# 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’’. To the contrary to the classical case, there are no any gravitational waves and radiation losses in the QGR, but “infinite” phase shifted oscillation of electric-like and magnetic-like gravitational fields, 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.

### Electrodynamic resonators

It is happened, that such physical values as capacitance and inductance have no interest in the modern Quantum electrodynamics. Furthermore, they are neglected even in classical electrodynamics, where electric and magnetic fields are dominated. The point is that, the last are not included in the evident form in the Maxwell equations, and so the resulting solutions includes fields only. Yes, sometimes these coefficients were obtained from the solutions of Maxwell equations, but it were very rarely, and relation to them were consideringly low. It is known too, that s.c. “field approach” in electrodynamics that considered “point charges” leads to the “fool infinities”, when the interaction radius trends to zero. Furthermore, these “fool infinities” are presented in the quantum electrodynamics too, where power methods are developed to compensate them. Contrary to the theoretical physics, the applied physics are widely used reactive parameters, such as capacitance and inductance, firstly in the electrotechnics and then in the applied radiotechnics. Now reactive parameters are widely used in the information technologies, which are based on the generation, transmission and radiation of electromagnetic waves of different frequencies.

The present day situation (without proper development of the theory of reactive parameters such as inductance, capacitance and electromagnetic resonator) brakes developments of information technologies and quantum computing. Note that, mechanical harmonic oscillator was considered in quantum mechanics in the early 30-ies of 20-th century, when the quantum theory is developed. However, the quantum consideration of the $LC - \$ circuit was started only by Louisell(1973) [1]. Since then, there were no practical examples of the quantum capacitance and inductance, therefore this approach did not obtain proper consideration. Theoretically correct introduction of the quantum capacitance, based on the density of states, first was presented by Luryi (1988) [2] for QHE. However, Luryi did not introduce the quantum inductance, and this approach was not considered in the quantum LC circuit and resonator. Year later, Yakymakha (1989) [3] considered an example of the series and parallel quantum $LC - \$ circuits (its characteristic impedances) during QHE explanation (integer and fractional). However in this work did not considering the Schrodinger equation for the quantum LC circuit.

For the first time, both quantum values, capacitance and inductance, were considered by Yakymakha (1994) [4], during spectroscopic investigations of MOSFETs at the very low frequencies (sound range). The flat quantum capacitances and inductances here had thicknesses about Compton wave length of electron and its characteristic impedance – the wave impedance of free space. And three year later, Devoret (1997) [5] presented complete theory of the Quantum LC Circuit (applied to the Josephson junction). Possible application of the quantum LC circuits and resonators in the quantum computation are considered by Devoret (2004) [6].

### Gravitational resonators

Due to McDonald[7] first who used Maxwell equations to describe gravity was Oliver Heaviside[8] The point is that in the weak gravitational field the standard theory of gravity could be written in the form of Maxwell equations[9] It is evident that in 19-th century there was no SI units, and therefore the first mention of the gravitational constants possibly due to Forward (1961)[10]

In the 80-ties Maxwell-like equations were considered in the Wald book of general relativity[11] In the 90-ties Kraus [12] first introduced the gravitational characteristic impedance of free space, which was detaled later by Kiefer [13], and now Raymond Y. Chiao[14] [15] [16] [17] [18] who is developing the ways of experimental determination of the gravitational waves.

## Classical gravitational resonator

In the general case classical gravitational resonator (CGR) is the cavity in the 3D-space. Therefore, CGR has infinite resonance frequencies, due to the three dimensions.

To the contrary of classical gravitational LC circuit, both electric-like and magnetic-like gravitational fields are displaced in the same volume of CGR. These oscillating of electric-like and magnetic-like gravitational fields in the classical case are like standing waves, that form gravitational waves, that could be radiated in the external world.

### Gravitational LC circuit

Gravitational voltage on gravitational inductance is:

$V_{GL} = -L_G\cdot \frac{d I_{GL}}{d t}. \$

Gravitational current through gravitational capacitance is:

$I_{GL} = C_G\cdot \frac{d V_{GL}}{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} \$
$\frac{d I_{GL}}{d t} = C_G\frac{d^2V_{GL}}{dt^2}. \$

Considering the following relationships for amplitudes of "voltages" and "currents":

$V_{GL} = V_{GC}; I_{GL} = I_{GC} \$

we obtain the following differential equation for gravitational oscillations:

$\frac{d^2 I_G}{dt^2} + \frac{1}{L_GC_G}I_G = 0. \quad \quad \quad \quad \quad (1) \$

Furthermore, considering the following relationships between "voltage" with "charges":

$q_G = C_GV_G \$

and "current" with "magnetic flux":

$\phi_G = L_GI_G \$

the above oscillation equation could be rewritten in the charge form:

$\frac{d^2 q_G}{dt^2} + \frac{1}{L_GC_G}q_G = 0. \quad \quad \quad \quad \quad (2) \$

This equation has the partial solution:

$q_G(t) = A_0 Sin (\omega_{LC}t) \$

where

$\omega_{LC} = \frac{1}{\sqrt{L_GC_G}} \$

is the resonance frequency, and

$\rho_{LC} = \sqrt{\frac{L_G}{C_G}} \$

is the gravitational characteristic impedance.

Note that, in the general case the gravitational charge($q_G \$) has the same dimensions as the gravitational mass ($m_G \$). For the sake of completeness we need to present the third differential equation for "rotor gravitational mass" (or flux) in the form:

$\frac{d^2 \phi_G}{dt^2} + \frac{1}{L_GC_G}\phi_G = 0, \quad \quad \quad \quad \quad (3) \$

where the following relationship between gravitational current and magnetic-like gravitational flux is considered:

$\phi_G = L_Gi_G. \$

## Quantum general approach

### Quantum gravitational LC circuit oscillator

Inductance momentum quantum operator in the electric-like gravitational mass space could be presented in the following form:

$\hat p_{GL} = -i\hbar \frac{d}{dq_G}, \quad \quad \quad \quad \quad \hat p_{GL}^* = i\hbar \frac{d}{dq_G},\quad \quad \quad \quad \quad (4a) \$

where $\hbar- \$ is reduced Plank constant, $\hat p_{GL}^*- \$ is the complex-conjugate momentum operator. Capacitance momentum quantum operator in the magnetic-like gravitational mass space could be presented in the following form:

$\hat p_{G\phi} = -i\hbar \frac{d}{d\phi_G}, \quad \quad \quad \quad \quad \hat p_{C\phi}^* = i\hbar \frac{d}{d\phi_G},\quad \quad \quad \quad \quad (4b) \$

where $\phi \$ is the induced magnetic charge. Considering the fact, that there are no free magnetic-like gravitational masses, but it could be immitated by electric-like gravitational mass current ($i_G \$):

$\phi_G = L_G\cdot iG, \$

we can introduce the third momentum quantum operator in the current form:

$\hat p_{Ci} = -\frac{i\hbar}{L_G} \frac{d}{di_G}, \quad \quad \quad \quad \quad \hat p_{Ci}^* = \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_{GLq} = -\frac{\hbar^2}{2L_G}\cdot \frac{d^2}{dq_G^2} + \frac{L_G\omega_0^2}{2}\hat q_G^2 \quad \quad \quad \quad \quad (5a) \$
$\hat H_{GC\phi} = -\frac{\hbar^2}{2C_G}\cdot \frac{d^2}{d\phi_G^2} + \frac{C_G\omega_0^2}{2}\hat \phi_G^2 \quad \quad \quad \quad \quad (5b) \$
$\hat H_{GCi} = -\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 $\omega_0 = \frac{1}{\sqrt{L_GC_G}} \$ is the resonance frequency. We consider the case without dissipation ($R_G = 0 \$). The only difference of the gravitational charge spaces and gravitational current spaces from the traditional 3D- coordinate space is that it are 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}{dq_G^2} + \frac{L_G\omega_0^2}{2}q_G^2\Psi = W\Psi \quad \quad \quad \quad \quad (6a) \$
$-\frac{\hbar^2}{2C_G}\frac{d^2 \Psi}{d\phi_G^2} + \frac{C_G\omega_0^2}{2}\phi_G^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_q = \frac{q_G}{q_{G0}}; \quad \quad q_{G0} = \sqrt{\frac{\hbar}{L_G\omega_0}}; \quad \quad \lambda_q = \frac{2W}{\hbar\omega_0} \quad \quad (7a) \$
$\xi_{\phi} = \frac{\phi_G}{\phi_{G0}}; \quad \quad \phi_{G0} = \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 $q_{G0} \$ is scaling "induced electric-like gravitational charge"; $\phi_{G0} \$ is scaling "induced magnetic-like gravitational charge" and $i_{G0} \$ is scaling "induced electric current". Then the Schrodinger equation will take the form of the differential equation of Chebyshev-Ermidt:

$(\frac{d^2}{d\xi^2} + \lambda - \xi^2)\Psi = 0. \$

The eigen values of the Hamiltonian will be:

$W_n = \hbar \omega_0(n + 1/2), \quad \quad n = 0,1,2,.. \$

where at $n = 0 \$ we shall have zero oscillation:

$W_0 = \hbar \omega_0/2. \$

In the general case the scaling charges could be rewritten in the form:

$q_{G0} = \sqrt{\frac{\hbar}{L_G\omega_0}} = \frac{m_{\alpha}}{\sqrt{4\pi \alpha }} \$
$\phi_{G0} = \sqrt{\frac{\hbar}{C_G\omega_0}} = \sqrt{\frac{\alpha}{\pi}}\cdot \frac{h}{m_{\alpha}}, \$

where

$m_{\alpha} = e\sqrt{\frac{\epsilon_G}{\epsilon_E}} = \frac{e}{\sqrt{4\pi G\epsilon_E}} = \sqrt{\alpha_E}\cdot m_P, \$

where $m_P = \sqrt{\frac{\hbar c}{G}}\$ is the Planck mass, first proposed by George Johnstone Stoney (1881), before the quantum theory was created.. This mass scale is now called the Stoney scale. Other designations are as follows: $m_P = \sqrt{\frac{\hbar c}{G}}\$ is the Planck mass, first proposed by George Johnstone Stoney (1881), before the quantum theory was created.. This mass scale is now called the Stoney scale; $\epsilon_G = \frac{1}{4\pi G} = 1.192708\cdot 10^9 kg/N\cdot m^2 \$ is the gravitoelectric permittivity (like electric constant); G is the gravitational constant, c is the speed of light in a vacuum, and $\mu_G = \frac{4\pi G}{c^2} = 9.328772\cdot 10^{-27} n\cdot s^2/kg \$ is the gravitomagnetic permeability (like magnetic constant); $h \$ is the Planck's constant, $\alpha_E = \frac{e^2}{2\epsilon_E hc}$ is the electric fine structure constant for electric charge quantum - $e \$(electron charge)

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 conciders elementary particles from the external point of view.

#### Planckion as quantum oscillator

Suppose that planckion has its mass defined by the quantum oscillations. Thus, its mass will be equivalent to the zero oscillations of the quantum gravitational LC circuit:

$0.5\hbar \omega_0 = m_Pc^2. \$

So, the oscillation length will be:

$x_0 = \frac{\lambda_P}{2\pi\sqrt{2n}}, \$

where $\lambda_P - \$ is the Compton wavelength of planckion. Further, let us consider that this gravitational mass is uniformly distributed on the sphere:

$S_P = 4\pi x_0^2 = \lambda_P^2/2\pi. \$.

Then we can find out the density of states for this mass:

$D_P = \frac{1}{S_P}\frac{1}{m_Pc^2} = \frac{m_P}{2\pi \hbar^2}. \$

Thus, presentation of the planckion gravitational mass as the zero oscillation of the harmonic oscillator yields to the uniform its distribution on the sphere or the $\lambda_P /2\pi \sqrt{2}$ radius.

#### Graviton as quantum oscillator

As is known, graviton momentum is defined as:

$p_g = \frac{\hbar \omega_g}{c}, \$

where $c - \$ is velocity of light. So, the "effective (energy) graviton mass" could be defined as:

$m_g = \frac{\hbar \omega_g}{c^2}. \$

Then the length scaling parameter of harmonic oscillator will be:

$x_0 = \sqrt{\frac{\hbar}{m_g\omega_g}} = \frac{\lambda_g}{2\pi}, \$

where $\lambda_g = \frac{2\pi c}{\omega_g}- \$ is the graviton wavelength.

### Gravitational resonator as quantum LC circuit

Due to Luryi density of states (DOS) approach we can define gravitational quantum capacitance as:

$C_{QG} = q_G^2\cdot D_{2D}\cdot S_G, \$

and quantum inductance as:

$L_{QG} = \phi_G^2\cdot D_{2D}\cdot S_G, \$

where $S_G - \$ resonator surface area, $D_{2D} = \frac{m}{\pi \hbar^2} - \$ two dimensional (2D) DOS, $q_G - \$ electric-like gravitational mass (or flux), and $\phi_G - \$ magnetic-like gravitational mass (or flux). Note that these fluxes should be defined afterward.

Energy stored on quantum capacitance:

$W_{CG} = \frac{q_G^2}{2D_{2D}S_G}. \$

Energy stored on quantum inductance:

$W_{LG} = \frac{\phi_G^2}{2D_{2D}S_G} = W_{CG}. \$

Resonator angular frequency:

$\omega_{QGR} = \frac{1}{\sqrt{L_{QG}C_{QG}}} = \frac{1}{q_G\phi_GD_{2D}S_G}. \$

Energy conservation law:

$W_{QGR} = \hbar \omega_{QGR} = \frac{1}{q_G\phi_GD_{2D}S_G} = W_{CG} = W_{LG}. \$

This equation can be rewritten as:

$q_G\phi_G = \hbar, \$

from which it is evident that these "gravity charges" are the fluxes, but not metallurgic charges.

Characteristic gravitational resonator impedance:

$\rho_{QG} = \sqrt{\frac{L_{QG}}{C_{QG}}} = \frac{\phi_G}{q_G} = 2\alpha \frac{\phi_{G0}}{e}=2\alpha R_{GH}, \$

where $\phi_{G0} = h/m_{\alpha} - \$ magnetic-like gravitational flux quantum. Considering above equations, we can find out the following electric-like and magnetic-like set of gravitational induced fluxes:

$q_G = \frac{m_{\alpha}}{\sqrt{4\pi \alpha}} \$
$\phi_G = \sqrt{\frac{\alpha}{\pi}}\frac{h}{m_{\alpha}} . \$

Note, that these values are not the real "metallurgic charges", but the maximal fluxes, that maintain the energy balance between resonator oscillation energy and total energy on capacitance and inductance

$\hbar \omega_{QGR} = W_{QGL}(t) + W_{QGC}(t). \$

Since capacitance oscillations are phase shifted ($\psi = \pi /2 \$) with respect to inductance oscillations, therefore we get:

$W_{QGL} = \begin{cases} 0, & \mbox{at }t=0; \psi=0\mbox{ and} t=\frac{T_{QR}}{2};\psi=\pi \\ W_{QL}, & \mbox{at }t=\frac{T_{QR}}{4};\psi=\frac{\pi}{4} \mbox{ and}t=\frac{3T_{QR}}{4};\psi=\frac{3\pi}{4} \end{cases} \$
$W_{QGC} = \begin{cases} W_{QC}, & \mbox{at }t=0; \psi=0\mbox{ and} t=\frac{T_{QR}}{2};\psi=\pi \\ 0, & \mbox{at }t=\frac{T_{QR}}{4};\psi=\frac{\pi}{4} \mbox{ and}t=\frac{3T_{QR}}{4};\psi=\frac{3\pi}{4} \end{cases} \$

where $T_{QR} = \frac{2\pi}{\omega_{QR}}- \$ is oscillation period.

## Applications

### Planckion resonator

$r_P = \frac{\lambda_P}{2\pi \sqrt{2}}. \$

Planckion surface scaling parameter:

$S_P = 4\pi r_P^2 = \frac{\lambda_P^2}{2\pi}. \$

Planckion angular frequency:

$\omega_P = \frac{m_Pc^2}{\hbar} = \frac{2\pi c}{\lambda_P}, \$

where $c - \$ is the velocity of light. Planckion density of states:

$D_P = \frac{1}{S_gW_g} = \frac{m_P}{2\pi \hbar^2}. \$

Standard DOS quantum resonator approach yields the following values for the graviton reactive quantum parameters:

$C_{QRP} = q_{G}^2D_gS_g = \frac{\alpha_G}{\alpha_E}\frac{\epsilon_G}{\lambda_P}\frac{\lambda_P^2}{2\pi} = \frac{\epsilon_G}{\lambda_P}S_P, \$

where $\alpha_E = \alpha_G \$ is considered, and

$L_{QRP} = \phi_{G}^2D_eS_e = 4\alpha_E\beta_G \frac{\mu_G\lambda_P}{2\pi} = \frac{\mu_G}{\lambda_P}S_P, \$

where $4\alpha_E\beta_G = 1 \$ is considered. Thus, s.c. "free planckion" could be considered as spherical quantum resonator which has radius $r_P \$ and thickness $\lambda_P \$.

### Graviton resonator

$r_g = \frac{\lambda_g}{2\pi}. \$

Graviton surface scaling parameter:

$S_g = 4\pi r_g^2 = \frac{\lambda_g^2}{\pi}. \$

Graviton angular frequency:

$\omega_g = \frac{2\pi c}{\lambda_g}, \$

where $c - \$ is the velocity of light. Graviton density of states:

$D_g = \frac{1}{S_gW_g} = \frac{\pi}{\lambda_ghc}. \$

Standard DOS quantum resonator approach yields the following values for the photon reactive quantum parameters:

$C_{QRg} = q_{G}^2D_gS_g = \frac{\alpha_G}{\alpha_E}\frac{\epsilon_G}{2\lambda_g}\frac{\lambda_g^2}{\pi} = \frac{\epsilon_G}{2\lambda_g}S_g \$

where $\alpha_E = \alpha_G \$ is considered, and

$L_{QRg} = \phi_{G}^2D_gS_g = 4\alpha_E\beta_G \frac{\mu_G\lambda_g}{2\pi} = \frac{\mu_G}{2\lambda_p}S_e. \$

where $4\alpha_E\beta_G = 1 \$ is considered. Thus, s.c. "free graviton" could be considered as spherical quantum resonator which has radius $r_g \$ and thickness $2\lambda_g \$.

## References

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2. Serge Luryi (1988). "Quantum capacitance device". Appl.Phys.Lett. 52(6). Pdf
3. Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p.91. ISBN 5-11-002309-3. djvu</
4. Yakymakha O.L., Kalnibolotskij Y.M. (1994). "Very-low-frequency resonance of MOSFET amplifier parameters". Solid- State Electronics 37(10),1739-1751 Pdf
5. Deboret M.H. (1997). "Quantum Fluctuations". Amsterdam, Netherlands: Elsevier. pp.351-386. Pdf
6. Devoret M.H., Martinis J.M. (2004). "Implementing Qubits with Superconducting Integrated Circuits". Quantum Information Processing, v.3, N1. Pdf
7. K.T. McDonald, Am. J. Phys. 65, 7 (1997) 591-2.
8. O. Heaviside, Electromagnetic Theory (”The Electrician” Printing and Publishing Co., London, 1894) pp. 455-465.
9. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, MA, 1955), p. 168, 166.
10. R. L. Forward, Proc. IRE 49, 892 (1961).
11. R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
12. J. D. Kraus, IEEE Antennas and Propagation. Magazine 33, 21 (1991).
13. C. Kiefer and C. Weber, Annalen der Physik (Leipzig) 14, 253 (2005).
14. Raymond Y. Chiao. "Conceptual tensions between quantum mechanics and general relativity: Are there experimental consequences, e.g., superconducting transducers between electromagnetic and gravitational radiation?" arXiv:gr-qc/0208024v3 (2002). [PDF
15. R.Y. Chiao and W.J. Fitelson. Time and matter in the interaction between gravity and quantum fluids: are there macroscopic quantum transducers between gravitational and electromagnetic waves? In Proceedings of the “Time & Matter Conference” (2002 August 11-17; Venice, Italy), ed. I. Bigi and M. Faessler (Singapore: World Scientific, 2006), p. 85. arXiv: gr-qc/0303089. PDF
16. R.Y. Chiao. Conceptual tensions between quantum mechanics and general relativity: are there experimental consequences? In Science and Ultimate Reality, ed. J.D. Barrow, P.C.W. Davies, and C.L.Harper, Jr. (Cambridge:Cambridge University Press, 2004), p. 254. arXiv:gr-qc/0303100.
17. Raymond Y. Chiao. "New directions for gravitational wave physics via “Millikan oil drops” arXiv:gr-qc/0610146v16 (2009). PDF
18. Stephen Minter, Kirk Wegter-McNelly, and Raymond Chiao. Do Mirrors for Gravitational Waves Exist? arXiv:gr-qc/0903.0661v10 (2009). PDF

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