Wednesday, September 13, 2017

F1 1980 - Separation and curvature

Whilst the primary cause of boundary layer separation is the severity of the adverse pressure gradient experienced during pressure recovery, curvature upstream of the pressure recovery region can also exert a significant influence. In this context, a useful rule-of-thumb to remember is that the thicker the boundary layer at the start of the pressure recovery region, the earlier separation will occur. The rate at which the thickness of the boundary layer on a flat surface increases with distance from the leading edge is generally used as a baseline, with respect to which the effects of curvature can be compared.

To understand the influence of curvature, let’s first introduce a distinction between 2-dimensional and 3-dimensional boundary layers. In a 2-dimensional boundary layer, the velocity profile and thickness of the boundary later vary only in a longitudinal direction, along the direction of streamwise flow. The boundary-layer velocity is a function only of height above the solid surface and longitudinal distance; it is therefore 2-dimensional. In contrast, in a 3-dimensional boundary layer the velocity profile and thickness vary in both a longitudinal and a lateral direction. 

Consider first a 2-dimensional boundary layer on a surface with either convex or concave curvature. Concave curvature increases the rate at which a boundary layer thickens (compared to a flat surface), whilst convex curvature either thins a boundary layer, or reduces the rate at which the thickness would otherwise increase.

One way to understand this is in terms of radial pressure gradients. For a flowfield to negotiate a curve, a pressure gradient develops which is directed towards the centre of the radius of curvature, balancing the centrifugal force associated with the curved flow.

A flowfield bounded by a concave curve is such that the centre of curvature is located inside the fluid itself, hence a pressure gradient develops which points upwards from the solid surface into the fluid, effectively trying to peel the boundary layer off the surface

In contrast, a flowfield bounded by a convex surface is such that the centre of curvature is located the ‘other side’ of the solid surface, hence a pressure gradient develops which points downwards onto the surface, effectively pushing the boundary layer onto it.

Hence, concave curvature is liable to trigger boundary layer separation, while convex curvature promotes boundary layer adhesion.

So much for the influence of curvature on a 2-dimensional boundary layer. Most actual flowfields tend to possess ‘crossflow’ velocity components in addition to streamwise components. Crossflow components point in a lateral direction. In the context of wings, this is often referred to as ‘spanwise flow’. The representation of separation under these circumstances requires the introduction of the aforementioned 3-dimensional boundary layers.

The crossflow velocity components correspond to the existence of crossflow pressure gradients. These pressure gradients will induce streamline curvature both inside the boundary layer attached to the solid surface, and in the adjacent outer-flow streamlines. The streamline curvature, however, will be greater inside the boundary layer. Hence, the skin-friction lines on the solid surface (otherwise known as the shear stress at the wall), have greater curvature than the streamlines just outside the boundary layer. (Understanding Aerodynamics, Doug McLean, Wiley, 2013, p88).

Inserting a bend or kink into the wall of a venturi tunnel will generate a radial crossflow pressure gradient, pointing towards the centre of the radius of curvature. The outer-flow streamlines will turn the corner due to this radial pressure gradient. The skin-friction lines on the ceiling of the tunnel, however, will turn the corner at a tighter angle.

The curvature of a surface will itself generate streamline curvature, but this effect is distinct from the streamline curvature generated by a crossflow pressure gradient. If an outer-flow streamline is projected onto a curve in the solid surface, the curvature at each point of that curve can be decomposed into a component which is parallel to the tangent plane of the surface at that point, and a component which is perpendicular to the tangent plane. The perpendicular component represents the part of the curvature which is due to the streamline simply following the extrinsic curvature of the surface in 3-dimensional space. In contrast, the parallel component represents the intrinsic curvature of the projected streamline due to a crossflow pressure gradient. If there is no crossflow, then the projected streamlines are geodesics of the surface, with zero intrinsic curvature. (McLean, p306-307).   

A similar but distinct type of curvature effect occurs when a solid is bounded by an axisymmetric surface, whose radius varies in a longitudinal direction. If the lateral extent of a surface tapers in a longitudinal direction, then successive lateral slices through the surface possess an increasingly smaller diameter. For example, in the special case of a cone-shaped surface, oriented with the tip of the cone pointing downstream, successive lateral slices through the surface of the cone have a smaller diameter. A boundary layer attached to such a surface will thicken at a faster rate than it would over a flat surface with the same streamwise pressure gradient, (McLean p124). This occurs as a consequence of the preservation of mass and the relative incompressibility of the air: the boundary layer air is forced to thicken as its lateral dimensions contract. This makes such a boundary layer more liable to detach.

Conversely, consider a surface which flares outwards with longitudinal direction, an extreme case of which would be a cone-shaped surface with its tip pointing upstream. The boundary layer on such a surface will either get thinner as the lateral extent of the surface increases, or its thickness will increase at a slower rate than it would on a flat surface in the same streamwise pressure gradient. Hence, a surface which spreads outwards promotes boundary layer adhesion.

In both cases the outer-flow streamlines are following longitudinal geodesics of the surface, and there is no pressure-driven crossflow, (ibid). A Formula 1 car, however, is rarely equipped with axisymmetric appendages. Rather, it exhibits reflection symmetry in a longitudinal plane, and as a consequence the flow around the nose and engine cover are special cases of ‘plane of symmetry’ flows (ibid., p125-126). In such flows, the boundary layer along the plane of symmetry resembles a 2-dimensional boundary layer, with no crossflow component, but either side of the symmetry plane there are crossflow components which either induce divergence or convergence.

In the case of a Formula 1 car, the flow over the nose will be a divergent plane-of-symmetry flow, and that over the engine cover will tend to be a convergent plane-of-symmetry flow. 

So, equipped with this understanding of the effects of curvature, let’s consider an example of its impact on F1 ground-effect aerodynamics. In 1980, some of the teams created vertical surfaces at the rear of the sidepods to partially seal the venturi tunnels from the effects of the rotating rear wheel. The motive for this may have been twofold: to enhance underbody performance, and also to reduce rear wheel lift and drag. However, these plates, when considered in horizontal cross-section, traced a sinuous curve which started with concave curvature, passed through a point of inflection, and ended with convex curvature. Hence, whilst such plates may have prevented the flow in the venturi tunnels from directly interacting with the rotating wheel, the geometrical restriction imposed by the presence of the wheel was in no way eliminated.

If a venturi tunnel entered a constriction towards the rear of the sidepod, then the reduced cross-sectional area would have a tendency to thicken the boundary layer. Moreover, at just this point, the initial concave curvature on the outer wall of the tunnel would also contribute towards thickening the boundary layer. Exacerbating matters yet further, the turbulent jet from the inner contact patch of the rotating rear wheel would be injected into this region of the underbody. All three factors, in conjunction, would have tended to promote boundary layer separation in this part of the underbody. The only mitigation here is that the cross-sectional constriction would have weakened the adverse pressure gradient.

As a specific example of the challenges in this region of the underbody, the Williams FW07B MKIV underwing, as specified in a design drawing from April 1980, contained a dashed outline of an alternative profile for the sinuous section of the outer wall as it passes inside the rear wheel. The rationale behind this is alluded to in a briefing note written by Patrick Head, dated 1st April 1980, (just in advance of the introduction of the MKIV underwing at the Belgian Grand Prix). Here, he notes that Williams would be “running the wide rear track with new rear plates and engine fairings plus a wheel fairing which will reduce leakage into the rear of the side wing and increase the velocities. A new side wing profile is also to be made with an altered profile in the defuser (sic) section to reduce proneness to separation.”

The alternative profile reduced the concave curvature, but it did so at the expense of beginning the transition further upstream, therefore sacrificing channel width. Hence, there was a trade-off here: concave curvature or convergence; both would have thickened the boundary layer.

Frank Dernie has since testified that “most people’s diffusers stopped at the rear suspension. It was very difficult to keep the flow attached any further back…I am told the Brabham BT49 never had attached flow rearward of the chassis because they never found a solution to keeping the flow attached after the sudden change of section.” (Motorsport Magazine, November 2004, X-ray Spec: Williams FW07, p77).

In fact, the initial underbody profile on the Williams FW07B in 1980 did attempt to extend the diffuser tunnels beyond the leading edge of the rear suspension. These gearbox enclosures and sidepod extensions appeared on the car during practice in Argentina, but serious porpoising problems were experienced, and the sidepods and underbodies were returned to 1979 MKIII specification for the race. The porpoising was attributed to the skirts jamming, hence the extensions were tried again in conjunction with the MKIII sidepods and underwing during practice in South Africa. They were, however, notable by their absence when the MKIV underwing made its debut in Belgium. 

FW07B venturi extensions, as seen at Kyalami. (Grand Prix International magazine)

F1 1980 - Nozzles and streamtubes

Let’s delve a little more deeply into the nature of ground-effect downforce. The underbody of a ground-effect car can be treated as a type of (subsonic) converging-diverging nozzle. Such a nozzle consists of a mouth, a throat, and a diffuser. The mouth consists of a duct with a contracting cross-section, which accelerates air into the narrowest section, the throat. In accordance with the Bernoulli effect, the pressure is at its lowest in the throat, and the airflow velocity is at its highest. The air then flows from the throat into the diffuser, a duct with an expanding cross-section, which decelerates the air, and thereby returns it towards the freestream pressure, a process referred to as ‘pressure recovery’.

To give an illustration of the relative proportions here, the MKIV underbody on the Williams FW07B had a throat about 30 inches (762mm) in length, compared with a mouth only about 10 inches (254mm) long. The diffuser was about 45 inches (1143mm) in longitudinal extent.

Pressure recovery is a delicate process because it creates an ‘adverse pressure gradient’. The pressure increases in the direction of flow, hence there is a force pushing against the flow in the diffuser. Such an adverse pressure gradient tends to promote separation of the boundary layer. When separation occurs, the boundary layer is released into the interior of the fluid, where it breaks up into turbulence. This reduces the effective cross-sectional area and flow capacity of the diffuser, which in turn reduces the low pressure upstream at the throat. Separation also transforms a portion of the mean-flow kinetic energy into turbulent kinetic energy, which eventually dissipates as heat energy. To avoid separation, the diffuser tends to be much longer than the mouth and throat, with a more gradual slope than that between mouth and throat.

At a fixed freestream velocity (determined by the car-speed), the steady-state mass-flow rate through this nozzle is determined by the area of the diffuser outlet (assuming there is no separation), and by the base pressure at the diffuser exit. The latter will be lower than the freestream pressure due largely to the low pressure created by the suction surface of the rear-wing, but also due to the low-pressure wake behind the car.

To understand this further, it’s useful to introduce the concept of a ‘streamtube’. This is defined by taking a closed loop in the flowfield, identifying the streamline which passes through each point of the loop, and extruding the loop along those streamlines. This defines the surface of the streamtube. By definition, because the surface of a streamtube is constructed from streamlines, the velocity field is tangent to the surface of the tube, hence no mass can flow through the surface. Moreover, in a steady flow the mass flow-rate is the same through any cross-section of the streamtube.

Now, whilst the underbody of a ground-effect car has a solid mouth, (defined in 1980 by the geometry of the sidepod inlets), the flow upstream of the mouth is not confined by solid walls. Instead, it is defined by the streamtube of the flow which enters each venturi tunnel.

At a fixed car-speed, the greater the exit area of the diffuser, and/or the lower the base pressure created by the rear-wing, the greater the cross-sectional area of the streamtubes feeding the sidepod inlets. The greater the cross-sectional area of the streamtube feeding the mouth of each venturi tunnel, the greater the contraction as the air enters the throat of the tunnel, hence the greater the acceleration of the air and the greater the pressure drop. Therefore, “the degree of expansion of the air in the diffuser rather than the physical dimensions of the mouth determines the effective contraction of air into the throat, hence the maximum airspeed that will be obtained,” (Ian Bamsey, The Anatomy and Development of the Sports Prototype Racing Car, Haynes, 1991, p63).

A principal concern in the design of the underbody mouth is the avoidance of separation. Depending upon the car-speed and the base-pressure, the stream-tubes entering the venturi tunnels may either expand or contract as they approach the mouth. There will be a stagnation line somewhere around the upper-lip of each mouth: flow below this line will enter the venturi duct, while flow above it will pass over the top of the sidepod. If the stream-tubes expand approaching the mouth of each tunnel, (as they might do at high car speeds), then the stagnation line might lie just inside the upper lip of the tunnel, and the external flow might separate as it accelerates over and around the upper lip. Conversely, if the stream-tubes contract approaching each mouth, the stagnation line might exist just outside the upper lip, and the flow might separate as it accelerates under that lip into the tunnel. The latter condition would inject turbulence into the throat of the underbody tunnel, leading to a significant loss of downforce.

Friday, August 11, 2017

Curved flow and the Arrows A3

After something of a sustained gestation period, the publication of F1 Retro 1980 is imminent, so it's a good opportunity to take a look at one of the more interesting aerodynamic experiments seen that season: the underbody venturi extensions on the Arrows A3 at Brands Hatch. 

This was the latest in a series of attempts to improve upon the original F1 ground-effect concept. In 1979, the Lotus 80 and the Arrows A2 had both attempted to extend the area of the underbody, but both had failed to reap the expected benefits.

The Lotus 80, in its initial configuration, featured skirts under the nose, and separate skirts extending all the way from the leading edge of the sidepods, inside the rear wheels, to the back of the car. The failure of the Lotus 80 is commonly attributed both to an ineffective skirt system, and an insufficiently rigid chassis.  

The Arrows A2 featured an engine and gearbox inclined at 2.5 degrees in an attempt to exploit the full width of the rear underbody. In its original configuration the A2 also dispensed with a conventional rear-wing, replacing it with a flap mounted across the rear-deck. The sidepod skirts were complemented by a parallel pair of skirts running inside the width of the rear wheels to the back of the car. Unfortunately, the higher CoG at the back entailed the car had to be run with a stiff rear anti-roll bar, detracting from the handling, (Tony Southgate - From Drawing Board to Chequered Flag, MRP 2010, p108).

The 1980 Arrows A3 was a more conventional car, with the engine and gearbox returned to a horizontal inclination. However, at Brands Match in 1980, Arrows experimented, like the initial Lotus 80, with skirts under the nose. Developed in the Imperial College wind-tunnel, the Arrows version of the idea had skirts suspended from sponsons attached to the lower edges of the monocoque, running back beneath the lower front wishbones to the leading edge of the sidepods. At the same event, the team also tried extending the rear underbody all the way to the trailing edge of the rear suspension, with bulbous fairings either side of the gearbox fairing. This was done with the avowed intention of sealing the underbody from the detrimental effects of rear wheel turbulence.

Sadly, although the nose-skirts were intended to cure understeer, it was reported that they actually exacerbated the understeer.

Now, many aerodynamic difficulties encountered in this era of Formula One were actually just a manifestation of inadequate stiffness in the chassis or suspension. However, for the sake of argument, let's pursue an aerodynamic hypothesis to explain why the nose-skirts on the A3 worsened its understeer characteristic.

The nose skirts on the Lotus 80 and Arrows A3 would have suffered from the fact that a Formula 1 car has to generate its downforce in a state of yaw. Thus, in a cornering condition, a car is subjected to a curved flow-field. This is difficult to replicate in a wind-tunnel, hence a venturi tunnel design which worked well in a straight-ahead wind-tunnel condition could have failed dramatically under curved flow conditions. To understand this better, a short digression on curved flow and yaw angles is in order.

The first point to note is that a car follows a curved trajectory through a corner, hence if we switch to a reference frame in which the car is fixed but the air is moving, then the air has to follow a curved trajectory. If we freeze the relative motion mid-corner, with the car pointing at a tangent to the curve, then the air at the front of the car will be coming from approximately the direction of the inside front-wheel, while the air at the back of the car will be coming from an outer direction.

That's the simplest way of thinking about it, but there's a further subtlety. The negotiate a corner, a car generates: (i) a lateral force towards the centre of the corner's radius of curvature; and (ii) a yaw moment about its vertical axis.

Imagine the two extremes of motion where only one of these eventualities occur. In the first case, the car would continue pointing straight ahead, but would follow a curved path around the corner, exiting at right-angles to its direction of travel. In the second case, it would spin around its vertical axis while its centre-of-mass continued to travel in a straight line.

In the first case, the lateral component of the car's velocity vector corresponds to a lateral component in the airflow over the car. The angle which the airflow vector subtends to the longitudinal axis of the car, is the same along the length of the vehicle.

In the second case, the spinning motion also induces an additional component to the airflow over the car. It's a solid body spinning about its centre of mass with a fixed angular velocity, and the tangential velocity of that spin induces an additional component to the airflow velocity along the length of the car. However, the further away a point is from the axis of rotation, the greater the tangential velocity; such points have to sweep out circles of greater circumference than points closer to the centre of mass, hence their tangential velocity is greater.

Curved-flow, side-slip and yaw-angle. (From 'Development methodologies for Formula One aerodynamics', Ogawa et al, Honda R&D Technical Review 2009).
Now imagine the two types of motion combined. The result is depicted above, in the left-part of the diagram. The white arrows depict the component of the airflow due to 'side-slip': the car's instantaneous velocity vector subtends a small angle to the direction in which its longitudinal axis is pointing. In the reference frame in which the car is fixed, this corresponds to a lateral component in the direction of the airflow which is constant along on the length of the car.

When the yaw moment of the car is included (indicated by the curved blue arrow about the centre-of-mass), it induces an additional airflow component, indicated by the green arrows. Two things should be noted: (i) the green arrows at the front of the car point in the opposite direction from the green arrows at the rear; and (ii) the magnitude of the green arrows increases with distance from the centre of mass. The front of the car is rotating towards the inside of the corner, while the rear of the car is rotating away, hence the difference in the direction of the green arrows. And, as we explained above, the tangential velocity increases with distance from the axis of rotation, hence the increase in the magnitude of the green arrows.

The net result, indicated by the red arrows, is that the yaw-angle of the airflow has a different sign at the front and rear of the car, and the magnitude of the yaw angle increases with distance from the centre-of-mass. (The red arrows in the diagram are pointing in the direction in which the car is travelling; the airflow direction is obtained by reversing these arrows).

So, to return to 1980, the Arrows A3 design trialed at Brands Hatch moved the mouth of the venturi tunnel forward to the nose of the car. The further forward the mouth, the greater the angle of the curved onset flow to the longitudinal axis of the car, and the further away it is from the straight-ahead condition. Hence, the curved flow might well have separated from the leading edge of the skirt on the side of the car facing the inside of the corner, injecting a turbulent wake directly down the centre of the underbody. In this respect, the conventional location of the venturi inlets on a 1980 F1 car, (i.e., behind the front wheel centreline), would have reduced yaw sensitivity.

Front-wings and rear-wings certainly have to operate in state of yaw, and do so with a relatively high level of success. However, such devices have a larger aspect-ratio than an underbody venturi, which has to keep its boundary layer attached for a much longer distance.

It should also be noted that the flow through the underbody tunnels, like that through any type of duct, suffers from ‘losses’ which induce drag. The energy budget of a flow-field can be partitioned into kinetic energy, pressure-energy, and ‘internal’ heat energy. Viscous friction in the boundary layers, and any turbulence which follows from separation in the duct, creates heat energy, and irreversibly reduces the sum of the mean-flow kinetic energy and the pressure energy.

These energy losses are proportional to the length of the duct, the average flow velocity through the duct, and inversely proportional to the effective cross-sectional diameter of the duct. Due to such losses, it is not possible for full pressure recovery to be attained in the diffuser and its wake, and this will contribute to the total drag of the car. Hence, whilst underbody downforce comes with less of a drag penalty than that associated with inverted wings in freestream flow, it is nevertheless true that the longer the venturi tunnels, and the greater the average velocity of the underbody flow, the greater the drag of the car. 
Moreover, the longer the mouth and throat of a venturi tunnel, the thicker the boundary layer at the start of the pressure recovery region, and the more prone it will be to separation in that adverse pressure gradient. All of which mitigates against a quick and easy gain from extending the area of the underbody.

Monday, August 07, 2017

Driverless cars and cities

Driverless cars are somewhat in the news this year, with Ford investing $1bn to meet their objective of launching a fleet of autonomous cars in 2021. Coincidentally, the July 2017 issue of 'Scientific American' features an article extolling the virtues of a driverless future in modern cities. The article is written by Carlo Ratti and Assaf Biderman, who work for something called the 'Senseable Lab' at the Massachusetts Institute of Technology. 

 A number of the claims made in the article are worth reviewing. Let's start with the following:

"On average, cars sit idle 96 percent of the time. That makes them ideal candidates for the sharing economy...The potential to reduce congestion is enormous...'Your' car could give you a lift to work in the morning and then, rather than sitting in a parking lot, give a lift to someone else in your family - or to anyone else in your neighbourhood or social media community...a city might get by with just 20 percent the number of cars now in use...fewer cars might also mean shorter travel times, less congestion and a smaller environmental impact."

A number of thoughts occur in response to these claims:

1) Ride-sharing would reduce the number of cars, not the number of journeys. Every journey which currently takes place would still take place, but in addition would be all the journeys made when a car needs to travel from the point where one passenger disembarks to the point where the next embarks. At present, each journey contains a passenger; with the proposed ride-sharing of driverless cars, there would be additional journeys in which the cars contain no passengers at all. All other things being equal, that would increase congestion and pollution, not reduce it. 

2) The modern technological world, including the GPUs and artificial neural networks which have created the possibility of driverless vehicles, has been built upon the wealth of a capitalist economy. Such an economy is driven by, amongst other things, the incentivization of private ownership. In particular, people like owning their own cars. It's not clear why a technological development alone, such as that of the driverless car, will prompt society to adopt a new model of shared ownership.

3) Not everyone lives in cities. Universities tend to be located in cities, hence many academics fall into the habit of thinking that everyone lives and works in cities. Many people live outside cities, and drive into them to their places of work. They drive into the cities from different places at the same time each morning. For such people, there needs to be a one-to-one correspondence between cars and passengers.

4) People like the convenience and efficacy of having a car parked adjacent to their home or place of work. If you're a parent, and your child falls ill at home, or there's an accident at school, you want to drive there immediately, not wait for a shared resource to arrive.

5) If cars are constantly in use, their components will degrade in a shorter period of time, so maintenance costs will greater, and the environmental impact of manufacturing new tyres, batteries etc. will be greater.

So that's just for starters. What else do our MIT friends have to say? Well, they next claim that "vacant parking lots could be converted to offer shared public amenities such as playgrounds, cafes, fitness trails and bike lanes."

Unfortunately, most car parks are privately owned, either by retail outlets or employers. If they become redundant, then those private companies will either extend their existing office space or floor space, or sell to the highest bidder. Car-parks are unlikely to become playgrounds.

The authors then claim that current traffic-light controlled intersections could be managed in the style of air traffic control systems: 

"On approaching an intersection, a vehicle would automatically contact a traffic-management system to request access. It would then be assigned an individualized time, or 'slot', to pass through the intersection.

"Slot-based intersections could significantly reduce queues and delays...Analyses show that systems assigning slots in real time could allow twice as many vehicles to cross an intersection in the same amount of time as traffic lights usually do...Travel and waiting times would drop; fuel-consumption would go down; and less stop-and-go traffic would mean less air pollution."

Sadly, this is a concept which seems to imagine that cities consist of grids of multi-lane highways. Most cities in the world don't. And in every city, the following 'min-max' principle of road-capacity applies:

For a sequence of interconnected roads, given the capacity (i.e., the maximum flow-rate) in each component, the capacity of the entire sequence is the minimum of those individual capacities. 

Hence, even if the capacity of every multi-lane intersection in a city is doubled, the capacity of a linked sequence is determined by the component with the lowest capacity. In many cities, multi-lane highways taper into single-lane roads, and it is the single-lane roads which limit the overall capacity. Doubling the capacity of intersections would merely change the spatial distribution of the queues.

So, all in all, not a positive advert for driverless cars.

Tuesday, July 04, 2017

Cosmology and entropy

After a hiatus of approximately seven years, I've just written a new paper, 'Cosmology and entropy: in search of further clarity'. Here's an extract:

It is a widespread belief amongst modern physicists that black holes, or their horizons, possess temperature and entropy. The putative black-hole temperature is inversely proportional to the surface area of the horizon, while the entropy is proportional to the surface area. In natural units, the entropy of a non-rotating black hole is (Penrose 2016, p271):
S_{bh} = \frac{1}{4} A = 4 \pi M^2 \,,
$$ where $A$ is the area and $M$ is the mass.

The concept that a black hole could be the bearer of entropy is often justified by claiming that the black-hole entropy compensates for the 'loss of information', or the 'lost degrees of freedom', associated with matter and radiation falling into the black hole, never to be seen again. Bekenstein's original argument went as follows:

"Suppose that a body carrying entropy $S$ goes down a black hole...The $S$ is the uncertainty in one's knowledge of the internal configuration of the body...once the body has fallen in...the information about the internal configuration of the body becomes truly inaccessible. We thus expect the black hole entropy, as the measure of the inaccessible information, to increase by an
amount $S$
," (Bekenstein 1973).

Presumably, the idea is that one loses both the actual entropy and the maximum possible entropy associated with these extinguished dimensions of phase-space. However, as Dougherty and Callender (2016) point out, Bekenstein-type arguments express an epistemic and operationalistic interpretation of entropy. They rightly complain that "The system itself doesn't vanish; indeed, it had better not because its mass is needed to drive area increase...there is no reason to believe that a body slipping past an event horizon would lose its compensation is necessary...we could observe the entropy of steam engines and the like that fall behind event horizons. Just jump in with them!"

We can make the objection more precise in general relativistic terms. For example, take the Oppenheimer-Snyder spacetime for a star collapsing to a black hole, or the Schwarzschild spacetime for a black hole itself. In each case, the spacetime is globally hyperbolic, hence it can be foliated by a one-parameter family of spacelike Cauchy hypersurfaces $\Sigma_t$, and the entire spacetime is diffeomorphic to $\mathbb{R} \times \Sigma$.

Each Cauchy surface is a complete and boundaryless 3-dimensional Riemannian manifold. There is no sense in which any Cauchy surface intersects the singularity. Each Cauchy surface which contains a region inside the event horizon also contains a region outside the horizon. Moreover, every inextendible causal curve in a globally hyperbolic spacetime $\mathbb{R} \times \Sigma$ intersects each Cauchy surface $\Sigma_t$ once and only once. Particles follow causal curves, hence because each particle will intersect each Cauchy surface exactly once, assuming that none of those particles form bound systems, it follows that no degrees of freedom are lost. The future may well be finite inside the event horizon, but that doesn't entail that any degrees of freedom are lost from the universe.

The entropy of one part of the universe can decrease, just as the entropy of a volume of water decreases when it transfers heat to some ice cubes immersed within it. Similarly, if a material system possessing entropy falls into a black hole, whilst the region of the universe exterior to the black hole loses entropy, the total entropy does not decrease from one spacelike Cauchy hypersurface to the next. To echo Dougherty and Callender, there is no reason for the event horizon of a black hole to possess entropy; there is simply no loss to compensate for.

Penrose, however, argues that "the enormous entropy that black holes possess is to be expected from...the remarkable fact that the structure of a stationary black hole needs only a very few parameters [mass, charge and angular momentum] to characterize its state. Since there must be a vast volume of phase space corresponding to any particular set of values of these parameters, Boltzmann's formula suggests a very large entropy," (2010, p179).

This appeal to the 'no-hair' theorem of black holes is based upon a sleight of hand: it is the space-time geometry of the stationary, asymptotically flat, vacuum solutions which are classified by just three parameters. Such vacuum solutions are useful idealisations for studying the behaviour of test particles in a black hole spacetime, but they do not represent the history of actual black-holes.

The spacetime of an actual black-hole contains the mass-energy which collapses to form the black hole, and any mass-energy which falls into the black-hole thereafter, including swirling accretion disks of matter and so forth. Hence, actual black holes are represented by variations of the Oppenheimer-Snyder spacetime, not the Schwarzschild space-time. As Dafermos and Rodnianski comment, "It is traditional in general relativity to 'think' Oppenheimer-Snyder but `write' maximally-extended Schwarzschild," (2013, p18).

Whilst the exterior region of a collapse solution is isometric to an exterior region of the vacuum solution, the difference in the interior solution makes all the difference in the world. Spacetimes which represent collapse to a black-hole are not classified by just three parameters; on the contrary, they are classified by a large number of parameters, characterising the specifics of the collapsing matter, including its entropy. The entropy of such black-hole spacetimes is possessed, not by the geometry of the black-hole horizon, but by the infalling mass-energy, just as it should be, (see Figure 1).

Conformal diagram of a black hole, including a pair of Cauchy
surfaces, $\Sigma_1$ and $\Sigma_2$. The shaded region represents the infalling
matter; the thin diagonal line represents the event horizon; and the jagged
line represents the singularity. Cauchy surface $\Sigma_2$ possesses a region
inside the event horizon, and a region outside the event horizon. The shaded
region possesses entropy; the horizon doesn't. (From Maudlin 2017)

Dougherty and Callender also draw attention to a number of conceptual contradictions associated with the notion that black-hole horizons possess entropy and temperature. For example:
  1. The increase in the area of a black-hole horizon, and therefore its purported entropy, is proportional to the mass-energy of the material which falls into the black-hole. Hence, if a massive object with a small entropy falls into the hole, it produces a large increase in black-hole entropy, whilst if a small object with a large entropy falls in, it produces a small increase in black-hole entropy.
  2. Entropy is an 'extensive' thermodynamic property, meaning that it is proportional to the volume of a system. In contrast, black-hole entropy is proportional to the area of the black-hole.
  3. Temperature is an 'intensive' thermodynamic property, meaning it is independent to the size of an object, yet if the size of a black-hole is increased, its temperature decreases.
  4. There is no 'equilibrium with' relationship in black-hole thermodynamics. Individual black-holes can be in equilibrium in the sense that the spacetime is stationary, but one black-hole cannot be in equilibrium with another.
  5. If two black-holes of the same area, and therefore with the same purported temperature, coalesce, then the area of the merged black-hole is greater than each of its progenitors, hence the purported entropy increases. In contrast, thermodynamics dictates that the coalescence of two entities at the same temperature is an isentropic process.
Even if it is accepted that black holes, or their horizons, possess entropy, a belief in black hole entropy is typically twinned with a belief in the eventual evaporation of black holes. For example, Penrose (2010,  p191) asserts that black holes will evaporate by Hawking radiation after the cosmic background radiation cools to a lower temperatures than the temperature of the holes. In this scenario, all the entropy in the universe eventually becomes radiation entropy. Hence, once again, it seems that the clumping of matter is nothing more than an intermediate state. If black holes can evaporate, then black holes are clearly not the ultimate means by which entropy is maximised.

An alternative scenario suggests that large black holes will not evaporate because there is a fundamental lower limit to the temperature of the cosmological radiation field, and this temperature is greater than the possible temperature of large black holes. The belief in such a lower limit is based upon the fact that a universe with a positive cosmological constant $\Lambda > 0$, such as ours currently appears to be, possesses a spacelike future conformal boundary, and the past light cone of each point on this future boundary defines an event horizon. It is then suggested that this event horizon possesses a temperature and an entropy, just as much as the event horizon of a black hole.

However, the reasons for believing that a cosmological event horizon possesses temperature and entropy are much weaker than those for believing a black hole possesses thermodynamic properties. The cosmological event horizon is entirely observer dependent, unlike the case of a black hole event horizon. Moreover, the region rendered unobservable by an event horizon is the region to the future of the event horizon, and in the case of the cosmological event horizon this is the region to the exterior of the past light cone. (In contrast, the
region to the future of the event horizon of a black-hole is the interior of the black hole). 

Penrose (2016, p278-279) points out that the region to the exterior will be of infinite volume if the universe is spatially non-compact, hence its entropy will also be infinite. It therefore makes no sense to interpret the (finite) entropy of a cosmological event horizon as the entropy/information of all the matter and radiation `lost' beyond that horizon.

Sunday, March 19, 2017

The fundamental fallacy of modern feminism

There is, within contemporary film and television, a prevailing fashion for portraying women, in various combinations and degrees, as physically strong, aggressive, competitive, risk-takers. The writers, actors, producers, and directors responsible, and their sympathetic media critics, believe that there is some form of entrenched, gender-based discrimination in society, which film and television can help to overturn. They regard themselves as agents of social-change, engaged on a type of quest.

It is a puzzling phenomenon because, far from being testimony to an industry driven by egalitarian values, it actually reveals a deep-seated dislike and contempt of femininity. These films and TV programmes portray female characters as good, or worthy of praise, in direct proportion to the extent to which their behaviour imitates that of men. It follows that masculinity, and the behaviour of men, is being assigned the highest value; masculinity is setting the standard by which female characters are to be judged.

So where does this fashion spring from? Part of the reason may be a strain of thought in feminist academia, which holds that the differences in male and female behaviour are purely contingent, and not rooted in biological differences between the sexes. It's no coincidence that this notion is largely promulgated by philosophers, psychologists, and sociologists;  i.e., those who lack a rigorous scientific education.

As something of an antidote, recall the principal scientific fact in this context: The human species has evolved by natural selection with sexual reproduction. As a consequence, sexual selection has operated, amplifying differences in appearance and behaviour between the sexes. Gendered humans experience reproductive success in proportion to the extent that they exhibit the appearance and modes of behaviour associated with their own sex. In this respect, humanity is just like many other animal species.

So what could make people think that the differences between the human sexes have anything other than a natural, biological explanation? The answer, it seems, is the concept of 'social conditioning'. In particular, this notion is presented as an independent explanatory alternative to biological explanations:

Are women's “feminine” traits the product of nature/biology or are they instead the outcome of social conditioning? (Feminist Ethics, Stanford Encyclopedia of Philosophy). conditioning creates femininity and societies...physiological features thought to be sex-specific traits not affected by social and cultural factors are, after all, to some extent products of social conditioning. Social conditioning, then, shapes our conditioning makes the existence of physical bodies intelligible to us by discursively constructing sexed bodies through certain constitutive acts. (Feminist perspectives on sex and gender, Stanford Encylopedia of Philosophy)

Not only are the differences in behaviour between the sexes attributed to social conditioning, but so also are the differences in appearance:

Uniformity in muscular shape, size and strength within sex categories is not caused entirely by biological factors, but depends heavily on exercise opportunities: if males and females were allowed the same exercise opportunities and equal encouragement to exercise, it is thought that bodily dimorphism would diminish (ibid.)

Now clearly, social conditioning exists. It is, for example, responsible for the differences in behaviour between "white working-class women, black middle-class women, poor Jewish women, wealthy aristocratic European women," (ibid). Moreover, women across all human societies are subject to different expectations than men. If women across all human societies have a set of shared characteristics (in a statistical sense), then those characteristics will correspond to a set of shared biological characteristics, and a shared stream of social conditioning.

The fallacy of modern feminism, however, is the implicit assumption that social conditioning is somehow independent of a biological explanation. It's clear from reading this type of material that the authors consider an explanation in terms of 'social causes' or 'social forces' to be an endpoint, rather than something in need of further explanation. The identification and discovery of a case of social conditioning is presented in triumph, as the culmination of the research.

Human society has emerged as a net consequence of the interactions between billions of biologically gendered individuals over thousands of generations. Society is not free-floating, it is tethered to the natural and biological world. All social phenomenon are ultimately explicable in terms of the biological processes from which they emerge. If men and women are subject to different social conditioning, then it is because men and women are biologically distinct. The differences in social conditioning are a response to the biological differences, and part of the sexual selection feedback loop which amplifies and controls those differences.

By presenting a false dichotomy between social explanations and biological explanations, modern feminists seem to have convinced a generation of film-makers and media types, not to mention a large fraction of the political classes, that the differences between men and women are social rather than biological. It's an important difference, because if you think the differences are merely social and contingent, then it follows that equality of outcome between the sexes, rather than mere equality of opportunity, is possible with the appropriate form of social re-engineering. In other words, it encourages a type of gender neo-Marxism.

Wednesday, December 14, 2016

Westworld and the mathematical structure of memories

The dominant conceptual paradigm in mathematical neuroscience is to represent the human mind, and prospective artificial intelligence, as a neural network. The patterns of activity in such a network, whether they're realised by the neuronal cells in a human brain, or by artificial semiconductor circuits, provide the capability to represent the external world and to process information. In particular, the mathematical structures instantiated by neural networks enable us to understand what memories are, and thus to understand the foundation upon which personal identity is built.

Intriguingly, however, there is some latitude in the mathematical definition of what a memory is. To understand the significance of this, let's begin by reviewing some of the basic ideas in the field.

On an abstract level, a neural network consists of a set of nodes, and a set of connections between the nodes. The nodes possess activation levels; the connections between nodes possess weights; and the nodes have numerical rules for calculating their next activation level from a combination of the previous activation level, and the weighted inputs from other nodes. A negative weight transmits an inhibitory signal to the receiving node, while a positive weight transmits an excitatory signal.

The nodes are generally divided into three classes: input nodes, hidden/intermediate nodes, and output nodes. The activity levels of input nodes communicate information from the external world, or another neural system; output notes transmit information to the external world or other neural systems; and the hidden nodes merely communicate with other nodes inside the network. 

In general, any node can possess a connection with any other node. However, there is a directionality to the network in the sense that patterns of activation propagate through it from the input nodes to the output nodes. In a feedforward network, there is a partial ordering relationship defined on the nodes, which prevents downstream nodes from signalling those upstream. In contrast, such feedback circuits are permitted in a recurrent network. Biological neural networks are recurrent networks.

Crucially, the weights in a network are capable of evolving with time. This facilitates learning and memory in both biological and artificial networks. 

The activation levels in a neural network are also referred to as 'firing rates', and in the case of a biological brain, generally correspond to the frequencies of the so-called 'action potentials' which a neuron transmits down its output fibre, the axon. The neurons in a biological brain are joined at synapses, and in this case the weights correspond to the synaptic efficiency. The latter is dependent upon factors such as the pre-synaptic neurotransmitter release rate, the number and efficacy of post-synaptic receptors, and the availability of enzymes in the synaptic cleft. Whilst the weights can vary between inhibitory and excitatory in an artificial network, this doesn't appear to be possible for synaptic connections.

Having defined a neural network, the next step is to introduce the apparatus of dynamical systems theory. Here, the possible states of a system are represented by the points of a differential manifold $\mathcal{M}$, and the possible dynamical histories of that system are represented by a particular set of paths in the manifold. Specifically, they are represented by the integral curves of a vector field defined on the manifold by a system of differential equations. This generates a flow $\phi_t$, which is such that for any point $x(0) \in \mathcal{M}$, representing an initial state, the state after a period of time $t$ corresponds to the point $x(t) = \phi_t(x(0))$.  

In the case of a neural network, a state of the system corresponds to a particular combination of activation levels $x_i$ ('firing rates') for all the nodes in the network, $i = 1,\ldots,n$. The possible dynamical histories are then specified by ordinary differential equations for the $x_i$. A nice example of such a 'firing rate model' for a biological brain network is provided by Curto, Degeratu and Itskov:

\frac{dx_i}{dt} = - \frac{1}{\tau_i}x_i + f \left(\sum_{j=1}^{n}W_{ij}x_j + b_i \right), \,  \text{for } \, i = 1,\ldots,n
$W$ is the matrix of weights, with $W_{ij}$ representing the strength of the connection from the $j$-th neuron to the $i$-th neuron; $b_i$ is the external input to the $i$-th neuron; $\tau_i$ defines the timescale over which the $i$-th neuron would return to its resting state in the absence of any inputs; and $f$ is a non-linear function which, amongst other things, precludes the possibly of negative firing rates. 

In the case of a biological brain, one might have $n=10^{11}$ neurons in the entire network. This entails a state-state of dimension $10^{11}$. Within this manifold are submanifolds corresponding to the activities of subsets of neurons. In a sense to be defined below, memories correspond to stable fixed states of these submanifolds.

In dynamical systems theory, a fixed state $x^*$ is defined to be a point $x^* \in \mathcal{M}$ such that $\phi_t(x^*) = x^*$ for all $t \in \mathbb{R}$. 

The concept of a fixed state in the space of possible firing patterns of a neural network captures the persistence of memory. Memories are stored by changes to the synaptic efficiencies in a subnetwork, and the corresponding matrix of weights $W_{ij}$ permits the existence of a fixed state in the activation levels of that subnetwork. 

However, real physical systems cannot be controlled with infinite precision, and therefore cannot be manoeuvred into isolated fixed points in a continuous state space. Hence memory states are better defined in terms of the properties of neighbourhoods of fixed points. In particular, some concept of stability is required to ensure that the state of the system remains within a neighbourhood of a fixed point, under the inevitable perturbations and errors suffered by a system operating in a real physical environment.

There are two possible definitions of stability in this context (Hirsch and Smale, Differential Equations, Dynamical Systems and Linear Algebra, p185-186):

(i) A fixed point is stable if for every neighbourhood $U$ there is a super-neighbourhood $U_1$ such that any initial point $x(0) \in U$ remains in $U_1$, and therefore close to $x^*$, under the action of the flow $\phi_t$.

(ii)  A fixed point is asymptotically stable if for every neighbourhood $U$ there is a super-neighbourhood $U_1$ such that any initial point $x(0) \in U$ not only remains in $U_1$ but $lim_{t \rightarrow \infty} x(t) = x^*$.

The first condition seems more consistent with the nature of human memory. The memories are not perfect, retaining some aspects of the original experience, but fluctuate with time, (and ultimately become hazy as the synaptic weights drift away from their original values). The second condition, however, is a much stricter condition. In conjunction with an ability to fix the weights of a subnetwork on a long-term basis, this condition seems consistent with the long-term fidelity of memory. 

At first sight, one might wish to design an artificial intelligence so that its memories are asymptotically stable fixed points in the possible firing rate patterns within an artificial neural network. However, doing so could well entail that those memories become as vivid and realistic to the host systems as their present-day experiences. It might become impossible to distinguish past from present experience. 

And that might not turn out so well...

Saturday, November 19, 2016

Neural networks and spatial topology

Neuro-mathematician Carina Curto has recently published a fascinating paper, 'What can topology tell us about the neural code?' The centrepiece of the paper is a simple and profound exposition of the method by which the neural networks in animal brains can represent the topology of space.

As Curto reports, neuroscientists have discovered that there are so-called place cells in the hippocampus of rodents which "act as position sensors in space. When an animal is exploring a particular environment, a place cell increases its firing rate as the animal passes through its corresponding place field - that is, the localized region to which the neuron preferentially responds." Furthermore, a network of place cells, each representing a different position, is collectively capable of representing the topology of the environment.

Rather than beginning with the full topological structure of an environmental space X, the approach of such research is to represent the collection of place fields as an open covering, i.e., a collection of open sets $\mathcal{U} = \{U_1,...,U_n \}$ such that $X  = \bigcup_{i=1}^n U_i$. A covering is referred to as a good cover if every non-empty intersection $\bigcap_{i \in \sigma} U_i$ for $\sigma \subseteq \{1,...,n \}$ is contractible. i.e., if it can be continuously deformed to a point.

The elements of the covering, and the finite intersections between them, define the so-called 'nerve' $\mathcal{N(U)}$ of the cover, (the mathematical terminology is coincidental!):

$\mathcal{N(U)} = \{\sigma \subseteq \{1,...,n \}: \bigcap_{i \in \sigma} U_i \neq \emptyset \}$.

The nerve of a covering satisfies the conditions to be a simplicial complex, with each subset $U_i$ corresponding to a vertex, and each non-empty intersection of $k+1$ subsets defining a $k$-simplex of the complex. A simplicial complex inherits a topological structure from the imbedding of the simplices into $\mathbb{R}^n$, hence the covering defines a topology. And crucially, the following lemma applies:

Nerve lemma: Let $\mathcal{U}$ be a good cover of X. Then $\mathcal{N(U)}$ is homotopy equivalent to X. In particular, $\mathcal{N(U)}$ and X have exactly the same homology groups.

The homology (and homotopy) of a topological space provides a group-theoretic means of characterising the topology. Homology, however, provides a weaker, more coarse-grained level of classification than topology as such. Homeomorphic topologies must possess the same homology (thus, spaces with different homology must be topologically distinct), but conversely, a pair of topologies with the same homology need not be homeomorphic. 

Now, different firing patterns of the neurons in a network of hippocampal place cells correspond to different elements of the nerve which represents the corresponding place field. The simultaneous firing of $k$ neurons, $\sigma \subseteq \{1,...,n \}$, corresponds to the non-empty intersection $\bigcap_{i \in \sigma} U_i \neq \emptyset$ between the corresponding $k$ elements of the covering. Hence, the homological topology of a region of space is represented by the different possible firing patterns of a collection of neurons.

As Curto explains, "if we were eavesdropping on the activity of a population of place cells as the animal fully explored its environment, then by finding which subsets of neurons co-fire, we could, in principle, estimate $\mathcal{N(U)}$, even if the place fields themselves were unknown. [The nerve lemma] tells us that the homology of the simplicial complex $\mathcal{N(U)}$ precisely matches the homology of the environment X. The place cell code thus naturally reflects the topology of the represented space."

This entails the need to issue a qualification to a subsection of my 2005 paper, 'Universe creation on a computer'. This paper was concerned with computer representations of the physical world, and attempted to place these in context with the following general definition:

A representation is a mapping $f$ which specifies a correspondence between a represented thing and the thing which represents it. An object, or the state of an object, can be represented in two different ways:

$1$. A structured object/state $M$ serves as the domain of a mapping $f: M \rightarrow f(M)$ which defines the representation. The range of the mapping, $f(M)$, is also a structured entity, and the mapping $f$ is a homomorphism with respect to some level of structure possessed by $M$ and $f(M)$.

$2$. An object/state serves as an element $x \in M$ in the domain of a mapping $f: M \rightarrow f(M)$ which defines the representation. 

The representation of a Formula One car by a wind-tunnel model is an example of type-$1$ representation: there is an approximate homothetic isomorphism, (a transformation which changes only the scale factor), from the exterior surface of the model to the exterior surface of a Formula One car. As an alternative example, the famous map of the London Underground preserves the topology, but not the geometry, of the semi-subterranean public transport network. Hence in this case, there is a homeomorphic isomorphism.

Type-$2$ representation has two sub-classes: the mapping $f: M \rightarrow f(M)$ can be defined by either (2a) an objective, causal physical process, or by ($2$b) the decisions of cognitive systems.

As an example of type-$2$b representation, in computer engineering there are different conventions, such as ASCII and EBCDIC, for representing linguistic characters with the states of the bytes in computer memory. In the ASCII convention, 0100000 represents the symbol '@', whereas in EBCDIC it represents a space ' '. Neither relationship between linguistic characters and the states of computer memory exists objectively. In particular, the relationship does not exist independently of the interpretative decisions made by the operating system of a computer.

In 2005, I wrote that "the primary example of type-$2$a representation is the representation of the external world by brain states. Taking the example of visual perception, there is no homomorphism between the spatial geometry of an individual's visual field, and the state of the neuronal network in that part of the brain which deals with vision. However, the correspondence between brain states and the external world is not an arbitrary mapping. It is a correspondence defined by a causal physical process involving photons of light, the human eye, the retina, and the human brain. The correspondence exists independently of human decision-making."

The theorems and empirical research expounded in Curto's paper demonstrate very clearly that whilst there might not be a geometrical isometry between the spatial geometry of one's visual field and the state of a subsystem in the brain, there are, at the very least, isomorphisms between the homological topology of regions in one's environment and the state of neural subsystems.

On a cautionary note, this result should be treated as merely illustrative of the representational mechanisms employed by biological brains. One would expect that a cognitive system which has evolved by natural selection will have developed a confusing array of different techniques to represent the geometry and topology of the external world.

Nevertheless, the result is profound because it ultimately explains how you can hold a world inside your own head.

Monday, November 14, 2016

Trump and Brexit

One of the strangest things about most scientists and academics, and, indeed, most educated middle-class people in developed countries, is their inability to adopt a scientific approach to their own political and ethical beliefs.

Such beliefs are not acquired as a consequence of growing rationality or progress. Rather, they are part of what defines the identity of a particular human tribe. A particular bundle of shared ideas is acquired as a result of chance, operating in tandem with the same positive feedback processes which drive all trends and fashions in human society. Alex Pentland, MIT academic and author of 'Social Physics', concisely summarises the situation as follows:

"A community with members who actively engage with each other creates a group with shared, integrated habits and beliefs...most of our public beliefs and habits are learned by observing the attitudes, actions and outcomes of peers, rather than by logic or argument," (p25, Being Human, NewScientistCollection, 2015).

So it continues to be somewhat surprising that so many scientists and academics, not to mention writers, journalists, and the judiciary, continue to regard their own particular bundle of political and ethical ideas, as in some sense, 'progressive', or objectively true.

Never has this been more apparent than in the response to Britain's decision to leave the European Union, and America's decision to elect Donald Trump. Those who voted in favour of these respective decisions have been variously denigrated as stupid people, working class people, angry white men, racists, and sexists.

To take one example of the genre, John Horgan has written an article on the Scientific American website which details the objective statistical indicators of human progress over hundreds of years. At the conclusion of this article he asserts that Trump's election "reveals that many Americans feel threatened by progress, especially rights for women and minorities."

There are three propositions implicit in Horgan's statement: (i) The political and ethical ideas represented by the US Democratic party are those which can be objectively equated with measurable progress; (ii) Those who voted against such ideas are sexist; (iii) Those who voted against such ideas are racist.

The accusation that those who voted for Trump feel threatened by equal rights for women is especially puzzling. As many political analysts have noted, 42% of those who voted for Trump were female, which, if Horgan is to be believed, was equivalent to turkeys voting for Christmas.

It doesn't say much for Horgan's view of women that he thinks so many millions of them could vote against equal rights for women. Unless, of course, people largely tend to form political beliefs, and vote, according to patterns determined by the social groups to which they belong, rather than on the basis of evidence and reason. A principle which would, unfortunately, fatally undermine Horgan's conviction that one of those bundles of ethical and political beliefs represents an objective form of progress.

In the course of his article, Horgan defines a democracy "as a society in which women can vote," and also, as an indicator of progress, points to the fact that homosexuality was a crime when he was a kid. These are two important points to consider when we turn from the issue of Trump to Brexit, and consider the problem of immigration. The past decades have seen the large-scale migration of people into Britain who are enemies of the open society: these are people who reject equal rights for women, and people who consider homosexuality to be a crime.

So the question is as follows: Do you permit the migration of people into your country who oppose the open society, or do you prohibit it?

If you believe that equal rights for women and the non-persecution of homosexuals are objective indicators of progress, then do you permit or prohibit the migration of people into your country who oppose such progress?

It's a well-defined, straightforward question for the academics, the writers, the journalists, the judiciary, and indeed for all those who believe in objective political and ethical progress. It's a question which requires a decision, not merely an admission of complexity or difficulty.

Now combine that question with the following European Union policy: "Access to the European single market requires the free migration of labour between participating countries."

Hence, Brexit.

What unites Brexit and Trump is that both events are a measure of the current relative size of different tribes, under external perturbations such as immigration. It's not about progress, rationality, reactionary forces, conspiracies or conservatism. Those are merely the delusional stories each tribe spins as part of its attempts to maintain internal cohesion and bolster its size. It's more about gaining and retaining membership of particular social groups, and that requires subscription to a bundle of political and ethical ideas.

However, the thing about democracy is that it doesn't require the academics, the writers, the journalists, the judiciary, and other middle-class elites to understand any of this. They just need to lose.

Sunday, September 18, 2016

Cosmological redshift and recession velocities

In a recent BBC4 documentary, 'The Beginning and End of the Universe', nuclear physicist and broadcaster Jim Al Khalili visits the Telescopio Nazionale Galileo (TNG). There, he performs some nifty arithmetic to calculate that the redshift $z$ of a selected galaxy is:
z = \frac{\lambda_o - \lambda_e}{\lambda_e} =
\frac{\lambda_o}{\lambda_e} - 1 \simeq 0.1\,,
$$ where $\lambda_o$ denotes the observed wavelength of light and $\lambda_e$ denotes the emitted wavelength. He then applies the following formula to calculate the recession velocity of the galaxy:
v = c z = 300,000 \; \text{km s}^{-1} \cdot 0.1 \simeq 30,000 \; \text{km s}^{-1} \,,
$$ where $c$ is the speed of light.

After pausing for a moment to digest this fact, Jim triumphantly concludes with an expostulation normally reserved for use by people under the mental age of 15, and F1 trackside engineers:


It's worth noting, however, that the formula used here to calculate the recession velocity is only an approximation, valid at low redshifts, as Jim undoubtedly explained in a scene which hit the cutting-room floor. So, let's take a deeper look at the concept of cosmological redshift to understand what the real formula should be.

In general relativistic cosmology, the universe is represented by a Friedmann-Roberston-Walker (FRW) spacetime. Geometrically, an FRW model is a $4$-dimensional Lorentzian manifold $\mathcal{M}$ which can be expressed as a 'warped product' (Barrett O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983):
I \times_R \Sigma \,.
$$ $I$ is an open interval of the pseudo-Euclidean manifold $\mathbb{R}^{1,1}$, and $\Sigma$ is a complete and connected $3$-dimensional Riemannian manifold. The warping function $R$ is a smooth, real-valued, non-negative function upon the open interval $I$, otherwise known as the 'scale factor'.

If we denote by $t$ the natural coordinate function upon $I$, and if we denote the metric tensor on $\Sigma$ as $\gamma$, then the Lorentzian metric $g$ on $\mathcal{M}$ can be written as
g = -dt \otimes dt + R(t)^2 \gamma \,.
$$ One can consider the open interval $I$ to be the time axis of the warped product cosmology. The $3$-dimensional manifold $\Sigma$ represents the spatial universe, and the scale factor $R(t)$ determines the time evolution of the spatial geometry.

Now, a Riemannian manifold $(\Sigma,\gamma)$ is equipped with a natural metric space structure $(\Sigma,d)$. In other words, there exists a non-negative real-valued function $d:\Sigma \times \Sigma
\rightarrow \mathbb{R}$ which is such that

$$\eqalign{d(p,q) &= d(q,p) \cr
d(p,q) + d(q,r) &\geq d(p,r) \cr
d(p,q) &= 0 \; \text{iff} \; p =q}$$ The metric tensor $\gamma$ determines the Riemannian distance $d(p,q)$ between any pair of points $p,q \in \Sigma$. The metric tensor $\gamma$ defines the length of all curves in the manifold, and the Riemannian distance is defined as the infimum of the length of all the piecewise smooth curves between $p$ and $q$.

In the warped product space-time $I \times_R \Sigma$, the spatial distance between $(t,p)$ and $(t,q)$ is $R(t)d(p,q)$. Hence, if one projects onto $\Sigma$, one has a time-dependent distance function on the points of space,
d_t(p,q) = R(t)d(p,q) \,.
$$Each hypersurface $\Sigma_t$ is a Riemannian manifold $(\Sigma_t,R(t)^2\gamma)$, and $R(t)d(p,q)$ is the distance between $(t,p)$ and $(t,q)$ due to the metric space structure $(\Sigma_t,d_t)$.

The rate of change of the distance between a pair of points in space, otherwise known as the 'recession velocity' $v$, is given by
v = \frac{d}{dt} (d_t(p,q)) &= \frac{d}{dt} (R(t)d(p,q)) \cr &= R'(t)d(p,q) \cr &=
\frac{R'(t)}{R(t)}R(t)d(p,q) \cr &= H(t)R(t)d(p,q) \cr &=
H(t)d_t(p,q)\,. }
$$ The rate of change of distance between a pair of points is proportional to the spatial separation of those points, and the constant of proportionality is the Hubble parameter $H(t) \equiv R'(t)/R(t)$.

Galaxies are embedded in space, and the distance between galaxies increases as a result of the expansion of space, not as a result of the galaxies moving through space. Where $H_0$ denotes the current value of the Hubble parameter, and $d_0 = R(t_0)d$ denotes the present 'proper' distance between a pair of points, the Hubble law relates recession velocities to proper distance by the simple expresssion $v = H_0d_0$.

Cosmology texts often introduce what they call 'comoving' spatial coordinates $(\theta,\phi,r)$. In these coordinates, galaxies which are not subject to proper motion due to local inhomogeneities in the distribution of matter, retain the same spatial coordinates at all times.

In effect, comoving spatial coordinates are merely coordinates upon $\Sigma$ which are lifted to $I \times \Sigma$ to provide spatial coordinates upon each hypersurface $\Sigma_t$. The radial coordinate $r$ of a point $q \in \Sigma$ is chosen to coincide with the Riemannian distance in the metric space $(\Sigma,d)$ which separates the point at $r=0$ from the point $q$. Hence, assuming the point $p$ lies at the origin of the comoving coordinate system, the distance between $(t,p)$ and $(t,q)$ can be expressed in terms of the comoving coordinate $r(q)$ as $R(t)r(q)$.

If light is emitted from a point $(t_e,p)$ of a warped product space-time and received at a point $(t_0,q)$, then the integral,
d(t_e) = \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \, ,
$$ expresses the Riemannian distance $d(p,q)$ in $\Sigma$, (equivalent to the comoving coordinate distance), travelled by the light between the point of emission and the point of reception. The distance $d(t_e)$ is a function of the time of emission, $t_e$, a concept which will become important further below.

The present spatial distance between the point of emission and the point of reception is:
R(t_0)d(p,q) = R(t_0) \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$ The distance which separated the point of emission from the point of reception at the time the light was emitted is:
R(t_e)d(p,q) = R(t_e) \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$ The following integral defines the maximum distance in $(\Sigma,\gamma)$ from which one can receive light by the present time $t_0$:
d_{max}(t_0) = \int^{t_0}_{0}\frac{c}{R(t)} \, dt \,.
$$ From this, cosmologists define something called the 'particle horizon':
R(t_0) d_{max}(t_0) = R(t_0) \int^{t_0}_{0}\frac{c}{R(t)} \, dt
$$ We can only receive light from sources which are presently separated from us by, at most, $R(t_0) d_{max}(t_0)$. The size of the particle horizon therefore depends upon the time-dependence of the scale factor, $R(t)$.

Under the FRW model which currently has empirical support, (the 'concordance model', with cold dark matter, a cosmological constant $\Lambda$, and a mass-energy density equal to the critical density), the particle horizon is approximately 46 billion light years. This is the conventional definition of the present radius of the observable universe, before the possible effect of inflationary cosmology is introduced...

To obtain an expression which links recession velocity with redshift, let us first return to the Riemannian/ comoving distance travelled by the light that we detect now, as a function of the time of emission $t_e$:
d(t_e) = \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$ We need to replace the time parameter here with redshift, and to do this we first note that the redshift can be expressed as the ratio of the scale-factor at the time of reception to the time of emission:
1+ z = \frac{R(t_0)}{R(t)} \,.
$$ Taking the derivative of this with respect to time (Davis and Lineweaver, p19-20), and re-arranging obtains:
\frac{dt}{R(t)} = \frac{-dz}{R(t_0) H(z)} \,.
$$ Substituting this in and executing a change of variables in which $t_o \rightarrow z' = 0$ and $t_{e} \rightarrow z' = z$, we obtain an expression for the Riemannian/comoving distance as a function of redshift:
d(z) = \frac{c}{R(t_0)} \int^{0}_{z}\frac{dz'}{H(z')} \, .
$$ From our general definition above of the recession velocity between a pair of points $(p,q)$ separated by a Riemannian/comoving distance $d(p,q)$ we know that:
v =  R'(t)d(p,q) \,.
$$ Hence, we obtain the following expression (Davis and Lineweaver Eq. 1) for the recession velocity of a galaxy detected at a redshift of $z$:
v = R'(t) d(z) = \frac{c}{R(t_0)} R'(t) \int^{0}_{z}\frac{dz'}{H(z')} \, .
$$ To obtain the present recession velocity, one merely sets $t = t_0$:
v = R'(t_0) d(z) = \frac{c}{R(t_0)} R'(t_0) \int^{0}_{z}\frac{dz'}{H(z')} \, .
$$ At low redshifts, such as the case of $z \simeq 0.1$, the integral reduces to:
 \int^{0}_{z}\frac{dz'}{H(z')} \approx \frac{z}{H(0)} =  \frac{z}{H(t_0)} \, .
$$ Hence, recalling that $H(t) \equiv R'(t)/R(t)$, at low redshifts one obtains Jim Al Khalili's:
v = cz \,.
$$ Boom...mathematics!