# Is the Planck force a truly "Planck unit"?

The Planck force appears to be defined as the ratio of the Planck energy to the Planck distance, $$F_P = E_P/l_P$$ that can be rewritten as $$F_P = \frac{ E_P }{ l_P} = \frac{ c^4 }{ G }.$$

Isn't it rather odd that it doesn't involve Planck's constant? Is there some other acceptable interpretation of the force?

According to Wikipedia: "In particle physics and physical cosmology, Planck units are a set of units of measurement defined exclusively in terms of four universal physical constants, in such a manner that these physical constants take on the numerical value of 1 when expressed in terms of these units. Originally proposed in 1899 by German physicist Max Planck, these units are a system of natural units because their definition is based on properties of nature, more specifically the properties of free space, rather than a choice of prototype object. They are relevant in research on unified theories such as quantum gravity."

• Related question: What is the significance of Planck force? May 10, 2022 at 10:20
• You might prefer to call it a geometrized unit.
– J.G.
May 10, 2022 at 11:27
• Note that the Planck velocity is just $c$ :) May 10, 2022 at 15:31
• Amended the question to show that Planck units do not necessarily have to involve Planck's constant.
– jim
May 11, 2022 at 14:17

It is roughly the force between two black holes of mass $$M$$, located at each other's event horizon, i.e. at a distance given by the Schwarzschild radius $$R=\frac{2GM}{c^2}$$. Of course Newtonian mechanics is not applicable here anymore. But we can still use it to get the order of magnitude for the gravitational force between the two black holes: $$F=\frac{GM^2}{R^2}=\frac{c^4}{4G}$$
Yes, there is some another interesting interpretation for $$c^4/G$$. In all static spherically symmetric perfect fluid solutions of Einstein field equations the pressure behavior near the central initial event horizon reads $$\begin{equation} p(r,\alpha_{c})=\frac{4}{\kappa}\cdot \frac{1}{r^2}-\frac{4}{3}~ \rho(0,\alpha_{c}) \cdot c^2+\mathcal{O}(r^2), \end{equation}$$ with $$\kappa\equiv 8\pi G/c^4$$ and $$\alpha_{c}$$ the critical compactness parameter ($$r_{S}/R \le 8/9$$) for a given solution. Remarkable, whereas the pressure near the singularity diverges, the force generated by it is finite and does not dependent of $$\alpha_{c}$$, thus is equal for all solutions. This force is inversely proportional to Einstein's gravitational constant and equal twice the Planck force $$\begin{equation} F(0,\alpha_{c})\equiv\lim_{r \to~0} p~4\pi\cdot r^{2}=\frac{16\pi}{\kappa}\equiv\frac{2c^{4}}{G}~. \end{equation}$$ The appearance of such a universal force upholds the idea of the maximal tension principle conjectured independently by Gibbons  and Schiller , which is also related to the dimension of the $$3+1$$ space-time  and is equivalent to the holographic principle . It is the force necessary to create an initial event horizon, or causally speaking, to generate a "crack" in the spacetime continuum. The different factor in the above derivation and that in referenced papers, here 2 and there 1/4, is due to the fact the former has been achieved by applying Newton's gravitation theory.
$$\begin{equation} F_{P} = \frac{c^3 M_{P}^2}{\hbar} \end{equation}$$