The Barber at the End of the Universe

You can't comb the hair on a hairy ball completely flat if the ball is completely hairy.

Not without having at least one hair go over another. If no hairs are allowed to cross over each other, you will always be left with a situation where at least one hair has to be left sticking up. You always need at least one tuft for the rest to lie flat.

This is called the "hairy ball" theorem. You can read more about it here:
https://en.wikipedia.org/wiki/Hairy_ball_theorem

The theorem states that "there is no non-vanishing continuous tangent vector field on even-dimensional n-spheres." It is sometimes stated in the form "every cow must have at least one cowlick."

The key terms here are "even dimensional" and "continuous." 

Let's consider a simple odd dimensional case. A 1-sphere. A circle. 

You can easily imagine how to lie the "hairs" on the circle flat. As the hairs get shorter - as we give our circle a very close shave - these hairs tend towards being tangents. We don't need to leave one sticking up. Odd dimensions. 

Now let's consider even dimensions, but with a discontinuous tangent vector field. A simple example is you! Consider your head to be a sphere! 

You can comb your hair flat, in this strictly mathematical sense, because if we consider your hair to be the tangent vector field, we can see it is not continuous. You don't have hair entirely covering your head (probably). It stops somewhere. You have a fringe: a discontinuity in the tangent vector field represented by your hair. So you don't need a tuft.    

So what? 

Well, this theorem interests me because of something I wrote about earlier. I was discussing black holes here:

http://moflomojo.blogspot.co.uk/2015/09/how-real-is-real-part-1.html 

I found myself wondering why the number of microstates of a hypothetical "unit" black hole should be the sum of the volumes of all even dimensional unit n-spheres, rather than simply 1, which one might have intuitively assumed to be the answer. 

Maybe the hairy ball theorem is the answer. Maybe we have to consider the tangent vector field of the event horizon of the black hole over all dimensions when evaluating the number of microstates.  

Given the event horizon of a black hole is the edge of space, and a location one can only reach at the end of time, due to General Relativistic considerations, this reminds me of the great Douglas Adams' novel "The Restaurant at the End of the Universe". 

The "tuftiness" we see as information content - microstates - manifested as the surface area of that event horizon, can be thought of having been styled by "The Barber at the End of the Universe".  

Which leaves me wondering about the status of that famous black hole theorem, the "no hair" theorem:

https://en.wikipedia.org/wiki/No-hair_theorem 

Maybe in the sense I discuss above we must consider black holes to be hairy. 

And God is their barber.