Space > 3D?

Can you provide arguments for why space has three dimensions? Hint: Are planetary orbits stable in a space of n dimensions, where n > 3? Is the hydrogen atom stable when n > 3?

Answer

There are several arguments for why space is not larger than three dimensions. Planetary orbits are not stable when n > 3, except for a circular orbit for n = 4, because the attractive force and the centripetal force both do not have the correct dependence on radial distance. In 1917 P. Ehrenfest showed that one needs to consider the Poisson equation for arbitrary dimensions to determine orbit stability. When the n ≥ 4 circular orbit for a body around a central mass becomes slightly perturbed, one can show that the comparison of the central force to the centripetal force for the orbit depends on the perihelion value r₁ and the aphelion value r₂ according to [1/2 – (n – 2)^–1]/r²₁ < [1/2 – (n – 2)^–1]/r²₂, which cannot be true for n = 4 and larger. In a 4-D space, a satellite launched from Earth toward the Sun would either fly away to infinity or spiral into the Sun.

The hydrogen atom is not stable when n > 3 because there is no energy minimum for n ≥ 5, which is shown using the indeterminancy principle that is, the Heisenberg uncertainty principle. For the case n = 4, the relativistic energy equation must be examined to show that no energy minimum is available and the atom is not stable. Several other physical phenomena would be unusual for n ≥4 dimensions.

There is no satisfactory propagation of sound waves or electromagnetic waves free of distortion and reverberation in spaces other than n = 1 and n = 3. Also, axial vectors such as the magnetic field and the angular momentum vectors do not exist in even-dimensional spaces.

The considerations could be extended to a universe with more than one time dimension! However, this matter and others we leave for future challenges.

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