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Let’s start with two point particles. Put them on one dimension, that is, a straight line. Generically they will always collide at some time, either in the past or the future. But in two dimensions (on a flat surface), or indeed in more dimensions, two particles will generically never collide. See the diagrams below; the arrows are sample random velocities of the particles.

Now think about two infinitely long lines, moving in some direction perpendicular to the line. In two dimensions, they have a point of intersection at any given time. In three dimensions, they will generically have an intersection point at some time, either in the past or future, at some point along their infinite length.

But not in four (spacial) dimensions. Each line’s motion with time describes a plane, and in four dimensions these need only meet at a point. But that point corresponds (generically) to different times for the two lines, so they never actually intersect.

A physical implication

This abstract argument has some use. One of the (many) possible string theoretic starting points of the universe is a very small ten-dimensional surface of a “sphere”, or some similar closed (essentially, go far enough and you get back to where you started) space. This will have strings living on the ten-dimensional surface, carrying energy in three ways: momentum of strings within directions transverse to the string; oscillations in the string; and winding of the string around the entire space, like wool in a ball of yarn. As the universe evolves, various mechanisms can make some of those directions “expand” to form our universe. The energy in the momentum and oscillation modes behaves as a simple energy density, which acts like a pressure toward the universe expanding. But the wound strings have the opposite effect: as the dimensions around which they are wound expand, their length increases. Since they have tension, this increases their energy, so they act against expansion of the dimensions around which they are wrapped.

So unless there is a mechanism to get rid of winding modes, the universe can’t expand to much above the string scale, which is tiny. But if strings intersect they can interact and exchange their attachment, as in the diagrams below show. Two strings swap at the intersection point, and so (since the strings are closed into loops behind the sphere) one string results. Thus intersections reduce the number of wrapped strings.

But now we’ve seen that string intersections don’t generically happen in four dimensions, so this can be used to argue why our universe could have at most three “large” spacial dimensions — which is of course exactly what we observe. There are problems with the argument — not least of which is that particles in quantum mechanics interact at non-zero seperations (for example, electromagnetically), so we might expect the same to be true for strings in four or more spacial dimensions. Nevertheless, it’s always cool to go from a simple mind puzzle to an argument for why the universe has three dimensions!

Exercise for the reader: By thinking a little more about my arguments for why lines in four dimensions don’t intersect, find the number of dimensions in which planes will/will not intersect, generically. Use this to provide a simple argument for why there are exactly seven continents on Earth. If you succeed in the latter task, collect an Ig Noble Prize.

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Because lurking beneath the surface of any grad student’s apparent calm are dramatic spikes in things such as work satisfaction, fear of advisors (or specifically, running into them and so needing to explain the progress one has / has not made), and perception of progress toward the Valhalla of PhD-dom (Valhalla since progress sometimes seems like a battle to the death, in order to enter paradise. And not because of the plentiful women, the thought hadn’t even crossed my mind).

In somewhat related news, I changed research direction entirely on Tuesday. The old direction was really going nowhere, so it’s something entirely new that we’re looking at now. You might think I’d be disappointed after a good many months’ work on the old area, but I’m actually quite pleased, since I had run out of ideas for the old project. And thank goodness I have no candidacy exams coming up soon!

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Keeping up with demand

In my opinion, Peak Oil is not our immediate problem. If only it were. Rather, the problems will start when supply growth can no longer match demand growth. Oil demand is growing rapidly, at both ends of the wealth spectrum. US oil consumption is still growing, while that of China is growing even faster. And I can’t see a way that this fact can change.

Per person, first world citizens use a number of times the oil of the bulk of the world’s population. Thus as places like China, and perhaps soon India, approach first world living standards, their oil consumption is going to dramatically increase. Developing countries around the world, especially China, are doing particularly well at the moment, so this is going to drive global demand growth for many years to come.

What could cause demand not to grow? Either getting richer will have to happen without increased oil demand, or getting richer itself will have to stop. Now the US is more oil efficient than China per dollar of GDP — but only by a factor of 2. This efficiency gain, if realised in its entirety, would slow China’s demand growth for only a few years. Is further efficiency possible? Absolutely — blogwaffe is full of good ideas. But are the people who matter interested? No.

The other option is stopping the process of getting richer. Note that this implies stopping GDP growth in the US too — but the effect is even greater for the developing world. In effect, it would mean the developed world saying to the developing world, “There’s not enough oil for all of us to be as rich as we are, so in order for us to stay rich, you’ll have to stay poor”. Besides being almost unenforceable in today’s global economy, this would probably lead to widespread war.

So demand growth is going to continue for the forseeable future. Thus we don’t need to maintain supply, we need to be continuously increasing supply. Thus if, for example, one claims that using the Canadian shale oil will in a few years/decades replace, say, Russian oil (which has smallish reserves), then one has to show that Canadian shale oil can produce a lot MORE than Russian oil has up to now. It’s not a case of finding reserves to replace exhausted fields, we need to be finding more and/or bigger reserves than ever before, all the time.

Facts on the ground show that the rate of discovery is, if anything, decreasing. But the point is that anything less than continuously increasing is not enough.

The Effect of Higher Prices

Yes, higher oil prices will lead to new oil sources being exploited. But this isn’t that helpful, for two reasons I can think of:

1. If oil prices are rising due to supply exhaustion, then because of continous growth in demand, we need new sources of supply immediately. New technologies and fields need a long time to come on line. The only way to make demand growth stop while we wait, is for prices to rise so high that global GDP growth stops — so this is what will happen (note that this relies on oil having a low price elasticity of demand). Such a price rise is disasterous, for the reasons below. We are not seeing much speculative investment in oil supply at higher prices, so this supply lag seems to be real problem.
2. Again, rising demand implies that the new sources that become available at higher prices need to be bigger than our current oil sources. And this needs to be true at every future price level. This doesn’t seem to be the case — Saudi Arabia’s huge oil reserves are already all available at the current price, for example.

More to the point is the economic effects of higher oil prices — because it isn’t just a simple overall slowing in growth rates. The oil industry constitutes a significant proportion of the world’s GDP. This is a rough indication of the proportion of the human race’s resources — labour, capital, research, risk mitigation insurance (explaining the instability premium in the current oil price), etc. — that are spent on oil extraction. Increased oil price results from a greater proportion of humanity’s resources being dedicated to oil extraction, and so less to other things. This leads to a drop in production of other products, and a corresponding drop in human welfare.

Different products leverage the oil price to differing degrees. Food, transport and energy are obviously highly dependent on the oil price, while the market price of CDs or movies depends on it very little. The people who experience the drop in welfare the greatest are those who spend the greatest amount of their income on products highly dependent on the oil price. This will be the poor of the world, who spend most of their money on food, transport and energy.

Oil price rises are thus a moral tragedy for the poor. But even worse, this means that the largest energy consumers, namely the rich, will “feel the pinch” comparatively later, and so will not reduce their oil demand till prices rise much higher. The poor will be starving long before drastically higher petrol prices stop the sale of SUVs.

So what does this mean?

I’ve argued that Peak Oil isn’t the immediate problem. Rather, we need to worry about supply growth slowing. Relying on new sources of oil to be found or become available at somewhat higher prices won’t help, as the rate of discovery needs to be constantly increasing — something it isn’t doing. Also, the time lag for supply to catch the endlessly growing demand will mean disaster already, and anyway price rises have the wrong demand-side effects, by hurting the poor rather than the largest consumers.

The only way out, seems to me to be:

1. Immediate, serious efforts to reduce oil consumption. This means energy efficiency drives such as raising the price of petrol in the rest of the world to Europe’s level, through taxation. Rather a gradual increase, allowing market forces to alleviate some of the suffering, than a sudden collapse when oil price inelasticity catches up with us.
2. Massive investment in alternatives to oil in the short term, such as biologically-derived plastics or fertilisers, or even bio-diesel.
3. Massive investment in long-term power alternatives, particularly fusion.
4. Stop gambling — even if you happen to believe we’ll keep finding ever larger amounts of oil, our civilisation is a lot to gamble! The Easter Islanders chopped down all their trees, and then starved when they couldn’t build canoes. Our situation is not too dissimilar.

In the end, it may be a race to see whether we develop fusion first, or whether the supply curve finally drops below the demand curve first. The result is somewhat important.

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The International Criminal Court is a recently created permanent organ, tasked with prosecuting crimes by individuals against humanity, such as war crimes and genocide. It’s a conceptual decendent of the Nuremburg trials, and more recently the courts that dealt with the war in the former Yugoslavia, and the Rwandan genocide. The obvious advantage of institutionalising it is that people contemplating crimes against humanity can see the personal risks that will imply for them.

The US, however, has been completely against the ICC. It very, very nearly suceeded in preventing it being created, and has since pressured numerous nations into bilateral agreements aimed at weaking the ICC. The chief reason given for this is that the US doesn’t want it’s soldiers in other countries subject to “politically motivated” prosecution. However, this seems impossible anyway: the ICC is only targeted at major crimes against humanity, which hopefully US soldiers will never get involved in. Furthermore, it will only prosecute individuals from countries who’s legal systems can’t or won’t prosecute individuals internally. Most to the point, however, is that since the US is not a signatory to the ICC accords, Americans can only be tried by Security Council resolution — which the US will always veto. Finally, none of the top five contributors of troops to international peacekeeping operations (Bangladesh, India, Pakistan, Nigeria and Ghana) objected to the ICC.

So the noteworthy news is that the US did NOT veto the Security Council resolution to refer Darfur to the ICC. The price of securing this, however, is that the resulting resolution explicitly exempts US citizens in the Suda from being tried at the ICC, or in any other nation’s courts. Also, the Security Council didn’t pay much attention to a Nigerian-led African Union suggestion, which is a pity. But overall, a victory, I think, for a global standard of conduct for leaders in sensitive areas. And hopefully a step in the direction of preventing future tragedies.

On the other hand, Zimbabwe’s president, Robert Mugabe, has successfully stolen another election. The election was obviously unfair; the only question is to the extent. All indications on the ground are that the opposition MDC had a good chance of winning, but instead the ruling ZANU-PF won over two thirds of the seats. Some of the techniques the government used included dominating airtime on it’s media monopoly; distributing food aid (required after the botched land reform process) to government supporters only; manning polling stations with the police and army; counting ballots at each station rather than at a more-easily-observed central location; preventing the millions of Zimbabweans who have been forced to leave (mostly to South Africa) from voting; and using a voters’ roll that has perhaps a double-digit percentage of voters who are dead, but yet seem to vote ZANU-PF to a surprising extent. Violence was much less common than last election, but people remember well the results then of voting for the opposition.

As the key country in the region, it is to South Africa’s shame that the South African observer mission declared the elections free and fair. This is part of the ongoing policy of “quiet diplomacy”, which has not so far worked. There are reasons that quiet diplomacy might be a good idea, but it’s now reached the stage of wilfull ignorance of reality. Most other observer missions were banned from the country.

I’ve been very excited about the wave of good governance and peace that has been spreading in Africa over the last few years, such as negotiated settlements in Rwanda, Burundi, Mozambique, Angola, Liberia, and southern Sudan; as well as African Union intervention in western Sudan (Darfur), Congo (DRC) and recently Togo, and a new emphasis on good governance. That South Africa has often been in the forefront of these activites makes Zimbabwe even more noteworthy, and a poor comparison to the actions of the West African states in Togo.

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As always happens, we had some crazy pictures over the course of the evening, and some inspired guessing. Which leads me to today’s…

Exercise for the reader: Determine the concept illustrated by each of the following hyper-realistic, emotional-response-tested graphics. Answers are on the next page.

(These pictures were all from the guys’ team, since those are the ones I remember. Also, all but one of them were successfully guesssed).

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Image 5:

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First, the terminology: classical mechanics is physics without discretisation — that is, anything that isn’t quantum. The cornerstones of classical mechanics are Maxwell’s electromagnetism, and Einstein’s Special and General Relativity. Quantum mechanics, or quantum theories in general, are theories which include the ideas (all related) of wave/particle duality; uncertainty relations between position and momentum; and discretisation of quantities such as momentum and energy. Point particles are particles with no spacial extent, ie. infintely small; all fundamental particles are either points or very, very close to point-size.

Let’s start with one of the simplest classical systems: an isolated, charged point particle. Coulomb’s law,

expresses the electric field due to the particle of charge Q, at a distance r. By considering the energy stored in a capacitor, it’s easy to show that the energy of an electric field is proportional to E². But the integral of this over all space is infinity! Yes, simple classical mechanics predicts that all isolated charged point particles have infinite energy. Add relativity, and the equivalence of mass and energy, and we should measure infinite mass for all point particles. Ooops – this is a serious problem! The infinity comes from extending the integral all the way to zero radius, so is called an ultraviolet divergence — ultraviolet light having a short wavelength.

Having shown a problem with classical mechanics, we can start adding quantum mechanics. Initially, things seem worse: by the uncertainty principle, probing very small lengths means a great uncertainty in the momentum of the particles doing the probing, and being probed. Since they cannot have velocity greater than the speed of light, this means great uncertainty in their mass. In fact, probe distances smaller than the Plank length, 1.616×10^-35, and a black hole develops — destroying what you were attempting to probe.

Catastrophic as this seems, it allows us to say that distances less than the Planck length make no sense — that is, there is a smallest definable distance. Point particles are equivalent to particles of this size, as we can’t tell them apart. Equivalently, we define an ultraviolet cutoff — a highest energy that a particle can have. This seems arbitrary, but it allows us to avoid the infinities we saw in the classical case, in a mathematically rigorous manner termed renormalisation, for all forces other than gravity. This is something we cannot do in classical mechanics, as classical mechanics has no “natural length scale”.

Some artifacts, however, remain. The energy of the field surrounding a particle (and so the observed particle energy) is now dependent on the exact length cutoff we choose. This means that physical masses for “bare” particles, without their surrounding fields, are very hard to define. So physics deals with observed particle masses.

The exception to the above is gravity. For various reasons, gravity is not renormalisable. One way to think about it is to note that gravity affects the curvature of spacetime, and so gravity changes the meaning of lengths near gravitational sources. We saw that probing small distances implies encountering strong gravitational fields, till eventually it’s hard to define what we mean by a standard length cutoff. This failure of renormalisability is means that we have no theory of gravity at short distances. So relativity and quantum mechanics are not yet compatible — we need a quantum theory of gravity.

String theory seems to be the best candidate at the moment. It is far from understood, but seems to include, even require gravity, in order to be consistently defined. One way to think about this MIGHT be the following (no promises that this is the correct picture): the problems above result from infinities extremely close to charges. If the particle is an extended object, like a string, then there is only a charge per unit length. Get sufficiently close, so that the string looks long in comparison, and the charge per unit length approaches zero. The infinite field strengths don’t cause the same problems.

What is the weirdest part of this story, for me? Around every single particle is a sea of particles which we can’t define properly. When we look closely, we see particles at the maximum possible energies we can define — if not even tiny black holes. We can’t even say what the actual particle under all this mess is, or what its mass is. But this all doesn’t matter to the rest of physics, science, even life. Everywhere other than the surface of black holes, we can completely ignore the complexity at these short lengths, and treat particles as nice, well behaved points. This seems to me to be an entirely non-trivial statement about the organisation of our universe.

Exercise for the reader: Do NOT, whatever you do, look too closely at anything. The universe might decide to show you what’s REALLY happening, and then we’re all in trouble.

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