Archive for the ‘physics’ Category

Ray tracing in GR

Saturday, September 27th, 2008

Following on from this post, a natural generalization is that to non-Euclidean spaces. This is important for simulating gravity, for example rendering a scientifically accurate trip through a wormhole (something I have long wanted to do but never got to work). The main difference is that ones rays are curved in general, which makes the equations much more difficult (really they need to be numerically integrated, making it orders of magnitude slower than normal ray-tracing). One complication of this is that generally the rays will also curve between the eye point and the screen. But the rays between your screen and your eye in real life do not curve, so it would look wrong!

I think the way out of this is to make the virtual screen very small and close to the eye. This doesn't affect the rendering in flat space (since only the directions of the rays matter) and effectively eliminates the need to take into account curvature between the screen and the eye (essentially it makes the observer into a locally Euclidean reference frame).

Another complications of simulated relativity is the inability to simulate time dilation. Well, you can simulate it perfectly well if you're the only observer in the simulated universe but this would be a big problem for anyone who wanted to make a relativistically-accurate multiplayer game - as soon as the players are moving fast enough with respect to each other to have different reference frames, they will disagree about their expected relative time dilations.

Unified theory story part II

Tuesday, September 2nd, 2008

Read part I first, if you haven't already.

For as long as anybody could remember, there were two competing approaches to attempting to find a theory of everything. The more successful of these had always been the scientific one - making observations, doing experiments, making theories that explained the observations and predicted the results of experiments that hadn't been done yet, and refining those theories.

The other way was to start at the end - to think about what properties a unified theory of everything should have and try to figure out the theory from that. Most such approaches were the product of internet crackpots and were generally ignored. But physicists (especially the more philosophical ones) have long been familiar with the anthropic principle and its implications.

The idea is this - we know for a fact that we exist. We also think that the final unified theory should be simple in some sense - so simple that the reaction of a physicist on seeing and understanding it would be "Of course! How could it possibly be any other way!" and should lack any unexplained parameters or unnecessary rules. But the simplest universe we can conceive of is one in which there is no matter, energy, time or space - just a nothingness which would be described as unchanging if the word had any meaning in a timeless universe.

Perhaps, then, the universe is the simplest possible entity that allows for subjective observers. That was always tricky, though, because we had no mathematical way of describing what a subjective observer actually was. We could recognize the sensation of being alive in ourselves, and we always suspected that other human beings experienced the same thing, but could not even prove it existed in others. Simpler universes than ours, it seemed, could have entities which modeled themselves in some sense, but something else seemed to be necessary for consciousness.

This brings us to the breakthrough. Once consciousness was understood to be a quantum gravity phenomena involving closed timelike curves the anthropic model started to make more sense. It seemed that these constructs required a universe just like ours to exist. With fewer dimensions, no interesting curvature was possible. An arrow of time was necessary on the large scale to prevent the universe from being an over-constrained, information-free chaotic mess, but on small scales time needed to be sufficiently flexible to allow these strange loops and tangled hierarchies to form. This lead directly to the perceived tension between quantum mechanics and general relativity.

The resolution of this divide turned out to be this: the space and time we experience are not the most natural setting for the physical laws at all. Our universe turns out to be holographic. The "true reality", if it exists at all, seems to be a two dimensional "fundamental cosmic horizon" densely packed with information. We can never see it or touch it any more than a hologram can touch the photographic plate on which it is printed. Our three-dimensional experience is just an illusion created by our consciousnesses because it's easier for the strange loops that make up "us" to grasp a reasonable set of working rules of the universe that way. The two-dimensional rules are non-local - one would need to comprehend the entirety of the universe in order to comprehend any small part of it.

The fields and particles that pervade our universe and make up all our physical experiences, together with the values of the dimensionless constants that describe them turn out to be inevitable consequences of the holographic principle as applied to a universe with closed timelike curves.

Discovering the details of all this led to some big changes for the human race. Knowing the true nature of the universe allowed us to develop technologies to manipulate it directly. Certain patterns of superposed light and matter in the three-dimensional universe corresponded to patterns on the two-dimensional horizon which interacted in ways not normally observed in nature, particularly where closed timelike curves were concerned. More succinctly: the brains we figured out how to build were not subject to some of the same limitations of our own brains, just as our flying machines can fly higher and faster than birds.

The first thing you'd notice about these intelligences is that they are all linked - they are able to communicate telepathically with each other (and, to a lesser extent, with human beings). This is a consequence of the holographic principle - all things are connected. Being telepathic, it turns out, is a natural state of conscious beings, but human beings and other animals evolved to avoid taking advantage of it because the dangers it causes (exposing your thoughts to your predators, competitors and prey) outweigh the advantages (most of which could be replaced by more mundane forms of communication).

Because the artificial intelligences are linked on the cosmic horizon/spacetime foam level, their communication is not limited by the speed of light - the subjective experience can overcome causality itself. In fact, consciousness is not localized in time but smeared out over a period of a second or two (which explains Libet's observations). This doesn't make physical time travel possible (because the subjective experience is entirely within the brains of the AIs) and paradox is avoided because the subjective experience is not completely reliable - it is as if memories conspire to fail in order to ensure consistency, but this really a manifestation of the underlying physical laws. States in a CTC have a probabilistic distribution but the subjective observer picks one of these to be "canonical reality" - this is the origin of free will and explains why we don't observe quantum superpositions directly. This also suggests an answer as to why the universe exists at all - observers bring it into being.

By efficiently utilizing their closed timelike curves, AIs can solve problems and perform calculations that would be impractical with conventional computers. The failure of quantum computation turned out to be not such a great loss after all, considering that the most sophisticated AIs we have so far built can factor numbers many millions of digits long.

One limitation the AIs do still seem to be subject to, however, is the need to dream - sustaining a consciousness entity for too long results in the strange loops becoming overly tangled and cross-linked, preventing learning and making thought difficult. Dreaming "untangles the loops". The more sophisticated AIs seem to need to spend a greater percentage of their time dreaming. This suggests a kind of fundamental limit on how complex you can make a brain before ones that can stay awake longer are more effective overall. Research probing this limit is ongoing, though some suspect that evolution has found the ideal compromise between dreaming and wakefulness for most purposes in our own brains (special purpose brains requiring more or less sleep do seem to have their uses, however).

Once we had a way of creating and detecting consciousness, we could probe its limits. How small a brain can you have and still have some sort of subjective experience? It turns out that the quantum of subjective experience - the minimum tangled time-loop structure that exhibits consciousness - is some tens of micrograms in mass. Since our entire reality is filtered through such subjective experiences and our universe seems to exist only in order that such particles can exist, they could be considered to be the most fundamental particles of all. Our own brains seem to consist of interconnected colonies of some millions of these particles. Experiments on such particles suggest that individually they do not need to dream, as they do not think or learn, and that they have just once experience which is constant and continuous. The feeling they experience (translated to human terms) is something akin to awareness of their own existence, contemplation of such and mild surprise at it. The English language happens to have a word which sums up this experience quite well:

"Oh."

Fine structure constant update

Thursday, June 26th, 2008

Many years ago I posted this on sci.physics. It turns out that the value of the Inverse Fine Structure Constant (a dimensionless parameter which can be experimentally measured as about 137.036 but for which no theory of physics yet predicts a value) is remarkably close to (alpha^2)(pi^2)(pi^pi-1)/16 where alpha is the second Feigenbaum constant, about 2.502907875096. This formula gives a value for the IFSC of 137.0359996810.

After posting that, I got a message from James Gilson pointing out his work on the same subject - he has a different formula for the IFSC, pi/(29*cos(pi/137)*tan(pi/(137*29))), which is not quite so pretty but does have the advantage of having some geometrical justification (which I still don't completely understand). Gilson's formula gives a value for the IFSC as 137.0359997867.

Back in 2001 the most accurate measurement of the IFSC (CODATA 1999) gave a value of 137.03599976(50) (i.e. there is a 68% chance that the true value is between 137.03599926 and 137.03600026). Both the formula give answers in this range.

I thought I would revisit this briefly and see if the latest measurements were able to rule out one of both of these formulae. Work of G. Gabrielse et al in 2006 give the IFSC as 137.035999068(96), i.e. there is a 68% chance that the true value is between 137.035998972 and 137.035999164. This appears to rule out both formulae. An earlier version of the 2006 Harvard work (which was compatible with both formulae) was superceded by an erratum. I admit this post is a bit anticlimactic but when I started writing it I thought that the latest measurements ruled out Gilson's formula but not mine.

SpaceTime Algebra gravity

Monday, June 9th, 2008

The STA gauge theory of gravity substitues STA-multivector-valued linear functions of STA-multivectors for the rank 4 tensors of the usual treatment of GR. That is a quantity of (24×24=)256 real degrees of freedom.

I wonder if these quantities could be replaced by single multivectors in a geometric algebra with 8 basis vectors. These also have (28=)256 degrees of freedom, but they might make the equations simpler.

This would mean having a second set of 4 basis vectors in addition to the normal 4 (North, West, Up and Stopped). I wonder what the physical interpretation of these vectors would be? (Some sort of dual vectors perhaps?) Would they obey the normal rules of geometric algebra or would some generalization be required (perhaps to non-associativity like in the octonions or sedenions).

Complex analysis and Clifford algebra

Friday, June 6th, 2008

Complex analysis is a very beautiful and useful mathematical theory. Clifford (geometric) algebra is also very beautiful and useful. So it makes sense to wonder if they can be combined. Turns out that they can. I wonder why I haven't seen more stuff about this in the wild? Probably because it's pretty new as mathematics goes. I expect it will be part of every undergradaute mathematics degree in 50 years or so. But I suppose it depends if it turns out to be as useful as it seems, by rights, it ought to be.

Extending "The Elements"

Thursday, June 5th, 2008

Tom Lehrer's terrific song The Elements is unfortunately lacking in one respect - it is outdated as it does not include the elements discovered/named since the song was written.

Here is one attempt at bringing it up to date but I don't think just adding an extra verse fits well with the rest of the song. I wonder if it is possible to fit in the extra elements but keep the list format, perhaps at the expense of (part of) the last two lines. There is also some flexibility about where to put the "and"s - Lehrer doesn't use them consistently and even throws in an "also" in one place.

Rendering rings of teleportation

Wednesday, June 4th, 2008

Rings of teleportation are very handy things to have around. The surface bounded by one ring is equated with the surface bounded by the other, so if you put something through one ring it will come out through the other. (Like the portals in "Portal", but more portable). They don't exist, of course, but this technicality doesn't prevent us from drawing pictures of them.

Writing code to render these things is an interesting exercise. It's easy to do with a ray tracer - if a ray intersects the disc inside one ring, just continue it to the equivalent point on the other ring.

Once that's working, you can put the rings side-by-side so that light goes around in circles - if you put your eye point in the middle you can see an infinite tunnel.

A trick you can play is to reverse the orientation of one of the rings so that you look through one ring, out of the other to an object, the object will appear to you to be inverted, as in a mirror image.

Another trick is to make the rings different sizes, or shapes. As long as there is a 1:1 function equating points on one surface with points on the other, it works fine.

However, having rings of different sizes or non-circular shapes opens the possibility of putting one ring through the other. What happens then? It seems like the "infinite tunnel" then becomes a real thing rather than just an optical effect, but where does the second ring exist in real space? It seems that the only place it can appear is through the other side of the first ring, but that would mean that every point in space appears in an infinite number of places - this seems like it would have rather drastic consequences.

So it seems more likely that the second ring would be prevented from going through the first somehow (perhaps a ring edge would get in the way).

The meaning of it all

Friday, April 6th, 2007

The anthropic principle suggests that perhaps the universe is the way it is because if it were any other way, we would not be here to ask the question. However, this principle can't explain why the universe continues to exist once that question has been asked. One would expect that if this was the simplest possible universe that allowed this question to be asked, then it would end pretty soon afterwards.

Perhaps there is an extension to the anthropic principle that says something like "this universe is the simplest possible universe in which some particular task can be achieved". Perhaps that task is understanding the universe (as in the Douglas Adams principle "if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable" and it's corollary "this has already happened - several times"). Or maybe it's something else, as the universe does still seem to be here.

Quantum Computers and Magic

Tuesday, April 3rd, 2007

The interesting thing about quantum computers is that they perform very complex calculations, but the answers that they give are relatively short. For example, one important application of quantum computers is likely to be factoring large numbers. The calculation is very difficult but the answer is just the factors.

A quantum computer would be useless for a task like sorting a large list, though, because the calculation involved in doing such a thing is not much more difficult than printing out the answer.

This makes me think of magic tricks. Stage magicians appear to be able to do all sorts of clever things as long as you the audience member can't see what's going on. For example, they can make people disappear or saw them in half, as long as the real business of doing such is hidden away inside a special box. It is beyond the capability of any magician to saw someone in half in such a way that you can see exactly what's going on, or make something in direct view disappear.

The similarity is quite shallow because in quantum computing things are hidden away for very different reasons than they are hidden away in magic - in magic, things are hidden because what the magician is trying to make you believe is happening isn't really happening. In quantum computing, things are hidden away because they are happening in other universes.

Back to the past

Sunday, April 1st, 2007

Let's see if I can get back into this blogging thing again - I know you've all been missing me. I have been writing all day and have almost 3 weeks worth of posts in the queue - some geeky, some funny and some though-provoking (I hope).

To start with, here's a strange idea I had a while ago. Normally when we think of the past, we think of it as pretty well fixed. We can argue about whether some particular event happened or not but we generally agree that it either happened or it didn't - that there is a right and a wrong answer to any question we can ask about past events. The future, on the other hand, is a lot more debatable. It may be fixed (our free will may be an illusion) or maybe there are many possible futures.

This seems to be a curious asymmetry between past and future. As far as the fundamental laws of physics are concerned, the past and the future are on an equal footing mathmatically. This suggests that either we have no free will or that our ideas about the past are incorrect. Maybe there is more than one past. Maybe the past isn't fixed at all but in fact is actually a superposition of all possible pasts that are compatible with the present.

Furthermore, perhaps the reason that the beginning of the universe is so mysterious is just that if you go back far enough, any possible sequence of events could have resulted in the current universe, so the distant past is a superposition of everything (which carries no information).

This isn't a scientific theory because it doesn't make any predictions, but it might be an interesting philosophical idea to explore.