16 December 2013

Spontaneous Dimensional Reduction

Spontaneous Dimensional Reduction


I have been reading a few papers by Steven Carlip on spontaneous dimensional reduction, also essentially the same here and most recently here.
Carlip is probably best known for his article in Scientific American on similar themes. There he played with a 2+1 dimension "Flatland" universe; here he is seriously proposing a 1+1 one.
It's not as crazy as it sounds. In fact, I find it quite promising.

In a nutshell

Spontaneous dimensional reduction is his idea that at the very smallest scales, space is 1-dimensional (so space-time is 1+1 dimensional). He brings together various lines of evidence that support this, including his own treatment of the Wheeler-deWitt equation at extremely small scales.
Discussing the last point, he suggests that spacetime at small scales "spends most of its time" in or near a Kasner solution, a anisotropic solution to general relativity that applies in 3 dimensions or more. He argues that Kasner solutions favor 1 dimension - strongly so if contracting, less strongly if expanding.
Elsewhere he argues that focussing effects dominate, albeit in a slightly different context. This would imply that the contracting state dominates, which is basically what he needs for this to work. To my knowledge he hasn't explicitly applied this to 1+1 dimensions - that puzzles me, since the two ideas of his seem to fit each other nicely.
Kasner solutions are vacuum solutions - solutions that only apply to empty space. Carlip argues that by looking at extremely small scales, spacetime is effectively flattened.
At larger scales, he says that expansion and contraction change repeatedly and choatically, the general idea being a Mixmaster universe or a BKL singularity. The familiar 3 spatial dimensions are built from 1-dimensional pieces not unlike tinkertoys are.


Carlip doesn't appear to cover some of the nice features of 1+1 dimensionality, but I will.

Scalar propagator for gravity

The first one he does mention: In 1+1, the gravitational propagator is a scalar. All the problems with renormalizing gravity come from it having a non-scalar propagator (in fact, a rank 2 tensor; the other fundamental forces have rank 1 tensor propagators (ie, vectors)). With a scalar propagator, they should all go away.
My guess is that the other fundamental forces might also see a solution without renormalization from this. Nobody really likes renormalization, it's just been a neccessary evil in quantum field theory. Presumably that'd happen at an intermediate scale that has 2+1 dimensions.

The hierarchy problem

The hierarchy problem asks, why is gravity so much weaker than the other forces? For instance, you can lift a brick against the pull of the entire earth. The electromagnetic forces of the molecular bonds in your hand and arm exceed the gravitational forces levied by the 6.6 sextillion ton earth.
This offers an answer. By way of background, gravity requires 3 dimensions in order to propagate: 1 dimension of travel and 2 transverse dimensions. That's because it's a spin 2 force, which is why its propagator is a rank 2 tensor.
So spontaneous dimensional reduction says that gravity can't propagate at all at small scales, only at large scales. This may be enough to explain the hierarchy problem. (That's me conjecturing this)
"But wait", you say. "If it can't propagate at small scales, how does it get anywhere at larger scales? That's like saying, I can't walk three feet but I can walk a mile. Surely the big journey is made of little journeys?"
Well, what Carlip suggests elsewhere (here he may be summarizing others' work) is that for reduced dimensions, what happens instead is that gravity rearranges the topology of space, presumably affecting the BKL or Mixmaster behavior. This may be enough to let it propagate.

The self-energy problem

In a nutshell, the self-energy problem is that if forces like the coulomb force go as 1/r2 and therefore diminish over distance, then at small distances they should increase over distance, becoming infinite at r=0.
But (me again) in a 1-dimensional space, that doesn't happen. Forces go as 1/r0, which is to say they are insensitive to distance. No self-energy problem.
Further, there's helpful logic in the other direction. Why does spacetime do this at small distances? Why a Kasner-like solution instead of a simpler isotropic solution? Because if it didn't, then there would be infinite forces at small distances. If we don't need renormalization, we can just say as a principle that energies can't be infinite and then we'll find that 1+1 dimensional Kasner-like spacetime is needed at small scales.

Potential for insight about dark energy

Everybody's heard this for years so I'll be brief: Dark energy, whatever it is, is making the universe expand faster.
If Carlip's theory wrt Kasner solutions is true, then at small scales space is constantly expanding and contracting. This suggests (me again) some relation to dark energy. Maybe it's as simple as whether contraction or expansion dominates at that scale, and by how much.


  1. To an outsider (for some purposes I'm a physics insider, but today I'm definitely an outsider) it seems the self-energy problem is why one supposes a singularity at the heart of a black hole. So, would I be right in inferring that under this approach black holes wouldn't have singularities?

  2. From what I understand, that's exactly right. There'd be some minute scale at which space was 1 dimensional, and at scales smaller than that, the force doesn't increase.

    There's a bit of a race going on there. The gravitational force wants to make space curve, but the logic here is that you zoom in close enough to flatten the curvature near zero to approximate a Kasner solution - but zooming in means the gravitational force is closer and therefore stronger, therefore you need to zoom in more than that, etc.

    Somewhere (don't have the reference at hand) the BKL singularity guys (Belinski, Khalatnikov and Lipschitz, if I haven't slaughtered their names) argue that flattening wins the race, or a similar race. This was their original motivation, I think. They investigated the BKL singularity to make sense of the big bang singularity; I didn't follow that in detail.

    I should qualify what I just said. Any Kasner solution has a singularity of a different sort. The Kasner metric picks out a particular time t0, and volume is proportional to time. In most solutions, two of the three dimensions go to zero at t0 and the third becomes infinite, so at t0 there's a line-like singularity. There's also one solution where one dimension vanishes at t=t0 and the other two dimensions don't change with time.

    So qualifiers aside, yes, that's right as I understand it.

    1. I've had a non-specific sense for years (hey, I like to poke at things to see where their weak points are) that special relativity might be wrong somewhere at the upper end (near c). (Maybe I just don't think the race between Achilles and the tortoise shouldn't /really/ stretch out indefinitely.) This new angle on self-energy puts me in mind of that, but I can't quite wrap my head around the connection --- and there's something very odd going on when one starts looking to punch holes in relativity using a theory that, if I'm following rightly, was grounded in the first place on relativity.