On wikipedia https://en.wikipedia.org/wiki/Planck_units it states that the angular-frequency is equal to the inverse of the planck time.

$$ \begin{align} \omega_p=\frac{1}{t_p}=\sqrt{\frac{c^5}{\hbar G}} \end{align} $$

However, on this page https://en.wikipedia.org/wiki/Angular_frequency, it states that the definition of angular frequency is as follows

$$ \begin{align} \omega=\frac{2\pi}{t} \end{align} $$

Thus, the Planck units, as they appear on wikipedia are wrong? Should the Planck angular frequency be

$$ \begin{align} \omega_P=2\pi \sqrt{\frac{c^5}{\hbar G}} \end{align} $$

This $2\pi$ factor would actually carry to many other units. The power would be $2\pi c^4/G$, etc.?

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    $\begingroup$ It really doesn't matter, because the specific values of the Planck units have no meaning; it's an order of magnitude thing. In any case, there's no reason that just because $\omega = 2 \pi / t$ we should have $\omega_p = 2 \pi / t_p$, since $t_p$ is not the period of anything specific. $\endgroup$ – knzhou Jan 12 '18 at 22:24
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    $\begingroup$ @knzhou It matters if your trying to balance your equations! $\endgroup$ – Alexandre H. Tremblay Jan 12 '18 at 22:26
  • $\begingroup$ If you want, you can think of $t_p$ as being defined as "the time it takes for an oscillator with angular frequency $\omega_p$ to move by 1 radian", in which case $t_p = 1 / \omega_p$. The point is, $t_p$ doesn't really have a specific definition, you can choose the numerical factors however you want. $\endgroup$ – knzhou Jan 12 '18 at 22:29

the "$t$" in

$$ \omega=\frac{2\pi}{t} $$

is the period of oscillation. the time required for one complete rotation of an angular vector. it's not just any old time, but a specific time. it is not the time it takes for the rotation of 1 radian.

where as the reason why $\omega_\mathrm{P}$ is used instead of "ordinary" frequency is because it's angular frequency (expressed as "$\omega$") that multiplies $\hbar$ and it is $\hbar$ that is normalized with Planck units.

i.e. 1 Planck Energy corresponds to a photon of angular frequency of 1, not to a photon with ordinary frequency (like "$f$" or "$\nu$") equal to 1.


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