Pointless space

This is one of my favourite diagrams. Produced by the virtuoso Swedish cosmologist, Max Tegmark, it displays part of the space of all mathematical structures. In particular, it shows which parts of that space are used by general relativity and quantum field theory. To unify relativity and quantum theory it will be necessary, amongst other things, to find a mathematical structure which generalises both of these structures. The obvious candidate already exists: non-commutative geometry. Quantum theory employs non-commutative algebras, relativity employs differential geometry, and non-commutative geometry employs non-commutative differential algebras.

One consequence of this approach is that one obtains an algebraic representation of space, rather than a manifold of spatial points. The conventional manifolds and geometries of relativity can be characterised in terms of the commutative algebra of functions defined upon them. In particular, each point of the underlying manifold corresponds to a special subalgebra called a maximal ideal. In non-commutative geometry, one introduces a non-commutative algebra of functions instead, and then one defines generalisations of the conventional geometrical objects in terms of this non-commutative algebra. There is no underlying manifold of points corresponding to a non-commutative algebra because a non-commutative algebra typically has no maximal ideals.

Whilst the French mathematician Alain Connes has attempted to introduce non-commutative geometry into the standard model of particle physics, Connes's basic construction seems rather ad-hoc to me. I am, however, a big fan of non-commutative geometry as an approach to quantum gravity. Parfionov and Zapatrin give a nice introduction to the idea, and the approach has been pursued in recent years by Michael Heller and his Polish colleagues. Oh, and modesty forbids...

Non-commutative geometry Michael Heller Max Tegmark

Fluorescent lights

There's a nice article in this week's New Scientist on the imminent replacement of incandescent light bulbs by compact fluorescent lamps (CFLs). Whilst the incandescent bulb only converts 5% of its electrical energy to light, the CFL converts 15%. CFLs are currently about 5 times more expensive than bulbs, but last 10 times as long, so already constitute both a cost and an environmental saving. What the article doesn't mention is that CFLs emit quite significant electromagnetic radiation in the radio frequency range. Those people I know who have tried CFLs in their own home, say that they interfere with other electrical devices such as the radio, and even have a tendency to spontaneously change television channel for you!

Fluorescent lights

Doctor Who returns!

My fellow time-lord returns to BBC1 tomorrow evening, although sadly, Billie Piper will no longer be his assistant. In fact, I thought the combination of Christopher Ecclestone and Billie Piper just about perfect. The Doctor should be slightly disturbing himself, not a pin-up like David Tennant.

Catherine Tate was very, VERY irritating on the Doctor Who Christmas special, so I'm hoping for a return to form with the new series.

Lithium

The first man-made nuclear reaction was produced by Cockroft and Walton in 1929, when the stable isotope, Lithium-7, was bombarded by accelerated protons, forming Beryllium-8, which underwent spontaneous fission to form two alpha particles.

However, the major geopolitical significance of lithium at present, pertains to its widespread use in batteries. Consumption of lithium increased by 4–5% per year between 2002 and 2005. The Wikipedia entry suggests that "Continued expansion in the portable electronic products market and commercialisation of hybrid electric vehicles using lithium batteries suggest growth of up to 10% per year in lithium carbonate consumption in this market through 2010...China may emerge as a significant producer of brine-based lithium carbonate towards the end of this decade. Potential capacity of up to 45,000 tonnes per year could come on-stream if projects in Qinghai province and Tibet proceed." So, whilst China continues to burn hydrocarbons, it will also be able to sell lithium to the environmentally-penitent, hybrid-vehicled West.

None of which was in Amy Lee's mind when she wrote 'Lithium':


The first Evanescence album, 'Fallen' (2003), was a brilliant gothic, post Nu-metal, post-Radiohead record. The band, however, were basically built around Amy Lee and Ben Moody, and Ben left the band "due to creative differences". Their second album, 'The Open Door' (2006) was a huge disappointment, and I'd be surprised if Evanescence do anything but fulfill their name, and fade away. Amy Lee, however, has a superb voice, and 'Lithium' is a great song, so perhaps Amy has a future.

An open letter to Oliver Letwin

Just before I hit my 18th birthday, I received a letter from John Redwood, my then-local MP, inviting me to join the Young Conservatives for one evening, where I could celebrate my coming-of-age with a complimentary bottle of champagne. It seems that the local Conservative party had access to the birth-dates and addresses of people in the area. On this occasion, I declined John's kind invitation.

John, however, was nothing if not persistent, and he arrived at our door during canvassing for the 1992 election. On this occasion, we pretended there was no-one in, and peered through the spy-hole at his local party assistants. As they moved off, my Dad heard John say, "Oh well, I don't blame them."

Some years have passed, and I now lie within Oliver Letwin's constituency. Not only that, but I think I might actually be on the electoral roll. There's the local elections coming up this May, and, Oliver, I know it's not a General Election, so you won't be standing yourself, but my birthday is the same day Margaret Thatcher came to power in 1979, and I wouldn't mind a free bottle of champagne to celebrate...

A thought experiment concerning money

Although money cannot be identified with any of its physical tokens, the curiosity is that money can be created and destroyed via the creation and destruction of its physical tokens. For example, a central bank can print more money to increase the money supply. And consider the following thought experiment: suppose that everyone in the world withdraws all the money they hold in bank accounts and investment accounts, and everyone with shares, or futures, or other financial options, also cashes them in. (Let us suppose, for the sake of argument, that there are sufficient bank notes to allow this). Then suppose that everyone builds a huge bonfire, and burns all their money. There would then be no money left in the world, and our economies would be crippled. The knowledge, skills and experience possessed by people would not have changed, and the value of the goods and services produced by people would not have changed, but every economy in the world would collapse.

Venus

As the Astronomer Patrick Moore once said, "If you were to stand on the surface of Venus, you would be simultaneously fried, poisoned, squashed and corroded."

The surface temperature on Venus is about 462 degrees Celsius (hot enough to melt lead), the atmosphere is composed of 97% carbon dioxide, the pressure is about 100 times the atmospheric pressure on Earth, and the clouds are composed of corrosive sulphuric acid.

Venus is a particularly interesting case study because planetary scientists believe that it originally possessed water, much like the Earth, but then underwent a runaway greenhouse effect, and all the water boiled away. The initial cause of this may simply have been Venus's closer proximity to the Sun. The higher temperatures increased the water vapour content in the atmosphere, and because water vapour is a greenhouse gas, this, in turn, increased the temperature further. As temperatures increased, greater and greater quantities of carbon dioxide would have been liberated into the atmosphere, which, of course, would also have amplified the temperature increase. As a consequence of the higher temperatures, the hydrogen atoms in water molecules were separated from the oxygen atoms, and the hydrogen atoms then escaped into space.

Whilst CO2 levels on Earth have never reached the current levels on Venus, the presence of life on Earth has acted to regulate the atmospheric content of CO2 on the Earth, as explained in this nice passage by James Lovelock:

Living organisms act like a giant pump. They continuously remove carbon dioxide from the air and conduct it deep into the soil where it can react with the rock particles and be removed. Conside a tree. In its lifetime it deposits tons of carbon gathered from the air into its roots, some carbon dioxide escapes by root respiration during its lifetime, and when the tree dies the carbon of the roots is oxidized by consumers, releasing carbon dioxide deep into the soil...There it comes into contact with, and reacts with, the calcium silicate of the rocks to form calcium carbonate and silicic acid. These move with the groundwater until it enters the streams and rivers, on their way to the sea. In the sea, the marine organisms continue the burial process by sequestering the silicic acid and calcium bicarbonate to form their shells. In the continuous rain of microscopic sea shells, the products of rock weathering - sedimented limestone and silica - and buried on the sea floor and eventually subducted by the movements of plate tectonics. (The Ages of Gaia, p127).

Whilst the atmosphere of Venus contains 300,000 as much carbon dioxide as the current atmosphere of the Earth, the Earth's crust contains almost as much CO2 chemically bound in the form of such limestone.

Gordon's fashion tips - Day 1

As demonstrated here by 1996 Formula One World Champion, Damon Hill, if you've got lots of grey hairs, it's a good idea to keep them cut short. Unless, of course, you're a wizard.

The gloaming

The weather here has been gorgeous the past couple of days, warm and sunny, with just a hint of cool. The sort of cool you get when a mouse opens its fridge to put some ice-cubes into its glass of lemonade.

Best of all, though, is the gloaming, around 9pm, when the hubbub of the day has subsided, and the trees, grasses and flowers exhale in unison, suffusing the air with vernal fragrance.

Skiing down the escalator

A Norwegian extreme-sports 'dude', called Arild, skis down the 100 metre escalator at The Angel underground station in North London:

Can anything move faster than the speed of light?

The answer is no, but there are some subtle concepts here. Relativity says that, in a vacuum, nothing can locally move faster than the speed at which zero-mass particles move. Light is composed of zero-mass particles called photons, hence this limit on the local speed of motion is typically expressed in terms of the speed of light. However, any particle of zero-mass should move at the same speed in a vacuum: if neutrinos are of zero mass, then they move at the same speed as the speed of light.

If there isn't a vacuum, then things can move faster than the speed of light, even on a local basis. For example, Cerenkov radiation, the blue-tinged light emitted by the water in nuclear reactors, is the shock-wave of radiation produced by a charged particle moving through an insulator at a speed greater than the speed of light in that insulator.

On a non-local basis, the time taken to travel between two spatial locations is dependent upon the geometry of the path taken between those two locations. If one creates or chooses the appropriate path, one can complete the journey before photons of light taking a different path. This is demonstrated by Miguel Alcubierre's model for a warp-drive. The basic idea of the warp drive is that it creates a bubble of compressed space which the space-ship travels within. The space-ship reaches its destination very rapidly because it travels a very short distance, not because it locally violates the speed of light.

Things can also change faster than the speed of light. For example, a pair of galaxies can recede from each other faster than the speed of light, under the expansion of the universe. The recession velocity v of a pair of galaxies separated by a distance d is given by

v = H d,

where H is the Hubble constant. Hence, any galaxies separated by a distance greater than c/H (where c is the speed of light), will recede at a rate greater than the speed of light. The recession velocity of the galaxies is due to the expansion of space between the galaxies, not due to the motion of the galaxies through space. There is no limit at all on the rate of change of spatial distance.