# Alternative Approach to Fundamental Equations in Equivalences with Defined Universal Constants

Nergis Üstoglu

Nergis.ust@hotmail.com

## Abstract:

The notation with unit coefficients formulates the relationship between macro spacetime and subatomic spacetime with defined constants. It is possible to basically express all forces with the Planck force coefficient. Transformable conjugate equivalences provide a simple description of the fundamental formulas as an alternative method.

**Keywords:** Black hole, compton force, conjugate equivalence, cosmological constant, escape velocity, expansion coefficient, gravitational constant, gravitational force, gravitational radius, Hubble parameter, mass constant, nuclear force, orbital frequency, planck charge, temperature

## 1. Introduction

In systems with constant energy, it is possible to express quantities with unit coefficients. The speed of light is the limit value and Planck units are taken as reference. At the system scale, the gravitational radius equivalent to the wave radius of the system particle is defined. The expansion coefficient at the mass radius determines the unit coefficients. At the minimum expansion coefficient, the system has a Schwarzschild radius.

In the equations, the Schwarzschild radius is defined by the Planck length coefficient. Speed is defined as a multiple of the speed of light can thus be expressed as

$$V = ca_0$$

Factor of $4\pi/3$ is added to the equations for spherical symmetries. By means of constants, equivalences are transformed so that the coefficients are equal and in the same ratio.

For an electron orbital,

<table><tr><td>Speed of light, c</td></tr><tr><td>Electron velocity coefficient, $\frac{\infty}{n}$</td></tr><tr><td>Mass constant, $\sigma$</td></tr><tr><td>Planck length, l</td></tr><tr><td>Schwarzschild radius, $R_s = 2sl$</td></tr></table>
$$ \frac{1 \cdot \sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} $$

$$ = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} $$

$$ = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} $$

$$ = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} $$

$$ = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} $$

$$ = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} $$

$$ = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} = \frac{\sigma^2 c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{3} $$

$$ 2. \sigma c^2 R_{s1}^2 R_{s2}^2 \alpha^6 n^6 = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi \sigma 3} $$

$$ = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi \sigma 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi \sigma 3} $$

$$ = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} $$

$$ = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi \sigma 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} $$
$$= \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3}$$

$$= \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3}$$

$$= \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi 3} = \frac{3\sigma c^2 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^6}{4\pi \sigma 3}$$

Planck Mass, $m_{pl}$

massive mass coefficient, $s_1$

orbital mass coefficient, $s_2$

$$m = \frac{\sigma R_s}{2} = s\sigma l = s m_{pl}$$

$$3 \cdot \frac{\sigma c^2 R_{s1}^2 R_{s2}^2 \alpha^2}{n^2} = \frac{3\sigma c^2 R_{s1}^2 R_{s2}^2 \alpha^6 n^4}{3 \alpha^4 n^6} = \frac{3c^2 4m_1 m_2 R_{s1} R_{s2} \alpha^6 n^4}{\sigma 3 \alpha^4 n^6}$$

$$c^2 = \frac{1}{\mu_0 \varepsilon_0}$$

$$= \frac{3c^2 4m_1 m_2 R_{s1} R_{s2} \alpha^6 n^4}{\sigma 3 \alpha^4 n^6} = \frac{3\sigma 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^4}{4\pi \mu_0 \varepsilon_0 3 \alpha^4 n^6} = \frac{3c^2 4m_{pl} m_2 s_1 R_{s1} R_{s2} \alpha^6 n^4}{\sigma 3 \alpha^4 n^6}$$

$$= \frac{3\sigma c^2 R_{s1}^2 R_{s2}^2 \alpha^6 n^4}{3 \alpha^4 n^6} = \frac{3m_2 2c^2 R_{s1}^2 R_{s2} \alpha^6 n^4}{3 \alpha^4 n^6} = \frac{3\sigma c^2 R_{s1}^2 R_{s2}^2 \alpha^6 n^4}{3 \alpha^4 n^6}$$

Planck's constant, $h = 2\pi\sigma cl^2$

$$= \frac{3\sigma 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^4}{4\pi \mu_0 \varepsilon_0 3 \alpha^4 n^6} = \frac{R_{s2}^2 3c^2 4m_1 m_1 \alpha^6 n^4}{\sigma 3 \alpha^4 n^6} = \frac{\alpha 2\pi\sigma cl^2 2s_1^2 4R_{s2}^2 c \alpha^5 n^4}{4\pi \alpha^4 n^6}$$

$$= \frac{3\sigma c^2 R_{s1}^2 R_{s2}^2 \alpha^6 n^4}{3 \alpha^4 n^6} = \frac{3\sigma 4\pi R_{s1}^2 R_{s2}^2 \alpha^6 n^4}{4\pi \mu_0 \varepsilon_0 3 \alpha^4 n^6} = \frac{3m_2 2c^2 R_{s1}^2 R_{s2} \alpha^6 n^4}{3 \alpha^4 n^6}$$

$$= \frac{3m_2 2c^2 R_{s1}^2 R_{s2} \alpha^6 n^4}{3 \alpha^4 n^6} = \frac{3m_2 2c^2 R_{s1}^2 R_{s2} \alpha^6 n^4}{3 \alpha^4 n^6} = \frac{3m_2 2c^2 R_{s1}^2 R_{s2} \alpha^6 n^4}{3 \alpha^4 n^6}$$
$$ = \frac{\sigma c l^2 2 s_1^2 4 R_{s2}^2 c \alpha^6 n^4}{2 \alpha^4 n^6} = \frac{3 m_{pl} 2 c^2 4 s_1^2 l R_{s2}^2 \alpha^6 n^4}{3 \alpha^4 n^6} = \frac{4 \pi \sigma c l^2 c^3 \sigma 4 s_1^2 R_{s2} 2 s_2 3 l \alpha^6 n^4}{4 \pi c^2 \sigma \alpha^4 n^6 3} $$

$$ \text{Gravitational constant, } G = \frac{c^2}{\sigma} $$

$$ R_{G0} = s_1 l n^2 $$

$$ 4. \frac{\sigma c^2 R_{s1} R_{s2}^2 \alpha^2}{n^2 4 R_{s1} R_{s2}} = \frac{\alpha^2 \sigma c^2 s_2 n^2 s_1 l}{2 n^4 s_1} = \frac{c^2 m_1 m_2 \alpha^2}{\sigma 2 s_1 l n^2} $$

$$ \mu_0 = \frac{1}{\varepsilon_0 c^2}, \quad R_E = \frac{n^2 l}{s_2 \alpha} $$

$$ = \frac{\alpha^2 c^2 m_1 m_2}{\sigma 2 s_1 l n^2} = \frac{\sigma 4 \pi \varepsilon_0 c^2 \alpha l^2 s_2 \alpha}{4 \pi \varepsilon_0 2 n^2 l} = \frac{\alpha^2 c^2 m_{pl} m_2 s_1 s_1 s_2}{\sigma 2 s_1 s_1 s_2 n^2 l} $$

$$ = \frac{\alpha^2 \sigma c^2 s_2 l}{n^2 2} = \frac{\alpha^2 m_2 c^2 3 s_1^4 s_2^2 2 l^2 4}{2 n^2 4 l^2 3 s_1^4 s_2^2 2} = \frac{\sigma c^2 s_2^2 \alpha^3 n^2 l}{2 n^4 s_2 \alpha} $$

$$ = \frac{s_2 4 \sigma c^2 \alpha 4 \pi \varepsilon_0 l^2 s_1 \alpha}{s_1 4 \pi \varepsilon_0 8 l n^2} = \frac{s_2 4 \alpha c^2 m_1 m_1 \alpha}{s_1 \sigma 8 s_1 l n^2} = \frac{s_2 4 \alpha h s_1 c \alpha}{s_1 16 \pi n^2 l} $$

$$ = \frac{s_2 4 \alpha \sigma c^2 \alpha^2 s_1^2 8 l}{s_1 n^2 64 s_1 \alpha} = \frac{s_2 4 \alpha \sigma c^2 4 \pi \varepsilon_0 l^2 s_1 \alpha}{s_1 4 \pi \varepsilon_0 8 l n^2} = \frac{\alpha^2 m_2 c^2 R_{s1} n^2}{n^4 2 R_{s1}} $$

$$ = \frac{\alpha^2 m_2 c^2}{2 n^2} = \frac{\alpha^2 m_2 c^2 3 n^4 2 s_1^2 l^2}{3 s_1^2 l^2 2 n^6 2} = \frac{\alpha^2 m_2 c^2 R_{s1}^2 n^4}{2 R_{s1}^2 n^6} $$

$$ \text{Reduced Planck constant, } \hbar = \sigma c l^2 $$

$$ = \frac{\alpha^2 \hbar s_2 c}{2 n^2 l} = \frac{3 s_2 \alpha^2 m_{pl} c^2 s_2 n^4}{s_2 3 n^6 2} = \frac{\alpha^2 \hbar c^3 4 \pi \sigma s_2 3 s_2}{4 \pi c^2 2 \sigma l n^2 s_2 3} $$

$$ 5. \frac{\sigma c^2 a_{20}^2 s_2 l}{2} = \frac{\alpha^2 \sigma c^2 s_2 n^2 s_1 l}{2 n^4 s_1} = \frac{\alpha^2 G m_1 m_2}{2 s_1 l n^2} $$

$$ Q_{pl} = \sqrt{4 \pi \varepsilon_0 \sigma c^2 l^2}, \quad Q = \sqrt{4 \pi \varepsilon_0 \sigma c^2 l^2} $$

$$ = \frac{\alpha^2 G m_1 m_2}{2 s_1 l n^2} = \frac{Q^2 s_2 \alpha}{4 \pi \varepsilon_0 2 n^2 l} = \frac{\alpha^2 G m_u m_2 s_1 s_2}{n^2 2 l} $$
$$ \Omega = \frac{2}{3} $$

$$ = \frac{\alpha^2 \sigma c^2 s_2 l}{2n^2} = \frac{\alpha^2 m_2 c^2 3 s_1^4 s_2^2 \Omega l^2 4}{\Omega n^2 4 l^2 6 s_1^4 s_2^2} = \frac{\sigma c^2 s_2^2 \alpha^3 n^2 l}{2 n^4 s_2 \alpha} $$

$$ \text{Planck time, } t_{pl} = \frac{l}{c} $$

$$ = \frac{s_2 4 Q^2 s_1 \alpha}{s_1 4 \pi \epsilon_0 n^2 8 l} = \frac{s_2 4 \alpha G m_1 m_1 \alpha}{s_1 8 s_1 l n^2} = \frac{s_2 4 \alpha h s_1 \alpha}{s_1 16 \pi n^2 t_{pl}} $$

$$ = \frac{s_2 4 \alpha \sigma c^2 \alpha^2 s_1^2 8 l}{s_1 n^2 64 s_1 \alpha} = \frac{s_2 4 \alpha Q_{pl}^2 s_1 \alpha}{s_1 n^2 4 \pi \epsilon_0 8 l} = \frac{\alpha^2 m_2 c^2 n^2 s_1 l}{n^4 2 s_1 l} $$

$$ = \frac{\alpha^2 m_2 c^2}{2n^2} = \frac{\alpha^2 m_2 c^2 3 \Omega n^4 s_1^2 l^2}{s_1^2 4 l^2 n^6} = \frac{\alpha^2 m_2 c^2 R_{s_1}^2 n^4}{2 R_{s_1}^2 n^6} $$

$$ \text{Planck temperature, } T_{pl} $$

$$ \text{Boltzmann constant, } k_B $$

$$ = \frac{\alpha^2 \hbar s_2}{2 n^2 t_{pl}} = \frac{3 \alpha^2 T_{pl} k_B s_2}{2 n^2 3} = \frac{\alpha^2 \hbar c^3 4 \pi s_2 k_B}{8 \pi G m_{pl} n^2 k_B} $$

$$ R_{Go} = \frac{s_1 l}{a_o^2} $$

$$ F_{Go} = \frac{\sigma c^2 s_2 a_o^4}{s_1} $$

$$ 6. \frac{\sigma c^2 a_{2o}^2 s_2 l}{2} = \frac{\alpha^2 \sigma c^2 s_2 a_o^4 s_1 l}{2 s_1 a_o^2} = \frac{\alpha^2 G m_1 m_2 a_o^2}{2 s_1 l} $$

$$ R_{su} = \frac{2l}{s_1^2 s_2} $$

$$ m_{ur} = m_2 a_o^2 $$

$$ = \frac{\alpha^2 G m_1 m_{ur}}{2 s_1 l} = \frac{Q^2 s_2 a_{2o}^2}{4 \pi \epsilon_0 2 l \alpha} = \frac{\alpha^2 G m_u m_{ur} s_1 s_1 s_2}{2 l} $$

$$ s_{ur} = s_2 a_o^2 $$

$$ = \frac{\alpha^2 F_{pl} s_{ur} l}{2} = \frac{\alpha^2 m_{pl} s_2 a_o^2 c^2 3 s_1^4 s_2^2 \Omega R_{su}^2}{\Omega 4 l^2 6} = \frac{F_{pl} a_{2o}^4 s_2^2 \alpha l}{2 \alpha s_2 a_{2o}^2} $$

$$ m_{pv} = s_1 \propto \sigma l = s_{pv} m_{pl} $$
$$= \frac{s_2 4 a_o^2 Q^2 s_{pv}}{s_1 4 \pi \epsilon_0 8 l} = \frac{s_2 4 \propto G m_1 m_{pv} a_o^2}{s_1 8 s_1 l} = \frac{s_2 4 \propto h s_1 \propto a_o^2}{s_1 16 \pi \propto t_{pl}}$$

$$= \frac{s_2 4 a_o^2 \sigma c^2 \propto s_1^2 8 l}{s_1 64 s_1 \propto} = \frac{s_2 4 \propto a_o^2 Q_{pl}^2 s_1 \propto}{s_1 4 \pi \epsilon_0 8 l} = \frac{\propto 2 m_2 c^2 a_o^4 s_1 l}{2 s_1 l a_o^2}$$

$$= \frac{\propto 2 m_2 c^2 a_o^2}{2} = \frac{\propto 2 m_2 c^2 3 a_o^6 \Omega s_1^2 4 l^2}{2 s_1^2 4 l^2 \Omega 3 a_o^4} = \frac{\propto 2 m_2 c^2 a_o^6 R_{s_1}^2}{2 R_{s_1}^2 a_o^4}$$

$$= \frac{\propto 2 \hbar s_2 a_o^2}{2 t_{pl}} = \frac{\propto 2 T_{pl} s_2 a_o^2 k_B}{2} = \frac{\propto 2 \hbar c^3 a_o^2 s_2 k_B}{2 G m_{pl} k_B}$$

$$7. E_1 a_o^2 = \frac{\propto 2 \sigma c^2 s_2 a_o^4 R_{Go}}{2 s_1} = \frac{\propto 2 G m_1 m_2}{2 R_{Go}}$$

$$= \frac{\propto 2 G m_1 m_{ur}}{2 s_1 l} = \frac{Q^2 s_2 \propto a_o^2}{4 \pi \epsilon_0 2 l} = \frac{\propto 2 G m_u m_{ur}}{R_{su}}$$

$$= \frac{\propto 2 F_{pl} s_{ur} 2 l}{4} = \frac{\propto 2 m_{ur} c^2 \Lambda R_{su}^2}{\Omega 6} = \frac{F_{E2} R_{CE}}{2 a_{2o}^2}$$

Cosmological constant, $\Lambda = \frac{3\Omega}{R_{su}^2}$

Classical radius, $R_{cE} = \frac{\propto l}{s_2}$

$$R_E = \frac{l}{s_2 \propto a_o^2} = \frac{R_{cE}}{a_{2o}^2}$$

$$\frac{s_2 4 \propto a_o^2 Q^2 s_1}{s_1 4 \pi \epsilon_0 8 l} = \frac{s_2 4 \propto 2 G m_1 m_1 a_o^2}{s_1 8 s_1 l} = \frac{s_2 4 \propto 2 h s_1 a_o^2}{s_1 16 \pi t_{pl}}$$

$$= \frac{s_2 4 a_o^2 \propto \sigma c^2 s_1^2 8 l}{s_1 64 s_1} = \frac{s_2 4 \propto a_o^2 Q_{pl}^2 s_1}{s_1 4 \pi \epsilon_0 8 l} = \frac{\propto 2 m_2 c^2 a_o^4 R_{Go}}{2 s_1 l}$$

Escape radius, $R_{Ges} = 2R_{Go}$
$$H_2 = \frac{ca_o^3}{R_{s1}}, \quad \Lambda_2 = \frac{3a_o^6\Omega}{R_{s1}^2}$$

$$= \frac{\alpha^2 m_2 c^2 a_o^2}{2} = \frac{\alpha^2 m_2 c^2 \Lambda_2 R_{ges}^2}{2\Omega 3} = \frac{\alpha^2 m_2 H_2^2 R_{ges}^2}{2}$$

$$= \frac{\alpha^2 \hbar s_2 a_o^2}{2t_{pl}} = \frac{\alpha^2 T_{pl} s_{ur} k_B}{2} = \frac{\alpha^2 \hbar c^3 s_2 a_o^2 k_B}{2Gm_{pl}k_B}$$

$$8. E_1 a_o^2 = \frac{\alpha^2 F_{Go} R_{Go}}{2} = \frac{\alpha^2 Gm_1 m_2}{2 R_{Go}}$$

$$= \frac{\alpha^2 Gm_1 m_{ur}}{2s_1 l} = \frac{Q^2}{4\pi\epsilon_0 2R_E} = \frac{\alpha^2 Gm_u m_{ur}}{R_{su}}$$

$$= \frac{\alpha^2 F_{pl} R_{sur}}{4} = \frac{\alpha^2 m_{ur} c^2 \Lambda R_{su}^2}{\Omega 6} = \frac{F_{2E} R_E}{2}$$

$$\frac{s_2 4 \propto a_o^2 Q^2}{s_1 4\pi\epsilon_0 R} = \frac{s_2 4 \propto Gm_1 m_1}{s_1 R} = \frac{s_2 4 \propto h\nu}{s_1}$$

Gravitational orbital acceleration, $g_o = \frac{c^2 a_o^4}{s_1 l}$

$$= \frac{s_2 4 \propto a_o^2 F_c R}{s_1} = \frac{s_2 4 \propto a_o^2 Q_{pl}^2}{s_1 4\pi\epsilon_0 R} = \frac{\alpha^2 m_2 g_o R_{Go}}{2}$$

$$= \frac{\alpha^2 m_2 a_o^2 c^2}{2} = \alpha^2 m_2 c^2 \Lambda_2 R_{Go}^2 = \frac{\alpha^2 m_2 H_2^2 R_{ges}^2}{2}$$

$$= \frac{\alpha^2 \hbar}{t_{ur}} = \alpha^2 T_{ur} k_B = \frac{\alpha^2 \hbar c^3 k_B}{2Gm_u k_B}$$

Temperature $T_{ur} = \frac{T_{pl} s_{ur}}{2}$

For hydrogen-like macroscopic objects,
speed coefficient, $a_0$

spin speed coefficient, $a_s$

macroscopic massive mass coefficient, $k_1$

orbital macroscopic mass coefficient, $k_2$

$$9 \cdot \frac{k_1 k_2 a_o^6 a_s^6 \sigma c^2 l}{k_1 k_2 a_s^4 a_o^4 2k_2} = \frac{h a_s^2 a_o^2}{2\pi t_{pl} k_2 2} = \frac{m_{pl} a_s^2 a_o^6 4\pi^2 k_1^2 l^2}{k_2 4\pi^2 k_1^2 t_{pl}^2 2 a_o^4}$$

$$= \frac{\sigma c^2 a_s^2 a_o^4 k_1 l}{2k_2 k_1 a_o^2} = \frac{G m_1 m_{pl} a_s^2 a_o^2}{k_2 2 k_1 l} = \frac{\sigma c^2 a_o^2 a_s^2 2l}{4k_2}$$

Planck charge, $Q_{pl}$

$m_1$ schwarzschild radius, $R_{s1} = 2k_1 l$

$$= \frac{G m_1 m_{pl} a_s^2 a_o^2}{k_2 2 k_1 l} = \frac{m_{pl} a_o^2 a_s^2 c^2 R_{s1}^2}{k_2 2 R_{s1}^2} = \frac{Q_{pl}^2 a_o^2 a_s^2}{4\pi\epsilon_0 2 l k_2}$$

$$= \frac{\propto h a_s^2 a_o^2}{2\pi t_{pl} k_2 \propto 2} = \frac{m_{pl} a_o^2 a_s^2 a_o^4 R_{s1}^2}{2 k_2 4 k_1^2 t_{pl}^2 a_o^4} = \frac{G m_1 a_s^2 a_o^2 m_{pl}}{k_2 2 k_1 l}$$

$$= \frac{m_{pl} a_s^2 c^2 a_o^4 k_1 l}{k_2 2 k_1 l a_o^2} = \frac{m_{pl} a_s^2 c^2 a_o^6 R_{s1}^2}{k_2 2 R_{s1}^2 a_o^4} = \frac{m_{pl} a_s^2 c^2 3 a_o^6 \Omega k_1^2 4 l^2}{k_2 2 k_1^2 4 l^2 \Omega 3 a_o^4}$$

$$= \frac{m_{pl} a_o^2 a_s^2 c^2}{k_2 2} = \frac{m_{pl} a_s^2 c^2 a_o^6 R_{s1}^2}{k_2 2 R_{s1}^2 a_o^4} = \frac{m_{pl} a_s^2 c^2 3 a_o^6 \Omega k_1^2 4 l^2}{k_2 2 k_1^2 4 l^2 \Omega 3 a_o^4}$$

$$= \frac{m_{pl} a_s^2 c^2 a_o^4 k_1 l}{k_2 2 k_1 l a_o^2} = \frac{T_{pl} a_s^2 a_o^2 k_B}{k_2 2} = \frac{m_{pl} a_s^2 a_o^2 c^2 R_{s1}^2}{k_2 2 R_{s1}^2}$$

$$= \frac{4\pi\hbar c^3 a_o^2 a_s^2 k_B}{4\pi G k_2 m_{pl} 2 k_B} = \frac{T_{pl} a_s^2 a_o^2 k_B}{k_2 2} = \frac{G m_1 m_{pl} a_s^2 a_o^2}{k_2 2 k_1 l}$$
$$= \frac{\propto h a_s^2 a_o^2}{2\pi t_{pl} k_2 \propto 2} = \frac{\sigma c^2 a_o^2 a_s^2 2l}{4k_2} = \frac{m_{pl} a_s^2 a_o^2 c^2 3\Omega k_1^2 4l^2}{k_2^2 2k_1^2 4l^2 \Omega 3}$$

$$= \frac{Q_{pl}^2 a_o^2 a_s^2}{4\pi\epsilon_0 2lk_2} = \frac{Gm_1 m_{pl} a_s^2 a_o^2 a_o^3}{a_o^3 k_2 2k_1 l}$$

$$10 \cdot \frac{a_o k_1 k_2 \sigma c^2 a_o^2 a_s^2 l}{a_o k_1 k_2 2k_2 \propto} = \frac{h a_s^2 a_o^2}{2\pi t_{pl} k_2 \propto 2} = \frac{m_{pl} a_s^2 a_o^6 4\pi^2 k_1^2 l^2}{\propto k_2 4\pi^2 k_1^2 t_{pl}^2 2a_o^4}$$

$$= \frac{\sigma c^2 a_s^2 a_o^4 k_1 l}{2 \propto k_2 k_1 a_o^2} = \frac{Gm_1 m_{pl} a_s^2 a_o^2}{\propto k_2 2k_1 l} = \frac{\sigma c^2 a_o^2 a_s^2 2l}{4 \propto k_2}$$

$$= \frac{Gm_1 m_{pl} a_s^2 a_o^2}{\propto k_2 2k_1 l} = \frac{m_{pl} a_o^2 a_s^2 c^2 R_{s1}^2}{\propto k_2 2R_{s1}^2} = \frac{a_o Q_{pl}^2 a_s^2 a_o}{4\pi\epsilon_0 2k_2 \propto l}$$

$$= \frac{a_o \hbar a_s^2 a_o}{t_{pl} k_2 \propto 2} = \frac{a_o m_{pl} a_o a_s^2 a_o^4 k_1^2 l^2}{2 \propto k_2 k_1^2 t_{pl}^2 a_o^4} = \frac{Gm_1 a_s^2 a_o m_{pl} a_o^2}{a_o \propto k_2 2k_1 l}$$

$$= \frac{a_o m_{pl} a_s^2 a_o c^2 a_o^2 2k_1 l}{\propto k_2 4k_1 la_o^2} = \frac{a_o m_{pl} a_s^2 a_o c^2 a_o^4 R_{s1}^2}{\propto k_2 2R_{s1}^2 a_o^4} = \frac{a_o m_{pl} a_s^2 a_o c^2 3a_o^4 \Omega k_1^2 4l^2}{\propto k_2 2k_1^2 4l^2 \Omega 3a_o^4}$$

$$= \frac{a_o m_{pl} a_o a_s^2 c^2}{\propto k_2 2} = \frac{m_{pl} a_s^2 c^2 a_o^6 R_{s1}^2}{\propto k_2 2R_{s1}^2 a_o^4} = \frac{m_{pl} a_s^2 c^2 3a_o^6 \Omega k_1^2 4l^2}{\propto k_2 2k_1^2 4l^2 \Omega 3a_o^4}$$

$$= \frac{m_{pl} a_s^2 c^2 a_o^4 k_1 l}{\propto k_2 2k_1 la_o^2} = \frac{T_{pl} a_s^2 a_o^2 k_B}{\propto k_2 2} = \frac{a_o^3 m_{pl} a_s^2 c^2 R_{s1}^2}{k_2 \propto a_o 2R_{s1}^2}$$
$$= \frac{a_o^3 4\pi\hbar c^3 a_s^2 k_B}{8\pi G k_2 a_o \propto m_{pl} k_B} = \frac{a_o^3 T_{pl} a_s^2 k_B}{\propto k_2 a_o 2} = \frac{a_o^3 G m_1 m_{pl} a_s^2}{\propto a_o k_2 2 k_1 l}$$

$$= \frac{a_o^3 \hbar a_s^2}{2 t_{pl} k_2 a_o} \propto \frac{a_o^3 \sigma c^2 a_s^2 2l}{4 k_2 \propto a_o} = \frac{m_{pl} a_s^2 a_o^2 c^2 3 \Omega k_1^2 4 l^2}{\propto k_2 2 k_1^2 4 l^2 \Omega 3}$$

$$= \frac{a_o^3 Q_{pl}^2 a_s^2}{4\pi\epsilon_0 2k_2 \propto a_o l} = \frac{G m_1 m_{pl} a_s^2 a_o^2 a_o^3}{a_o^3 \propto k_2 2k_1 l}$$

$$\text{orbital particle mass coefficient, } k_v = \frac{a_s^2}{\propto k_2}, \quad m_v = k_v m_{pl}$$

$$R_{Go} = \frac{k_1 l}{a_o^2}, \quad t_{Go} = \frac{2\pi k_1 t_{pl}}{a_o^3}$$

$$11. \frac{k_1 k_2 \sigma c^2 a_o^2 a_s^2 l}{a_o n_q 2 k_2 \propto} = \frac{h k_v a_o^2}{2 \pi t_{pl} 2} = \frac{m_v 4 \pi^2 R_{Go}^2}{t_{Go}^2 2}$$

$$= \frac{\sigma c^2 k_v a_o^4 R_{Go}}{2 k_1} = \frac{G m_1 m_v a_o^2}{2 k_1 l} = \frac{F_{pl} a_o^2 a_s^2 2l}{4 \propto k_2}$$

$$k_r = \frac{a_s^2 a_o}{\propto k_2}, \quad k_g = \frac{a_s^2 a_o^2}{\propto k_2}$$

$$= \frac{G m_1 m_g}{2 k_1 l} = \frac{m_g c^2 R_{s1}^2}{2 R_{s1}^2} = \frac{a_o Q_{pl}^2 k_r}{4 \pi \epsilon_0 2 l}$$

$$t_r = \frac{k_1 t_{pl}}{a_o^2}$$

$$= \frac{a_o \hbar a_s^2 a_o}{2 t_{pl} k_2 \propto} = \frac{a_o m_r R_{Go}^2}{2 t_r^2} = \frac{G m_1 m_r a_o^2}{a_o 2 k_1 l}$$

$$H_r = \frac{c a_o^2}{2 k_1 l}, \quad \Lambda_r = \frac{3 a_o^4 \Omega}{k_1^2 4 l^2}$$

$$= \frac{a_o m_r c^2 a_o^2 R_{Ges}}{4 k_1 l} = \frac{a_o m_r H_r^2 R_{Ges}^2}{2} = \frac{a_o m_r c^2 \Lambda_r R_{Ges}^2}{2 \Omega 3}$$

$$R_{Ges} = \frac{R_{s1}}{a_o^2}$$

$$H_v = \frac{c a_o^3}{2 k_1 l}, \quad \Lambda_v = \frac{3 a_o^6 \Omega}{k_1^2 4 l^2}$$
$$ = \frac{a_o m_r c^2}{2} = \frac{m_v H_v^2 R_{Ges}^2}{2} = \frac{m_v c^2 \Lambda_v R_{Ges}^2}{\Omega 6} $$

$$ = \frac{m_v c^2 a_o^4 R_{G_o}}{2 k_1 l} = \frac{T_{pl} k_v a_o^2 k_B}{2} = \frac{a_o^3 m_h c^2 R_{s1}^2}{2 R_{s1}^2} $$

$$ = \frac{a_o^3 \hbar c^3 k_h k_B}{2 G m_{pl} k_B} = \frac{a_o^3 T_{pl} k_h k_B}{2} = \frac{a_o^3 G m_1 m_h}{2 k_1 l} $$

$$ = \frac{a_o^3 \hbar a_s^2}{2 t_{pl} k_2 a_o} \propto \frac{a_o^3 F_{pl} a_s^2 2l}{4 \propto k_2 a_o} = \frac{m_g c^2 \Lambda_g R_{s1}^2}{\Omega 6} $$

$$ = \frac{a_o^3 Q_{pl}^2 k_h}{4 \pi \epsilon_0 2l} = \frac{G m_M m_g a_o^2}{a_o^2 R_{sM}} $$

$$ 12. \frac{k_1 k_2 E_v}{a_o n_q} = \frac{h v_v}{2} = \frac{m_v 4 \pi^2 R_{G_o}^2}{t_{G_o}^2 2} $$

$$ = \frac{F_{Gvo} R_{G_o}}{2} = \frac{G m_1 m_v}{2 R_{G_o}} = \frac{F_{pl} R_{sg}}{4} $$

$$ = \frac{G m_1 m_g}{R_{s1}} = \frac{m_g H_0^2 R_{s1}^2}{2} = \frac{a_o Q_{pl}^2}{4 \pi \epsilon_0 2 R_{Cr}} $$

$$ = \frac{a_o \hbar}{2 t_r} = \frac{a_o m_r R_{G_o}^2}{2 t_r^2} = \frac{a_o G m_1 m_r}{2 a_o^2 R_{G_o}} $$

$$ = a_o m_r a_{res} R_{Ges} = \frac{a_o m_r H_r^2 R_{Ges}^2}{2} = \frac{a_o m_r c^2 \Lambda_r R_{Ges}^2}{\Omega 6} $$

$$ k_h = \frac{a_s^2}{\propto k_2 a_o} $$

$$ \Lambda_g = \frac{3 \Omega}{R_{s1}^2} $$

$$ m_M \text{ schwarzschild radius, } R_{sM} = \frac{2 k_1 l}{a_o^3} $$

$$ m_r \text{ acceleration, } a_{es} = \frac{c^2 a_o^2}{4 k_1 l} $$
$$ = \frac{a_o m_r c^2}{2} = \frac{m_v H_v^2 R_{Ges}^2}{2} = m_v c^2 \Lambda_v R_{Go}^2 $$

$$ = \frac{m_v g R_{Go}}{2} = T_{ges} k_B = \frac{a_o^3 m_h H_0^2 R_{s1}^2}{2} $$

$$ = \frac{a_o^3 \hbar c^3 k_B}{2Gm_1 k_B} = a_o^3 T_{hes} k_B = \frac{a_o^3 Gm_1 m_h}{R_{s1}} $$

$$ = \frac{a_o^3 \hbar}{t_h} = \frac{a_o^3 F_{pl} R_{sh}}{4} = m_g c^2 \Lambda_g R_{Go}^2 $$

$$ = \frac{a_o^3 Q_{pl}^2}{4\pi\varepsilon_0 2R_{Ch}} = \frac{Gm_M m_g}{a_o^2 R_{Mes}} $$

$m_v$ acceleration, $g = \frac{c^2 a_o^4}{4k_1 l}$

$$ R_{Mes} = \frac{R_{sM}}{a_o^2} $$

In equivalences, the Schwarzschild radius[1] formula emerges as

$$ \textit{Massive mass Schwarzschild radius, } R_{s1} = 2k_1 l, R_{s1} = 2s_1 l $$

$$ V = \frac{c}{n} = a_o c $$

$$ \frac{Gm_1 m_2 a_o^2}{R_{s1}} = \frac{m_2 c^2 a_o^2}{2} $$

$$ \frac{2Gm_1}{R_{s1}} = c^2 $$

The equations reveal the ratio, $\frac{c^2}{\sigma}$ as the Gravitational constant.

$$ \frac{c^2 m_1 m_2 a_o^2}{\sigma 2 s_1 l} = \frac{m_2 c^2 a_o^2}{2} $$

Macroscopic massive mass coefficient, $k_1$
$$ \text{Planck mass, } m_{pl} = \sigma l $$

$$ \text{Massive mass, } m_1 = k_1 \sigma l = k_1 m_{pl} = \frac{\sigma R_{s1}}{2} $$

$$ \frac{2Gm_1}{R_{s1}} = c^2 $$

$$ \frac{2G\sigma k_1 l}{2k_1 l} = c^2 $$

$$ G = \frac{c^2}{\sigma} $$

The gravitational constant, G is constant regardless of radius and is related to the mass constant σ and the square of the speed of light c.

$$ \sigma \approx 1.35 \times 10^{27} \text{kg/m} $$

The electric charge is obtained as

$$ \text{Elementary charge, } Q = \sqrt{\sigma c^2 \propto 4\pi\varepsilon_0 l^2} $$

$$ \text{Planck charge, } Q_{pl} = \sqrt{\sigma c^2 4\pi\varepsilon_0 l^2} $$

In equivalences, the Planck's constant [2] emerges defined by the mass constant.

$$ h = 2\pi\sigma cl^2 $$

Where massive mass $m_1$ is the proton $m_p$ and orbital mass $m_2$ is the electron $m_e$, the radiuses are expressed as

$$ \text{Electron mass coefficient, } s_e \approx 4.19 \times 10^{-23} $$

$$ m_e \text{ orbital speed coefficient, } a_{2o} = \propto a_o $$

$$ \text{Classical electron radius, } R_{cE} = \frac{\alpha l}{s_e} $$

$$ R_{cE} \approx 2.8 \times 10^{-15} m $$

$$ R_E = \frac{l}{s_e \propto a_o^2} $$

The Bohr radius [3] is expressed as
$$ \frac{l}{s_e \alpha} \approx 5.3 \times 10^{-11} m $$

It is possible to formulate the gravitational radius, both in the macroscopic world and the microscopic world, by the coefficient of unit planck length or the Schwarzschild radius expansion coefficient. In the equations, the orbital gravitational radius is as

$$ \begin{aligned} E_1 a_o^2 &= \frac{\alpha^2 c^2 m_1 m_2 a_o^2}{\sigma 2 s_1 l} = \frac{\alpha^2 m_2 c^2 a_o^2}{2} \\ \frac{E_1 a_o^2}{\alpha^2} &= \frac{G m_1 m_2}{2 R_{Go}} = \frac{m_2 V^2}{2} \end{aligned} $$

The value of the gravitational orbital radius in the Sun-earth system is

$$ \begin{aligned} \text{Solar mass coefficient, } k_S &\approx 0.91 \times 10^{38} \\ \text{Earth orbital speed coefficient, } a_E &\approx 9.9 \times 10^{-5} \end{aligned} $$

$$ \begin{aligned} R_{Go} &= \frac{k_1 l}{a_o^2} = \frac{k_S l}{a_E^2} \\ R_{Go} &\approx 1.5 \times 10^{11} m \end{aligned} $$

The Schwarzschild radius expansion coefficient of the system is

$$ \frac{1}{a_E^2} $$

In the equivalences, the gravitational radius is quantized by Planck's constant and is defined as

$$ \begin{aligned} \text{Earth mass coefficient, } k_E &\approx 2.74 \times 10^{32} \\ \text{Mass particle coefficient, } k_v &= \frac{a_s^2}{k_2 \alpha} = \frac{a_s^2}{k_E \alpha} \end{aligned} $$

$$ \begin{aligned} \frac{k_S k_E E_v}{a_E n_q} &= \frac{h v_v}{2} = \frac{m_v 4 \pi^2 R_{Go}^2}{2 t_{Go}^2} = \frac{G m_S m_v a_E^2}{2 k_S l} \\ \frac{h}{2 t_{Go}} &= \frac{m_v 4 \pi^2 R_{Go}^2}{2 t_{Go}^2} \\ \frac{h k_v a_o^2}{2 \pi t_{pl}^2} &= \frac{m_v 4 \pi^2 R_{Go}^2}{2 t_{Go}^2} \end{aligned} $$
$$ \frac{h}{2} = \frac{m_v 4\pi^2 R_{Go}^2}{2t_{Go}} $$

$$ h = m_v V 2\pi R_{Go} $$

$$ 2\pi R_{Go} = \frac{h}{m_v V} $$

The orbit of the Earth around the Sun is equal to the de Broglie wavelength [4] of the particle $m_v$.

In the equations, at the $R = \frac{8l}{s_1}$ boundary, the forces in the atomic nucleus are as

$$ \text{Proton mass coefficient, } s_p \approx 0.77 \times 10^{-19} $$

$$ \text{Proton mass, } m_p = s_p \sigma l = s_p m_{pl} $$

$$ m_{pv} = m_p \propto $$

$$ m_{pv} \approx 1.2 \times 10^{-29} \text{kg} $$

$$ E_1 = \frac{\sigma c^2 \propto s_2 l}{2} $$

$$ E_1 \approx 13.6 \text{ eV} $$

$$ E_1 a_o^2 = \frac{s_2 4 Q^2 s_v a_o^2}{s_1 4\pi\epsilon_0 8l} = \frac{s_2 4 \propto G m_1 m_1 \propto a_o^2}{s_1 8s_1 l} = \frac{s_2 4 \propto a_o^2 Q_{pl}^2 s_1 \propto}{s_1 4\pi\epsilon_0 8l} $$

$$ \frac{\sigma c^2 \propto s_{pl} a_p^2}{2} = \frac{4Q^2 s_p \propto a_p^2}{4\pi\epsilon_0 8l} = \frac{4 \propto G m_p m_p \propto a_p^2}{8s_p l} = \frac{4 \propto a_p^2 Q_{pl}^2 s_p \propto}{4\pi\epsilon_0 8l} $$

System unit charge is the Planck charge. At the gravitational radius equal to the Planck charge radius, the mass form is observed. In this case, the gravitational force radius and the Compton force radius of the system radial particle are equal.

If the gravitational radius, reduced Compton wavelength and electric radius are equalized, the following result is obtained.

$$ \frac{\sigma c^2 s_p a_o^2 l}{8} = \frac{a_o^2 Q^2 s_p}{\propto 4\pi\epsilon_0 8l} = \frac{G m_p m_p a_o^2}{8s_p l} = \frac{a_o^2 Q_{pl}^2 s_p}{4\pi\epsilon_0 8l} $$

The orbital velocity coefficient $a_p$ is given by

$$ R_{Go} = R_C $$

$$ \frac{s_p l}{a_o^2} = \frac{l}{s_p} $$
$$a_o^2 = s_p^2$$

$$a_p^2 \approx 0.59 \times 10^{-38}$$

$$a_p \approx 0.77 \times 10^{-19}$$

The Planck mass coefficient of the proton is equal to the orbital velocity coefficient.

At the $R = \frac{8l}{s_p}$ boundary, the forces are compared as

$$F_{Go} = \frac{Gm_p m_p a_o^4}{64s_p^2 l^2} = \frac{a_p^2 Q^2 s_p^2}{\propto 4\pi\epsilon_0 64l^2} = \frac{a_p^2 Q_{pl}^2 s_p^2}{4\pi\epsilon_0 64l^2}$$

$$F_{Go} = \frac{a_p^2 F_E}{\propto} = a_p^2 F_C$$

$$F_G = \frac{\sigma c^2 s_p^2 a_p^2}{64} = \frac{\sigma c^2 a_p^4}{64} = \frac{\sigma c^2 s_r^2 a_p^2}{64} = \frac{F_{pl} a_p^6}{64s_p^2}$$

$$F_E = \frac{\sigma c^2 \propto s_p^2}{64} = \frac{\sigma c^2 \propto a_p^2}{64} = \frac{\sigma c^2 \propto s_r^2}{64} = \frac{F_{pl} \propto a_p^4}{64s_p^2}$$

$$F_C = \frac{\sigma c^2 s_r^2}{64} = \frac{\sigma c^2 a_p^2}{64} = \frac{F_{pl} a_p^4}{64s_1^2}$$

$$F_C \approx 10000N$$

$$\frac{F_{Go}}{F_E} \approx 0.8 \times 10^{-36}$$

$$\frac{F_{Go}}{F_C} = a_p^2$$

$$\frac{F_{Go}}{F_C} \approx 0.6 \times 10^{-38}$$

$$\frac{F_E}{F_C} \approx \frac{1}{137}$$

The ratio of Compton force to gravitational force is the expansion coefficient.

Compton force, $F_C$ is approximately 38 times larger than the gravitational force and approximately 137 times larger than the electric force.
In equivalences, the mass of a radial particle whose reduced Compton wavelength $R_c$ is equal to the gravitational radius is given as

$$R_{Go} = \frac{s_p l}{a_p^2}, \quad s_r = \frac{a_p^2}{s_p}$$

$$R_{Cr} = \frac{\hbar}{m_r c}$$

$$R_{cr} = \frac{\sigma c l^2}{s_r \sigma l c}$$

$$R_{cr} = \frac{l}{s_r} = \frac{s_p l}{a_p^2} = \frac{l}{s_p}, \quad m_r = \frac{a_p^2 m_{pl}}{s_p} = m_p$$

In a macroscopic hydrogen-like system such as the Sun-earth system, the radial particle of light-speed charge is as

Radial particle mass coefficient, $k_r$

$$m_r = \frac{m_{pl} a_o^2}{k_1} = \frac{m_{pl} a_E^2}{k_S}$$

$$m_r \approx 2.3 \times 10^{-54} kg$$

In subatomic and macroscopic systems, the Schwarzschild radius expansion coefficient of the gravitational radius is related to the massive mass and radial particle coefficients.

$$\frac{s_1 l}{a_o^2} = \frac{l}{s_r}$$

$$\frac{1}{a_o^2} = \frac{1}{s_1 s_r}$$

$$a_p^2 = s_p s_r, \quad a_E^2 = k_S k_r$$

In the equations, the relationship between the coefficients in the Sun-earth system is as

$$k_v = \frac{a_s^2}{k_2 \propto}$$

$$\frac{k_1 k_2 E_v}{a_o n_q} = \frac{h v_v}{2} = \frac{G m_1 m_v}{2 R_{Go}} = \frac{m_v 4 \pi^2 R_{Go}^2}{2 t_v^2}$$

$$\frac{h v_v}{2} = \frac{m_v 4 \pi^2 R_{Go}^2}{2 t_{Go}^2}$$
$$ \frac{h k_v a_o^2}{2 \pi t_{pl} 2} = \frac{m_v a_o^6 4 \pi^2 R_{Go}^2}{4 \pi^2 k_1^2 t_{pl}^2 2} $$

Earth's spin speed coefficient, $a_s$

$$ \frac{2 \pi t_{pl}}{k_v a_o^2} = \frac{2 \pi k_1 t_{pl}}{a_o^3} $$

$$ \frac{2 \pi k_E \propto t_{pl}}{a_s^2 a_E^2} = \frac{2 \pi k_S t_{pl}}{a_E^3} $$

$$ \frac{a_s^2}{k_E \propto} = \frac{a_E}{k_S} $$

$$ a_s^2 \approx 2.2 \times 10^{-12} $$

$$ a_E = k_v k_S $$

$$ a_E^2 = k_r k_S $$

The expansion coefficient associated with the sun's mass and radial particle mass coefficients is as

$$ \frac{1}{a_E^2} = \frac{1}{k_r k_S} $$

The force ratios for $m_r$ particles in the Sun - $m_r$ system are as follows

$$ E_v = \frac{\sigma c^2 a_o^2 a_s^2 l}{2 k_2 \propto} = \frac{\sigma c^2 a_o^3 l}{2 k_1} $$

$$ E_v = \frac{a_o \hbar}{2 t_r} = \frac{G m_1 m_r}{a_o 2 R_{Go}} = \frac{a_o Q_{pl}^2}{4 \pi \varepsilon_0 2 R_{Cr}} = \frac{a_o m_r c^2}{2} $$

$$ \frac{E}{a_o} = \frac{\hbar}{2 t_r} = \frac{Q_{pl}^2}{4 \pi \varepsilon_0 2 R_{Cr}} = \frac{m_r c^2}{2} $$

$$ F_C = \frac{Q_{pl}^2}{4 \pi \varepsilon_0 R_{Cr}^2} = \sigma c^2 k_r^2 = \frac{\sigma c^2 k_r a_o^2}{k_1} = \frac{F_{pl} a_o^4}{k_1^2} $$

$$ F_{Go} = \frac{\sigma c^2 a_o^6}{k_1^2} = \frac{\sigma c^2 k_r a_o^4}{k_1} = \sigma c^2 k_r^2 a_o^2 = \frac{F_{pl} a_o^6}{k_1^2} $$
$$F_{Go} = \frac{Gm_S m_r}{R_{Go}^2} = \frac{a_E^2 Q_{pl}^2}{4\pi\epsilon_0 R_{Cr}^2}$$

$$\frac{F_{Go}}{F_C} = a_E^2$$

$$\frac{F_{Go}}{F_C} \approx 9.8 \times 10^{-9}$$

In maroscopic space and subatomic space, the ratio of $F_C$ to $F_{Go}$ is equal to the expansion coefficient. The reduced Compton wavelength of a radial particle at the speed of light is equal to the gravitational radius.

The force ratios at the $R_E$ boundary in the electron orbit are as

$$R_E = \frac{\propto l}{s_2 a_{2o}^2} = \frac{l}{s_2 \propto a_o^2}$$

$$F_E = \frac{\sigma c^2 a_{2o}^4 s_2^2}{\propto} = \frac{F_{pl} a_{2o}^4 s_2^2}{\propto}$$

$$E_1 a_o^2 = \frac{\sigma c^2 \propto^2 s_2 l a_o^2}{2} = \frac{\propto^2 G m_1 m_2 a_o^2}{2 s_1 l} = \frac{Q^2 s_2 a_{2o}^2}{4 \pi \epsilon_0 2 l \propto} = \frac{a_{2o}^2 Q_{pl}^2 s_2}{4 \pi \epsilon_0 2 l}$$

The distance at which the gravitational radius is equal to the reduced Compton wavelength is the mass radius.

$$R_{Go} = \frac{s_1 l}{a_o^2}, \quad R_C = \frac{l}{s_2}$$

$$R_{Go} = R_C$$

$$\frac{s_p l}{a_e^2} = \frac{l}{s_e}$$

$$a_e^2 = s_p s_e$$

The radial particle at the speed of light is the electron.

$$m_r = \frac{m_{pl} a_o^2}{s_1}$$

$$m_r = \frac{m_{pl} s_p s_e}{s_p} = m_e$$

At the electron radius $R_{Go} = \frac{s_p l}{a_e^2}$, the proton Schwarzschild radius expansion coefficient is as
$$ \frac{1}{a_e^2} = \frac{1}{s_1 s_r} = \frac{1}{s_p s_e} $$

The force formulas of the radial particle in the expansion of the mass radius are as

$$ a_{2o} = \alpha a_o = \alpha a_e $$

$$ E_1 a_o^2 = \frac{\alpha^2 G m_1 m_2 a_o^2}{2 s_1 l} = \frac{a_{2o}^2 Q^2 s_2}{4 \pi \epsilon_0 2 l} = \frac{a_{2o}^2 Q_{pl}^2 s_2}{4 \pi \epsilon_0 2 l} = \frac{a_{2o}^2 m_2 c^2}{2} $$

$$ \frac{E_1 a_o^2}{a_{2o}^2} = \frac{\alpha^2 G m_p m_e}{a_{2o}^2 2 R_{Go}} = \frac{Q^2 s_e}{\alpha 4 \pi \epsilon_0 2 l} = \frac{Q_{pl}^2 s_e}{4 \pi \epsilon_0 2 l} = \frac{m_e c^2}{2} $$

$$ F_{Go} = \frac{G m_p m_e a_e^4}{s_p^2 l^2} = \frac{\sigma c^2 s_e a_e^4}{s_p} = \frac{\sigma c^2 a_e^6}{s_p^2} = \sigma c^2 s_r^2 a_e^2 = \frac{F_{pl} a_e^6}{s_p^2} $$

$$ F_C = \frac{Q_{pl}^2 s_e^2}{4 \pi \epsilon_0 l^2} = \frac{\sigma c^2 s_e a_e^2}{s_p} = \sigma c^2 s_e^2 = \sigma c^2 s_r^2 = \frac{F_{pl} a_e^4}{s_p^2} $$

$$ F_E = \frac{Q^2 s_e^2}{4 \pi \epsilon_0 l^2} = \frac{\sigma c^2 s_e a_e^2}{s_p} = \sigma c^2 \propto s_e^2 = \sigma c^2 s_r^2 \propto \frac{F_{pl} \propto a_e^4}{s_p^2} $$

$$ \frac{F_E}{F_C} \approx \frac{1}{137} $$

$$ \frac{F_G}{F_E} = \frac{s_p s_e}{\alpha} = \frac{a_e^2}{\alpha} $$

$$ \frac{F_G}{F_E} \approx 0.44 \times 10^{-39} $$

$$ \frac{F_G}{F_C} \approx a_e^2 $$

$$ a_e^2 \approx 3.2 \times 10^{-42} $$

The ratio of the Compton force associated with a radial particle at the speed of light to the gravitational force is the expansion coefficient.

Radial particles are as

$$ s_p = \frac{a_p^2 m_{pl}}{s_p} , \quad m_r = \frac{a_E^2 m_{pl}}{k_s} , \quad m_e = \frac{a_e^2 m_{pl}}{s_p} $$

Radial forces are as
$$ \sigma c^2 s_p^2, \sigma c^2 k_r^2, \sigma c^2 s_e^2 $$

While the particle velocity coefficient is constant in subatomic space, the escape radius must be 2 times the orbital radius. The mass radius is 4 times the orbital radius with the same velocity coefficient.

$$ \frac{E_1 a_o^2}{\alpha^2} = \frac{G m_1 m_2}{2 R_{Go}} = \frac{G m_1 m_2}{R_{Ges}} = \frac{2 G m_1 m_2}{R_{ms}} = \frac{m_2 V^2}{2} $$

$$ \frac{E_1 a_o^2}{\alpha^2} = \frac{G m_1 m_2}{R_{Ges}} = \frac{m_2 H_2^2 R_{Ges}^2}{2} $$

$$ \frac{E_1 a_o^2}{\alpha^2} = \frac{G m_1 m_2 a_o^2}{R_{s1}} = \frac{m_2 H_2^2 R_{s1}^2}{2 a_o^4} $$

At the escape radius $R_{Ges} = \frac{R_{s1}}{a_o^2}$, $a_o$ is the escape velocity coefficient; at the orbital radius $R_{Go} = \frac{s_1 l}{a_o^2}$, $a_o$ is the orbital speed coefficient.

The density, $\rho$ in the system with escape velocity coefficient $a_o$ and escape radius $R_{Ges} = \frac{R_{s1}}{a_o^2}$ is as

$$ \rho = \frac{3 \sigma a_o^6}{4 \pi 8 s_1^2 l^2} $$

For the proton, the density at the mass radius $R_{ms} = \frac{4s_p l}{a_p^2}$ is as

$$ \frac{1}{a_p^2} \approx \frac{1}{0.59 \times 10^{-38}} $$ $$ R_{ms} \approx 0.84 \times 10^{-15} m $$

$$ \rho = \frac{3 \sigma a_p^6}{4 \pi 64 s_p^2 l^2} $$ $$ \rho \approx 7 \times 10^{17} \text{kg m}^{-3} $$

The density is proportional to the 6th power of the velocity coefficient when the expansion coefficient is $\frac{1}{a_o^2}$.
In the Sun - $m_v$ system, at a distance where the expansion coefficient is $\frac{1}{a_E^2}$, the particle with orbital velocity coefficient $a_o$ is $m_v$. In the equivalences, the number of mass particles $m_v$ is $n_q$. The mass of the particle, $m_v$, is related to the coefficients of the electron and proton.

$$n_q = \frac{k_1 k_2}{a_o}$$

$$\mathbf{n}_q \quad m_v = m_2$$

$$\frac{m_{pl}s_p s_e n_q}{n} = m_2$$

$$n = \frac{n_q m_{pl} s_p s_e}{m_E}$$

$$n = \frac{k_S k_E m_{pl} s_p s_e}{a_E k_E m_{pl}}$$

$$n = \frac{k_S s_p s_e}{a_E}$$

$$n a_E = k_S s_p s_e$$

$$n \approx 3$$

$$3a_E = \text{constant}$$

$$m_v = \frac{m_{pl}s_p s_e}{n} = \frac{m_{pl}a_e^2}{3}$$

$$m_v \approx 2.3 \times 10^{-50} \text{kg}$$

The mass of the Earth is as much as the quantum number $n_q$ of mass particles.

$$\frac{m_{pl}s_p s_e k_S k_E}{3a_E} = m_E$$

$$m_E \approx 6 \times 10^{24} \text{kg}$$

If the orbital speed of the particle $m_v$ is the speed of light and the system expansion coefficient is 1, the system particle is $m_h$.

$$n a_o = 3 a_E$$

$$n = 3a_E$$

$$n = 3a_E \Rightarrow m_v = \frac{m_{pl}s_p s_e}{n} = \frac{m_{pl}s_p s_e}{3a_E} = m_h$$

$$m_v = a_E m_h$$
In equivalences, the following relationship exists.

$$k_v = \frac{a_E}{k_S}$$

$$a_E m_h = \frac{a_E m_{pl}}{k_S}$$

$$m_h = \frac{m_{pl}}{k_S}$$

In equivalences, $m_h$ is the particle associated with the black hole mass, and the inverse of the particle mass coefficient is the black hole Schwarzschild radius and mass coefficient.

$$m_S = \frac{m_{pl}}{k_h} = \frac{a_E 3 m_{pl}}{a_e^2}$$

$$m_S \approx 2 \times 10^{30} kg$$

The relationship between the subatomic mass coefficient and the Solar mass coefficient is as

$$s_p s_e = \frac{3 a_E}{k_S}$$

In this case at another measuring radius, if the Solar mass is constant, either the orbital mass particle $m_v$ coefficient varies, or maybe the proton and electron mass coefficient values may vary depending on the expansion coefficient of the measuring radius.

In the Sun-$m_v$ system, the Planck length coefficient of the orbital gravitational radius is related to the subatomic coefficient and the orbital velocity coefficient and is formulated as

$$R_{Go} = \frac{R_{s1}}{2a_o^2} = \frac{a_E 3l}{a_e^2 a_E^2} = \frac{3l}{a_e^2 a_E}$$

$$E_v = \frac{F_{Go} R_{Go}}{2} = \frac{G m_1 m_v a_o^2}{2 k_1 l} = \frac{h k_v a_o^2}{2 \pi t_{pl} 2} = \frac{T k_B}{2} = \frac{m_v c^2 a_o^2}{2}$$

According to the subatomic and macroscopic velocity coefficients, the above equivalences are expressed as follows. The coefficient ratios in the equivalences are equal.

$$E_v = \frac{\sigma c^2 k_v a_o^4 R_{Go}}{k_1 2} = \frac{G m_1 a_e^2 m_{pl} a_o^2}{3 k_1 l 2} = \frac{h a_e^2 a_o^2}{2 t_{pl} 6\pi} = \frac{T_{pl} k_v a_o^2 k_B}{2} = \frac{m_v c^2 a_o^2}{2}$$
$$E_v = \frac{\sigma c^2 a_e^2 a_e^2 a_E^4 3 l}{3 a_E 3 a_e^2 a_E} = \frac{G 3 a_E m_{pl} a_e^2 m_{pl} a_E a_e^2}{2 a_e^2 3 l 3} = \frac{h a_e^2 a_E^2}{2 t_{pl} 6 \pi} = \frac{T_{pl} a_e^2 a_E^2 k_B}{2 3} = \frac{a_e^2 m_{pl} c^2 a_E^2}{3 2}$$

The gravitational orbital force $F_{Go}$ for the orbital particle $m_v$ is as

$$F_{Go} = \frac{\sigma c^2 k_v a_o^4}{k_1} = \frac{\sigma c^2 a_e^4 a_E^3}{9}$$

The expansion coefficient of the mass radius of the proton, when the electron is a mass particle, is as

$$R_{ms} = \frac{4 s_{pl}}{a_p^2}$$

$$\frac{R_{ms}}{4 a_{\ddot{g}}^2} = \frac{l}{s_e}$$

$$a_{\ddot{g}}^2 = \frac{s_e}{s_p}$$

proton schwarz yarışçı, $R_{sp} = 2s_{pl}$

$$R_{Go} = \frac{R_{ms}}{4 a_{\ddot{g}}^2} = \frac{s_{pl}}{a_p^2 a_{\ddot{g}}^2} = \frac{s_{pl}}{s_p^2 a_{\ddot{g}}^2} = \frac{s_{pl}}{a_e^2}$$

This radius is equal to the electron reduced Compton wavelength and the expansion coefficient at the electron mass radius is as

$$R_{Go} = \frac{R_{sp}}{2 a_e^2} = \frac{l}{s_e}$$

$$a_e^2 = s_p s_e$$

$$\frac{1}{a_e^2} \approx \frac{1}{3.2 \times 10^{-42}}$$

In equivalences, the Universe - $m_{ur}$ system is as

$$\text{Universe mass, } m_u = \frac{m_{pl}}{s_p a_e^2} = k_u m_{pl}$$

$$k_u \approx 0.4 \times 10^{61}$$

$$s_{ur} = s_e a_p^2 = s_p a_e^2$$
$$m_{ur} = a_p^2 s_e m_{pl} = \frac{m_{pl}}{k_u}$$

$$E_1 \approx 13.6 \text{ eV}$$

$$\frac{E_1 a_p^2}{\alpha^2} = \frac{G m_p m_e a_p^2}{2 s_p l} = \frac{G m_p m_{ur}}{2 s_p l} = \frac{G m_u m_{ur}}{R_{su}} = \frac{G m_u m_{ur} a_e^2 a_p^2}{s_p l 2} = \frac{G m_p m_{ur}}{s_p^2 a_e^2 R_{su}} = \frac{G m_p m_{ur} a_p^2 s_e}{s_p^2 a_e^2 l 2}$$

$$= \frac{m_{ur} c^2 \Lambda_{ur} R_{su}^2}{\Omega 6} = \frac{m_{ur} c^2 a_e^4 a_p^4 R_{su}^2}{2 a_p^2 l^2 \Omega 6} = \frac{m_{ur} H_o^2 R_{su}^2}{2} = \frac{\hbar}{t_{ur}} = \frac{\hbar s_e a_p^2}{2 t_{pl}} = T_{es} k_B$$

$$= \frac{T_{pl} s_e a_p^2 k_B}{2} = \frac{\hbar c^3 k_B}{2 G m_u k_B} = \frac{\hbar c^3 a_p^2 s_e k_B}{2 G m_{pl} k_B} = m_{ur} g_{es} R_{su} = \frac{m_{ur} c^2 a_p^2 s_e R_{su}}{4 l} = \frac{m_{ur} c^2}{2}$$

The radius at which the reduced Compton wavelength $R_C = \frac{l}{s_e a_p^2}$ of particle $m_{ur}$ is equal is as

$$R_{uGo} = \frac{s_p l}{a_e^2 a_p^2} = \frac{s_p l}{s_{ur} a_p} = k_u l = \frac{l}{s_e a_p^2}$$

$$R_{Ges} = 2R_{uGo}$$

$$R_{Ges} \approx 1.3 \times 10^{26} m$$

The orbital radius of the Earth is as

$$R_{Go} = \frac{s_p l 3}{s_{ur} a_E}$$

The mass $m_u$ associated with the particle $m_{ur}$ emerges in the equations as

$$\frac{\hbar c^3 a_p^2 s_e k_B}{2 G m_{pl} k_B} = \frac{\hbar c^3 k_B}{2 G m_u k_B}$$

$$m_u = \frac{m_{pl}}{a_p^2 s_e} = \frac{m_{pl}}{a_e^2 a_p}$$

$$m_u \approx 0.9 \times 10^{53} kg$$

$$m_{ur} \approx 0.5 \times 10^{-68} kg$$

$$g_{es} = \frac{c^2 a_p^2 s_e}{4l}, \quad g_o = \frac{c^2 a_p^2 s_e}{l}$$
$$ \frac{\mathbf{m}_{ur} \mathbf{g}_{es} \mathbf{R}_{ges}}{k_B} = T_{es} $$

$$ \frac{a_p^2 s_e \sigma l g_{es} 2l}{s_e a_p^2 k_B} = T_{es} $$

$$ \frac{g_{es} 2 \sigma c l^2}{k_B c} = T_{es} $$

$$ \frac{g_{es} 2 \hbar}{k_B c} = T_{es} $$

$$ \frac{E_1 a_p^2}{\alpha^2 k_B} = \frac{\hbar c^3 k_B}{2 G m_u k_B} = \frac{g_{es} 2 \hbar}{k_B c} = T_{es} $$

At the escape radius, it differs from Unruh temperature [5] and Hawking temperature [6] by a factor of $\frac{1}{4\pi}$.

The cosmological constant [7] and Hubble constant [8] are as

$$ \Lambda = \frac{3\Omega}{R_{su}^2} = \frac{3a_p^4 s_e^2 \Omega}{4l^2} = \frac{a_p^4 s_e^2}{2l^2} \quad , \quad H_o = \frac{c}{R_{su}} = \frac{ca_p^2 s_e}{2l} $$

$$ \Lambda \approx 1.1 \times 10^{-52} m^{-2} \quad , \quad H_o \approx 2.3 \times 10^{-18} s^{-1} $$

$$ \rho_{ur} = \frac{3\sigma s_e^2 a_p^4}{4\pi 8l^2} $$

$$ \rho_{ur} \approx 9 \times 10^{-27} \text{ kg m}^{-3} $$

The mass radius density $\rho_{mur}$ is 1/8 of the escape radius density $\rho_{ur}$.

$$ \rho_{mur} \approx 1.1 \times 10^{-27} \text{ kg m}^{-3} $$

$$ \rho_{ur} = \frac{3\sigma s_e^2 a_p^4}{4\pi 8l^2} $$

$$ \frac{\rho_{ur}}{\Lambda_{ur}} = \frac{\sigma}{8\pi\Omega} $$
$$ \rho_{ur} = \frac{c^2 \sigma \Lambda_{ur}}{8\pi c^2 \Omega} $$

$$ G = \frac{c^2}{\sigma} $$

$$ \rho_{ur} = \frac{c^2 \Lambda_{ur}}{8\pi G \Omega} $$

The relationship between the radial particle time $t_{ro}$ at the speed of light in the Sun $-m_r$ system and the Hubble time $t_{ur}$ in the Universe $-m_{ur}$ system is as

$$ E_1 = \frac{\sigma c^2 \propto s_e l}{2}, \quad E_v = \frac{\sigma c^2 a_E^3 l}{2k_S} = \frac{\sigma c^2 a_E^2 a_e^2 l}{6} $$

$$ \frac{E_1 a_p^2}{\propto^2} = \frac{3 a_p E_v}{a_E^2} = \frac{G m_p m_{ur}}{a_p^2 a_e^2 R_{su}} = \frac{G m_p m_e a_p^2}{2 S_1 l} = \frac{G m_u m_{ur}}{R_{su}} = \frac{\hbar}{t_{ur}} = \frac{a_p 3 \hbar}{a_E 2 t_{ro}} $$

$$ \frac{2t_{ro}}{t_{ur}} = \frac{3a_p}{a_E} $$

$$ H_o = \frac{3a_p}{2a_E t_{ro}} $$

In equivalences, in the black hole form, the gravitational energy increases by $\frac{1}{a_o^3}$. In this case, the particle of the system is $m_h$. In the temperature formula, the mass of the black hole with which the $m_h$ particle is associated emerges as

$$ m_h = k_h m_{pl} = \frac{a_s^2 m_{pl}}{\propto k_E a_E} = \frac{m_{pl}}{k_S} = \frac{a_e^2 m_{pl}}{3a_E} $$

$$ \text{Schwarzschild radius of the Sun, } R_{ss} = k_S 2l = \frac{6l a_E}{a_e^2} $$

$$ t_h = \frac{2t_{pl}k_E a_E}{a_s^2} \propto 2k_S t_{pl} = \frac{6a_E t_{pl}}{a_e^2} $$

$$ R_{sh} = \frac{2l}{k_S} = \frac{a_e^2 2l}{3a_E} $$

$$ E_v = \frac{a_o^3 \hbar c^3 k_B}{2Gm_1 k_B} = \frac{a_o^3 \hbar}{t_h} = a_o^3 T_h k_B = \frac{a_o^3 Q_{pl}^2}{4\pi\varepsilon_0 2 R_{Ch}} $$

$$ = \frac{a_o^3 G m_1 m_h}{R_{s1}} = \frac{a_o^3 m_h c^2}{2} = \frac{a_o^3 F_{pl} R_{sh}}{4} $$
$$ \frac{E_v}{a_E^3} = \frac{\hbar c^3 a_e^2 k_B}{2G3a_E m_{pl} k_B} = \frac{\hbar a_e^2}{6a_E t_{pl}} = \frac{T_{pl} a_e^2 k_B}{6a_E} = \frac{Q_{pl}^2 a_e^2}{4\pi\epsilon_0 6a_E l} $$

$$ = \frac{G3a_E m_{pl} a_e^2 m_{pl} a_e^2}{a_e^2 3a_E 6a_E l} = \frac{a_e^2 m_{pl} c^2}{3a_E 2} = \frac{F_{pl} a_e^2 2l}{4 a_E 3} $$

$$ R_{ss} = \frac{6la_E}{a_e^2} $$

$$ R_{ss} \approx 3 \times 10^3 m $$

$$ m_h = \frac{a_e^2 m_{pl}}{3a_E} $$

$$ m_h \approx 2.3 \times 10^{-46} kg $$

In the equations, the resulting solar mass associated with the particle $m_h$ is as

$$ \frac{\hbar c^3 k_B}{2Gm_S k_B} = \frac{\hbar c^3 a_e^2 k_B}{2G3a_E m_{pl} k_B} $$

$$ m_S = \frac{3a_E m_{pl}}{a_e^2} $$

$$ m_S \approx 2 \times 10^{30} kg $$

In the sun-$m_v$ energy system, another particle is $m_g$. Different states are possible for the particle. Particle $m_g$ may be particle $m_h$ of another system. If the reduced Compton wavelength of the radial particle at the speed of light or the de Broglie radius of the orbital particle is equal to the gravitational radius, the mass particle is observed.

$$ m_g = \frac{a_E^3 m_{pl}}{k_S} = \frac{a_S^2 a_E^2 m_{pl}}{\alpha k_E} = \frac{a_e^2 a_E^2 m_{pl}}{3} $$

$$ m_g \approx 2.3 \times 10^{-58} kg $$

$$ 2\pi R_C = \frac{2\pi k_S l}{a_E^3} = \frac{6\pi l}{a_e^2 a_E^2} $$

$$ 2\pi R_C \approx 1 \times 10^{16} m $$
$$ \frac{k_1 k_2 E_v}{a_o n_q} = \frac{G m_s m_g}{R_{ss}} = \frac{G m_s m_g a_E^2}{a_E^2 a_E^2 R_{sA}} = \frac{G m_A m_g}{a_E^2 R_{gG}} = \frac{m_g V^2}{a_E^2 2} $$

$$ = \frac{G m_M m_g}{R_{sM}} = \frac{m_g c^2}{2} = \frac{Q_{pl}^2 a_E^3}{4 \pi \epsilon_0 2 k_S l} = \frac{G m_M m_v a_E^2}{R_{sM}} $$

$$ R_{Gg} = \frac{\hbar}{m_g a_E c} , \quad R_{sM} = \frac{\hbar}{m_g c} $$

$$ R_{sA} = \frac{R_{ss}}{a_E^2} = \frac{2k_S l}{a_E^2} , \quad R_{Gg} = \frac{R_{sA}}{a_E^2} = \frac{2k_S l}{a_E^4} , \quad R_{sM} = \frac{2k_S l}{a_E^3} $$

$$ m_A = \frac{R_{sA} \sigma}{2} = \frac{k_S m_{pl}}{a_E^2} = \frac{\alpha k_E m_{pl}}{a_S^2 a_E} = \frac{3 m_{pl}}{a_E^2 a_E} $$

$$ m_A \approx 2 \times 10^{38} kg $$

$$ m_M \approx 2 \times 10^{42} kg $$

The minimum Schwarzschild radius of the unit Planck force in the Sun-Earth system is as

$$ n_q = \frac{k_S k_E}{a_E} , \quad k_g = \frac{a_S^2 a_E^2}{\alpha k_E} = \frac{a_E^3}{k_S} = \frac{a_E^2 a_E^2}{3} $$

$$ \frac{k_1 k_2 \sigma c^2 a_o^2 a_s^2 l}{a_o n_q 2 k_2 \alpha} = \frac{G m_1 m_v a_o^2}{2 k_1 l} = \frac{F_{pl} R_{sg}}{4} $$

$$ n_q E_v = \frac{G m_s m_E a_E^2}{2 k_S l} = \frac{F_{pl} a_E^2 a_s^2 n_q 2 l}{4 \alpha k_E} $$

$$ \frac{F_{pl} a_E^2 a_s^2 n_q 2 l}{4 k_E \alpha} = \frac{F_{pl} n_q R_{sg}}{4} $$

$$ \frac{n_q R_{sg}}{2} \approx 4.3 \times 10^{-11} m $$

It is possible to formulate the mass at this Schwarzschild radius $\frac{n_q R_{sg}}{2}$ by the square of the velocity coefficient and the Earth mass, or subatomic mass coefficients
$$n_q k_g l \sigma = m_E a_E^2 = \frac{n_q s_p s_e a_E^2}{3}$$

$$n_q k_g l \sigma \approx 6 \times 10^{16} kg$$

If the spatial range, $n_q R_{sg}$ increases by the expansion coefficient, the earth Schwarzschild radius is obtained.

$$R_{SE} = \frac{n_q R_{sg}}{a_E^2} = n_q k_v 2l$$

$$R_{SE} \approx 8.8 \times 10^{-3} m$$

$$m_E = \frac{n_q R_{sg} \sigma}{a_E^2 2} = \frac{n_q k_v 2l \sigma}{2}$$

At the Sun-$m_v$ energy level, the associated escape radius of the smallest particle $m_g$ is $R_{ss} = 2k_v l$. The ratio of the cosmological constant $\Lambda$ associated with the particle $m_{ur}$ to $\Lambda_g$ is as

$$\frac{k_S k_E E_v}{a_E n_q} = \frac{G m_S m_g}{R_{ss}} = \frac{m_g H_g^2 R_{ss}^2}{2} = m_g c^2 \Lambda_g R_{Go}^2 = \frac{m_g c^2 \Lambda_g R_{ss}^2}{\Omega 6}$$

$$\Lambda_g = \frac{3\Omega}{k_S^2 4l^2} = \frac{a_e^4 \Omega}{12 a_E^2 l^2}$$

$\Lambda_g$ associated with the particle $m_g$ is constant and independent of expansion.

$$\frac{\Lambda_g}{\Lambda} = \frac{a_e^4 \Omega 4l^2}{12 a_E^2 l^2 3 a_p^4 s_e^2 \Omega} = \frac{1}{9 a_E^2 a_p^2}$$

At the escape radius $R_{ges} = \frac{2k_v l}{a_E^2}$, $\Lambda_v$ and $\Lambda_r$ are as

$$\Lambda_v = \frac{3 a_E^6 \Omega}{k_S^2 4l^2} = \frac{a_E^4 a_e^4 \Omega}{12l^2} , \quad \Lambda_r = \frac{3 a_E^4 \Omega}{k_S^2 4l^2} = \frac{a_E^2 a_e^4 \Omega}{12l^2}$$
If the system particle is $m_v$, the relationship between the gravitational force and the density at the escape radius is as

$$F_{ges} = \frac{\sigma c^2 a_E^5}{4k_S^2} = \frac{\sigma c^2 a_E^3 a_e^4}{36}$$

$$\rho_{es} = \frac{3\sigma a_E^6}{4\pi 8k_S^2 l^2} = \frac{\sigma a_E^4 a_e^4}{96\pi l^2}$$

$$\rho_{es} = \frac{3\sigma c^2 a_E^6}{c^2 8\pi l^2 4k_S^2} = \frac{\sigma c^2 a_E^4 a_e^4}{c^2 8\pi 12l^2}$$

$$\rho_{es} = \frac{3a_E F_{ges}}{c^2 8\pi l^2}$$

$$\hbar = \sigma c l^2, \quad G = \frac{c^2}{\sigma}$$

$$F_{ges} = \frac{\rho_{es} \hbar 8\pi G}{3c a_E}$$

As the force decreases, the density decreases. The force at the escape radius is one quarter of the orbital force.

$$F_{Go} = \frac{\sigma c^2 a_E^5}{k_S^2} = \frac{\sigma c^2 a_E^3 a_e^4}{9}, \quad \rho_o = 8\rho_{es}$$

$$F_{Go} = \frac{\rho_o \hbar 4\pi G}{3c a_E}$$

$$\frac{F_{ges}}{F_{Go}} = \frac{\rho_{es}^2}{\rho_o} = \frac{1}{4}$$

$$\Lambda_v = \frac{3 a_E^6 \Omega}{k_S^2 4l^2} = \frac{a_E^4 a_e^4 \Omega}{12l^2}$$

$$F_{ges} = \frac{\Lambda_v \hbar c}{3\Omega a_E}$$

At the escape radius, $a_E$ is the escape velocity coefficient.

$$\frac{\Lambda_v \hbar c}{3\Omega a_E} = \frac{\rho_{es} \hbar 8\pi G}{3V_{es}}$$

$$\rho_{es} = \frac{c^2 \Lambda_v}{\Omega 8\pi G}$$
If the particle $m_r$ of the system is at the speed of light in the escape radius $R_{Ges} = \frac{2k_S l}{a_E^2} = \frac{6l}{a_e^2 a_E}$ and orbital radius $R_{Go} = \frac{k_S l}{a_E^2} = \frac{3l}{a_e^2 a_E}$, the acceleration is as

$$a_{es} = \frac{c^2 a_E^2}{4k_S l} = \frac{c^2 a_e^2 a_E}{12l} \quad , \quad a_o = \frac{c^2 a_E^2}{k_S l} = \frac{c^2 a_e^2 a_E}{3l}$$

$a_{es}$ is the acceleration at the escape radius, $a_o$ is the acceleration at the orbital radius.

$$R_{Go} = R_{Cr}$$

$$E_v = a_o m_r a_{res} R_{Ges} = \frac{a_o m_r c^2 \Lambda_r R_{Ges}^2}{\Omega 6}$$

$$\frac{E_v}{a_o} = m_r a_{res} R_{Ges} = \frac{m_r c^2 \Lambda_r R_{Ges}^2}{\Omega 6}$$

$$\Omega = \frac{2}{3}$$

$$m_r a_{res} = \frac{m_r c^2 \Lambda_r R_{Ges}}{4}$$

$$m_r a_{res} = \frac{\sigma l a_E^2 c^2 \Lambda_r R_{Ges}}{4k_S}$$

$$m_r a_{res} = \frac{\sigma c l^2 c a_E^2 \Lambda_r 2k_S}{4k_S a_E^2}$$

$$m_r a_{res} = \frac{\hbar c \Lambda_r}{2}$$

$$m_r a_{res} = \frac{\sigma c l^2 c \Lambda_r}{2}$$

$$m_r a_{res} = \frac{4\pi\epsilon_0\sigma c^2 l^2 \Lambda_r}{4\pi\epsilon_0 2}$$

$$m_r a_{res} = \frac{Q_{pl}^2 \Lambda_r}{4\pi\epsilon_0 2}$$
$$a_{res} = \frac{\hbar c \Lambda_r}{m_r 2} = \frac{\sigma c l^2 k_S c \Lambda_r}{a_E^2 \sigma l 2} = \frac{l k_S c^2 \Lambda_r}{a_E^2 2} = \frac{R_{Cr} c^2 \Lambda_r}{2}$$

$$\frac{Q_{pl}^2 \Lambda_r}{4\pi\epsilon_o 2} = \frac{R_{Cr} m_r c^2 \Lambda_r}{2} = \frac{\hbar c \Lambda_r}{2}$$

The Hubble constant associated with the Schwarzschild radius $R_{SS}$ in the Sun-$m_g$ and Sun-$m_h$ systems is as

$$H_g = \frac{c}{2k_S l} = \frac{ca_e^2}{6a_E l}, \quad H_h = \frac{c}{2k_S l} = \frac{ca_e^2}{6a_E l}$$

In the Sun-$m_v$, Sun-$m_r$ systems, $H_v$ and $H_r$ at the radius $R_{Ges} = \frac{6l}{a_e^2 a_E}$ are as

$$H_v = \frac{ca_E^3}{2k_S l} = \frac{ca_e^2 a_E^2}{6l}, \quad H_r = \frac{ca_E^2}{2k_S l} = \frac{ca_e^2 a_E}{6l}$$

At the escape radius in the equations, the temperature $T_{res}$ and acceleration $a_{res}$ relationship of particle $m_r$ is as

$$\frac{E_v}{a_E} = m_r a_{res} R_{Ges} = T_{res} k_B = \frac{T_{ro} k_B}{2} = \frac{m_r a_{ro} R_{Go}}{2}$$

$$T_{ro} = \frac{a_E^2 T_{pl}}{k_S} = \frac{a_e^2 a_E T_{pl}}{3}$$

$$a_{res} = \frac{c^2 a_E^2}{4k_S l}, \quad a_{ro} = \frac{c^2 a_E^2}{k_S l}$$

$$\frac{m_r a_{res} R_{Ges}}{k_B} = T_{res}$$

$$\frac{a_E^2 \sigma l a_{res} 2k_S l}{k_S k_B a_E^2} = T_{res}$$

$$\frac{a_{res} 2\hbar}{k_B c} = T_{res}$$
The ratio of system acceleration $a_{res}$ to planck acceleration $a_{pl}$ gives the Planck temperature coefficient of the system

$$ \frac{a_{res} 2\hbar}{k_B c} = T_{res} $$

$$ \frac{a_{res} \sigma l c^2 2l}{k_B c^2} = T_{res} $$

$$ \frac{a_{res} m_{pl} c^2 2l}{k_B c^2} = T_{res} $$

$$ \frac{a_{res} T_{pl} k_B 2l}{k_B c^2} = T_{res} $$

$$ \frac{a_{res} 2T_{pl}}{a_{pl}} = T_{res} $$

In the equations, the period of the Earth and the mass particle $m_v$ at earth orbital speed around the Sun is as

Earth spin speed, $ca_s \approx 1600 \text{km/hr}$

$$ a_s = \frac{a_e^2 \sqrt{n_q}}{3} \propto \frac{k_2 a_E}{k_1} $$

$$ a_s^2 \approx 2.2 \times 10^{-12} $$

$$ \frac{h v_v}{2} = \frac{m_v 4 \pi^2 R_{Go}^2}{t_{Go}^2 2} $$

$$ \frac{h a_s^2 a_E^2}{2 \pi t_{pl} k_E \propto 2} = \frac{m_v a_E^6 4 \pi^2 R_{Go}^2}{4 \pi^2 k_s^2 t_{pl}^2 2} $$

$$ \frac{2 \pi k_E \propto t_{pl}}{a_s^2 a_E^2} = \frac{2 \pi k_s t_{pl}}{a_E^3} $$

$$ t_{Go} = \frac{2 \pi t_{pl}}{k_v a_E^2} = \frac{6 \pi t_{pl}}{a_e^2 a_E^2} $$

$$ t_{Go} \approx 370 \text{ days} $$

The orbital frequency for the Sun-$m_v$ system emerges in the equations as
$$v = \frac{a_e^2 a_E^2}{6\pi t_{pl}} = \frac{a_E^3}{2\pi k_S t_{pl}} = \frac{a_S^2 a_E^2}{2\pi \propto k_E t_{pl}}$$

$$v \approx 3 \times 10^{-8} \text{Hz}$$

The particles and Earth orbital frequency can be formulated by the fine- structure constant and the spin velocity coefficient.

In the Sun-$m_r$ system, radial time $t_r$ is as

$$E_v = \frac{a_o \hbar}{t_r 2} = \frac{a_o m_r R_{Go}^2}{2 t_r^2} = \frac{a_o Q_{pl}^2}{4 \pi \epsilon_0 2 R_{Cr}}$$

$$\frac{E_v}{a_E} = \frac{\hbar}{t_r 2} = \frac{m_r R_{Go}^2}{2 t_r^2} = \frac{Q_{pl}^2}{4 \pi \epsilon_0 2 R_{Cr}}$$

$$\frac{\hbar}{t_r 2} = \frac{m_r R_{Go}^2}{2 t_r^2}$$

$$\frac{\hbar a_s^2 a_E}{t_{pl} k_E \propto 2} = \frac{m_r a_E^4 R_{Go}^2}{k_S^2 t_{pl}^2 2}$$

$$t_r = \frac{\propto t_{pl} k_2}{a_s^2 a_E} = \frac{k_S t_{pl}}{a_E^2} = \frac{3 t_{pl}}{a_e^2 a_E}$$

$$t_r = \frac{t_{pl}}{k_r}$$

$$t_r \approx 8 \text{ minutes}$$

## 2. Conclusion

In equivalences consisting of equal coefficients in equal ratio, a spacetime defined to the expansion coefficient emerges by the universal constants formulated from mass constant, reference units and coefficients. The gravitational radius is defined by the Schwarzschild radius expansion coefficient.

Planck length expansion coefficient of macroscopic spacetime determines particle mass. As the Schwarzschild radius expansion coefficient increases, the mass of the system particle decreases.

The reference force is the Planck force and it is possible to basically formulate all forces with the Planck force expansion coefficient.

The wavelength of a radial particle at the speed of light defines the Planck charged particle. If the gravitational radius is equal to the system wave radius, the mass form is observable and is
proportional to the Planck constant. In hydrogen-like systems, the ratio of Compton force to gravitational force is equal to the expansion coefficient.

The orbital velocity coefficient determines the subatomic elementary particle unit mass coefficient. In the subatomic and macroscopic world, it may be possible to define physical quantities based on a fundamental factor. In the Sun-earth system, the Planck length expansion coefficient of the measurement space is formulated with the subatomic elementary particles and the orbital velocity coefficient determined by the expansion coefficient.

The ratio of the Hubble time to the Sun-earth system radial time is equal to 1/3 of the ratio of the earth orbital velocity to the proton orbital velocity.

Convertible equations can provide fundamental formulas relating between subatomic spacetime and macroscopic spacetime.

The force radius is defined by the expansion coefficient of the system. The instantaneous change of the velocity coefficient of the subatomic system can influence instantly the behavior and radius of the system. In equivalences, the relationship between expansion of spacetime and the mass particle can explain difficult-to-understand phenomena such as non-locality and wave particle duality, fine-tuned universe, as a new interpretation.

**Declaration of interests:**

The author declare no competing interests.

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