Gravity Without an Explicit Newton Constant: A Review of Planck–Compton Forms

This review demonstrates that the Newton gravitational constant $G$ need not appear explicitly in gravitational formulae when the Planck length $\ell_p$ and the reduced Compton wavelength of the gravitating mass are taken as the primary length variables. The substitutions $G=\frac{\ell_p^2c^3}{\hbar}$, $M=\frac{\hbar}{\bar{\lambda}_M c}$, $GM=\frac{c^2\ell_p^2}{\bar{\lambda}_M}$ remove $G$ from a broad collection of Newtonian, relativistic, black-hole, astrophysical, and cosmological equations. The principal claim of the paper is precise: no formula in the survey requires $G$ as an explicit input parameter once the same empirical gravitational content is represented by $\ell_p$ and, where a mass occurs, by $\bar{\lambda}_M$. This becomes more than a notational convenience if $\ell_p$ can be determined operationally without first inserting a tabulated value of $G$. Cohen (1987) had pointed out the resulting circularity: expressing $G$ through Planck units is of limited operational value if those same Planck units can only be obtained from a prior value of $G$ \cite{Cohen1987}. A 2017 study addressed this problem by presenting particle, Cavendish-style, and related routes for determining the Planck length without prior numerical knowledge of $G$ \cite{Haug2017Planck}; later work developed additional mechanical and metrological implementations \cite{Haug2020Spring,Haug2022Independent}. We also show that $\frac{\ell_p}{\bar{\lambda}_M}=t_p\omega_C$, $\omega_C=\frac{c}{\bar{\lambda}_M}$, so the central Planck--Compton ratio is the reduced Compton angular frequency accumulated over one Planck time. This supplies a natural quantum-scale interpretation of gravitational strength in both Newtonian and Einsteinian formulae, although a discrete spectrum or a complete theory of quantum gravity is not established by this identity alone.