Below is a curated list of the 10 most significant equations in Flux Field Theory (FFT), each playing a critical role in unifying classical, quantum, and cosmological phenomena. Click on each equation title to toggle its description.
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Aether Field Dynamics
\[ \Box \rho_A - k \rho_A - \kappa \rho_A^3 e^{-\rho_A^2/\rho_c^2} + \frac{\kappa}{\rho_c^2} \rho_A^5 e^{-\rho_A^2/\rho_c^2} + \eta f'(\rho_A, \rho_c) \partial_t \rho_A = \beta(\rho_m) \rho_m \]
Governs the evolution of the Aether Field \( \rho_A \), driving its interactions with mass-energy density \( \rho_m \).
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Aether Field Distribution
\[ \rho_A(r) = \min\left( \frac{\beta(\rho_m) \rho_m(r)}{m_A^2} e^{-r/R_A}, 1.5 \times 10^{-5} \right) \, \text{GeV}^4 \]
Describes the spatial distribution of \( \rho_A \), peaking at \( \rho_c \) in dense regions and decaying over the characteristic scale \( R_A \).
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Normalized Aether Field
\[ \phi(\mathbf{x}, t) = \frac{\rho_A(\mathbf{x}, t)}{\rho_c} \]
Defines the normalized field \( \phi \), unifying classical and quantum scales, with \( \rho_c = 1.5 \times 10^{-5} \, \text{GeV}^4 \).
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Normalized Field Dynamics
\[ \Box \phi - k \phi - \kappa \rho_c^2 \phi^3 e^{-\phi^2} + \kappa \rho_c^2 \phi^5 e^{-\phi^2} + \eta f'(\rho_c \phi, \rho_c) \partial_t \phi = \beta(\rho_m) \frac{\rho_m}{\rho_c} \]
Governs the evolution of \( \phi \), enabling the description of classical motion via field gradients.
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Mass Modification
\[ M_{FFT} = M \left( 1 + \gamma_A \frac{\rho_{A,source}}{\rho_{A,local}} e^{-r/R_A} \right) \]
Modifies mass based on the local Aether Field, with \( \gamma_A = 2.43 \times 10^{-4} \).
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Gravitational Force Modification
\[ F_{FFT} = G \frac{M_1 M_2}{r^2} \left( 1 + \gamma_A \frac{\rho_{A,1}}{\rho_{A,2}} e^{-r/R_A} \right) \]
Adjusts gravitational force to account for \( \rho_A \) effects, resembling a Yukawa correction.
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Emergent Gravity Path Integral
\[ Z = \int \mathcal{D}\rho_A \, e^{i \int d^4 x \sqrt{-g} \left[ \frac{1}{2} (\partial_\mu \rho_A)(\partial^\mu \rho_A) - V(\rho_A) + \mathcal{L}_{int}(\rho_A) \right]} \]
Describes how gravity emerges from \( \rho_A \) quantum fluctuations.
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Emergent Metric
\[ g_{\mu\nu}^{eff}(x) = \langle \rho_A(x) \rho_A(x) \rangle g_{\mu\nu} \]
Defines the emergent spacetime metric from \( \rho_A \) correlations.
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Effective Einstein Equation
\[ R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda_R(\mu) g_{\mu\nu} = 8 \pi G_{eff} T_{\mu\nu} \]
The effective field equation for emergent gravity, with \( G_{eff} = \frac{1}{M_{Pl,eff}^2} \).
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Higgs Mass Shift
\[ m_H^2(\rho_A) = 2 \lambda v_\phi^2 + \eta \rho_A^2 \]
Describes the density-dependent Higgs mass shift due to \( \rho_A \).