How can geometrized units have more than one constant equal to 1?

Question

I can understand how you could manipulate units to make a certain constant equal to $1$, like $c$ or $G$, et cetera. But how can you make it so two constants (in this case $c$ and $G$) are equal to $1$? You can equate one of them to $1$ but surely when you try to equate the other to $1$ you change the value of the first one?

Answer 1

When we set $c=1$, you can think of it as a redefinition of the second in terms of the meter: we solve $299792458 \, \text{m/s} = 1$ for s in terms of m. Now all time quantities will be measured in m.

You can also define $G=1$ at the same time. Notice that $G$ has units of $\text{kg}^{-1} \, \text{m}^3 \, \text{s}^{-2}$. You are imposing another condition but also have another unit. So now, with s already in terms of m, solve $6.674 \times 10^{-11} \, \text{kg}^{-1} \, \text{m}^3 \, \text{s}^{-2} = 1$ for kg, and you will also have all your masses measured in units of m. That's why they're called metric units.

You can go further and also define $\hbar =1$, the Planck constant. Now you have to solve the two previous equations and $1.0546 \times 10^{-34} \, \text{J} \, \text{s} = 1$, with $1 \, \text{J} = 1 \, \text{kg} \, \text{m}^2 \, \text{s}^{-2}$, at the same time for kg, m, s, and then all the masses, lengths and times will be adimensional. These are the the natural units. At the end of your calculations, if you want to express some quantity again in SI units, you have to multiply it by the appropriate powers of $c$, $G$ and $\hbar$, but now using their SI values, so you can get a time in seconds, a length in meters or a mass in kg.

Answer 2

There are two radically different concepts of what "a system of natural units" is:

1) In the SI system we need 3 base units (m, kg, s) (representing 3 base quantities (L, M, T) = length, mass, time) to form all other mechanical quantities. In a "natural system of units" one chooses three other base units representing 3 other base quantities, namely "natural" constants. For example you can choose (c, h, G) as base units. All other units can then be expressed in these base units.

• all equations stay the same, for example $E = mc^2$ stays as it is
• note that $c \neq 1$ (same as kg $ \neq 1$)
• note that the numerical value of $c$ = 1,
• which makes those equations containing fundamental constants simple to calculate
• the units of mass, time, and length become combinations of $c, h, G$.

2) Some physicists do the same as above but in a sloppy way where they set $c=1$.

• Equations now change, $E = mc^2$ becomes $E = m$,
• which is confusing for "normal" people
• dimensions get lost

So the answer is: since you start with 3 base units (in mechanics) you can substitute those with 3 natural constants. Your worry only comes into play when you want to use a forth natural constant as a base unit.

Answer 3

The most Natural way to base units on is to base them on the Planck length, the Planck time and Planck mass wich are Universal units. They are not based on human measures or alien measures. Every scientific thinking being in the universe would agree on how big these units are. So you can set the Planck length, the Planck time and the Planck mass equal to one. And because all other units are based on those three (except the charges of truly elementary particles, which you can nevertheless also set equal to one), you can make $h$, $c$, and $G$ one as well. It would be very impractical, though, because when you asked me for example how tall I am I had to answer you: "I´m $1.80*10^{35}$ Planck lengths tall", or when you asked for my age I had to answer: "I´m $1,6*10^{49}$ Plank times old". Indeed not so practical!