Units

Newton and Einstein form

“Wavelength”-form

Mass form

Gravitational constant

$G\approx 6.67408\times {10}^{-11}$

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

Newton’s gravitational force

$F=G\frac{m_1{m}_2}{r^2}$

$F=\frac{\hslash c}{r^2}\frac{l_p^2}{{\overline{\lambda }}^2}$

$F=\frac{\hslash c}{r^2}\frac{m^2}{m_p^2}$

Escape velocity

$v_e=\sqrt{\frac{2Gm}{r}}$

$v_e=\frac{l_p}{\overline{\lambda }}c\sqrt{2}$

$v_e=\sqrt{2}\frac{m}{m_p}c$

Orbital velocity

$v_o=\sqrt{\frac{Gm}{r}}$

$v_o=\frac{l_p}{\overline{\lambda }}c$

$v_o=\frac{m}{m_p}c$

Gravitational parameter

$\mu =Gm$

$\mu =\frac{l_p^2}{\overline{\lambda }}{c}^2$

$\mu =\frac{m}{m_p}{l}_p{c}^2$

Gravitational acceleration field

$g=\frac{Gm}{r^2}$

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

$g=\frac{m^2}{m_p^2}\frac{c^2}{\overline{\lambda }}$

Gravitational time dilation

$t_0=t_f\sqrt{1-\frac{2Gm}{r{c}^2}}=\sqrt{1-\frac{v_e^2}{c^2}}$

$t_o=t_f\sqrt{1-2\frac{l_p^2}{{\overline{\lambda }}^2}}$

$t_o=t_f\sqrt{1-2\frac{m^2}{m_p^2}}$

Orbital time dilation

$t_0=t_f\sqrt{1-3\frac{Gm}{r{c}^2}}=\sqrt{1-\frac{3}{2}\frac{v_e^2}{c^2}}$

$t_o=t_f\sqrt{1-3\frac{l_p^2}{{\overline{\lambda }}^2}}$

$t_o=t_f\sqrt{1-3\frac{m^2}{m_p^2}}$

Light deflection

$\delta =\frac{4Gm}{c^{2}r}$

$\delta =4\frac{l_p^2}{{\overline{\lambda }}^2}$

$\delta =4\frac{m^2}{m_p^2}$

Schwarzschild Radius^a

$r_s=\frac{2Gm}{c^2}$

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

$r_s=2l_p\frac{m}{m_p}$

Gravitational redshift GR

${\lim }_{r\to +\infty }z\left(r\right)=\frac{1}{\sqrt{1-\frac{\frac{2Gm}{r}}{c^2}}}-1$

${\lim }_{r\to +\infty }z\left(r\right)=\frac{1}{\sqrt{1-2\frac{l_p^2}{{\overline{\lambda }}^2}}}-1$

$\frac{1}{\sqrt{1-2\frac{m^2}{m_p^2}}}-1$

Gravitational redshift ABS^b

${\lim }_{r\to +\infty }z\left(r\right)=\frac{Gm}{c^{2}r}$

${\lim }_{r\to +\infty }z\left(r\right)=\frac{l_p^2}{{\overline{\lambda }}^2}$

$\frac{m^2}{m_p^2}$