Flux Field Theory (FFT)

Quantum Gravity Equations in FFT

Flux Field Theory (FFT) proposes that gravity is not a fundamental force but an emergent phenomenon arising from quantum fluctuations of the Aether Field \( \rho_A \). This section highlights the most important equations related to emergent gravity in FFT, which connect quantum field theory to cosmological dynamics.

1. Path Integral for Emergent Gravity:
   \[ Z = \int \mathcal{D}\rho_A \, e^{i \int d^4 x \sqrt{-g} \left[ \frac{1}{2} (\partial_\mu \rho_A)(\partial^\mu \rho_A) - V(\rho_A) + \mathcal{L}_{int}(\rho_A) \right]} \]

   This path integral describes how quantum fluctuations of \( \rho_A \) give rise to emergent gravitational effects at macroscopic scales. The interaction term \( \mathcal{L}_{int}(\rho_A) \) introduces self-interactions that lead to spacetime correlations.

2. Emergent Metric:
   \[ g_{\mu\nu}^{eff}(x) = \langle \rho_A(x) \rho_A(x) \rangle g_{\mu\nu} \]

   The effective metric \( g_{\mu\nu}^{eff} \) emerges from the expectation value of \( \rho_A \) correlations, defining a classical spacetime geometry from quantum fluctuations.

3. Effective Einstein Equation:
   \[ R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda_R(\mu) g_{\mu\nu} = 8 \pi G_{eff} T_{\mu\nu} \]

   This equation governs emergent gravity, where \( \Lambda_R(\mu) \) is the running gravitational coupling, and \( G_{eff} = \frac{1}{M_{Pl,eff}^2} \) is the effective gravitational constant derived from \( \langle \rho_A \rangle \).

4. Inflation Driven by \( \rho_A \):
   \[ S_{inflation} = \int d^4 x \sqrt{-g} \left[ \frac{M_{Pl}^2}{2} R + \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V_{FFT}(\phi) \right] \]

   Describes cosmic inflation driven by \( \rho_A \) vacuum fluctuations, with \( \phi = \rho_A / \rho_c \) acting as an effective inflaton field, testable via CMB power spectrum shifts.

These equations highlight FFT’s approach to quantum gravity, where the Aether Field \( \rho_A \) bridges quantum field theory and general relativity, offering a novel perspective on the emergence of spacetime and gravitational interactions.