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

[PeterDonis]

I don't know if you are aware that there was an interim period when the meter was defined in terms of wavelengths of a standard light emission, with the second (as now) being defined in terms of period of a standard emission.

Yes, I'm aware of that.

IMO, the newest definition is just streamlining the wavelength definition by virtue of using speed and period in place of wavelength and period. Any two of these are equivalent to all three. Any one of these is NOT equivalent to all three.

Hmm. I see what you're saying. I still don't think it's the same as defining "separate units" by using, say, a standard meter stick for distance and a cesium clock for time, but I'll agree that the physical speed of light does provide a second standard.

 

[vanhees71]

Again, the fundamental constants $c$, $\hbar$, and $G$ are mere conversion factors between units. Setting them to 1, makes all quantities dimensionless, and everything is measured in "natural units". That's of course impractical to handle. Thus one defines various different systems of units depending on the application you are working on.

You can see this on the example of electromagnetics. There the only fundamental constant appearing in the equations is the speed of light, $c$ (i.e., the phase velocity of electromagnetic waves in a vacuum). However, due to practicality the SI has chosen to introduce an additional unit, the Ampere for electric currents (to be changed very soon by defining the elementary charge, but that doesn't matter here too much), which introduces additional conversion factors, namely $\epsilon_0$ and $\mu_0$. The relation to the physical units is $\mu_0 \epsilon_0=1/c^2$.

 

[Herman Trivilino]

This is traditionally called beta. It is dimensionless, and, for a given object, is the same in all systems of units. For precisely this reason, it is not a speed. On expects that a speed expressed in meters/second should be a different number from the same speed expressed in feet per hour. Thus, there is a difference between a speed expressed in units where c=1 (it still has units, e.g. light seconds per second), while beta has no units.

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!

 

[vanhees71]

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.

This is a no-brainer. We (in the high-energy heavy-ion community) use this natural system of units all the time, and it works: There we have $\hbar=c=k_{\text{B}}=1$. There's only one unit left, usually GeV. For convenience we also use fermi (fm) for lengths and times. The key to go from one to the other base unit is $\hbar c=0.197 \text{GeV} \text{fm}$. That's all you need in this field.

Of course the choice of the system of units is arbitrary and thus semantics. A change from one to another system doesn't change the physics.

 

[burakumin]

Are you aware of cgs units, which are often used in electromagnetics? In cgs units the unit of electric charge is the statcoulomb. The statcoulomb is not a base unit, but a derived unit equal to $1 cm^{3/2} g^{1/2} s^{-1}$. So the dimensionality of the statcoulomb is $L^{3/2} M^{1/2} T^{-1}$, which differs from the dimensionality of electric charge in SI. So the dimensionality of electric charge is a matter of convention.

No I didn't but as far as I understand this is equivalent to stating that the electric constant is a number. So I think this example and the debate about $c$ can be summed up with a single question: To what extent can we consider fundamental physical constants like $c, \varepsilon_0, h, G, e, m_e, \dots$ as naturally identifiable to (dimensionless) numbers (and when we can, to which numbers precisely)? The natural choice would be to set them to 1 (but by the way, which one exactly? Should we set $h = 1$ or $\hbar = 1$ ? $\varepsilon_0 = 1$ or $\mu_0 = 1$ ?).
I think @PeterDonis's remark about the fine structure constant has important consequences. This means in particular that $\frac{e^2}{\varepsilon_0 \cdot h}$ is a speed and it's entirely made up of fundamental constants. Now $\frac{G \cdot m_e^2}{h}$ is also one. And then $\frac{1}{h} \cdot \sqrt[n+m]{ \frac{G^m \cdot m_e^{2m} \cdot e^{2n}}{\varepsilon_0^n} }$ are also speeds for all $n, m$ in $\mathbb{Z}^*$.
So if we have to identifies fundamental constants to numbers, it suddently becomes unclear (at least to me) which one of these many non-equivalent combinations should be set to 1.

 

[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).