Physics/Essays/Fedosin/Selfconsistent gravitational constants

Self-consistent gravitational constants are complete sets of fundamental constants, which are self-consistent and define various physical quantities associated with gravitation. These constants are calculated in the same way as electromagnetic constants in electrodynamics. This is possible because in the weak field equations of general relativity are simplified into equations of gravitoelectromagnetism, similar in form to Maxwell's Equations. Similarly, in the weak field approximation equations of covariant theory of gravitation turn into equations of Lorentz-invariant theory of gravitation (LITG). LITG equations are Maxwell-like gravitational equations, which are similar to equations of gravitoelectromagnetism. If these equations are written with the help of self-consistent gravitational constants, there is the best similarity of equations of gravitational and electromagnetic fields. Since in 19-th century there was no International System of Units, the first mention of gravitational constants was possibly due to Forward (1961).

Definition
Primary set of gravitational constants is:

1. First gravitational constant: $$~c_g $$, which is the speed of gravitational waves in vacuum;

2. Second gravitational constant: $$~\rho_{g} $$, which is the gravitational characteristic impedance of free space.

Secondary set of gravitational constants is:

1. Gravitoelectric gravitational constant (like electric constant): $$~\varepsilon_g = \frac{1}{4\pi G } = 1.192708\cdot 10^9 \mathrm {kg \cdot s^2 \cdot m^{-3}}, $$ where $$~ G $$ is the gravitational constant.

2. Gravitomagnetic gravitational constant (like vacuum permeability): $$~\mu_g = \frac{4\pi G }{ c^2_{g}}.$$ If the speed of gravitation is equal to the speed of light, $$~ c_{g}=c,$$ then $$~\mu_{g0} = 9.328772\cdot 10^{-27} \mathrm {m / kg}. $$

Both, primary and secondary sets of gravitational constants are selfconsistent, because they are connected by the following relationships:


 * $$~\frac{1}{\sqrt{\mu_g\varepsilon_g}} = c_g, $$


 * $$~\sqrt{\frac{\mu_g}{\varepsilon_g}} = \rho_{g} = \frac{4\pi G }{c_g}. $$

If $$~ c_{g}=c,$$  then gravitational characteristic impedance of free space be equal to:
 * $$~ \rho_{g0} = \frac{4\pi G }{c} =2.796696\cdot 10^{-18} \mathrm {m^2/(s\cdot kg)}. $$

In Lorentz-invariant theory of gravitation the constant $$~ \rho_g $$ is contained in formula for vector energy flux density of gravitational field (Heaviside vector):
 * $$~ \mathbf{H} = -\frac{ c^2_g }{4 \pi G } \mathbf{\Gamma }\times \mathbf{\Omega} = -\frac{ c_g }{\rho_g }\mathbf{\Gamma }\times \mathbf{\Omega},$$

where:
 * $$~ \mathbf{\Gamma }$$ is gravitational field strength or gravitational acceleration,
 * $$~ \mathbf{\Omega}$$ is gravitational torsion field or simply torsion field.

For plane transverse uniform gravitational wave, in which for amplitudes of field strengths holds $$ ~ \Gamma = c_g \Omega $$, may be written:
 * $$~H = \frac{ \Gamma^2 }{\rho_g }.$$

A similar relation in electrodynamics for amplitude of flux density of electromagnetic energy of a plane electromagnetic wave in vacuum, in which $$ ~ E = c B $$, is as follows:
 * $$~S = \frac{ E^2 }{Z_0 },$$

where $$~ \mathbf {S} = \frac {\mathbf{E}\times \mathbf{B} }{\mu_0} = \frac {c}{Z_0}\mathbf{E}\times \mathbf{B} $$ – Poynting vector, $$~ E $$ – electric field strength, $$~ B $$ – magnetic flux density, $$~ \mu_0 $$ – vacuum permeability, $$~ Z_0 = c \mu_0 $$ – impedance of free space.

Gravitational impedance of free space $$~\rho_{g0} $$ was used in paper to evaluate the interaction section of gravitons with the matter.

Connection with Planck mass and Stoney mass
Since gravitational constant and speed of light are included in Planck mass $$ m_P = \sqrt {\frac {\hbar c} {G}} \ $$, where ~ \hbar – reduced Planck constant or Dirac constant, then gravitational characteristic impedance of free space can be represented as:
 * $$~ \rho_{g0} = \frac{2h}{m_{P}^2} $$,

where $$~ h$$ – Planck constant.

There is Stoney mass, related to  elementary charge $$ ~ e $$ and electric constant $$~ \varepsilon_0$$:
 * $$~m_S = e\sqrt{\frac{\varepsilon_g}{\varepsilon_0}} = \frac{e}{\sqrt{4\pi G \varepsilon_0}} $$.

Stoney mass can be expressed through the Planck mass:
 * $$~m_S = \sqrt{\alpha}\cdot m_P $$,

where $$~ \alpha $$ is the electric fine structure constant.

This implies another expression for gravitational characteristic impedance of free space:
 * $$~ \rho_{g0} = \alpha \cdot \frac{2h}{m_{S}^2} $$.

Newton law for gravitational force between two Stoney masses can be written as:
 * $$~F_g = \frac{1}{4\pi \varepsilon_g}\cdot \frac{m_{S}^2}{r^2}= \alpha_g \cdot \frac{\hbar c}{r^2}. $$

Coulomb's law for electric force between two elementary charges is:
 * $$~F_e = \frac{1}{4\pi \varepsilon_0}\cdot \frac{e^2}{r^2}= \alpha \cdot \frac{\hbar c}{r^2}. $$

Equality of $$~F_g$$ and $$~F_e$$ leads to equation for the Stoney mass $$~m_S = e\sqrt{\frac{\varepsilon_g}{\varepsilon_0}},$$ that was stated above. Hence the Stony mass may be determined from the condition that two such masses interact via gravitation with the same force as if these masses had the charges equal to the elementary charge and only interact through electromagnetic forces.

Connection with fine structure constant
The electric fine structure constant is:
 * $$~\alpha = \frac{e^2}{2\varepsilon_0 hc}.$$

We can determine the same value for gravitation so: $$~\alpha_g = \frac{m_{S}^2}{2\varepsilon_g hc}=\alpha ,$$ with the equality of the fine structure constants for both fields.

On the other hand, the gravitational fine structure constant for hydrogen system at the atomic level and at the level of star is also equal to fine structure constant:
 * $$~\alpha = \frac{G_s M_p M_e}{\hbar c}=\frac {G M_{ps} M_{\Pi } }{\hbar_s C_s}=\frac {1}{137.036}$$,

where $$~G_s $$ – strong gravitational constant, $$~M_p $$ and $$~M_e $$ – the mass of proton and electron, $$~ M_{ps} $$ and $$~ M_{\Pi } $$ – mass of the star-analogue of proton and the planet-analogue of electron, respectively, $$~ \hbar_s $$ – stellar Dirac constant, $$~ C_s $$ – characteristic speed of stars matter.

Strong gravitational torsion flux quantum
The magnetic force between two fictitious elementary magnetic charges is:
 * $$F_m = \frac{1}{4\pi \mu_0}\cdot \frac{ q_m^2}{r^2} = \beta \cdot \frac{\hbar c}{r^2}, \ $$

where $$ q_m = \frac{h}{e} \ $$ is the magnetic charge, $$ \beta = \frac {\varepsilon_0 h c}{2 e^2} = \frac {\pi \hbar}{c \mu_0 e^2}$$ is the magnetic coupling constant for fictitious magnetic charges.

The force of gravitational torsion field between two fictitious elementary torsion masses is:
 * $$F_{\Omega} = \frac{1}{4\pi \mu_{g0}}\cdot \frac{m_{\Omega }^2}{r^2} = \beta_g\cdot \frac{\hbar c}{r^2}, \ $$

where $$\beta_g = \frac {\varepsilon_g h c}{2 m_S^2} = \frac {\pi \hbar}{c \mu_{g0} m_S^2} \ $$ is the gravitational torsion coupling constant for the gravitational torsion mass $$ m_{\Omega } \ $$.

In the case of equality of the above forces, we shall get the equality of the coupling constants for magnetic field and gravitational torsion field:
 * $$\beta = \beta_g = \frac{1}{4\alpha}, \ $$

from which the Stoney mass $$ m_S \ $$ and the gravitational torsion mass could be derived:
 * $$ m_S = e \cdot \sqrt {\frac{\mu_o}{\mu_{g0}}} = \frac{e}{\sqrt {4 \pi \varepsilon_0 G}}. \ $$


 * $$ m_{\Omega } = q_m \cdot \sqrt {\frac{\mu_{g0}}{\mu_o }} = \frac{h \sqrt {4 \pi \varepsilon_0 G}}{e }=\frac {h}{m_S} . \ $$

Instead of fictitious magnetic charge $$ q_m= h/e $$ the single magnetic flux quantum $Φ_{0} = h/(2e) ≈ 2.068 Wb$ has the real meaning in quantum mechanics. On the other hand at the level of atoms the strong gravitation operates and we must use the strong gravitational constant. So we believe that the strong gravitational torsion flux quantum there should be important:


 * $$ \Phi_\Gamma = \frac{h }{2 e }\sqrt {\frac {4 \pi \varepsilon_0 G_s M_e}{M_p}} = \frac{h}{2 M_p} = 1.98 \cdot 10^{-7}$$ m2/s,

which is related to proton with its mass $$ M_p $$ and to its velocity circulation quantum.