> Are some [constants] more fundamental than others? [...] one useful choice has been just three: h, c and G, collectively representing relativity and quantum theory. In 1899, Max Planck, who founded quantum physics, examined the relations among h, c and G and the three basic aspects or dimensions of physical reality: space, time, and mass.
I thought there were 5 "Planck units" (h, c, G, the electron charge, and Boltzmann's constant) matching 5 dimensions of physical reality (space, time, mass, also charge and temperature). But I don't understand two things:
* Why is Boltzmann's constant there, temperature being a statistical property more akin to the fine structure constant than the other 4 Planck units?
* Why do physicists nowadays prefer to use the electron charge instead of Coulomb's constant as Planck did? Coulomb's constant and the gravitational constant G slot into similar looking formulas for electromagnetism and gravity respectively, and pairing them seems more "fundamental" than using the electron charge.
> [...] matching 5 dimensions of physical reality (space, time, mass, also charge and temperature).
I'm not quite sure I understand that, but I'll try to answer your questions anyway:
Yes, Boltzmann's constant is more or less just a conversion factor between temperature and energy. If we measured temperature in joules, we wouldn't need Boltzmann's constant. But then the same is true for c; we could measure time in meters or distance in seconds.
If you have defined the basic units of mechanics (time, distance, and mass), the value of G is fixed. But not the value (and even the dimension!) of Coulomb's constant and of electric charges. You can use a system of units where Coulomb's constant is 1 and electric charge then has dimension mass^(1/2)length^(3/2)time^(-1), while in SI units it has dimension current*time. Basically, you can make Coulomb's constant have any value and any dimension you want. Using just Coulomb's constant or just an electric charge in such a formula wouldn't make much sense. On the other hand, something like the fine structure constant (which is practically the square of the elementary charge measured in natural units) is a quantity that makes sense inpendently of your system of units.
I guess the reason for that is that mass has meaning outside of gravity (in Newton's second law), but electric charge is confined to electromagnetism. There's also the thing with electric charge being quantized to integer multiples of the elementary charge e (apart from quarks for which it is an integer multiple of e/3, but they usually don't appear isolated), but mass not being quantized.
I think the similarities between electromagnetism and gravity are at least partly superficial. Coulomb's law looks a lot like Newton's law, but that's because both the electrostatic and the classical gravitational potential obey a poisson equation (in the statical limit and to the lowest order). And that's about the simplest equation such a potential can obey. I don't think there's much more to it than that (perhaps apart from both being long-ranged because their symmetries are not broken).
> both being long-ranged because their symmetries are not broken
I guess this is the property of both that appears similar, and it just seems more than coincidental that two of the four (or five) fundamental constants of nature describe these at least partly similar interactions.
When doing calculations the electric charge of a thingy is usually placed into some varible and when using electron charge versions of constants you can usually cancel them out. So less writing, typing and calculating; scientists are lazy.
I thought there were 5 "Planck units" (h, c, G, the electron charge, and Boltzmann's constant) matching 5 dimensions of physical reality (space, time, mass, also charge and temperature). But I don't understand two things:
* Why is Boltzmann's constant there, temperature being a statistical property more akin to the fine structure constant than the other 4 Planck units?
* Why do physicists nowadays prefer to use the electron charge instead of Coulomb's constant as Planck did? Coulomb's constant and the gravitational constant G slot into similar looking formulas for electromagnetism and gravity respectively, and pairing them seems more "fundamental" than using the electron charge.