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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".
Thoughts
Date: 2025-09-06 06:03 am (UTC)People use a lot of things as natural units, more loosely. A foot in measurement is roughly the size of a large human foot, which is a useful size to work with. An astronomical unit is the distance from Earth to Sun. 1 G is Earth's gravity. It makes me suspect that, in a 1D universe, if it had life, they would come up with something that made sense to them there, which might not even be perceivable from the outside.
Re: Thoughts
Date: 2025-09-07 05:39 am (UTC)But yes "natural" here is in the sense of, like, something that even aliens should be able to agree on (perhaps up to some factors of small integers or π), which I think necessitates physical constants. And the thing about units like "earth's gravity" isn't just that to an alien they'd seem provincial -- it's that they're not even as constant as one might like! Earth's gravity varies -- not enough for most purposes but enough to matter for some -- depending on where you're standing. It's only a good enough standard when you don't need that high precision. Even units like the AU have been redefined in terms of the metric system, which in turn has been redefined in more universal terms (mostly in terms of the physical constants above; the second is the only one that uses something weirder, due to G being too hard to measure precisely), in order to deal with this problem; even if normally we just think of it as "the length of the semi-major axis of earth's orbit".
Re: Thoughts
Date: 2025-09-07 06:51 am (UTC)That would work in a 2D universe, but 1D? Hmm. Maybe they'd space out in a line.
>>But yes "natural" here is in the sense of, like, something that even aliens should be able to agree on (perhaps up to some factors of small integers or π), which I think necessitates physical constants. <<
Numbers are good. Does π work in 1D or does it need 2D to make a circle? I guess you could have the number itself in 1D but you wouldn't know what it does -- that it makes a circle. *laugh* Which is exactly the same as why polydimensional math is hard in 3D, you can't see what it does without a lot of finagling such as hyperbolic crochet.
>>And the thing about units like "earth's gravity" isn't just that to an alien they'd seem provincial <<
It's like the difference between historic and metric measures. In a lab, metric is useful because it's precise and in the base 10 that most modern societies favor. But for many other purposes, human-scale measures are more convenient. On Earth, 1G and 1 AU are useful; and they're useful around the galaxy for humans. But in dealing with interspecies relations, abstract constants will be much less argumentative and more objectively useful.
>> it's that they're not even as constant as one might like!<<
LOL yes.
Re: Thoughts
Date: 2025-09-17 11:16 pm (UTC)I mean the problem here is pure numbers don't have dimensions, they're no good as units for dimensioned quantities!