The Planck Mass Illusion: Why a Famous Gravity Formula is Circular

By John Gavel

For years, I used a neat little formula that seems to “explain” why gravity is so weak. I instinctively kept coming back to it, because it seems so simple:

\[ \alpha_G = \left(\frac{m_p}{m_{\text{Planck}}}\right)^2 \]

It looks elegant.

It looks fundamental.

It looks like it reveals something deep about nature.

But it doesn’t.

Eventually I realized it’s circular.

And I didn’t see it at first — because I implicitly assumed something that felt so natural that I never questioned it:

I assumed the underlying definition was fundamental, when in fact it was only conventional.

This mistake has nothing to do with infinities, nothing to do with the continuum, nothing to do with determinism or discreteness. It’s not a renormalization issue. It’s not a quantum-gravity issue.

It’s simply a logical issue.

Let me explain.

The Planck Mass Is Not Fundamental

The Planck mass is defined — not derived — by the equation:

\[ m_{\text{Planck}}=\sqrt{\frac{\hbar c}{G}} \]

This definition:

• does not involve infinity
• does not involve a continuum
• does not involve any limiting process
• does not come from a physical mechanism

It is just a unit-system construction — a way of combining three constants into a mass scale.

Nothing more.

So the circularity has nothing to do with how the universe behaves. It comes entirely from how the quantity was defined.

The Circularity Is About Derivational Direction

The algebra is perfectly valid.
The topology of the equations allows the manipulation.
There is no contradiction.

But the direction of the derivation is wrong.

You cannot use a quantity defined using \(G\) to derive \(G\).

That’s the circularity.

It’s like defining:

\[ x=\frac{1}{y} \]

and then “deriving”:

\[ y=\frac{1}{x} \]

Algebraically allowed.
Logically meaningless.

What This Does Not Mean

This does not mean the Planck mass is useless.

Planck units are extraordinarily valuable. They provide natural scales for discussing quantum gravity, black holes, cosmology, and high-energy physics.

The issue is not that the mathematics is wrong.

The issue is that it is easy to forget where a quantity came from.

Once a definition becomes familiar enough, the mind naturally begins to treat it as a fundamental object in its own right rather than as a quantity constructed from earlier assumptions.

When that happens, a simple identity can start to feel like an explanation.

But the formula itself does not explain gravity’s weakness.

At most, it expresses that weakness in a compact and elegant way.

The Explicit Circularity

Write both sides of the gravitational coupling.

Left side (physical definition):

\[ \alpha_G=\frac{Gm_p^2}{\hbar c} \]

Right side (Planck mass definition substituted):

\[ \alpha_G= \left( \frac{m_p} {\sqrt{\hbar c/G}} \right)^2 = \frac{Gm_p^2}{\hbar c} \]

Put them together:

\[ \frac{Gm_p^2}{\hbar c} = \frac{Gm_p^2}{\hbar c} \]

This is an identity, not a derivation. I thought I was looking at a deep relation between proton mass and quantum gravity; I was actually just looking at a mirror.

Once written explicitly, the circularity becomes obvious.

Conclusion

The weakness of gravity cannot be “explained” by the ratio \(m_p/m_{\text{Planck}}\), because the Planck mass itself is defined using \(G\). Any formula that expresses \(G\) in terms of the Planck mass is simply the definition inverted.

The algebra works.
The units look natural.
The numbers look meaningful.

But the logic is circular.

And once you see it, you can’t unsee it.

The way I see it now, gravity is weak because it requires many independent symmetry channels to be activated simultaneously. Gravity is weak because it couples only to the color-neutral part of matter, and the color-neutral state requires simultaneous activation of many independent SU(3) symmetry channels. The two-body nature of gravity doubles this requirement. The multiplicative cost of activating all these channels, combined with geometric propagation factors, produces the observed gravitational coupling — without invoking Planck units or circular definitions.