How does the speed of light determine the size of material bodies?

[Aidyan]

I'm not talking about relativistic effects, just solid state physics. I'm wondering what determines the size of objects at the particle, atomic and molecular level, and especially how the constant c enters the equations? I suppose it depends from the strength of the molecular bonds which determine the size of the molecular lattices. But this in turn depends from the electric forces, i.e. the electric permittivity, i.e. the speed of light.

 

[mfb]

It depends on the definition of "size".

If you use a ruler as length definition, you cannot see an effect - as your ruler will always have the length of 1 ruler.
Our current meter definition is "the distance light travels in a specific time". If you change the speed of light, the material might get modified, but your length scale changes as well. Is that the effect you are interested in?

Electron orbitals (and therefore atoms, molecules and all solid and liquid objects) have a size that scales with the Bohr radius. It includes the speed of light, so if you change this you also change the size of all atoms and molecules - but you also change the length of your ruler (in the same direction), and the length of a meter with our current definition (in the opposite direction).

 

[Aidyan]

Yes, I didn't think about that, it is quite obvious indeed. In fact, Bohr's radius scales with $a_{0}\sim\varepsilon_{0}$. If that would have had another value, say $x*\varepsilon_{0}$, with $x$ some factor, Bohr's radius would be $x$ times $a_{0}$. But we would not see any size change because our ruler of length $l$ would also be rescaled to $x*l$. So far so good.

But I'm wondering then if we would live in a different universe with a different c, would we notice nothing different? Would the speed of light be rescaled or might it itslef even remain the same for an observer in such a universe? Since, in our universe, $c=\frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}}$, in a different universe, it scales with $c'=\frac{c}{\sqrt{x}}$. But maybe clocks would tick also differently (perhaps slower just by a factor of $\sqrt{x}$?!). Since a second is defined according to the frequency of the transition between the two hyperfine levels of Cs atoms etc., one should examine how that would change. But I don't know if and how that would be the case... am a bit confused at this point.

 

[BruceW]

If instead we lived in a universe with a different c, then as long as all other fundamental constants (for example Planck's constant and the gravitational constant) were also rescaled appropriately, then we would not be able to tell the difference. For example, the fine structure constant is:
$$\alpha = \frac{e^2}{4\pi \varepsilon_0 \hbar c}$$
This is a dimensionless number, so if we were in a different universe, it would only look the same as our universe if the fine structure constant was the same in that universe. So, we can find out how we must scale the other constants, given that we want to change $c$ to $c'$ (and I'll use a prime for the other 'new' constants in the new universe). So anyway, for both universes to look the same, we must have $\alpha =\alpha'$
$$\frac{e^2}{4\pi \varepsilon_0 \hbar c} = \frac{{e'}^2}{4\pi \varepsilon_0 ' \hbar ' c'}$$
and, rearranging, gives:
$$\frac{c'}{c} = \frac{{e'}^2 \hbar \varepsilon_0}{e^2 \hbar ' \varepsilon_0 '}$$
So here, we have at least one rule for how certain constants would need to scale so that our universe looks the same. Now, if instead they did not scale like this, then our universe would not look the same. In the example I gave, if the fine structure constant was different, we would have different physics, since the 'strength' of the electromagnetic force would be different. There are other dimensionless constants, which would also need to stay the same in the new universe if we want the new universe to look the same, but the fine structure constant is the best example I could think of.

 

[mfb]

The problem is that there is no unit-independent way to measure the speed of light.
It is perfectly valid to ask about a universe with different dimensionless constants like the fine-structure constant*. These constants can be measured in the universe, and you can examine their value. This is not true for the speed of light. And there are natural unit systems where the speed of light is exactly 1 - the speed of light cannot change at all in these units.

*there are ~25 of them in the Standard Model (if you include neutrino masses and mixing). Changing one of them always leads to different physics, but just a few of them influence the size of objects.