The flying fish model:
quantum spinor dynamics
and gravity


The purpose of the Flying Fish model is to investigate the interactions between quantum spinor dynamics and gravity. He investigates how quantum fields can affect space-time through their interactions, creating localized temporal variations and gravitational effects. These fields obey the Dirac equation:

\[(i \gamma^\mu \nabla_\mu – m) \psi = 0\]

where \(\gamma^\mu\) represents the gamma matrices for spinors in curved spacetime, \(\nabla_\mu\) is the covariant derivative, and \(m\) is the spinor field mass. This equation is the basis for describing the interactions in the model. Spinor fields can create coherent oscillations between different spin states. This oscillatory behavior, similar to the Josephson effect observed in superconductors [14], enables complex dynamics in quantum systems that respond to initial conditions and external magnetic fields. Under certain conditions, these systems can spontaneously transition into new quantum states [2].


The time gradient

The Flying Fish model introduces the concept of space-time gradients created by quantum transitions in spinor fields. The time gradient, represented by the symbol \(\mathcal{G}_t\), involves interactions between the spinor field energy and topological stabilization [13]. It is defined as:

\[\mathcal{G}_t = T_0 \left( 1 + \alpha_s \frac{\rho_{\text{spinor}}(r) \cdot L_P}{c^2 \cdot (1 + \beta \mathcal{T}(x))} \right)\]

where:
– \(T_0\): reference time (time flow in the absence of quantum transitions),
– \(\alpha_s\): dimensionless coefficient quantifying the effect of the spinor field energy density on the time flow,
– \(\rho_{\text{spinor}}(r)\): local spinor field energy density,
– \(L_P\): Planck length,
– \(\beta\): coefficient determining the stabilizing effect of quantum knots,
– \(\mathcal{T}(x)\) is a factor that modulates the effect of the spinor field energy density [9

The topological invariant \(\mathcal{T}(x)\) adapts based on local time gradients:

\[\mathcal{T}(x) = \mathcal{T}_0 \cdot \exp\left(-\frac{\mathcal{G}_t}{T_0}\right).\]

The stabilizing effect of quantum knots weakens in regions with extreme time gradients, thus preserving the overall coherence of spacetime. To integrate the effect of spinor fields, the model modifies the equations of general relativity with dynamical contributions of spinor fields and topological invariants to include a detailed definition of \(f(\rho_{\text{spinor}}, \mathcal{G}_t)\), which captures the effect of the spinor energy density and time gradients:

\[G_{\mu \nu} + \alpha_s \cdot \rho_{\text{spinor}} \cdot \left( 1 + \frac{\mathcal{G}_t}{T_0} \right) + \mathcal{ T}(x) = \frac{8 \pi G}{c^4} T_{\mu \nu}.\]

here:
– \(\alpha_s \cdot \rho_{\text{spinor}} \cdot \left( 1 + \frac{\mathcal{G}_t}{T_0} \right)\) quantifies the effect of the spinor energy density on the curvature of spacetime,
– \(\mathcal{T}(x)\) stabilization invariant representing quantum knots.

This modification minimizes the impact of quantum transitions while ensuring compatibility with the strong equivalence principle [4]. After this modification, it is possible to analyze the spatial geometric properties of spacetime affected by these phenomena. The function for the curvature of spacetime, defined as:

\[P^2 = \int \left(G_{\mu\nu} + f(\rho_{\text{spinor}}, \mathcal{G}_t) + \mathcal{T}(x)\right) dx^\mu dx^\nu,\]

allows the calculation of the resulting geodesic distance between points in this modified spacetime. This relation captures the contributions to the curvature resulting from the energy density of the spinor fields and the effect of quantum knots. The function \(P^2\) provides a geometric interpretation of the modified spacetime and can be used to analyze the local or global effects of the spinor field dynamics on the structure of spacetime. This approach allows for the explicit study of the spatial aspects of the interactions within the model.


Energy transfer between states of spinor fields

The energy density \(\rho_{\text{spinor}}(r)\) of the spinor field is critical in generating spacetime gradients. It is expressed as:

\[\rho_{\text{spinor}}(r) = \langle \bar{\psi}(r) \gamma^0 \psi(r) \rangle\]

where:
– \(\bar{\psi} = \psi^\dagger \gamma^0\) is the spinor conjugate,
– \(\langle \cdots \rangle\) denotes the quantum expectation value [13].

Energy transfer occurs due to transitions between accelerated and decelerated states of the spinor field, as it adjusts to changes in the local flow of time. This energy exchange is quantified by:

\[\Delta E = \hbar \omega \cdot \left( \frac{\mathcal{G}_t – T_0}{T_0} \right)\]

where:
– \(\mathcal{G}_t\) is the unified time gradient, representing the local variation in time flow [14],
– \(T_0\) is the reference time in the absence of quantum transitions,
– \(\hbar \omega\) characterizes the energy associated with the quantum transition.

This formulation describes how the energy of the spinor field dynamically adapts to the local space-time conditions. At the event horizon, the interaction between spinor fields and extreme time gradients leads to intense energy exchanges. The stabilizing effect \(\mathcal{T}(x)\) mitigates the accumulation, but also facilitates the release of the stored energy in the form of quantum radiation. These phenomena can be interpreted as the emergence of a Hawking firewall, which represents a region of high energy density near the event horizon. The energy transferred between spinor field states increases with the time gradient (\(\mathcal{G}_t\)). The relation for \(\Delta E\) is defined in the basic part of the model and serves as the basis for this extension. The energy density of the spinor fields (\(\rho_{\text{spinor}}(r)\)) and the time gradients (\(\mathcal{G}_t\)) contribute to the energy accumulation, described as follows:

\[E_{\text{accum}} = \int_{r_{\text{horizon}}} \rho_{\text{spinor}}(r) \cdot \mathcal{G}_t \, dr.\]

The stabilization invariant \(\mathcal{T}(x)\) dynamically responds to extreme time gradients, as described in the model stabilization section. This limits fluctuations and regulates the energy accumulation. The subsequent formation of the firewall is the result of the energy accumulation and release. The energy density near the event horizon is given by:

\[\rho_{\text{firewall}}(r) = \rho_{\text{spinor}}(r) \cdot \left( 1 + \frac{\mathcal{G}_t}{T_0} \right) \cdot \mathcal{T}(x),\]

and the total energy of the firewall is:

\[E_{\text{firewall}} = \int_{r_{\text{horizon}}} \rho_{\text{firewall}}(r) \, dr.\]


Equations for gravitational and inertial mass

In the model, gravitational and inertial mass are emergent properties that depend on local spatiotemporal conditions and the stabilizing effects of quantum knots. These weights are expressed as:

\[M_{\text{grav}} = \frac{1}{2 \alpha_s \cdot \rho_{\text{spinor}}} \left( 1 – \frac{\mathcal{G}_t}{T_0} \right) \cdot (1 + \delta \mathcal{T}(x))\]

\[M_{\text{inert}} = \frac{r c^4}{\rho_{\text{spinor}}} \left( \frac{\mathcal{G}_t}{T_0} – 1 \right) \cdot (1 + \delta \mathcal{T}(x))\]

where:
– \(\mathcal{G}_t\) is the unified time gradient,
– \(\delta\) represents the stabilizing influence of quantum knots on gravitational and inertial mass,
– \(\mathcal{T}(x)\) is the topological invariant representing quantum knots,
– \(\rho_{\text{spinor}}\) is the energy density of the spinor field,
– \(T_0\) is the reference time in the absence of quantum transitions,
– \(r\) is a spatial factor,
– \(c\) is the speed of light.

These equations show that mass is not an intrinsic property but a dynamical result of interactions between spinor fields. Quantum nodes stabilize these interactions by reducing fluctuations and maintaining coherence in the system. This relation can be formalized using the effective gravitational Hamiltonian:

\[\hat{H}_{\text{grav}} = \int d^3x \, \left( \rho_{\text{spinor}} \cdot \mathcal{G}_t \cdot \mathcal{T} (x) \right)\]

where:
– \(\hat{H}_{\text{grav}}\) is the effective gravitational Hamiltonian associated with the energy density of the spinor field.

The \(\alpha_s\) parameter quantifies how the energy density from quantum transitions in spinor fields affects the gravitational acceleration. This relationship is given by:

\[\alpha_s = \frac{g_{\text{spinor}}}{\rho_{\text{spinor}}}\]

where:
– \(g_{\text{spinor}}\) is the gravitational acceleration influenced by the energy density of the spinor field,
– \(\rho_{\text{spinor}}\) is the local energy density of the spinor field.

Higher \(\rho_{\text{spinor}}\) energy densities amplify gravitational effects in specific regions, while \(\alpha_s\) provides a direct measure of how quantum transitions and spinor fields affect their dynamics. This ensures that the model accurately reflects the interplay between quantum effects and gravitational phenomena.


Planck scale in the model

In this model, the Planck length is the natural limit where quantum effects create an extreme time gradient [10]. The high energy density of spinor fields in this scale leads to a slowing down of the flow of time, effectively freezing the dynamics of quantum processes and making them unobservable by classical methods. Consequently, quantum processes in this region appear to be macroscopically static. This extreme time dilation is not stable. The high energy density at the Planck scale induces quantum fluctuations that can lead to the creation and annihilation of virtual particles in accordance with the Heisenberg uncertainty principle. These fluctuations are described by the expression [12]:

\[\langle (\Delta \phi)^2 \rangle = \frac{\rho_{\text{spinor}}}{E} \cdot \left(1 + \frac{\mathcal{G}_t}{T_0}\right).\]

For extreme conditions where nonlinear effects are needed:

\[\langle (\Delta \phi)^2 \rangle = \frac{\rho_{\text{spinor}}}{E} \cdot \exp\left(\frac{\mathcal{G}_t}{T_0}\right).\]

where \(\delta \phi\) represents the quantum fluctuation of the field, and \(\hbar\) is the reduced Planck constant.

These quantum effects affect space-time at the smallest scales and form the basis for the quantum nature of gravitational phenomena. Adding energy increases their stability, this stabilization is reflected in the lifetime of quantum structures:

\[\tau = \tau_0 \cdot \left( 1 + \frac{\rho_{\text{spinor}} \cdot E \cdot L_P^4}{\hbar \cdot c \cdot M_P} \cdot \frac{\mathcal{G}_t}{T_0} \right)\]

where:
– \(\tau\) is the lifetime of the structure,
– \(\tau_0\) is the base lifetime without added energy,
– \(E\) is the added energy,
– \(M_P\) is the Planck mass,
– \(c\) is the speed of light,
– \(\mathcal{G}_t\) is the unified time gradient.


Conclusion

In the Flying Fish model, spinor fields and their quantum transitions play a key role in generating gravitational effects. Quantum knots stabilize this dynamic, maintain coherence, and allow gravity to emerge as a dynamic expression of spinor field interactions. This approach considers gravity to be a consequence of changes in the space-time flow rather than an inherent property of matter in accordance with quantum principles and maintaining compatibility with established gravitational laws.


Design of the experimental part of the Flying Fish model

The aim of the experiment is to determine whether coherent magnetic oscillations, generated by closely spaced harmonic frequencies, can accelerate quantum precession in spinor fields. By generating pulse frequencies, the experiment could induce time dilation effects similar to weak gravitational fields [11]. Helmholtz coils will generate precise magnetic oscillations to modulate spinor fields, while NMR spectroscopy will measure changes in proton precession. The selected frequencies are:

– First frequency: \( f_1 = 127.740000 \, \text{MHz} \),
– Second frequency: \( f_2 = 127.74000563 \, \text{MHz} \),
– Third frequency: \( f_3 = 127.74001126 \, \text{MHz} \).

These frequencies are selected to produce interference (beat) frequencies within the Hz range, specifically around the desired value of 5.63 Hz. The differences between these close frequencies generate low-frequency components that control the quantum transitions of the spinor fields:

\[f_{\text{beat1}} = f_2 – f_1 = 5.63 \, \text{Hz}, \quad f_{\text{beat2}} = f_3 – f_2 = 5.63 \, \text{Hz}, \quad f_{\text{beat3}} = f_3 – f_1 = 11.26 \, \text{Hz}.\]

These low-frequency beat components are critical for achieving the desired modulation of the spinor field dynamics, facilitating quantum transitions that align with the local time gradients, leading to time dilation effects similar to those experienced in Earth’s gravitational field. The coherent manipulation of these frequencies is essential to exploring the relationship between quantum transitions in spinor fields and their impact on local gravitational behavior.


Generating magnetic pulses

Helmholtz coils will be used to generate the magnetic oscillations, ensuring a homogeneous magnetic field that interacts precisely with the spinor fields. The three coils will generate harmonic frequencies \( f_1 \), \( f_2 \), and \( f_3 \), creating coherent magnetic oscillations crucial for influencing the dynamics of the spinor fields. The magnetic pulses will be synchronized with a precision of ±0.1 radians to ensure accurate modulation of the spinor fields. The pulse settings are as follows:

– Pulse duration: 10 microseconds,
– Interval between pulses: 20 microseconds,
– Number of cycles per pulse: 100 cycles per pulse.

By creating interference between the frequencies \( f_1 \), \( f_2 \), and \( f_3 \), low-frequency beat components will be generated, which are essential for achieving the desired spinor state transition. The differences between these frequencies will produce beat frequencies in the Hz range, specifically 5.63 Hz, enabling controlled modulation of spinor flipping and the associated quantum transitions. These low-frequency components are tuned to a target frequency, calculated based on experimental conditions, to match the gravitational effects induced by the spinor fields, analogous to the weak gravitational acceleration observed in orbit (about 8.7 m/s²).


Sample and environment preparation

The material to be tested will be titanium hydride (TiH2) or palladium hydride (PdHx), as these materials efficiently bind hydrogen atoms and provide a sufficient number of protons. The experiment will be performed in a cryogenic environment with the temperature maintained below 10 K to ensure the stability of the quantum states and to minimize thermal noise. The sample will be placed in the center of the experimental setup, between three Helmholtz coils that will create a homogeneous magnetic field.

To verify the predictions of the Flying Fish model, precise measurements of the temporal gradients and their effects on local dynamics are needed. One possible approach involves detecting changes in proton precession frequencies due to changes in the temporal gradient. The relationship is expressed as:

\[\Delta \omega = \omega_{\text{prec}} – \omega_L \propto \mathcal{G}_t,\]

where:
– \(\Delta \omega\): Change in the precession frequency caused by the spinor field dynamics,
– \(\omega_{\text{prec}}\): Adjusted precession frequency affected by the time gradient,
– \(\omega_L\): Larmor frequency, which represents the fundamental precession frequency without external influence,
– \(\mathcal{G}_t\): Time gradient.

This proportional relationship determines that changes in \(\mathcal{G}_t\) directly affect the proton precession frequencies. By generating controlled magnetic fields via Helmholtz coils, it is possible to induce specific time gradients. Using NMR spectroscopy [7], these frequency shifts can be precisely measured, providing experimental confirmation of the influence of spinor field dynamics on the properties of local spacetime. After the magnetic fields are turned off, the gravitational field should return to its original state, confirming that the observed gravitational changes are directly related to the modulation. A successful experiment would confirm the model’s assumption that quantum transitions in spinor fields are responsible for the observed gravitational variations.


Applications

This experiment has the potential to develop technology for controlling gravitational fields, which could impact object manipulation, space exploration, and energy optimization. By creating stable conditions that change gravitational fields in controlled environments, the Flying Fish model lays a foundation for future gravity manipulation technologies, including applications in quantum material engineering and fusion reactions.


Author

Madala Roman I madala@modifiedplanck.com I October, 2024


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