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sniffnoyIf you're unfamiliar with the concept, the idea of natural units is that you can create a "natural" system of units, determined by physical constants, by declaring that certain dimensioned physical constants should be equal to 1. E.g., when doing relativity, one often sets c=1, causing length and time to have the same dimensions; the idea is that by extending this convention to other physical constants, you fix enough things that you get a natural length, a natural time, etc.
The best known of these is Planck units, and when I say "natural units", I really mean "Planck units and slight variants", because the other systems of natural units seem much less natural. Planck units work by setting c=1 (from relativity), ℏ=1 (from quantum physics), and G=1 (from general relativity). Personally, I'd prefer setting κ=1, where κ=8πG/c4 is Einstein's gravitational constant, since that's the constant that actually appears in Einstein's equation (presumably more fundamental than Newton's law of gravity!), but G is the version that's caught on (partly because Planck units predate general relativity![0]).
Setting these three constants to 1 gets you natural units of length, time, and mass. You can then also include temperature by setting kB=1 (from thermodynamics), and include electric charge by setting... oogh, this one's tricky. Planck's choice was to set the Coulomb constant to 1, but the naturality of that choice is questionable; OTOH, it's not really clear what the correct choice should be; there's several good candidates (vacuum permittivity, elementary charge). Well -- I'm not going to discuss temperature or electricity, so let's ignore that.
Anyway, the key to the whole system is the fact that c, ℏ, and κ have dimensions that are linearly independent (under multiplication), so you can in fact get the isolated dimensions of length, time, and mass out of them. But... what if we changed the number of spatial dimensions[1] of the universe?[2] The dimensions of κ are curvature/density (or energy density, whatever, we're setting c=1). The dimensions of curvature are inverse area, 1/L². But the dimensions of density are M/Ld, where d is the number of spatial dimensions. Which means that, in a 1-dimensional universe, κ and ℏ would (up to length/time conversions, since we're setting c=1) have the same dimensions as each other, making it impossible to extract natural units from them!
But wait, there's a way to rescue this! Remember the electricity mess above? If you're not getting natural units from ℏ and G, this changes from a problem to a solution -- you could set both the elementary charge to 1, and either the Coulomb constant or ε0 to 1, depending on your preference!
Except... this doesn't work either! Once you set c=1 and e=1, you find that ℏ and the Coulomb constant have the same dimensions. You'd need G here to make things work, but with only 1 spatial dimension, G doesn't help you.
So: In a world with only one spatial dimension, there would be no natural units.
(Yeah OK the systems of natural units based around the electron or proton masses would still work fine but c'mon those clearly aren't that natural, I did say I was only considering slight variants of Planck units. :P )
-Harry
[0]Actually, IINM, Planck units also predate the switch to ℏ from h=2πℏ, but that change got made... the G to κ change never became popular, partly because I think switching from G to κ never got that popular in other contexts either. The version using κ instead of G is sometimes known as "reduced Planck units". So, yeah.
[1]Yes, this post uses the word "dimension" in two different senses a bunch. Don't get confused!
[2]I'm assuming there's always exactly 1 time dimension; if we allow changing that, then what I really mean is "two total dimensions", rather than "1 spatial dimension".
The best known of these is Planck units, and when I say "natural units", I really mean "Planck units and slight variants", because the other systems of natural units seem much less natural. Planck units work by setting c=1 (from relativity), ℏ=1 (from quantum physics), and G=1 (from general relativity). Personally, I'd prefer setting κ=1, where κ=8πG/c4 is Einstein's gravitational constant, since that's the constant that actually appears in Einstein's equation (presumably more fundamental than Newton's law of gravity!), but G is the version that's caught on (partly because Planck units predate general relativity![0]).
Setting these three constants to 1 gets you natural units of length, time, and mass. You can then also include temperature by setting kB=1 (from thermodynamics), and include electric charge by setting... oogh, this one's tricky. Planck's choice was to set the Coulomb constant to 1, but the naturality of that choice is questionable; OTOH, it's not really clear what the correct choice should be; there's several good candidates (vacuum permittivity, elementary charge). Well -- I'm not going to discuss temperature or electricity, so let's ignore that.
Anyway, the key to the whole system is the fact that c, ℏ, and κ have dimensions that are linearly independent (under multiplication), so you can in fact get the isolated dimensions of length, time, and mass out of them. But... what if we changed the number of spatial dimensions[1] of the universe?[2] The dimensions of κ are curvature/density (or energy density, whatever, we're setting c=1). The dimensions of curvature are inverse area, 1/L². But the dimensions of density are M/Ld, where d is the number of spatial dimensions. Which means that, in a 1-dimensional universe, κ and ℏ would (up to length/time conversions, since we're setting c=1) have the same dimensions as each other, making it impossible to extract natural units from them!
But wait, there's a way to rescue this! Remember the electricity mess above? If you're not getting natural units from ℏ and G, this changes from a problem to a solution -- you could set both the elementary charge to 1, and either the Coulomb constant or ε0 to 1, depending on your preference!
Except... this doesn't work either! Once you set c=1 and e=1, you find that ℏ and the Coulomb constant have the same dimensions. You'd need G here to make things work, but with only 1 spatial dimension, G doesn't help you.
So: In a world with only one spatial dimension, there would be no natural units.
(Yeah OK the systems of natural units based around the electron or proton masses would still work fine but c'mon those clearly aren't that natural, I did say I was only considering slight variants of Planck units. :P )
-Harry
[0]Actually, IINM, Planck units also predate the switch to ℏ from h=2πℏ, but that change got made... the G to κ change never became popular, partly because I think switching from G to κ never got that popular in other contexts either. The version using κ instead of G is sometimes known as "reduced Planck units". So, yeah.
[1]Yes, this post uses the word "dimension" in two different senses a bunch. Don't get confused!
[2]I'm assuming there's always exactly 1 time dimension; if we allow changing that, then what I really mean is "two total dimensions", rather than "1 spatial dimension".
