I How do we justify "Natural" Units

[quickAndLucky]

How is it that when using "natural" units we drop the units themselves. I understand that you can arbitrarily change the magnitude of a parameter by choosing a new unit. For example Oliver R. Smoot is exactly 1 smoot tall.

However, in natural units with [c]=[h/(2π)]=1 the "smoot" part is left off. Furthermore, leaving off the units allows us to conflate speed and action, which to me have always been different physical quantities.

How do we justify this as it seems like its common practice?

 

[Dale]

I don’t understand. Units don’t need to be justified. They are a convention, you simply use them (or not, whatever your preference).

 

[DrClaude]

I don’t understand. Units don’t need to be justified. They are a convention, you simply use them (or not, whatever your preference).

I think that the OP is perplexed by the fact that if you calculate something using $c \approx 3 \times 10^8\, \mathrm{m/s}$, then $c$ has dimensions of length/time, but if you use natural units, then $c$ is dimensionless. Where have the units gone?

To answer the OP, the thing is that dimensioned fundamental constants can be seen as conversion factors between different dimensions. Take mass for instance: in the SI system, it has units of kg, but by the relation $E=mc^2$, we have that mass is just a form of energy, so we could also express mass as an energy, which is done all the time in particle physics, where masses are often given in MeV. So the constant $c^2$ appears as a conversion factor between mass and energy.

Similarly, it is arbitrary that we measure temperature in units of kelvin, instead of simply stating the temperature as an energy. The Boltzmann constant is the factor that allows to convert between the two (if you look closely, in thermodynamics you will rarely find $T$ by itself, whereas $kT$ is ubiquitous).

 

[Dale]

For example Oliver R. Smoot is exactly 1 smoot tall.

However, in natural units with [c]=[h/(2π)]=1 the "smoot" part is left off.

Perhaps the confusion is that not only the size of the units, but also the dimension of the unit is a matter of convention. For example, the dimension of charge is different in MKS and CGS units. In MKS it has its own dimension, Q, and in CGS it is $\sqrt{L^3 M T^{-2}}$

 

[quickAndLucky]

I like the examples of temperature and mass measured as energy and get

dimensioned fundamental constants can be seen as conversion factors between different dimensions

but one of the confusing things with natural units is that the fundamental constants are unitless. Also it is intuitively clear that temperature and energy should be related quantities as they are essentially the macro and micro versions of the same physical phenomina. But it is not clear to me that h bar and c are the same physical quantity.

 

[Dale]

But it is not clear to me that h bar and c are the same physical quantity.

Don’t look at converting h into c, look at the things that they each convert. For c, it is used to convert time into length, and when you work with spacetime it is clear that those should be the same. It is also used to convert E into B, and since E and B are both part of the overall electromagnetic field they should also have the same units. Similarly for h.

 

[DrClaude]

But it is not clear to me that h bar and c are the same physical quantity.

As @Dale pointed out, they are not. There can be some confusion coming from the sloppiness of writing

[c]=[h/(2π)]=1

This is simply a shorthand for $c=1$ and $\hbar = 1$, not a statement about a relation between $c$ and $\hbar$.

I confused a student a few days ago by writing something like $P(+) = P (-) = 1/2$, which brought up the question "Why should the two probabilities be equal?" In general the two probabilities I was considering are not related, but they just had the same value for the situation at hand.

 

[f95toli]

Also, note that when some people say "natural units" they simply mean setting "unimportant" parameters equal to 1 since they do not affect the qualitative properties of the solution. A good example would be to set the mass equal to 1 in a mechanics problem where you are really only interested in solving the resulting differential equation.
Some theorists/mathematician love doing this and it can be very confusing.

 

[bobob]

How do we justify this as it seems like its common practice?

You can justify different ways for different reasons. Sometimes, it's just convenient. Sometimes there is a good physical reason for doing so to try to identify which units are fundamental and which are derived. When doing relativity, you are talking about geometry so that all of your coordinates should be on equal footing. (You use the same units for measuring, length, width and height. If you did not, then there would be a conversion constant that is physically meaningless to deal with.) We measure distance and time with different instruments and therefore we have c as a conversion constant, but it's physically meaningless in relativity. You can just as easily use meters to measure time and in that case, velocities are dimensionless.

 

[Herman Trivilino]

How is it that when using "natural" units we drop the units themselves. I understand that you can arbitrarily change the magnitude of a parameter by choosing a new unit. For example Oliver R. Smoot is exactly 1 smoot tall.

However, in natural units with [c]=[h/(2π)]=1 the "smoot" part is left off.

One smoot of time would be the time it takes light to travel a distance of 1 smoot. Thus the speed of light is one smoot per smoot, or simply 1.

Furthermore, leaving off the units allows us to conflate speed and action, which to me have always been different physical quantities.

What I did above was conflate distance and time by measuring them in the same unit, even though I fully realize that distance and time are different physical quantities.

How do we justify this as it seems like its common practice?

There's nothing wrong with it, so in that sense it needs no justification. But in fact the justification is that it's convenient to do it.