Can c be set equal to 1 in certain systems of units?

[Dale]

(and when we can, to which numbers precisely)?

(but by the way, which one exactly? Should we set h=1h=1h = 1 or ℏ=1ℏ=1\hbar = 1 ? ε0=1ε0=1\varepsilon_0 = 1 or μ0=1μ0=1\mu_0 = 1 ?).

it suddently becomes unclear (at least to me) which one of these many non-equivalent combinations should be set to 1.

In all of these, we choose whichever is most convenient for the application we have in mind. That is the great thing about a good convention: it makes things easier.

 

[PAllen]

The central issue in this thread for me is whether it's possible, by a choice of systems of units, to set $c$ identically equal to the dimensionless $1$. Note, for example, that that is what (at least appears to me that) Taylor Wheeler do in Spacetime Physics.

For example, if we choose a system where distance is measured in meters and time in light meters (the time it takes light to travel a distance of one meter, then $c=1 \mathrm {\ meter\ per\ light\ meter}$. On the other hand, if we choose a system where distance is measured in meters and time in meters (the time it takes light to travel a distance of one meter) then $c=1$.

If I interpret what you're saying correctly, it can all be seen as a matter of semantics. If we insist as you do that $\beta$ is not a speed, then I need to give it another name, such as dimensionless speed, or whatever term seems suitable, to make my point. My central issue then becomes whether we have a system of units where the speed $v$ of light is $1 \mathrm{\ unit\ of\ distance\ per\ unit\ of\ time}$ or whether we have a system where the dimensionless speed $\beta$ of light is $1$.

Of course, it all seems silly when reduced to a matter of semantics. But it seems that's all there is to it!

I think an earlier post by Peter put this best. Some fundamental constants are dimensionless no matter what units you use. Furthermore, their value is independent of units. Most physicists accept that these are the only true fundamental constants. For other constants, different systems of units determine both the value and the units of the constant. Thus, there are systems of units where c is a dimensionless 1, and others where it has a value of 1 with dimensions. In contrast, the fine structure constant is about 137 AND dimensionless in ALL systems of units. But c remains in the category of dimensionful constant, because its value and dimensions are not independent of unit choice.

Beta versus speed is similar: beta is dimensionless, and has the same value (for a given object), in all systems of units. Speed of that object will have different dimensions and values depending on system of units. You can construct systems of units where speed has the same value as beta and is dimensionless. However speed remains in the category of dimensionful parameter (because it has dimensions in many systems of units), while beta is a dimensionless parameter (because this feature is independent of units).