0

I was curious to know the logic behind “Planck Units”, I read this question but did not understand it. Do you have a better (simpler) explanation for setting $c = G = \hbar = 1$?

Qmechanic
  • 201,751
zeynel
  • 217
  • 3
    Could you be more specific what you didn't understand about the answers to the question you link? – ACuriousMind Nov 30 '22 at 11:30
  • He introduces his own variables «ℓ, , and to stand for the length, time, and mass units in this system, respectively...» It seems this complicates things. I'd like to know how you justify setting c = 299792458 m/s = 1, this looks like saying 1 = 299792458 – zeynel Nov 30 '22 at 11:42
  • 3
    But that is exactly what the linked answer explains. Just because you think the actual explanation is complicated doesn't mean it's wrong! Perhaps see also https://physics.stackexchange.com/q/224232/50583 – ACuriousMind Nov 30 '22 at 11:53
  • 3
    @zeynel "looks like saying 1=299792458" no. When you equate 100cm and 1m do you say it is like saying 100=1? I don't think so. – Jeanbaptiste Roux Nov 30 '22 at 16:40
  • 1
    I understand that this is related to Planck units https://en.wikipedia.org/wiki/Planck_units. Physicists created new sets of units so that G, c, hbar becomes 1. But they express this as $c=G=\hbar=1$ is a shorthand way of writing it. Otherwise there is no mystery. It is like Astronomical unit which has a value of AU=1. – zeynel Nov 30 '22 at 17:28
  • 1
    Well, for one, they assume that, say, G is not dependent on the value of c or hbar, and vice versa, etc. That way you can set them all to be 1. It makes the math easier, but is it correct? No one knows, yet. And, on top of that, the log of 1 is 0, which disallows fractal interpretations of the dimension of a system. IMHO, that's bad, because we know that fractals reign. – shawn_halayka Nov 30 '22 at 19:43
  • 1
    @shawn_halayka Well this is what I don't understand. If you set $c=G=\hbar=1$ you basically eliminate (temporarily?) the constants from equations. But if I understand correctly, the answers say something different, they say that a new unit is created so that, in that unit, the constants appear as unity. – zeynel Dec 01 '22 at 08:08
  • Yes, but when it comes to actual astrophysics, one generally does not set the fundamental constants to unity. It's the theorists who do so. But that said, setting c = G = hbar = 1 makes things simple, in terms of orbits: Like, if you want to do unit speed while orbiting a unit mass gravitating object, at a distance of unity, the acceleration required to sustain a circular orbit is.... you guessed it: unity. Pretty simple! – shawn_halayka Dec 01 '22 at 17:38

2 Answers2

10

The numeric values for dimensionful quantities are actually their ratio with some arbitrary standard that has same dimension. So when you say your length $L=10$ meters it means $L$ is ten times longer than some stick which was called a "meter". Alternatively you can say that $L$ is 100 times longer than another stick called "decimeter". Obviously this is by no means same as proclaiming $100=10$.

In your example in the comments, saying that $c=1$ is not like saying $1 = 299792458$. Instead it means saying that $c= 299792458$m/s=$1\,[299792458\,\mathrm{meters]/second}$. So putting $c=1$ is equivalent to defining a new length unit (call it "zeynel") which is equal to

1 zeynel = 299792458 meters

Then the speed of light $c=1$ zeynel/second. So to summarize, you never set dimensionful quantities to 1 (or any other number for that matter), instead you pick a new set of "standard sticks" in the units of which, those quantities have a numeric value 1 (like $c=1$ zeynel per second).

John
  • 3,481
  • This makes sense. The shorthand way of expressing and omitting the background of new units make it confusing. – zeynel Nov 30 '22 at 17:36
1

Quoting from the Appendix on Units and Dimensions in J.D. Jackson's Classical Electrodynamics, it is important to understand "The arbitrariness in the number of fundamental units and in the dimensions of any physical quantity in terms of those units".

Max Planck similarly noted:

The fact that when a definite physical quantity is measured in two different systems of units it has not only different numerical values, but also different dimensions has often been interpreted as an inconsistency that demands explanation, and has given rise to the question of the 'real' dimensions of a physical quantity. … it is clear that this question has no more sense than inquiring into the 'real' name of an object"

The logic behind Planck units is the same as for most systems of units: they are convenient for use in a particular context. Planck units simplify some equations and relationships frequently used in some areas of physics. This is the same reason why there used to be multiple systems of electromagnetic units in common use, such as esu, emu, and cgs, that each made different equations and calculations simpler in some context, but confused generations of students. For example, only the SI system had an independent unit for electric charge (the Coulomb), while the others defined electric charge in different ways in terms of length, time, and mass.

In a unit system where "$c=1$", $c$ serves as a conversion constant between length and time, e.g. 299792458 metres = 1 second. Similarly, setting $\hbar=1$ provides a conversion constant between energy and time. Given $\hbar=c=1$, setting $G=1$ essentially defines the base unit. If you calculate an energy to be "$1$" in Planck units, that is equal to $1.2209\times10^{19}$ GeV or $1.9561$ GJ. It is certainly confusing at first, but eventually become "natural" if you work with it every day.

David Bailey
  • 11,172