Re: This Week's Finds in Mathematical Physics (Week 222)

John Baez wrote:
Loll and company believe they're seeing spacetimes that are *exactly*
2-dimensional in the limit of very small distance scales, *exactly* 4-dimensional in the limit of very large scales, with a continuous change in dimension in between.

Note that this does *not* mean that you have a smooth manifold of either dimension at any scale. But at least it doesn't rule 4-dimensional space time *out*, any other number would have.

My main worry about their work is that it uses a fixed slicing of
spactime by timelike slices. So, there's a danger that their procedure breaks Lorentz-invariance, even in the continuum limit which they are attempting to compute. I would like to find a way
around this problem!

This is only a problem if the slices have any physical effects, hopefully they don't.

You can make a pot of any shape whatsoever out of thin strips of clay. It at least seams reasonable that you could build any causal Lorentzian 4 manifold from oriented 3d strips.

Assuming that the whole space *is* a 4-d smooth manifold at large scales, there is no guarantee that the built in space-like slices will be smooth. In fact they really can't be because they seem to have the wrong dimension.

So if the built in slices or time coordinates have any physical effects at large scales, Lorentz invariance would be the least of the problems. The physics would not even be smooth.

What I found a bit odd in the papers was the amount of effort they devoted to determining the properties of the space like slices. If the slicing has no physical effects, and I agree that it had better not, then what do they expect their properties to tell them about physics?

I suspect that they looked at the properties of the slices because the *could*. The slices were right there in the implementation. It isn't easy extracting information about the behavior of the classical limit. I'm not entirely sure what the right properties to measure are, but here is one suggestion:

Place two particles at a random point. Allow them to follow a *timelike* random walk on the surface. Record the probability of them meeting again as a function of the proper times along the two paths (not the paper's "proper time"). See how it scales for intervals much longer than the simplexes, but much shorter than the whole surface.

The spacelike slices do serve one function, they remove from consideration manifolds that have *no* spatial slices.

I think GR does have solutions that can't be sliced that way. They would be good things to look at for testing the theory. Something quantum would have to happen as you approached such a situation, it couldn't continue acting classically.

Ralph Hartley

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