Modern Newton:

Alternative:

Mass

$M=\frac{\hslash }{{\overline{\lambda }}_M}\frac{1}{c}\left(\text{kg}\right)$

$\overline{M}=\frac{l_p}{c}\frac{l_p}{{\overline{\lambda }}_M}$ (collision-time, see [29] )

Non observable (contains $GMm$ )

Gravitational constant

$G,\left(G=\frac{l_p^2{c}^3}{\hslash }\right)$

$c^3$

Gravity force

$F=G\frac{Mm}{R^2}\left(\text{kg}\cdot \text{m}\cdot {\text{s}}^{-2}\right)$

$F=c^3\frac{\overline{M}\overline{m}}{R^2}\left(\text{m}\cdot {\text{s}}^{-1}\right)$

Observable predictions, identical for the two methods: (contains only $GM$ )

Gravity acceleration

$g=\frac{GM}{R^2}=\frac{c^2}{R^2}\frac{l_p^2}{{\overline{\lambda }}_M}$

$g=\frac{c^3\overline{M}}{R^2}=\frac{c^2}{R^2}\frac{l_p^2}{{\overline{\lambda }}_M}$

Orbital velocity

$v_o=\sqrt{\frac{GM}{R}}=c\sqrt{\frac{l_p^2}{R{\overline{\lambda }}_M}}$

$v_o=\sqrt{\frac{c^3\overline{M}}{R}}=c\sqrt{\frac{l_p^2}{R{\overline{\lambda }}_M}}$

Time dilation

$T_R=T_f\sqrt{1-{\sqrt{\frac{2GM}{R}}}^2/c^2}=T_f\sqrt{1-\frac{2l_p^2}{R{\overline{\lambda }}_M}}$

$T_R=T_f\sqrt{1-{\sqrt{\frac{2c^3\overline{M}}{R}}}^2/c^2}=T_f\sqrt{1-\frac{2l_p^2}{R{\overline{\lambda }}_M}}$

Gravitational red-shift

$z=\frac{\sqrt{1-\frac{2GM}{R_1{c}^2}}}{\sqrt{1-\frac{2GM}{R_2{c}^2}}}-1=\frac{\sqrt{1-\frac{2l_p^2}{R_1{\overline{\lambda }}_M}}}{\sqrt{1-\frac{2l_p^2}{R_2{\overline{\lambda }}_M}}}-1$

$z=\frac{\sqrt{1-\frac{2c^3\overline{M}}{R_1{c}^2}}}{\sqrt{1-\frac{2c^3\overline{M}}{R_2{c}^2}}}-1=\frac{\sqrt{1-\frac{2l_p^2}{R_1{\overline{\lambda }}_M}}}{\sqrt{1-\frac{2l_p^2}{R_2{\overline{\lambda }}_M}}}-1$

Gravitational red-shift

$z_{\infty }\left(r\right)\approx \frac{GM}{c^{2}R}=\frac{l_p^2}{R{\overline{\lambda }}_M}$

$z_{\infty }\left(r\right)\approx \frac{c^3\overline{M}}{c^{2}R}=\frac{l_p^2}{R{\overline{\lambda }}_M}$

Gravitational deflection (GR)

$\delta =\frac{4GM}{c^{2}R}=\frac{4}{R}\frac{l_p^2}{{\overline{\lambda }}_M}$

$\delta =\frac{4c^3\overline{M}}{c^{2}R}=\frac{4}{R}\frac{l_p^2}{{\overline{\lambda }}_M}$

Advance of perihelion

$\frac{6\pi GM}{a\left(1-e^2\right)c^2}=\frac{6\pi }{a\left(1-e^2\right)}\frac{l_p^2}{{\overline{\lambda }}_M}$

$\frac{6\pi c^3\overline{M}}{a\left(1-e^2\right)c^2}=\frac{6\pi }{a\left(1-e^2\right)}\frac{l_p^2}{{\overline{\lambda }}_M}$

Indirectly/“hypothetical” observable predictions: (contains only $GM$ )

Escape velocity

$v_e=\sqrt{\frac{2GM}{R}}=c\sqrt{2\frac{l_p^2}{R{\overline{\lambda }}_M}}$

$v_e=\sqrt{\frac{2c^3\overline{M}}{R}}=c\sqrt{2\frac{l_p^2}{R{\overline{\lambda }}_M}}$

Schwarzschild radius

$r_s=\frac{2GM}{c^2}=2\frac{l_p^2}{{\overline{\lambda }}_M}$

$r_s=\frac{2c^3\overline{M}}{c^2}=2\frac{l_p^2}{{\overline{\lambda }}_M}$