Equations of Motion for the Flow Fluctuation Field δF

 

Equations of Motion for the Flow Fluctuation Field δF

To derive the equations of motion for the fluctuation field δF from my proposed action, I analyze each term in the effective action Γ[g, δF], treating the emergent metric g₍μν₎ as a fixed background during variation.

1. Full Action Recap

The effective action is

$$\Gamma[g,\delta F] =\int d^4x\,\sqrt{-g}\Bigl[\underbrace{\tfrac12\,g^{\mu\nu}\,\partial_\mu\delta F\,\partial_\nu\delta F}_{\text{Kinetic}} -\underbrace{V(\delta F)}_{\text{Potential}} -\underbrace{\tfrac\lambda2\!\int d^4y\,\sqrt{-g(y)}\,K(x-y)\,\delta F(x)\,\delta F(y)}_{\text{Nonlocal Interaction}}\Bigr].$$

2. Variation of the Action

The Euler–Lagrange equation, $\delta\Gamma/\delta\delta F(x)=0$, breaks into three pieces.

• Kinetic Term

$$\Gamma_{\rm kin} =\tfrac12\!\int d^4x\,\sqrt{-g}\,g^{\mu\nu}\,\partial_\mu\delta F\,\partial_\nu\delta F \;\;\Longrightarrow\;\; \frac{\delta\Gamma_{\rm kin}}{\delta\delta F(x)} =-\partial_\mu\!\bigl(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\delta F\bigr) =-\nabla_\mu\nabla^\mu\delta F.$$

• Potential Term

$$\Gamma_{\rm pot} =-\int d^4x\,\sqrt{-g}\;V(\delta F), \quad V(\delta F)=\tfrac\alpha2\bigl(\delta F^2-F_0^2\bigr)^2 \;\;\Longrightarrow\;\; \frac{\delta\Gamma_{\rm pot}}{\delta\delta F(x)} =-\sqrt{-g}\,V'(\delta F) =-\sqrt{-g}\,2\alpha\bigl(\delta F^3-\delta F\,F_0^2\bigr).$$

• Nonlocal Interaction

$$\Gamma_{\rm int} =-\tfrac\lambda2\!\int d^4x\,\sqrt{-g(x)}\!\int d^4y\,\sqrt{-g(y)}\,K(x-y)\,\delta F(x)\,\delta F(y) \;\;\Longrightarrow\;\; \frac{\delta\Gamma_{\rm int}}{\delta\delta F(x)} =-\lambda\,\sqrt{-g(x)}\!\int d^4y\,\sqrt{-g(y)}\,K(x-y)\,\delta F(y).$$

3. Combined Equation of Motion

Putting all pieces together, dividing through by $\sqrt{-g(x)}$, gives

$$\boxed{ \nabla_\mu\nabla^\mu\,\delta F -\,V'(\delta F) -\,\lambda\!\int d^4y\,\sqrt{-g(y)}\,K(x-y)\,\delta F(y) \;=\;0. }$$

Or, in expanded form,

$$\frac1{\sqrt{-g}}\partial_\mu\!\bigl(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\delta F\bigr) +2\alpha\,\delta F\,(F_0^2-\delta F^2) -\lambda\!\int d^4y\,\sqrt{-g(y)}\,K(x-y)\,\delta F(y) =0.$$

4. How This Fits Together

• Double-well potential $V(\delta F)$ drives fluctuations toward $\pm F_0$.

• Covariant Laplacian $\nabla_\mu\nabla^\mu\,\delta F$ encodes local alignment of flows.

• Kernel $K(x-y)$ implements Lorentz-invariant, causal interactions across the flow network.

• Emergent metric $g_{\mu\nu}$ itself is sourced by flow fluctuations (via $\langle\partial\delta F\,\partial\delta F\rangle$), closing the self-consistent loop: fluctuations shape geometry, geometry guides fluctuations.

5. Key Special Cases

• Flat spacetime & no nonlocality ($g_{\mu\nu}\to\eta_{\mu\nu},\,\lambda=0$):

  $$\partial_\mu\partial^\mu\,\delta F +2\alpha\,\delta F\,(F_0^2-\delta F^2) =0,$$
  

  i.e. the standard nonlinear Klein–Gordon with spontaneous symmetry breaking.

• Nonlocal memory effects arise from $\int K(x-y)\,\delta F(y)\,dy$.

• Gravitational coupling emerges when one sets $g_{\mu\nu}=\langle\partial\delta F\,\partial\delta F\rangle$, making the flow dynamics and spacetime geometry fully intertwined.