The hierarchy problem revolves around the fact that we don't have a way to explain why gravity is so much weaker than the other forces. It may just be a coincidence, but since physicists tend to understandably not like coincidences very much, we treat it as a "problem" and therefore look for a solution.
One of the proposed solutions is that gravity spreads out in some extra dimensions, and how this would explain the hierarchy problem is clear to me. My question is, would this solve the problem? From what I reckon, this would still lead to gravity to be treated differently from the other forces, and the question would just translate from "why is gravity weaker than the others" to "why does gravity span more dimensions than the others". Is there something I'm missing? Do other proposed solutions also fall into this eventual fallacy?
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1 Answers
The hierarchy problem asks why the weak scale is so much smaller than the Planck scale. There's a specific, technical reason this is a problem, called "technical naturalness," which is delicate to explain properly. Roughly speaking it boils down to an expectation that unknown physics at the Planck scale should contribute via quantum effects to the weak scale we observe. Just by naive dimensional analysis there is no obvious reason why a sum of different terms, each of which is of order the Planck scale, should cancel amongst each other to produce a much smaller scale (the weak scale we observe).
Anyway, the extra dimensional solution essentially boils down to saying: the world really has 5 dimensions, and the true 5 dimensional Planck scale actually is approximately the same size as the weak scale today. Then there is no hierarchy problem, because there is no hierarchy between the weak scale and the true gravitational scale. The cleverness in the idea is finding a way to warp the extra dimension, and the way our four dimensional world sits inside of it, to produce an effective four-dimensional Planck constant that is much bigger than the weak scale.
If you want to learn more, you should look at the original papers, or subsequent reviews. Here are two classics:
https://arxiv.org/abs/hep-ph/9905221 https://arxiv.org/abs/hep-ph/9803315
It is actually a very elegant idea, but is moving out of fashion as a solution to the hierarchy problem (as are essentially all solutions to the hierarchy problem) as data from the LHC have ruled out significant parts of the interesting parameter space of these models.
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