From a purely mathematical view, the Planck length comes about when you normalize the speed of light (c), the Boltzmann constant (kb), the gravitational constant (G) and the reduced Planck constant (h - bar) to a value of one. You then multiply these together with powers so that you end up with a unit of length. This then works out so that the Planck length is 1.616255 x 10^-35 m. So with that it is an interesting dimensional analysis problem.
But the Planck length is the smallest meaningful length in out universe. At the very least, it is at lengths shorter than this that our current understanding of physics break down. So it seems that Planck length is an actual physical property of the universe.
Those two definitions seem like they have nothing in common so why are they the same value? Are they related or is it merely a coincidence? Is there some deeper meaning that normalizing the basic constants of the universe to a value of one unlocks other basic constants in our universe?
And the same goes for the other Planck units as well (suitably reinterpreted). Perhaps most obviously in the case of Planck mass, which is about 22 micrograms. Clearly neither the largest nor smallest meaningful mass.
Equations are often expressed in Planck units because it simplifies expressions by avoiding the need for physical constants. The Planck length is just the smallest indivisible unit of length for equations that are expressed in Planck terms.
Because the other physical constants are derived indirectly from the limits of certain physical equations, the Planck length naturally correlates to phenomena that are related to those limits.
As I understand it, things can be smaller than the Planck length, but we have no theory of physics adequate to discuss phenomena at that scale. Are things smaller than that “meaningful”? I think that’s a semantic question.
I’m not sure why the Bolzmann constant entered into this. It’s hardly an obvious fundemental property of the universe… it seems to come from thermodynamics, which is a sort of emergent concept?
Surely there are more elementary and universal constants:
Speed of light.
Gravitational constant.
Charge on the electron.
Ratio of electromagnetic to gravitational force.
Ratio of proton to electron mass.
Planck’s constant.
Fine stucture constant (about 137).
And of course that’s without taking into account the strong and weak forces which defeated Einstein’s quest for a unified field theory…?
You need the Boltzmann constant if you want to be able to define a “Planck temperature”. G, c, and ℏ, between them, give you mass, length, and time, and everything that’s a combination of those (energy, momentum, speed, etc.). But there’s only one good choice of a constant that combines those with temperature units.
That can’t be a list of elementary constants, because it’s not independent. Once you have c, Planck’s constant, and the electron charge, for instance, you have the fine structure constant. And that, plus the force ratio and the mass ratio on your list, are all dimensionless, making them useless for establishing units.
Einstein was never really in on the quest for unified field theories, and the Weak Interaction is, in fact, one of the triumphs of unified field theories: The theory unifying the Electromagnetic and Weak Interactions is 100% fully developed, and agrees stunningly well with every experiment that’s ever been done on it. The Strong Interaction is more difficult, but there’s no fundamental difficulty with it, either, and there are definite indications that it can be unified with Electroweak, too. The big difficulty in theoretical physics, which Einstein (along with everyone else) fruitlessly pursued, was reconciling gravity with quantum mechanics, which might or might not involve unifying it with the other interactions.
My impression was that Einstein started on a quest for unification before it was clearly understood that the weak and color forces existed, and he also wanted to use unification to eliminate the core indeterminacy in quantum theory - so basically he wanted a unified EM/gravity without indeterminacy. This was way off from the rest of the community’s idea of what “unification” was all about, so Einstein became disconnected from the rest of the workers in the field.
Reconciling general relativity with quantum mechanics is not the same thing as unifying gravity with the other forces. There are a number of models that attempt to both at once, but there are also models that unify other forces without gravity, and models that attempt to reconcile gravity with QM without unifying anything.
To the best of my knowledge, Einstein was never particularly active in the effort to unify gravity with electromagnetism. That was mostly the work of Kaluza and Klein.
Albert Einstein spent the last thirty years of his life on a fruitless quest for a way to combine gravity and electromagnetism into a single elegant theory.
Einstein was motivated by an intellectual need to unify the forces of nature. He felt very strongly that all of nature must be described by a single theory. “The intellect seeking after an integrated theory cannot rest content with the assumption that there exist two distinct fields totally independent of each other by their nature,” Einstein said in his Nobel lecture in 1923.
In addition, he believed there was a link between the need to resolve apparent paradoxes of quantum mechanics and the need to unify electromagnetism and gravity. Einstein always insisted that quantum mechanics could be derived from some more complete theory. For Einstein, who was never satisfied with the weirdness and randomness inherent in quantum theory, any acceptable unified field theory had to have quantum mechanics as a consequence