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In the Planck unit, $G=1$ as dimensionless. The fine structure constant is also dimensionless. I had an impression that predicting the speed of light is not meaningful, since it has unit/dimension and it would be more meaningful to predict some quantity that is dimensionless, e.g., fine structure constant, even that is somehow numerology.

On top of that, is there any difference between $G=1$ and fine structure constant prediction? No difference aspect: the former is by construction as 1, difference aspect: $G=1$ still carries some unit ?

ElementSegment
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1 Answers1

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Are they actually dimensionless in natural units?

For example, I've seen a meter of time described as the time it takes light to travel 1 meter. So the speed of light is (1 meter of distance/1 meter of time)=1. It's implied that time is expressed in terms of units of $t_m$=1 meter of time. $c= 1 m/ 1t_m=1$. $c=1$ implies scaling your units for time a certain way so that the numerical value of the speed is unity, but dimensions are implies. $G=1$ implies a certain scaling of distance, mass, and force.

By definition the fine structure constant is the ratio of the speed of an electron in the ground state of an electron to the speed of light. No matter what units you use for speed, they cancel out, so the fine structure constant is truly dimensionless $\alpha \approx 1/137$.

I'd think you can no more say $\alpha=1$ than you can say $\pi=1$.

R. Romero
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  • By construction is a valid operation. It is to say 1 = 1. 2 = 2. Thus, I think both $G=1$ and $\alpha \approx 137$ are equally dimensionless. But, I am not sure if I miss anything. – ElementSegment Aug 20 '23 at 14:52