necessarily ideal. There is also, of course, the question of laser energy consumption. One might wish to power F1 lasers with the energy stored by kinetic energy recovery systems (KERS), and such an aspiration will be assisted both by the gradual decrease in laser energy requirements over time, and the gradual increase in the energy harvested by KERS.

Nevertheless, given the heat energy injected into the airflow by lasers, it is perhaps at the rear of the car where they could most usefully be employed. Red Bull, of course, have re-introduced exhaust-blown diffusers to the sport this year, and the point about these is that they ultimately use heat energy generated within the engine for aerodynamic purposes. Craig Scarborough explains how Red Bull may be using retarded ignition in qualifying this year to maintain the flow of exhaust gases even when the driver is off the throttle. Perhaps a single KERS-powered laser inserted into the exhaust tract of the engine could achieve the same effect here...

Viscoelasticity and F1 tyres

Formula 1 aerodynamics is all about utilising a viscous fluid, to maximize the forces generated by a viscoelastic solid.

As Mark Hughes explained in last week's Autosport, an F1 tyre generates grip by two different mechanisms: physical grip and chemical adhesion. The latter is only triggered when the tyre reaches a certain critical temperature, and to reach that temperature, the physical grip must be used to repeatedly load and unload the tyre, the deformation cycle thereby heating up the carcass of the tyre.

However, if a rubber tyre is represented as an elastic solid, there is a puzzle here, for the theory of elasticity says that an elastic solid is non-dissipative; in other words, no heat is generated as a result of loading and unloading an elastic material. There is no net work done on a perfectly elastic substance during a load cycle. If rubber was genuinely elastic, it would deform, and then return to its initial configuration, at its initial temperature. Tyre friction between the tyre and the road surface would certainly generate heat, even if the tyre were an elastic solid, but if rubber were genuinely elastic, the carcass of the tyre would not heat up as a result of going through repeated load cycles.

To put the puzzle into context, and to understand the answer, we need to consider three types of material defined within continuum mechanics: elastic solids, viscous fluids, and viscoelastic solids.

An elastic solid undergoes elastic deformation up to a yield point, and thereafter undergoes a degree of irreversible plastic deformation, (until it finally fractures). Under plastic deformation, net work will be performed on the solid, and that net work will go into heating it up. In some respects, a solid undergoing plastic deformation behaves like an incompressible viscous fluid.

A viscous fluid, conversely, can be thought of as a solid with no elasticity and no yield point; it is something which flows under the action of an applied force, and which possesses internal resistance to shear forces.

A viscoelastic solid responds to an applied stress by undergoing simultaneous elastic deformation and viscous flow. There is a viscous flow for all stress levels, unlike plastic deformation, which only occurs in elastic solids after a yield point has been attained. The viscous flow produces heat. When the stress is removed, the material will return to its initial configuration via a different curve on a stress-strain graph. This phenomenon is referred to as hysteresis.

To reiterate, perfectly elastic materials do not dissipate energy as heat when a stress is applied and removed; there is zero net work done on an elastic material over a load cycle. The work done deforming the elastic solid will be stored as strain energy, and when the external stress is removed, the strain energy will be used to do work on the environment, resulting in zero net work being done on the elastic solid over a load cycle.

Now, rubber is slightly unusual in that it will heat up when deformed. In most elastic materials, the strain energy will be stored in the electrostatic bonds between molecules. In rubber, whilst some of the strain energy is stored in such bonds, a component of the strain energy is stored in the form of thermal energy, and when the external stress is removed, and the rubber returns to its original configuration, it will adiabatically cool.

If the unloading curve on a stress-strain graph followed the same path as the load curve, then the material would return to its initial temperature at the end of a load cycle. In contrast, a viscoelastic material dissipates a quantity of heat energy equal to the area enclosed by the curves on the stress-strain graph, and this is equal to the net work done on the material. Rubber is a viscoelastic solid. (This is also nicely explained in Pat Symonds's trilogy of articles on the science of F1 tyres, featured in the April/May/June issues of RaceTech Magazine).

Formula 1 aerodynamicists attempt to use the flow of a viscous substance over the car to maximize the downforce on the viscoelastic contact surfaces; this enables the drivers to corner at the speeds necessary to load and unload the viscoelastic contact surfaces to the point at which they dissipate sufficient heat that the chemical adhesion of the contact surfaces is then maximized.

It's all about resisting the flow.

The F1 Fans' Forum

I thought I'd touch the hem of F1 today by attending the inaugural F1 Fans' Forum at the British Academy of Film and Television Arts in London's Piccadilly.

The idea of this was to provide a Question Time format in which fans could pose questions to a selection of key participants from the sport, comprising McLaren Team Principal Martin Whitmarsh, Lotus Racing Team Principal Tony Fernandes, Mercedes race engineer Jock Clear, Ferrari Press Officer Luca Colajanni, and Force India test and reserve driver Paul di Resta.

The overall concept here is a good one, for F1 fans are genuine financial stakeholders in the sport: the income streams upon which both the teams and Bernie Ecclestone's business are dependent, are ultimately predicated upon the existence of a huge global television audience. Alienate these people, then, at your peril.

Today's event was chaired in a polished fashion by ex-ITV F1 commentator, James Allen, and it was noticeable that every member of the panel was able to organise their thoughts and speak in an impressively coherent and intelligent manner. F1 folk clearly become very well trained in answering any and all questions popped at them.

There were no big surprises in any of the answers, although Martin Whitmarsh took the opportunity to endorse the plan for driver-adjustable rear wings next year, and there was wide general support for the return of F1 test days at Silverstone. One member of the audience made an interesting suggestion that each team could be given a finite fuel allocation for the season, to use as they see fit; teams with more fuel-efficient engines might therefore be able to gain testing miles as a reward for their environmental-friendliness.

PR-impresario Nav Sidhu was much in evidence, looking much like a finalist in The Apprentice, and I also spotted ex-F1 racing editor, Matt Bishop, looking rather well fed in his role as head of McLaren communications. Jonathan Legard and Ted Kravitz were also in attendance, Ted looking rather jolly, Jonathan looking quite animated. And, rather surprisingly, long-time F1 engineer Frank Dernie turned up, Frank suggesting from the back of the hall, in typically contrarian style, that there's no evidence whatsoever that increased mechanical grip makes for better racing.

All in all, it was a perfectly fine event. One suggestion for next time, to spice things up a little, would be to remove the pre-vetting of questions...

Sources of turbulence in F1

Whilst this season's F1 World Championship is shaping up into a fascinating pentahedral battle between Jenson Button, Lewis Hamilton, Sebastien Vettel, Mark Webber, and Fernando Alonso, the sport's underlying aerodynamic problems remain. The turbulent wake created by a Formula 1 car, and the loss of downforce induced in a following vehicle, continue to mitigate against good racing. The recommendations of the FIA's Overtaking Working Group, implemented for the 2009 season, were intended to both reduce downforce, and to reduce the amount of turbulence to which a following car's downforce-generating devices would be sensitive. These recommendations were, of course, promptly undermined by the development of double and multi-deck diffusers. Hence, the regulations for the 2011 season seek to improve the opportunities for overtaking by banning multi-deck diffusers, and by also introducing driver-adjustable rear wings.

Perhaps, however, the regulations could go still further, and to this end it's worth noting that the rear end of a Formula 1 car generates turbulence by several different mechanisms. Any wing profile will, of course, generate a turbulent wake, but more generally, any surface will possess a boundary layer, and when that boundary layer detaches, it will inevitably create turbulence. In addition, it may be that modern F1 aerodynamics create some degree of Kelvin-Helmholtz turbulent instability. If two parallel adjacent layers of airflow have different velocities, then the velocity shear will induce such turbulent instability. The different levels of a multi-deck diffuser may well create some degree of Kelvin-Helmholtz instability, unless they discharge their airflows at exactly the same velocity.

Note that this is a different source of turbulence from that created by wing section profiles, which induce turbulence because they require a circulatory airflow component to operate. To understand this, consider the idealised situation where the airflow around a wing is a superposition of (i) a uniform 'freestream' flow from left to right, where the streamlines are parallel straight lines, and (ii) a pure circulatory flow, where the streamlines are anti-clockwise concentric circles.

Taking the sum of the velocity vector fields for the uniform and circulatory flow at each point, the anti-clockwise circulation is added to the freestream velocity below the wing, and subtracted from it above. Hence, the airflow beneath the wing will be faster than that above, and in accordance with the Bernoulli principle, the pressure beneath the wing will be lower than that above. This pressure differential causes a net downward force. The presence of circulation is also consistent with the use of Newton's third law ('action equals reaction') to explain the creation of downforce, because the circulatory flow adds an upward component to the airflow in the wake of the wing, corresponding to the upward deflection of air, and a downward reaction force.

If multi-deck diffusers do indeed induce Kelvin-Helmholtz instability, then whilst this particular source of turbulence will be eliminated next year, the teams will continue to use the airflow over the rear deck and beam wing to help pull the airflow out of a single diffuser, and these airflows presumably have different velocities. If so, it will remain a source of Kelvin-Helmholtz turbulent instability, and will continue to mitigate against overtaking.

So why not ban beam wings?

Initial thoughts on turbulence

Whilst a general theory of turbulence continues to elude mathematical physics, the phenomenon can actually be characterised in rather simple terms:

Turbulence is an intermediate state of a fluid between laminar flow and thermodynamic equilibrium.

Thermodynamic equilibrium is the state of maximum entropy, the state in which no information is embedded in the fluid. In such a state, all the energy that may once have been carried by the streamlines of the fluid, has been dissipated into thermal energy, the random heat energy of the molecules in the fluid. The direction of motion of the molecules in thermodynamic equilibrium is isotropically distributed, and the speed of motion of the molecules possesses the Maxwell-Boltzmann distribution.

In contrast, laminar flow is a low-entropy state of a fluid, in which the fluid carries both directional information, and speed of motion information, defined by the streamlines and trajectories of the fluid flow velocity field.

Turbulent flow is intermediate between laminar flow and thermodynamic equilibrium because it constitutes a flow regime in which the directional information of laminar flow has been degraded, but the speed of motion information has been at least partially retained. In this respect, turbulent flow is defined by its possession of the following two characteristics:

(i) Chaotic motion.
(ii) Vorticity.

In chaotic motion, the distance between a pair of particle trajectories, initially close together, can diverge as a function of time. This property destroys the 'parallel' nature of the particle trajectories in laminar flow.

'Vorticity' is the fancy word for rotation in a fluid, and turbulent flow is typically characterised by a cascade of vortices of different sizes. Each vortex transfers its angular momentum to vortices of a smaller size, which then, in turn, transfer their angular momentum to yet smaller vortices. (In the case of turbulent aerodynamics, one can think of the cascade as a type of pneumatic clockwork mechanism). The smallest vortices are sufficiently small that the viscosity of the fluid is able to transform their rotational energy into heat energy.

Chaotic motion and vorticity both clearly degrade the directional information carried by laminar fluid flow. However, the distribution of particle speeds carried by the laminar flow can be partially preserved in turbulent motion. Hence, it is in this sense that turbulent motion is an intermediate state between laminar flow and thermodynamic equilibrium.

Axioms for the many-worlds interpretation

Quantum theory is conventionally thought to be basis-independent. The state of a quantum system can be represented by a vector Ψ in a special type of vector space, called a Hilbert space H, and a basis is simply a collection of vectors {ψ} which enable any element to be decomposed as a linear combination of those basis vectors:

Ψ = c₁ ψ₁ + ⋅ ⋅ ⋅ + c_n ψ_n.

The elements of the general linear group GL(H) transform from one basis to another, and by virtue of being basis-independent, the same quantum state Ψ can be expressed as a different linear combination in a different basis {ψ'}:

Ψ = c₁' ψ₁' + ⋅ ⋅ ⋅ + c_n' ψ_n'.

In this sense, quantum theory can be said to be a GL(H)-invariant theory.

The many-worlds interpretation of quantum theory, however, suggests that: (a) there is a process called decoherence which selects a preferred basis; and (b) the universe splits into the branches selected by decoherence. Thus, whilst quantum theory per se does not identify a branching structure for the universe, the many-worlds interpretation does. A measurement-like interaction, as a special case of decoherence, can be said to extrude a collection of branches from a GL(H)-invariant structure, much like a rose-bush emerging from a thicket of brambles.

To elaborate, let us attempt to define some axioms for the many-worlds interpretation of quantum theory:

(i) Quantum theory is fundamental and universal.

(ii) A pure quantum state provides a maximal specification of the state (or history, in the Heisenberg picture) of a physical system.

(iii) Each type of physical system is represented by a unitary representation of the local space-time symmetry group on a Hilbert Space H.

(iv) The time evolution of a physical system in a local reference frame is represented by a continuous one-parameter group of unitary linear transformations U(t):H → H of the Hilbert space. This corresponds to the representation of the time-translation subgroup of the local space-time symmetry group.

(v) The interaction Hamiltonian between a macroscopic system and its environment is such that any macroscopic observable commutes with the interaction Hamiltonian.

(vi) Approximate GL(H) symmetry-breaking selects a preferred basis in the Hilbert space. Given a superposition of macroscopically distinguishable eigenstates of a macroscopic observable, the interaction between the macroscopic system and its environment is such that the reduced state of the macroscopic system evolves very rapidly towards a state which is empirically indistinguishable from a mixture of the states which were initially superposed. In other words, the eigenbasis of the macroscopic observable almost diagonalizes the reduced density operator. This process is referred to as decoherence. In effect, a preferred basis is selected by the interaction Hamiltonian. Given that there is a one-one mapping between the bases of a Hilbert space and the general linear group GL(H), decoherence approximately breaks the GL(H) symmetry of quantum theory.

(vii) Each decohering macroscopic state can be treated as a Gaussian wave-packet Ψ_(x,p), of mean position 〈x〉 and mean momentum 〈p〉. A free Gaussian wave-packet initially minimises the position and momentum uncertainty. i.e., Δx Δ p = 1/2 ℏ. By virtue of being a wave-packet, the mean 〈x〉 of the position probability distribution will move with a velocity equal to the mean 〈p〉/m of the velocity probability distribution. A free Gaussian wave-packet is such that Δp remains constant, but Δx increases with time, a process referred to as the spreading of the wave-packet. Macroscopic Gaussian wave-packets, however, are constrained from spreading due to the continual interaction of the macroscopic system with its environment.

(viii) For each measurement-like interaction, the universe branches, and it is the branching which transforms potentiality into actuality. The branches are those selected by decoherence, and each branch realises one and only one of the states in the mixed state produced by decoherence.

(ix) The squared modulus of the complex amplitudes in the initial superposition correspond to the relative frequencies with which the different outcomes occur in most branches of the universe.

A frozen universe?

Philosopher of Physics Craig Callender discusses the arguments for considering that time is an illusion in the June edition of Scientific American. Particularly striking is the following analogy Callender draws between time and money:

We might describe the variation in the location of a satellite around Earth in terms of the ticks of a clock in my kitchen, or vice versa. What we are doing is describing the correlations between two physical objects, minus any global time as intermediary...Instead of saying a baseball accelerates at 20 meters per second per second, we can describe it in terms of the change of a glacier...Time becomes redundant. Change can be described without it.

This vast network of correlations is neatly organized, so that we can define something called "time" and relate everything to it, relieving ourselves of the burden of keeping track of all those direct relations...Money, too, makes life easier than negotiating a barter transaction every time you want to buy coffee. But it is an invented placeholder for the things we value, not something we value in and of itself. Similarly, time allows us to relate physical systems to one another without trying to figure out how a glacier relates to a baseball. But it, too, is a convenient fiction that no more exists fundamentally in the natural world than money does.

In terms of direct relations between physical objects, one could say that x generations of bacteria reproduce in one's intestine for every rotation of the Earth, and one could exchange y cups of coffee for an iPod. In terms of more abstract concepts, in the first case one could say how many seconds it takes for one rotation of the Earth, and how many seconds it takes for the bacteria in one's intestine to reproduce, and in the second case one could express the value of a cup of coffee in pounds, and the value of an iPod in pounds.

However, whilst it might well be possible to describe change without time, a static universe is a universe without time or change, hence the eliminability of time does not entail that the universe is static. Consider again the economic analogy. Money is a common means of expressing the relative values of different goods and services. If we refer to goods and services as economic objects, then money can be said to be an abstraction from the network of direct relative values of all the various pairs of economic objects. Time, by analogy, is a common means of expressing the relative change of different pairs of physical objects.

If time is to physical objects as money is to economic objects, then it must be an abstraction from a network of direct relations between pairs of physical objects. And what is that direct relation, if it isn't the relative amount of change? Conversely, relative change is to time as relative value is to money. The notion of money makes no sense without the concept of value, and the notion of time makes no sense without the concept of change.

As Callender asserts, [relative] change can be described without time, just as one can imagine an economy which operates without money. However, an economy without money is clearly not an economy in which economic objects have no value, and a timeless universe is not necessarily a universe without change. To eliminate change, and to reduce it to mere correlations between variables, an independent argument is required.

Here, Callender turns to canonical quantum gravity, in which the wave-function of the universe is represented by an apparently time-independent solution to the Wheeler-DeWitt equation. It is this fact which has been the primary spur behind the modern arguments for a static universe. To reconcile the time-independence of the wave-function of the universe with our perception of change, the concept of intrinsic time has been proposed.

The wave function Ψ in quantum theory is considered to be a function of various degrees of freedom: Ψ(x₁,...,x_j,...x_n). (In quantum cosmology, there are an infinite number of such degrees of freedom, but to keep things simple, let us suppose that there are only a finite number). The idea of intrinsic time is to identify at least one degree of freedom x_j, which behaves like a clock, and can be used as a surrogate time variable. Thence, one can denote x_j as t, and treat the wave-function as a time-dependent function Ψ_t(x₁,...,x_j-1,x_j+1,...x_n) of the remaining degrees of freedom.

On this view, time is an internal, approximate, emergent property of certain physical systems.

Spontaneous symmetry breaking in F1

Team-mates crashing into each other; team-management revealing a hitherto covert favouritism for one of their drivers; and a Grand Prix winner who appears indifferent to the rituals of victory, nurturing a belief that he was tricked by his team-mate: yes, Sunday's Turkish Grand Prix was a proper race.

Both Red Bull and McLaren purport to provide their drivers with equality of opportunity, but this intra-team driver symmetry will always be broken under racing conditions, for equality of outcome is neither possible, nor desirable. In the case of the Turkish Grand Prix, the root cause of the symmetry-breaking was the 2010 sporting regulations, which prohibit refuelling during a race. As Mark Hughes explains, filling up with 10kg less fuel gives a lap-time benefit of around 0.33s. From the evidence of Sunday's race, the top teams are therefore trimming their fuel loads to the absolute minimum, and giving themselves very little margin for error.

Hence, fuel consumption was an issue for both Red Bull and McLaren. It was this fact which created the opportunity for a momentary performance disparity between Webber and Vettel, and a similar disparity between Hamilton and Button a few laps later.

Whilst Hamilton and Webber fought tooth-and-nail in the first stint, Vettel and Button were able to watch from a convenient distance, staying within range, but conserving their brakes and fuel. Thus, when Mark Webber was forced to turn his engine settings down at mid-race, Sebastian Vettel was able to continue running his at a higher level for three vital laps, taking 0.3 secs out of Webber on two consecutive laps, and then using his superior power to catch Webber's slipstream on the run to Turn 12.

It is possible that there was no team favouritism here, for Vettel may genuinely have been able to conserve his fuel in the first part of the race, and the power discrepancy between the drivers at this point may be attributable to the differing circumstances they faced in the race. However, there can be no doubt that the ensuing accident was caused by Sebastian Vettel. Webber was perfectly within his rights to squeeze his team-mate onto the dirty part of the track, and Vettel was perfectly within his rights to go for the gap. At this point, however, team-principal Christian Horner could reportedly be seen mouthing "Move! move!" on the pitwall. Given that Webber was squeezing Vettel, rather than vice-versa, this can only be interpreted as an expostulation directed towards Webber.

Horner's bias became rather more overt after the race, when he argued that the principal cause of the accident had been the fact that [Webber] hadn't given Vettel sufficient room. Both Horner and Helmut Marko, (Red Bull owner Dietrich Mateschitz's representative within the team), voiced the opinion afterwards that Vettel needed to pass Webber at this stage, otherwise Hamilton would have passed Vettel. Although Vettel himself had only another lap available at the higher engine setting, it seems that the Red Bull team-management believed that he offered them their best chance of victory on this occasion.

Circumstances, then, have conspired to expose and amplify the Red Bull team's underlying favouritism towards Sebastian Vettel. It is difficult to imagine that Mark Webber will wish to remain with the team in these circumstances, hence the individual who suffers most from the collision between the Red Bull cars may ultimately be Felipe Massa...

Meanwhile, over at McLaren, Lewis Hamilton may feel that he can no longer trust his team-mate. In this respect, there was an interesting tell-tale conversation between Hamilton, Button and Webber, caught by the TV cameras, just before the post-race podium ceremony. Webber briefly recounted his accident with Vettel, and when Hamilton responded that "Yeah, he did the same to me," Button jumped to the conclusion that Lewis was referring to their own intra-team battle. "All they told me was..." Button immediately began to protest, before Hamilton interjected "No, no, no," and explained that he was complaining about Vettel's behaviour when Lewis attempted to overtake the Red Bull. This exchange suggests that Button felt he had done something about which Lewis could be expected to complain, and duly had his defence prepared.

So what was the problem? Well, Button it seems had been saving fuel from lap 20, and while Lewis took the fight to the Red Bulls, thereby triggering the very collision which put the McLarens first and second, Jenson had been able to sit back, watching it unfold, ready to pounce. After the Red Bulls had taken themselves out, McLaren told both drivers to save fuel, and according to Chief Engineer Tim Goss, set a 1m31s lap-time target for both of them. It's possible that Lewis interpreted this to be an instruction that the McLaren drivers should stop racing, and coast to the finish. If so, then it's equally clear that Button didn't share this interpretation. In fact, Jenson also claimed afterwards that he had not received a target lap-time instruction from the team.

When Jenson overtook Lewis, he came from a long way back, and didn't even need to use the full benefit of Lewis's slipstream as he zoomed down the outside. It's difficult to reach a conclusion other than that Button had a higher engine setting at this moment in time. Jenson presumably thought that he was entitled to do this by virtue of the fact that he had been saving fuel from an earlier point in the race. Lewis, therefore, was "surprised" to see Jenson come past him, and clearly thought that Button had tried to trick him out of victory.

Perhaps McLaren will be able to smooth over this mis-understanding in the days to come, and one duly expects some form of words to be issued to that effect. Nevertheless, in Lewis's mind, it's likely that an irreversible switch has been thrown. For both Red Bull and McLaren in 2010, things will never be the same again.