Deriving the Planck Length, Time, and Mass

[Saketh]

How are the Planck length, Planck time, and Planck mass "derived"?

I know their values, but I don't understand where we get them from.

 

[rbj]

it's similar to how the "Newton" was chosen to be exactly the force that would accelerate 1 kg of mass at a rate of 1 m/s². in the SI or cgs systems, they start with a sort of arbitrary (from a universal POV) definition of unit length, unit mass, and unit time, and given those definitions, they naturally derive a unit of velocity, unit momentum, unit force, unit energy, etc.

then, given that set of units (pre-1960) physicists went out and measured a bunch of stuff including 3 universal constants intrinsic to free-space, that is the speed of E&M propagation $c$, Planck's constant $\hbar$, and the universal gravitational constant $G$.

remember that these constants take on the numbers that they do because of the 3 arbitrary base units we came up with to measure them. so if we chose to, we could choose the 3 base units (and adjust the consequential derived units) so that those three constants $c$, $\hbar$, and $G$ all take on the value "1" in terms of those units. that is what Planck units are.

check it out in the Wikipedia. at least the hard-core POV pushers haven't f*cked that one up too much.

 

[Saketh]

I once saw a problem which asked to "derive" the Planck units. Your explanation makes sense -- that the units are just a standard -- but are the units a logical outgrowth of a previous scientific result? Or are they nothing more than a standard?

 

[rbj]

I once saw a problem which asked to "derive" the Planck units. Your explanation makes sense -- that the units are just a standard -- but are the units a logical outgrowth of a previous scientific result? Or are they nothing more than a standard?

the units are more "conceptual" than a standard, although i suppose one could treat them as a standard if they want to have a Cavendish-like apparatus for measuring $G$, a Watt-balance for measuring $\hbar$ and a Michaelson-like thing (including a mirror on a mountain some distance away) for measuring $c$ in terms of existing temporary standards like a cesium-clock, platinum-iridium prototype meter bar, and prototype mass. if we measure these constants in terms of the temporary standards, then we can say how those temp standards stack up in relation to the Planck units.