Saturday, February 21, 2009

How String Theory Gets Rid Of Singularities

Okay, I admit up front I am probably glossing over a lot of details, (real string theorists please don't complain too loudly) but on a basic level I found a very profound explanation why string theory removes singularities from theories like GR form Cumrun Vafa, a very prominent string theorist at Harvard.

String theory has a duality where, if the planck length = 1, then the physics at a scale R is equivalent to physics at a scale 1/R. One way to see this is to remember that our universe, and most string models, is compact meaning the spaces have boundaries, even if at infinity. In string theory strings either exist wrap around the boundary, or they don't. (Think of all the ways to draw a string on a Torus. The strings either wind around the boundary or they don't.).

One can show in Fourier space, normal strings have energy ~n/R. Strings that wind around the boundary have energy ~m*R. Hence, every state at energy scale R represented by a normal string is equivalently described by a winding string at at 1/R. Therefore the particle/energy spectrum at energy scale R is equivalent to 1/R where winding strings now produce the normal string modes, and visa versa.

Now for defining distance. Recall with GR you get coordinate singularities. Singularities which exist only because at that point you are using bad coordinates. If you do a coordinate transformation, the singularity at that point goes away.

So if we use coordinates based off of light rays, as we usually do, we get good results until we run into singularities often at R=0. However, if for distances shorter than the planck length, we define a new coordinate distance based of the winding dual to light rays, then the singularity disappears because the physics at 1/R is equivalent to R. Hurray, no more singularity!

At this point Vafa points out it is therefore impossible to know if we really exist on scales greater then the plank length and are choosing to base distances off of light rays, or if we are actually living at distances much smaller than the planck length and are basing our distance measurements off of the winding mode counter part to light rays. :)

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