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Section on Wikipedia dealing with the possibility of different dimensions.

When reading this section it feels like there's a giant elephant in the room that is not addressed. For example, here's a quote from the section:

In 1920, Paul Ehrenfest showed that if there is only one time dimension and greater than three spatial dimensions, the orbit of a planet about its Sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy. Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a wave impulse will travel at different speeds. If there are $5+2k$ spatial dimensions, where $k$ is a whole number, then wave impulses become distorted. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only with three dimensions of space and one of time. Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.

Okay, but this assumes that in the hypothetical universe with only one time dimension and >3 spatial dimensions, the laws of physics as we know them remain valid. How do we know they remain valid? After all, if these theories were formulated in 3+1 dimensions, it can't be surprising then that they work best in 3+1 dimensions. Besides, if we're allowing the number of dimensions to vary, we're presumably also allowing the laws of physics to vary.

Concrete example: suppose it's proven that in GR, with 1 time dimension and >3 spatial dimensions, the orbit of a planet about its Sun cannot remain stable. That still isn't fatal because $f(R)$ gravity can conceivably replace GR in those universes and one of the theories might have stable orbits.

In other words, for the arguments in the quote to be convincing against extra spatial dimensions, they need to show that there is no possible theory in >3 spatial dimensions where the orbit of a planet around a Sun is stable - something which is presumably very hard if not impossible.

I'm wondering if I'm missing something.

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    $\begingroup$ Counterexample: one motivation for string/brane theory was the suggestion that gravity works the same way in the extra dimensions but that the Standard Model interactions are stuck on the 1+3-dimensional "surface" that we live on. Search term is "hierarchy problem." $\endgroup$
    – rob ♦
    Sep 16, 2019 at 1:33
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    $\begingroup$ "How do we know they remain valid?" We don't, but so what? The point of the mental exercise is to try and understand the laws of physics better by studying what happens when you change something -- in this case, the dimension. If in a hypothetical universe with $d\neq4$ the laws were different, that would be irrelevant. What we want to do is to understand our laws, and we do that by assuming they hold in other $d$. Whether this is actually true is beyond the point. $\endgroup$
    – AccidentalFourierTransform
    Sep 16, 2019 at 1:44
  • $\begingroup$ Allure, I think that you have the question "backwards". The extra dimensions were introduced into more modern physics mathematical models to explain observations that can't be explained by the current 4 dimensions (e.g., 3D + time). $\endgroup$
    – David White
    Sep 16, 2019 at 1:45
  • $\begingroup$ @AccidentalFourierTransform well, the laws of physics we have are formulated in 3+1 dimensions, so it shouldn't be surprising if they work best in 3+1 dimensions. In fact I don't even find it surprising that they fail to work in other dimensions. The section reads to me as though it assumes this degree of freedom $N_{dim} = 4$, and then concludes that therefore, $N_{dim} = 4$ is privileged, which doesn't sound like much of a result. $\endgroup$
    – Allure
    Sep 16, 2019 at 2:11
  • $\begingroup$ @DavidWhite the works cited in the quoted paragraph predate string theory though. I realize that today questions of this sort are in string theory (which is why I tagged as such), but I don't per se see how the question asked in the OP necessarily involves modern physics. $\endgroup$
    – Allure
    Sep 16, 2019 at 2:14

2 Answers 2

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You're pointing out the weakness of any anthropic argument. Any claim that something about nature had to be the way it is, for structures like atoms or life to exist, can be countered by stating that we have insufficient imagination. This is a genuine objection, and in many cases perhaps the most important one.

However, the arguments you've cited are on more solid ground than they look. The point is that in physics, once you formulate a theory, you often automatically get theories in any dimension. For example, the postulates of Newtonian mechanics don't change between dimensions, which is why all high school courses begin with one-dimensional dynamics problems. Similarly, the postulates of quantum mechanics are totally independent of spatial dimension. In both cases the structures of the theories are rigid; it's hard to bend them without breaking them, getting something utterly alien.

In other words, mathematically these ideas seem more fundamental than $d$, which seems to be an arbitrary number tacked on at the end. That's why it's interesting and surprising to find that, in fact, if you change $d$ (while keeping the surrounding framework intact) everything falls apart. Of course this is nowhere close to an airtight mathematical proof, and it was never intended to be. Physical insights rarely are.

If you're curious, what really powers most of the conclusions you cited is special relativity. The geometrical structure of special relativity follows by postulating a spacetime metric $$\eta = \text{diag}(-1, 1, 1, 1).$$ To accommodate more spatial dimensions, you just add more ones at the end. To accommodate more time dimensions, you just add more $-1$'s at the beginning. So then you can ask, assuming the postulates of special relativity hold, would physics work if the number of dimensions were changed in this way? Again, this is an interesting question to ask if you treat relativity as a deeper fact that the number of dimensions.

It turns out relativity is so powerful that we get a lot of mileage out of changing $\eta$ alone. For example, the very simplest equation of motion for a field is $$\eta^{\mu\nu} \partial_\mu \partial_\nu \phi = 0$$ which is called the wave equation, and describes fields propagating at the speed of light. So you can analyze the wave equation in an arbitrary number of dimensions, which is what was done in the second equation you cited. The wave equation also can be used to derive how fields fall off with distance (e.g. inverse square in 3 spatial dimensions), allowing one to analyze orbits in arbitrarily many dimensions, either classically (your first example) or in quantum mechanics (your fourth example). Your second example is a little weaker; Weyl showed that electromagnetism only has a certain cute theoretical property, conformal invariance, in $3$ spatial dimensions. But this property can again be determined solely from the structure of special relativity.

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  • $\begingroup$ Well, using relativity as a framework, we also have the cosmological constant. If this were zero, we'd have a simpler theory, but observations indicate it's not zero. GR is easily modified to include it. How can we know there's no analogous modification that would allow planets to orbits suns in higher dimensions, even assuming GR? It's not just cosmological constants either - every $f(R)$ gravity theory is a similar modification, and they could have solutions with stable orbits in higher dimensions. $\endgroup$
    – Allure
    Sep 17, 2019 at 5:01
  • $\begingroup$ @Allure Of course, we don't. There are always assumptions for simplicity. The arguments just show that physics in $d \neq 3$ spatial dimensions, assuming the framework stays fixed, is either more complicated or doesn't support specific structure that we're used to in $d = 3$. $\endgroup$
    – knzhou
    Sep 17, 2019 at 6:00
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    $\begingroup$ @Allure Again, in physics and in science in general, we never ever prove anything beyond doubt, either experimentally or theoretically. You can always conclude anything you want, the only question is how complicated you will let your analysis become. Despite this seemingly devastating blow, science still works excellently anyway, which to me is one of the greatest miracles of all time -- possibly the miracle. These heuristic arguments for $d = 3$ should be seen as tiny pieces that may or may not someday fit into an enormous puzzle we are all assembling. $\endgroup$
    – knzhou
    Sep 17, 2019 at 6:02
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Those relationships were originally formulated in 3+1 dimensions because at the time, it was fundamentally believed that our universe actually contained only 3+1 dimensions.

However, almost immediately thereafter, mathematical physicists then began asking questions about what if we missed counting one or more of those dimensions; what would the consequences be? This is not a philosophical issue- the math can and was explicitly worked out by Kaluza, Klein, Ehrenfest, Weyl, and others, and compared by experiment and observation with the universe we happen to inhabit.

The conclusion: things fundamentally do not work out properly in the case where there are (as you pointed out) two time dimensions, four spatial dimensions, etc. when we then try to slot in the known physical laws we discovered in the 3+1 case.

I think the point you are missing is this: if for example we squeeze in an extra unseen dimension of space, then the 1/r^2 law for newtonian gravity no longer holds because gravity would then "leak into" the 4th dimension and thereby change the strength of gravity back here in the 3 dimensions we know about. It is in principle possible to rewrite our constituitive laws of physics as derived in 3+1 so they would still furnish for example stable gravitational orbits in the 4+1 case but then they wouldn't yield the right answers for 3+1 anymore.

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