Non observable (contains GMm or $\tilde{M}\tilde{m}$ )

1873 modified Newton and forward:

“Original” Newton:

Gravity force

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

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

Mass must be

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

$\tilde{M}=l_p\frac{l_p}{{\overline{\lambda }}_M}$ (collision time, see [21] )

Gravitational constant

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

$c=1$

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

Gravity acceleration

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

$g=\frac{\tilde{M}}{R^2}=\frac{1}{R^2}\frac{l_p^2}{{\overline{\lambda }}_M}$

Orbital velocity

$v_o=\sqrt{\frac{GM}{R}}=c{l}_p\sqrt{\frac{1}{R{\overline{\lambda }}_M}}$

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

Orbital time

$T=\frac{2\pi R}{\sqrt{\frac{GM}{R}}}=\frac{2\pi \sqrt{\overline{\lambda }{R}^3}}{c{l}_p}$

$T=\frac{2\pi R}{\sqrt{\frac{\tilde{M}}{R}}}=\frac{2\pi \sqrt{\overline{\lambda }{R}^3}}{l_p}$

Velocity ball Newton cradle

$v_{out}=\sqrt{2\frac{GM}{R^2}H}=\frac{c{l}_p}{R}\sqrt{\frac{H}{\overline{\lambda }}}$

$v_{out}=\sqrt{2\frac{\tilde{M}}{R^2}H}=\frac{l_p}{R}\sqrt{\frac{H}{\overline{\lambda }}}$

Periodicity Pendulum (clock)

$T=2\pi \sqrt{\frac{L}{g}}=2\pi R\sqrt{\frac{L}{GM}}=\frac{2\pi R}{c{l}_p}\sqrt{L\overline{\lambda }}$

$T=2\pi \sqrt{\frac{L}{g}}=T=2\pi R\sqrt{\frac{L}{\tilde{M}}}=\frac{2\pi R}{l_p}\sqrt{L\overline{\lambda }}$

Frequency Newton spring

$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}=\frac{1}{2\pi R}\sqrt{\frac{GM}{x}}=\frac{c{l}_p}{2\pi R}\sqrt{\frac{1}{\overline{\lambda }x}}$

$f=\frac{1}{2\pi }\sqrt{\frac{k}{\tilde{m}}}=\frac{1}{2\pi R}\sqrt{\frac{\tilde{M}}{x}}=\frac{l_p}{2\pi R}\sqrt{\frac{1}{\overline{\lambda }x}}$

Gravitational red-shift

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

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

Observable predictions (from GR): (contains only GM or only $\tilde{M}$ )

Time dilation

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

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

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

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

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

Quantum analysis:

Constants needed

G, $\hslash$ , and c or $l_p$ , $\hslash$ , and c

$l_p$ and indirectly c, but $c=1$

Variable needed

one for mass size

one for mass size