Physics/Essays/Fedosin/Quantum Gravitational Resonator

Quantum Gravitational Resonator (QGR) – closed topological object of the three dimensional space, in the general case – ‘’cavity’’ of arbitrary form, which has definite ‘’surface’’ and ‘’thickness’’. The QGR can have “infinite” phase shifted oscillations of gravitational field strength and gravitational torsion field, due to the quantum properties of QGR.

History
Considering that the theory of the gravitational resonator is based on the Maxwell-like gravitational equations and Quantum Electromagnetic Resonator (QER), therefore the QGR history is close connected with the QER history.

Gravitational LC circuit
The gravitational LC circuit can be composed by analogy with the electromagnetic LC circuit, and gravitational field strength and gravitational torsion field oscillate in the circuit as a result of oscillating mass current.

Gravitational voltage on gravitational inductance is:
 * $$V_{gL} = -L_g\cdot \frac{d I_{gL}}{d t}. \ $$

Gravitational mass current through gravitational capacitance is:
 * $$I_{gC} = C_g\cdot \frac{d V_{gC}}{d t}. \ $$

Differentiating these equations with respect to the time variable, we obtain:
 * $$\frac{d V_{gL}}{d t} = -L_g \frac{d^2I_{gL}}{dt^2}, \qquad \frac{d I_{gC}}{d t} = C_g \frac{d^2V_{gC}}{dt^2}.$$

Considering the following relationships for voltages and currents:
 * $$V_{gL} = V_{gC} =V_g, \qquad I_{gL} = I_{gC}=I_g ,\ $$

we obtain the following differential equations for gravitational oscillations:
 * $$~\frac{d^2 I_g}{dt^2} + \frac{1}{L_g C_g}I_g = 0, \qquad \frac{d^2 V_g}{dt^2} + \frac{1}{L_g C_g}V_g = 0. \quad \quad \quad \quad \quad (1) \  $$

Furthermore, considering the following relationships between voltage and mass, current and flux of gravitational torsion field:
 * $$ m = C_g V_g, \qquad \Phi = L_g I_g $$

the above oscillation equation can be rewritten in the form:
 * $$\frac{d^2 m}{dt^2} + \frac{1}{L_g C_g} m = 0. \quad \quad \quad \quad \quad (2) \ $$

This equation has the partial solution:
 * $$ m(t) = m_0 \sin (\omega_g t) ,\ $$

where
 * $$\omega_g = \frac{1}{\sqrt{L_g C_g}} \ $$

is the resonance frequency, and
 * $$\rho_{LC} = \sqrt{\frac{L_g}{C_g}} ,\ $$

is the gravitational characteristic impedance.

For the sake of completeness we can present the differential equation for the flux of gravitational torsion field in the form:
 * $$\frac{d^2 \Phi}{dt^2} + \frac{1}{L_g C_g}\Phi = 0. \quad \quad \quad \quad \quad (3) \ $$

The realization of gravitational LC circuit is described in a section of maxwell-like gravitational equations.

Quantum gravitational LC circuit oscillator
Inductance momentum quantum operator in the electric-like gravitational mass space can be presented in the following form:
 * $$\hat p_{gm} = -i\hbar \frac{d}{dm}, \quad \quad \quad \quad \quad \hat p_{gm}^* = i\hbar \frac{d}{dm},\quad \quad \quad \quad \quad (4a) \ $$

where $$\hbar \ $$ is reduced Plank constant, $$\hat p_{gm}^* \ $$ is the complex-conjugate momentum operator, $$ m \ $$ is the induced mass.

Capacitance momentum quantum operator in the magnetic-like gravitational mass space can be presented in the following form:
 * $$\hat p_{g\Phi} = -i\hbar \frac{d}{d\Phi}, \quad \quad \quad \quad \quad \hat p_{g\Phi}^* = i\hbar \frac{d}{d\Phi},\quad \quad \quad \quad \quad (4b) \ $$

where $$\Phi \ $$ is the induced torsion field flux, which is imitated by electric-like gravitational mass current ($$i_g \ $$):
 * $$\Phi = L_g \cdot i_g. \ $$

We can introduce the third momentum quantum operator in the current form:
 * $$\hat p_{gi} = -\frac{i\hbar}{L_g} \frac{d}{di_g}, \quad \quad \quad \quad \quad  \hat p_{gi}^* = \frac{i\hbar}{L_g} \frac{d}{di_g},\quad \quad \quad \quad \quad (4c) \ $$

These quantum momentum operators defines three Hamilton operators:
 * $$\hat H_{gLm} = -\frac{\hbar^2}{2L_g}\cdot \frac{d^2}{dm^2} + \frac{L_g \omega_0^2}{2}\hat m^2 \quad \quad \quad \quad \quad (5a) \ $$
 * $$\hat H_{gC\Phi} = -\frac{\hbar^2}{2C_g}\cdot \frac{d^2}{d\Phi^2} + \frac{C_g \omega_0^2}{2}\hat \Phi^2 \quad \quad \quad \quad \quad (5b) \ $$
 * $$\hat H_{gLi} = -\frac{\hbar^2\omega_0^2}{2L_g}\cdot \frac{d^2}{di_g^2} + \frac{L_g \omega_0}{2}\hat i_g^2, \quad \quad \quad \quad \quad (5c) \ $$

where $$\omega_0 = \frac{1}{\sqrt{L_g C_g}} \ $$ is the resonance frequency. We consider the case without dissipation ($$R_g = 0 \ $$). The only difference of the gravitational charge spaces and gravitational current spaces from the traditional 3D- coordinate space is that it is one dimensional (1D). Schrodinger equation for the gravitational quantum LC circuit could be defined in three form:
 * $$-\frac{\hbar^2}{2L_g}\frac{d^2 \Psi}{dm^2} + \frac{L_g\omega_0^2}{2}m^2\Psi = W\Psi \quad \quad \quad \quad \quad (6a) \ $$
 * $$-\frac{\hbar^2}{2C_g}\frac{d^2 \Psi}{d\Phi^2} + \frac{C_g\omega_0^2}{2}\Phi^2\Psi = W\Psi \quad \quad \quad \quad \quad (6b) \ $$
 * $$-\frac{\hbar^2\omega_0^2}{2L_g}\frac{d^2 \Psi}{di_g^2} + \frac{L_g\omega_0}{2}i_g^2\Psi = W\Psi. \quad \quad \quad \quad \quad (6c) \ $$

To solve these equations we should to introduce the following dimensionless variables:
 * $$\xi_m = \frac{m}{m_0}; \quad \quad m_0 = \sqrt{\frac{\hbar}{L_g\omega_0}}; \quad \quad \lambda_m = \frac{2W}{\hbar\omega_0} \quad \quad (7a) \ $$
 * $$\xi_{\Phi} = \frac{\Phi}{\Phi_0}; \quad \quad \Phi_0 = \sqrt{\frac{\hbar}{C_g\omega_0}}; \quad \quad \lambda_{\Phi} = \frac{2W}{\hbar\omega_0} \quad \quad (7b) \ $$
 * $$\xi_i = \frac{i_g}{i_{g0}}; \quad \quad i_{g0} = \sqrt{\frac{\hbar \omega_0}{L_g}}; \quad \quad \lambda_i = \frac{2W}{\hbar\omega_0}. \quad \quad (7c) \ $$

where $$m_0 \ $$ is scaling induced electric-like gravitational mass; $$\Phi_0 \ $$ is scaling induced gravitational torsion field flux and $$i_{g0} \ $$ is scaling induced mass current.

Then the Schrodinger equation will take the form of the differential equation of Chebyshev-Ermidt:
 * $$\left(\frac{d^2}{d\xi^2} + \lambda - \xi^2\right)\Psi = 0. \ $$

The eigen values of the Hamiltonian will be:
 * $$W_n = \hbar \omega_0(n + 1/2), \quad \quad n = 0,1,2,\dots \ $$

where at $$n = 0 \ $$ we shall have zero oscillation:
 * $$W_0 = \hbar \omega_0/2. \ $$

In the general case the scaling mass and torsion flux can be rewritten in the form:
 * $$m_0 = \sqrt{\frac{\hbar}{L_g\omega_0}} = \frac{m_P}{\sqrt{4\pi }}= \sqrt {\frac {\hbar c}{4 \pi G}} ,\ $$
 * $$\Phi_0 = \sqrt{\frac{\hbar}{C_g\omega_0}} = \frac{h}{m_P \sqrt{\pi} } = \sqrt {\frac {4 \pi G \hbar }{ c } }, \ $$

where $$m_P \ $$ is the Planck mass, $$c  \ $$ is the speed of light, $$G  \ $$ is the gravitational constant.

These three equations (4) form the base of the nonrelativistic quantum gravidynamics, which considers elementary particles from the intrinsic point of view. Note that, the standard quantum electrodynamics considers elementary particles from the external point of view.

Gravitational resonator as quantum LC circuit
Due to Luryi density of states (DOS) approach we can define gravitational quantum capacitance as:
 * $$C_g = m_g^2\cdot D_{2D}\cdot S_g, \ $$

and quantum inductance as:
 * $$L_g = \Phi_g^2\cdot D_{2D}\cdot S_g, \ $$

where $$S_g \ $$ is the resonator surface area, $$D_{2D} = \frac{m_0}{\pi \hbar^2} \ $$ is two dimensional (2D) DOS, $$m_0  \ $$ is the carrier mass, $$m_g \ $$ is the induced gravitational mass, and $$\Phi_g \ $$ is the gravitational torsion field flux.

Energy stored on quantum capacitance is:
 * $$W_{Cg} = \frac{m_g^2}{2 C_g } = \frac{1}{2D_{2D}S_g}. \ $$

Energy stored on quantum inductance is:
 * $$W_{Lg} = \frac{\Phi_g^2}{2 L_g }= \frac{1}{2D_{2D}S_g} = W_{Cg}. \ $$

Resonator angular frequency is:
 * $$\omega_{gR} = \frac{1}{\sqrt{L_g C_g}} = \frac{1}{m_g \Phi_g D_{2D}S_g}. \ $$

Energy conservation law for zero oscillation is:
 * $$W_{gR} = \frac {1}{2} \hbar \omega_{gR} = \frac{\hbar }{2 m_g \Phi_g D_{2D}S_g} = W_{Cg} = W_{Lg}. \ $$

This equation can be rewritten as:
 * $$m_g \Phi_g = \hbar . \ $$

Characteristic gravitational resonator impedance is:
 * $$ \rho_g = \sqrt{\frac{L_g}{C_g}} = \frac{\Phi_g}{m_g} = 2 \alpha \frac{\Phi_{g0}}{m_S}= \rho_{g0},  \ $$

where $$ \alpha \ $$ is the fine structure constant, $$\Phi_{g0} = h/m_S  \ $$ is the gravitational torsion flux quantum, $$ h  \ $$ is the Planck constant,  $$m_S  \ $$ is the Stoney mass, $$ \rho_{g0}\ $$ is the gravitational characteristic impedance of free space.

Considering above equations, we can find out the following induced mass and induced gravitational torsion flux:
 * $$m_g = \frac{m_S}{\sqrt{4\pi \alpha}} ,\ $$
 * $$\Phi_g = \sqrt{\frac{\alpha}{\pi}}\frac{h}{m_S} . \ $$

Note, that these induced quantities maintain the energy balance between resonator oscillation energy and total energy on capacitance and inductance
 * $$\hbar \omega_{gR} = W_{gL}(t) + W_{gC}(t). \ $$

Since capacitance oscillations are phase shifted ($$\psi = \pi /2 \ $$) with respect to inductance oscillations, therefore we get:
 * $$W_{gL} = \begin{cases} 0, & \mbox{at }t=0; \psi=0\mbox{ and}\, t=\frac{T_R}{2};\psi=\pi \\ W_L, & \mbox{at }t=\frac{T_R}{4};\psi=\frac{\pi}{4} \mbox{ and} \, t=\frac{3T_R}{4};\psi=\frac{3\pi}{4} \end{cases} \ $$


 * $$W_{gC} = \begin{cases} W_C, & \mbox{at }t=0; \psi=0\mbox{ and} \, t=\frac{T_R}{2};\psi=\pi \\ 0, & \mbox{at }t=\frac{T_R}{4};\psi=\frac{\pi}{4} \mbox{ and} \, t=\frac{3T_R}{4};\psi=\frac{3\pi}{4} \end{cases} \ $$

where $$T_R = \frac{2\pi}{\omega_{gR}} \ $$ is the oscillation period.

Planckion resonator
Planckion radius is:
 * $$r_P = \frac{\lambda_P}{2\pi }, \ $$

where $$\lambda_P = \frac {h}{m_P c} \ $$ is the Compton wavelength of planckion, $$c  \ $$ is the speed of light, $$ m_P  \ $$ is the Planck mass.

Planckion surface scaling parameter is:
 * $$S_P = 2\pi r_P^2 = \frac{\lambda_P^2}{2\pi}. \ $$

Planckion angular frequency is:
 * $$ \omega_P = \frac{m_Pc^2}{ \hbar} = \frac{2\pi c}{\lambda_P}. \ $$

Planckion density of states is:
 * $$D_P = \frac{1}{S_P W_P} = \frac {1}{ S_P \hbar \omega_P } = \frac{m_P}{2\pi \hbar^2}. \ $$

Standard DOS quantum resonator approach yields the following values for the gravitational reactive quantum parameters:
 * $$C_P = m_g^2 D_P S_P = \frac {m_S^2 }{4\pi \alpha} \frac{m_P}{2\pi \hbar^2} \frac{\lambda_P^2}{2\pi} = \frac{ \varepsilon_g \lambda_P }{2\pi } = \frac {m_P}{4\pi c^2}, \ $$

where $$ \varepsilon_g = \frac {1}{4\pi G} \ $$ is the gravitoelectric gravitational constant in the set of selfconsistent gravitational constants, and
 * $$L_P = \Phi_g^2 D_P S_P = \frac {\Phi_0^2}{4 \pi \beta} D_P S_P = \frac {\alpha h^2}{\pi m_S^2 } D_P S_P = \frac {\alpha h^2}{\pi m_S^2 }\frac{m_P}{2\pi \hbar^2} \frac{\lambda_P^2}{2\pi} = \frac{\mu_g \lambda_P }{ 2\pi }, \ $$

where $$ \mu_g = \frac {4 \pi G }{ c^2} \ $$ is the gravitomagnetic gravitational constant of selfconsistent gravitational constants, $$ \beta = \frac {1}{4\alpha} \ $$ is the gravitational torsion coupling constant, which is equal to magnetic coupling constant.

Thus, s.c. free planckion can be considered as discoid quantum resonator which has radius $$ r_P \ $$.

Bohr atom as a gravitational quantum resonator
The gravitational quantum capacitance for Bohr atom is:
 * $$C_\Gamma = m^2_R  D_B S_B = \varepsilon _\Gamma a_B, \  $$

where $$ a_B \ $$ is the Bohr radius, $$S_B = \pi a_B^2 \ $$ is the flat surface area, $$ ~ m_R = \frac { \sqrt {m_p m_e}}{2 \sqrt \pi } $$ is the induced mass, $$ D_B= \frac {m_e}{\pi \hbar^2} \ $$ is the density of states, $$ \varepsilon _\Gamma = \frac {1 }{4 \pi \Gamma } $$ is the gravitoelectric gravitational constant of selfconsistent gravitational constants in the field of strong gravitation, $$\Gamma $$ is the strong gravitational constant, $$ m_p \ $$ and $$ m_e \ $$ are masses of proton and electron.

The gravitational quantum inductance is:
 * $$ L_\Gamma = \phi_\Gamma ^2 D_B S_B = \mu _\Gamma a_B, \  $$

where $$ \mu_\Gamma = \frac {4 \pi \Gamma }{ c^2 } $$ is the gravitomagnetic gravitational constant of selfconsistent gravitational constants in the field of strong gravitation, and the induced gravitational torsion flux is:
 * $$ \phi_\Gamma = \frac {\alpha h }{\sqrt {\pi m_p m_e } }= \frac {2 \alpha \sqrt{m_e} }{ \sqrt {\pi m_p } } \sigma_e = \frac {2 \alpha \sqrt{m_p} }{ \sqrt {\pi m_e } } \Phi_\Gamma = \frac {2 \sqrt{m_p} }{ \alpha \sqrt {\pi m_e } } \Phi_\Omega ,$$

where $$ \sigma_e \ $$ is the velocity circulation quantum, $$ \Phi_\Gamma = \frac{h}{2 m_p} $$ is the strong gravitational torsion flux quantum, which is related to proton with its mass $$ m_p $$.

Here the strong gravitational electron torsion flux for the first energy level is:
 * $$ \Phi_{\Omega } = \Omega S_B = \frac { \mu_\Gamma m_e }{4 \pi a_B } \sigma_e = \frac { \Gamma m_e }{c^2 a_B }\sigma_e  = \frac { \Gamma h }{2c^2 a_B }= \frac { \pi  \alpha \Gamma m_e }{c }= \alpha^2 \Phi_\Gamma, \ $$

where $$ \Omega \ $$ is the gravitational torsion field of strong gravitation in electron disc.

The gravitational wave impedance is:
 * $$ \rho_\Gamma = \sqrt {\frac{ L_\Gamma }{ C_\Gamma }}= \sqrt{\frac {\mu _\Gamma }{\varepsilon _\Gamma }} = \frac{4\pi \Gamma }{c} = 6.346\cdot 10^{21}\, \mathrm {m^2/(s\cdot kg)}. \ $$

The resonance frequency of gravitational oscillation is:
 * $$\omega_\Gamma = \frac{1}{\sqrt{L_\Gamma C_\Gamma }} = \frac{ c}{a_B} = \frac {\omega_B}{\alpha }, \ $$

where $$\omega_B = \frac{c \alpha }{ a_B } \ $$ is the angular frequency of electron rotation in atom.

For the quantum gravitational resonator approach we can derive the following maximal values for the energies stored on capacitance and inductance:
 * $$W_C = \frac{m_R^2}{2 C_\Gamma } = \frac{\hbar \omega_B }{2} = \frac{\alpha \hbar \omega_\Gamma }{2}= W_B, \ $$
 * $$W_L = \frac{ \phi_\Gamma ^2}{2 L_\Gamma } = W_B .\ $$

The energy $$ W_B \ $$ of the wave of strong gravitation in the electron matter has the same value as in case of rotating electromagnetic wave, and can be associated with the mass:
 * $$m_{Bmin} = \frac{W_B}{c^2} = \frac{\hbar \omega_B}{2 c^2} = \frac{\alpha^2}{2}m_e << m_e, \ $$

which could be named as the minimal mass-energy of the quantum resonator.

One way to explain the minimal mass-energy $$m_{Bmin}$$ is the supposition that Planck constant can be used at all matter levels including the level of star. As a result of the approach one should introduce different scales such as Planck scale, Stoney scale, Natural scale, with the proper masses and lengths. But such proper masses do not relate with the real particles.

Another way recognizes the similarity of matter levels and SPФ symmetry as the principles of matter structure where the action constants depend on the matter levels. For example there is the stellar Planck constant at the star level that describes star systems without any auxiliary mass and scales.

Reference Books

 * Johnstone Stoney, Phil. Trans. Roy. Soc. 11, (1881)
 * Stratton J.A.(1941). Electromagnetic Theory. New York, London: McGraw-Hill.p.615. djvu
 * Детлаф А.А., Яворский Б.М., Милковская Л.Б.(1977). Курс физики. Том 2. Электричество и магнетизм (4-е издание). М.: Высшая школа, "Reference Book on Electricity"  djvu
 * Гольдштейн Л.Д., Зернов Н.В. (1971). Электромагнитные поля. 2- издание. Москва: Советское Радио. 664с. "Electromagnetic Fields" djvu