Monday, 2 July 2012

Challenging Dimensions

It's time to talk about ... The Fifth Dimension! <Cue Tw- oh, I've already made this joke.

Yes, I want to talk about a popular model of theoretical physics that isn't supersymmetry: extra dimensions.  Actually, this is really three classes of models, with different motivations and features.  In subsequent posts, I will get into those details.  But here I want to talk about the idea in a very general context.

In my earlier post, I defined the number of dimensions of a thing as how many different coordinates you need to uniquely identify a point on that thing.  I contrasted the two numbers needed to locate a point on your computer screen, to the three numbers (e.g. longitude, latitude and altitude) to find a point near the Earth, to the four numbers to additionally specify the time an event happened.  So if there are five dimensions, then we would need five numbers to locate objects in space and time.

And yet, we seem to get by perfectly well with three spatial and one time coordinate; that was the very example I just used.  How can this be resolved?

The answer is that the extra dimension must be very small.  Now, thinking in four spatial dimensions is very hard.1  So let me illustrate things with an analogy.  Imagine a really huge piece of paper; say, the size of a country, or maybe large enough to cover the whole planet.  The important thing is that the thickness of the paper is unchanged, still less than a millimetre; only the length and breadth have been made big.

Now imagine that within that piece of paper live a species of intelligent creatures, of a size comparable to humans.  They cannot see or observe outside the paper in any way; the entirety of their existence is within it.  And lastly, imagine that despite being self-aware they are unable to measure any length smaller than a few centimetres.  They just don't have either the innate ability or the tools to do so.

What will those creatures think about their world?  They will think it is two dimensional, like the inhabitants of Edwin Abbot Abbot's Flatland.  If you doubt this, ask yourself what observation they could make that would reveal the third dimension to them.  For example, they would measure the same distance between any two objects, independent of those objects' positions in the width of the paper.
Two situations unresolvable to the paper-dwellers.  The paper is outlined in black, and the red lines join two different points.  In the top case, the points are at different locations in the third dimension; in the bottom case, the same.  The distances differ by a few millimetres, to small for the paper dwellers to measure.

The analogy is that instead of a sheet of paper with two large and one small spatial dimensions, we have a Universe with three large and one small.  We are the creatures trapped inside the 'paper', unable to measure lengths small enough.  The fourth spatial dimension (fifth dimension over all) must be small, smaller than the smallest length scales we have probed.  Since we have examined sub-nuclear scales, we are talking about really small dimensions here.

One thing this means is that if there is a fifth dimension, we could not hope to travel through it.  In particular, it wouldn't serve as a type of Hyperspace for space travel, at least not in any of the mainstream ideas.

Lastly, let me note that in all the above, I have spoken of a single extra dimension.  But there is no reason to limit us to only one!  Indeed, in string theory we would need six (or maybe seven) extra dimensions, all very, very small.  Even in models more relevant to the LHC, there are several with six dimensions and a few with more.  As we add more and more, the conceptual image gets harder and harder, but the general principles remain the same.

1. Even thinking in three dimensions is often difficult.

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