Is the exact definition of the Planck units important?

I think it's all just order-of-magnitude stuff and factors of π$\pi$ etc. are unimportant, but would be happy to be corrected.

Having said that, if someone defined a Planck time tp$t_p$ and then defined a Planck length to be anything other than ctp$c{t}_p$ then on the face of it they would be being a bit bizarre.

I've been a little disappointed by the changes made in the last couple years to the Wikipedia articles on Planck Units. There are pretty good reasons for defining Planck units as "rationalized Planck units". These units would result in these four dimensionful scaling factors to be set to one:

Specifically, in terms of any other system of units, these Rationalized Planck Units are:

natural unit of time:tpnatural unit of length:ℓpnatural unit of mass:mpnatural unit of electric charge:qp=ℏ4πGc5−−−−−−√=ℏ4πGc3−−−−−−√=ℏc4πG−−−−√=ϵ0ℏc−−−−√$\begin{array}{rl}\text{natural unit of time:}{t}_{\mathrm{p}}& =\sqrt{\frac{\hslash 4\pi G}{c^5}}\\ \\ \text{natural unit of length:}{\ell }_{\mathrm{p}}& =\sqrt{\frac{\hslash 4\pi G}{c^3}}\\ \\ \text{natural unit of mass:}{m}_{\mathrm{p}}& =\sqrt{\frac{\hslash c}{4\pi G}}\\ \\ \text{natural unit of electric charge:}{q}_{\mathrm{p}}& =\sqrt{{ϵ}_0\hslash c}\end{array}$

Planck units help us to understand which constants of nature are truly fundamental and which are simply a consequence of human-defined units of physical quantity.

The current and mostly undisputed position of physicists publishing about it is that only the dimensionless universal constants are fundamental.

Trialogue on the number of fundamental constants

Comment on time-variation of fundamental constants

How fundamental are fundamental constants?

The current enumeration of fundamental constants (those that are both universal and dimensionless) is 26; twenty-five for the Standard Model and one for General Relativity.

Now, it might be easier to think that G$G$ is a fundamental constant, but it isn't. I will insist that G=14π$G=\frac{1}{4\pi }$ and choose my units for that to be the case, and that is that. There is no meaning to the question "What if G$G$ were different?" And that's because it's not dimensionless and is merely a reflection on what units we use to express it. If you use the rationalized Planck Units, G=14π$G=\frac{1}{4\pi }$, Gauss's Law will equate gravitational flux density with gravitational field strength (making the conversion factor between the two concepts is equal to 1$1$), and with c=1$c=1$, then the speed of propagation and characteristic impedance of gravitational radiation 4πGc$\frac{4\pi G}{c}$ are both equal to 1$1$.

Now there is a dimensionless measure that describes what you mean by the strength of gravity and that is the Gravitational Coupling Constant

which is very similar to the Fine Structure Constant,

except it's about gravitation and not about EM. For some reason (likely political), the Wikipedia page about it has been removed (even from the Wikipedia archive) but there are other sites that have mirrored that page that still exist.

It's not hard to see that the fine-structure constant, α$\alpha$, is simply the square of the elementary charge to Planck charge ratio. It's a dimensionless number that, using historical Planck units, is e=α−−√≈0.08542454$e=\sqrt{\alpha }\approx 0.08542454$. But using rationalized Planck units, e=4πα−−−√≈0.30282212$e=\sqrt{4\pi \alpha }\approx 0.30282212$ . The elementary charge is well within an order of magnitude from the natural unit of charge (that result in wave speed c=1ϵ0μ0−−−√=1$c=\sqrt{\frac{1}{{ϵ}_0{\mu }_0}}=1$ and characteristic impedance Z0=μ0ϵ0−−√=1$Z_0=\sqrt{\frac{{\mu }_0}{{ϵ}_0}}=1$).

The gravitational coupling constant is the square of the ratio of the electron mass to the Planck mass. And, again, using rationalized Planck units, the electron mass is about 1.4837077×10−22$1.4837077\times {10}^{-22}$. Compare that to 0.30282212$0.30282212$ and we can see that the gravitational attraction of two electrons in free space is far, far less than the electrostatic repulsion of the same. That is the dimensionless expression of the strength of gravity (in comparison to another fundamental force, EM) and gravity is, indeed, very weak. But, from the POV of Planck units, it's really because the mass of an electron is very very small while the charge of an electron is not very small at all.

Indeed, Frank Wilczek puts it quite well in Scaling Mount Planck: A view from the bottom (Physics Today 2001):

We see that the question [posed] is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in natural (Planck) units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13quintillion)].

All of the other particle masses in the Standard Model can be expressed in terms of the electron mass, as dimensionless values.

That is the number to be considering in fine-tuning arguments.

Likewise, the Coulomb Constant (ke=14πϵ0$k_{\mathrm{e}}=\frac{1}{4\pi {ϵ}_0}$) is not fundamental. But, similarly, the strength of EM is represented by the Fine Structure Constant, which is the square of the ratio of the elementary charge to the Planck charge (or better yet, the rationalized Planck charge). That number is fundamental.

It's just not accurate to be making the case that life would be different if the Newton constant G$G$ were different or if the Coulomb constant ke$k_{\mathrm{e}}$ were different or if the Planck constant ℏ$\hslash$ were different or even if c$c$, the speed of causality, were different. All they need to be is real, positive, and finite and, if the dimensionless universal constants (those 26 constants enumerated by John Baez) remain unchanged, there is no way that any of us would know the difference.

But the main reason why rationalized Planck units are better than the historically-defined Planck units is that every inverse-square law should be expressed with 4π$4\pi$ in the denominator next to the r2$r^2$ to make it ready to rock 'n roll with Gauss's law. Whether it's EM or gravitational radiation, in natural units, flux density is the same thing as field strength (in a vacuum where the relative permittivity and relative permeability are both 1) and both the speed of propagation (the speed of causality) and the characteristic impedance of free space should be simply 1.

Consider this YouTube video Why is the speed of light what it is? Maxwell equations visualized, which I mostly thought was informative and decently accurate. But at time index 9:55, Arvin Ash asks "Why are μ0${\mu }_0$ and ϵ0${ϵ}_0$ these values?" and answers "No one knows why. They're just the constants of nature."

That answer is just false. And since it's key to the title of the lesson on YouTube, I was both surprised and alarmed by it. Especially regarding μ0${\mu }_0$, which has an exactly defined mathematical value (or at least did before May 20, 2019).

So, of course we know exactly why μ0=4π10−7${\mu }_0=4\pi {10}^{-7}$. The 4π$4\pi$ comes from Gauss's Law (4π$4\pi$ steradians in a sphere) and the 10−7${10}^{-7}$ comes from how the Ampere was originally defined (and remained so until 2019). It's the 2×10−7$2\times {10}^{-7}$ newtons per meter force on the infinite wires spaced one meter apart. So it's just false to say, regarding μ0${\mu }_0$, that "No one knows why" it takes that value.

Then c$c$ and ϵ0${ϵ}_0$ (which is a scaled reciprocal of the Coulomb Constant) are directly related. It shouldn't surprise anyone that the leading 3 in c$c$ and the leading 9 in the Coulomb constant are related. So it's just one or the other. Let's stick with c$c$.

The real question is about how these anthropometric units of the second and the meter got defined. Of course they are based on some terrestrial measures (the circumference of the Earth and the speed of rotation of the Earth) and some anthropometric divisions made by humans (10,000,000 meters and 86,400 seconds).

Why was that arc length (from North pole to equator) divided by 10 million instead of, say, 1 million? It's because a unit length that is 30-something feet long is not as practical to people as one that is about the length of us. A meter is about as big as we are.

Why was the solar day divided by about 100,000 instead of, say, 1000? It's because a second is in the ballpark of the smallest time that we can perceive in our consciousness. Maybe a little bit longer, but in the ballpark. It's a measure in how fast we perceive things and think about things. We can have about one or two or maybe three thoughts per second. But if we were tiny like a fly, even though our consciousness might be much simpler, we would be perceiving events on the scale of 0.02 second or even faster. It's likely not coincidental that a second is on the scale of our heart rate and breathing rate.

So instead of asking "Why does light travel 299,792,458 meters in the time of one second?", we should be asking "Why does light travel about 108${10}^8$ human body lengths in the time it takes a human to think a thought?"

Because the first question is exactly the same as asking "Why does light travel one Planck Length in the time of one Planck Time?" along with the questions "Why are there about 1035${10}^{35}$ Planck Lengths in a meter?" and "Why are there about 1043${10}^{43}$ Planck Times in a second?"

Let's look at the first question. It's like asking "Why are there about 1035${10}^{35}$ Planck Lengths in the size of a conscious sapient being like humans?"

That can be broken into three questions: "Why are there about 1025${10}^{25}$ Planck Lengths in the size of atoms (the Bohr radius)?" That is a question for physicists.

Then "Why are there about 105${10}^5$ atoms in the across the width of a typical biological cell?" That is a question for micro-biologists.

Then "Why are there about 105${10}^5$ biological cells across the length (in one dimension) of beings like us?" That's a question for biologists to answer.

Similarly we can search for answers to "Why there are about 1043${10}^{43}$ Plank Times in the time it takes us to think a thought?" It might begin with the Rydberg constant and the frequencies of atomic spectra, leading to the rate of chemical reactions, of enzyme catalyzed reaction, metabolism, and eventually the rate of functioning of organs including muscles and neurology.

So if we can answer why there are about 1035${10}^{35}$ Planck lengths across a complex biological being like us (which gives us a sense of scale of length) and why there are about 1043${10}^{43}$ Planck times in the complete functions of our biology (which gives us a sense of scale of time), and those are asking about dimensionless values, then we'll understand why it takes light about one second to move about 108${10}^8$ meters. All the rest of it is simply about historical accident in how units were defined.