Taking ispiration from this question units and nature .

The natural Plank units set the gravitational costant $G$, the planck costant $\hbar$, the speed of light $c$, the Boltzmann costant $k_B$ and the Coulomb costant $(4πε_0)^{−1}$ to be equal to 1.

From wikipedia http://en.wikipedia.org/wiki/Planck_units

Frank Wilczek puts it succinctly: We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)].

Now my question is: suppose there is another universe where the speed of light (for example) is much slower, say $\tilde{c}=10^{-6}c$. Relativistic effects would very likely be much easier to detect, they would probably be part of everyday life. Physicists in this universe would anyway use Planck units and set $\tilde{c}=1$.

What would be the difference between their universe and ours? Would they measure a different proton mass?

And if instead of the speed of light this universe differed in any other fundamental constant, could we distinguish this case from the previous one?

  • $\begingroup$ Since these constants are used in so many equations, it is probably near impossible to list all the differences. But my hunch is that protons as we know them wouldn't exist. My question is slightly different: would the equivalence principle still hold? If not, we're in for a truly wacky universe. $\endgroup$ – biziclop Oct 28 '14 at 13:50
  • $\begingroup$ This question is vacuous since the assertion that one can compare "universes" with "different" constants is unfounded. The claim $\bar c = 10^{-6}c$ makes only sense if there is a unit system common to both universes, and this is not established. $\endgroup$ – ACuriousMind Oct 28 '14 at 15:35
  • $\begingroup$ @ACuriousMind If we accept that the laws of physics are universal, then the question makes perfect sense and isn't opinion-based at all. You just have to do the calculations. Since the energy of the photon directly depends on the speed of light for example, what would happen if the energy of the photon was bigger or smaller? If anything, the question is too broad. $\endgroup$ – biziclop Oct 28 '14 at 17:35