Standard:

Alternative:

Mass

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

$M_g=t_p\frac{l_p}{{\overline{\lambda }}_M}$ (collision-time)

Energy

$E=M{c}^2$ (joule)

$E_g=M_g{c}_g=l_p\frac{l_p}{{\overline{\lambda }}_M}$ (collision-length)

Gravitational constant

$G=\frac{l_p^2{c}^3}{\hslash }$

$c_g$

Gravity force

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

$F=c_g\frac{E_g{E}_g}{R^2}\text{m}/\text{s}$

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

Gravity acceleration

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

$g=c_g^2\frac{E_g}{R^2}=\frac{c_g^2{l}_p}{R^2}\frac{l_p}{{\overline{\lambda }}_M}$

Orbital velocity

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

$v_o=c_g\sqrt{\frac{E_g}{R}}=c_g\sqrt{\frac{l_p}{R}\frac{l_p}{{\overline{\lambda }}_M}}$

Orbital time

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

$T=\frac{2\pi \sqrt{R^3}}{c_g\sqrt{E_g}}=\frac{2\pi \sqrt{{\overline{\lambda }}_M{R}^3}}{c_g{l}_p}$

Velocity ball Newton cradle

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

$v_{out}=\frac{c_g}{R}\sqrt{2E_{g}H}=\frac{c_g}{R}\sqrt{2H{l}_p\frac{l_p}{{\overline{\lambda }}_M}}$

Periodicity pendulum (clock)

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

$T=\frac{2\pi R}{c_g}\sqrt{\frac{L}{E_g}}=\frac{2\pi R}{c_g{l}_p}\sqrt{L{\overline{\lambda }}_M}$

Frequency Newton spring

$f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}=\frac{1}{2\pi R}\sqrt{\frac{GM}{x}}=\frac{c_g}{2\pi R}\sqrt{\frac{l_p}{x}\frac{l_p}{{\overline{\lambda }}_M}}$

$f=\frac{c_g}{2\pi R}\sqrt{\frac{E_g}{x}}=\frac{c_g}{2\pi R}\sqrt{\frac{l_p}{x}\frac{l_p}{{\overline{\lambda }}_M}}$

Gravitational red shift

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

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

Time dilation

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

$T_R=T_f\sqrt{1-\frac{2E_g}{R}}=T_f\sqrt{1-\frac{2l_p}{R}\frac{l_p}{{\overline{\lambda }}_M}}$

Gravitational deflection (GR)

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

$\delta =\frac{4E_g}{R}=\frac{4l_p}{R}\frac{l_p}{{\overline{\lambda }}_M}$

Advance of perihelion

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

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

Micro lensing

$\theta =\sqrt{\frac{4GM}{c^2}\frac{d_s-d_L}{d_s{d}_L}}=2\sqrt{l_p\frac{l_p}{{\overline{\lambda }}_M}\frac{d_s-d_L}{d_s{d}_L}}$

$\theta =2\sqrt{E_g\frac{d_s-d_L}{d_s{d}_L}}=2\sqrt{l_p\frac{l_p}{{\overline{\lambda }}_M}\frac{d_s-d_L}{d_s{d}_L}}$

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

Gravitational parameter

$\mu =GM=c^2{l}_p\frac{l_p}{{\overline{\lambda }}_M}$

$\mu =c_g^2{E}_g=c_g^2{l}_p\frac{l_p}{{\overline{\lambda }}_M}$

Two body problem

$\mu =G\left(M_1+M_2\right)=c^2{l}_p\frac{l_p}{{\overline{\lambda }}_1}+c^2{l}_p\frac{l_p}{{\overline{\lambda }}_2}$

$\mu =c_g^2\left(E_{g,1}+E_{g,2}\right)=c_g^2{l}_p\frac{l_p}{{\overline{\lambda }}_1}+c_g^2{l}_p\frac{l_p}{{\overline{\lambda }}_2}$

Cosmology ( ${\overline{\lambda }}_c$ : reduced Compton wavelength Friedmann critical universe mass $M_c$ ) (contains only $G{M}_c$ )

Cosmological red shift

$z_H\approx \frac{d{H}_0}{c}=\frac{1}{\frac{2G{M}_c}{c^{2}d}}=\frac{d{\overline{\lambda }}_c}{2l_p^2}$

$z_H\approx \frac{R}{E_g}=\frac{d{\overline{\lambda }}_c}{2l_p^2}$

Hubble constant

$H_0=\frac{c^3}{2G{M}_c}=\frac{{\overline{\lambda }}_c}{2t_p{l}_p}$

$H_0=\frac{c}{E_g}=\frac{{\overline{\lambda }}_{c}c}{2l_p^2}=\frac{{\overline{\lambda }}_c}{2t_p{l}_p}$

Hubble radius

$R_H=\frac{c}{H_0}=\frac{2G{M}_c}{c^2}=\frac{2c{t}_p{l}_p}{{\overline{\lambda }}_c}$

$R_H=E_g=\frac{2l_p^2}{{\overline{\lambda }}_c}=\frac{2c_g{t}_p{l}_p}{{\overline{\lambda }}_c}$

Quantum analysis:

Constants needed

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

$l_p$ and $c_g$ , for some phenomena only $l_p$ ( $c_g=c$ )

Variable needed

one for mass size

one for mass size