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As has been mentioned in other posts, Kaluza originally didn't require the 5th dimension to be curled up/compactified.

So how exactly would our 4D world emerge from a non-compactified 5D manifold? I would imagine it would be a submanifold/cross-section. But that wouldn't really make sense, because then there should be all manner of particles crossing through our cross-section, and hence appearing and suddenly disappearing. That is, there's no reason to suppose there would be any 4D submanifold in which a whole bunch of matter would co-propagate, but that's what we have after all. And if the 5th dimension weren't distinguished geometrically, why shouldn't we perceive it?

Or, it could have to do with the cylinder condition. According to this, the metric is independent of $x^5$, such that the manifold is "stationary" along that direction: $e_5$ is a Killing vector. Thus, any particle trajectory must actually have infinite copies, because you can translate it to any value of $x^5$. Then, the single trajectory we perceive would be the set of all intersections of those true 5D geodesic trajectories with our 4D slice. And we wouldn't perceive the 5th dimension because nothing changes along it anyway. And maybe our 4D world is the particular slice that is everywhere orthogonal to the Killing vector.

But that doesn't quite square up, because apparently the cylinder condition is not a fundamental postulate of the theory; it was added more because it made the math tractable and still seemed to work. Right? Or is it an intrinsic part of it? Because it seems like, in general, we need to have 2D matter flows in 5D, rather than 1D trajectories, such that their intersections with 4D are the observed 1D trajectories. Is this on the right track?

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  • $\begingroup$ I mean this non-snarkily, though without specific expertise -- maybe these just actually are problems with the theory? It didn't catch on, after all, and not every conjecture that comes along has every conceptual hole plugged right away. $\endgroup$
    – jwimberley
    Aug 9, 2021 at 1:03
  • $\begingroup$ The cylinder condition is necessary to get a universe that resembles our universe. As Wikipedia says, "Standard 4D physics seems to manifest the cylinder condition, and the corresponding simpler mathematics". Compactification isn't required, but it (kind of) explains where the 5th dimension is hiding. $\endgroup$
    – PM 2Ring
    Aug 9, 2021 at 1:27
  • $\begingroup$ @jwimberley K-K theory gathered initial interest as a "theory of everything" because it united gravity & electromagnetism, but it got abandoned because it didn't cover the nuclear forces. Decades later, it became the basis of string theory, by adding enough extra dimensions to cover the more complicated symmetries of the strong & weak nuclear forces. Also see physics.stackexchange.com/a/339686/123208 $\endgroup$
    – PM 2Ring
    Aug 9, 2021 at 1:32
  • $\begingroup$ I suspect this is unrelated to Kaluza's ideas, hence I think this is a comment and not an answer, but modern ideas of branes inspired by string theory have led to extra-dimensional models where the extra dimensions are "large", such as the ADD model, Randal-Sundrum model, DBI inflation, the DGP model, and cascading gravity. These share the feature that Standard Model particles are confined to the brane (4D submanifold), but gravity can "leak" into the bulk (5+ dimensional space). $\endgroup$
    – Andrew
    Aug 9, 2021 at 2:03

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