Saturday, May 30, 2009

Essays in Idleness 01: Singularities and Data Storage

I posit the following theorem: that cosmic phenomena that have one or more attributes that are infinite (or approaching infinity quickly enough to be considered effectively infinite) resist direct observation.

An example of just such a phenomenom is a black hole, where the density of matter--and, as a result the strength of the gravitational field--at its core is infinite or effectively infinite, causing a singularity (i.e. a point where the rules of physics pertaining to space/time do not hold [1]).

Let's assume for a moment that I could directly observe the core of a black hole, the singularity itself. Since the density of matter is infinite, I am effectively observing an infinite amount of matter. So far, so good. I can simply jot down on my notepad that the quantity of matter is ∞ and move on.

Now what if I want to make observations about one or more characteristics of the matter? Say, I want to write down a simple binary value (i.e. true or false) that represents whether each particle of matter is a boson, which I determine by noting whether it has integer spin (as opposed to a fermion, which has half-integer spin [2]). Now I have a problem, because my notepad will have to be infinitely large, in order to contain one bit of information about each particle of matter in the singularity.

Ah-ha, you may now be saying, but the universe is infinite, so your notepad can be as well. Which leads us to the difficult realisation that the universe is likely *not* infinite at all, but rather finite and expanding or contracting somewhere between 0 and twice the speed of light. Of course, proving whether the universe is infinite is well outside the scope of this essay (though there are pointers to information on this topic, should the intrepid reader wish [3]). So let us take the easy way out and pretend that we know for sure that the universe is finite (see footnote A if you'd rather not take the easy way out), and there we have a proof of my theorem: you cannot store an infinite amount of data anywhere in the finite universe, so direct observation of a singularity is theoretically impossible.

This may rankle some readers, who would tell me that they can observe something, say a bird singing in a tree, without storing any data about the experience. I, of course, would claim that they are wrong: any observation results in data storage somewhere, be it on your retina, various places in your brain, and quite possibly on Twitter's or Blogger's servers as you share your experience with others after the fact.

In the case of the bird singing, there is a staggeringly large but potentially finite amount of data available. The way that you deal with this as a human observer is to capture the experience at a relatively low resolution: you consider only the notes the bird is singing, the colours of its plumage, the tree it is sitting in, the local weather conditions, how it makes you feel, etc. In the case of observing a singularity, you have no such luxury, since you can lower the resolution to one single data point and still have to record that one bit of data for an infinite amount of matter.

The way this problem is dealt with in real life is to observe not the singularity itself, but the effects of the singularity on the portion of its surroundings that lie outside of its event horizon. From this, and using the abstract tools of mathematics, we can determine much about the makeup of the singularity.

In our theoretical discussion, there may be another solution: create a second singularity to record the data garnered from directly observing the first. If you find a way to do that, do post it in the comments.


A: What if the universe is infinite? You could potentially store your data, though you'd have to be careful about not writing it over stars, planets, etc. This would lead to storage fragmentation, which would be most difficult to cope with, given the limitations imposed by the speed of light. Your best bet is to stream your data by flicking a light on and off, with every tick (defined as say, a number of milliseconds equal to the atomic weight of hydrogen or some other well-known finite number--using π here would be a very bad idea) representing one particle of matter.