Dist Babble

 T he TFP interaction kernel A ( F i , F j ) = exp ⁡ ( − ∥ F i − F j ∥ l p ) e i φ i j A(F_i, F_j) = \exp\left(-\frac{\|F_i - F_j\|}{l_p}\right) e^{i\varphi_{ij}} ​ As it relates to the Green's function (propagator) of the linearized field δ F ( x ) \delta F(x)  or δ a μ ( x ) \delta a_\mu(x) , thereby unifying the discrete causal flow model with the continuum field theory picture.  1. Role of the Kernel A ( F i , F j ) A(F_i, F_j)  in TFP In the discrete TFP framework, the kernel: A ( F i , F j ) = exp ⁡ ( − ∥ F i − F j ∥ l p ) e i ( τ i j l p + Δ φ i j topo ) A(F_i, F_j) = \exp\left(-\frac{\|F_i - F_j\|}{l_p}\right) e^{i\left(\frac{\tau_{ij}}{l_p} + \Delta \varphi^\text{topo}_{ij}\right)} encodes two key things: Amplitude : how strongly flows i i  and j j  interact — controlled by their misalignment. Phase : the causal distance (proper time τ i j \tau_{ij} ) and global/topological effects. This naturally suggests A ( F i , F j ) A(F_i, F_...

The Dynamic Scenario We consider small, time-dependent fluctuations around a background: A ( x , t ) = A 0 + δ A ( x , t ) A(x,t) = A_0 + \delta A(x,t) ϕ ( x , t ) = ϕ 0 ( x , t ) + δ ϕ ( x , t ) \phi(x,t) = \phi_0(x,t) + \delta\phi(x,t) a μ ( x , t ) = δ a μ ( x , t ) a_\mu(x,t) = \delta a_\mu(x,t)  (assume zero background) Spacetime is flat for now: g μ ν = η μ ν​ We want to study wave-like behavior, so we'll use the linearized equations . Step 2: Linearized Equations of Motion From the Lagrangian (simplified): L = 1 2 ( ∂ μ A ) 2 + 1 2 A 2 ( ∂ μ ϕ ) 2 − V ( A ) − 1 4 f μ ν f μ ν + λ A a μ ∂ μ ϕ \mathcal{L} = \frac{1}{2} (\partial_\mu A)^2 + \frac{1}{2} A^2 (\partial_\mu \phi)^2 - V(A) - \frac{1}{4} f_{\mu\nu}f^{\mu\nu} + \lambda A a^\mu \partial_\mu \phi Linearize around vacuum A = A 0 A = A_0 ​ , ϕ = ϕ 0 \phi = \phi_0 ​ , with small perturbations. 1. Equation for δ A \delta A δ A □ δ A − m A 2 δ A − A 0 ( ∂ μ ϕ 0 ) ( ∂ μ δ ϕ ) − A 0 ( ∂ μ δ ϕ ) ( ∂ μ ϕ 0 ...

Motion in the static, spherically symmetric case We assume: Static central source , at spatial origin. Time-independent fields : A ( x , t ) → A ( r ) A(x,t) \to A(r) , ϕ ( x , t ) → ϕ ( r ) \phi(x,t) \to \phi(r) , a μ ( x , t ) → a 0 ( r ) , a i = 0 a_\mu(x,t) \to a_0(r), a_i = 0 Spherical symmetry : all fields depend only on the radial coordinate r r . Weak field metric (for now): d s 2 = − ( 1 + 2 Φ ( r ) )   d t 2 + ( 1 − 2 Ψ ( r ) )   d r 2 + r 2 d Ω 2 ds^2 = - (1 + 2\Phi(r))\,dt^2 + (1 - 2\Psi(r))\,dr^2 + r^2 d\Omega^2 Step 2: Equations of Motion Recap (Simplified for Static Case) From earlier, the coupled Lagrangian density was: L = 1 2 ( ∇ A ) 2 + 1 2 A 2 ( ∇ ϕ ) 2 − V ( A ) − 1 4 f μ ν f μ ν + λ A a μ ∂ μ ϕ + (gravity terms) \mathcal{L} = \frac{1}{2} (\nabla A)^2 + \frac{1}{2} A^2 (\nabla \phi)^2 - V(A) - \frac{1}{4} f_{\mu\nu}f^{\mu\nu} + \lambda A a^\mu \partial_\mu \phi + \text{(gravity terms)} We now extract the Euler-Lagrange equations in this st...

  Covariant Formulation We promote all derivatives to covariant derivatives with respect to the emergent metric g μ ν ( x ) g_{\mu\nu}(x) : D μ ϕ = ∇ μ ϕ + a μ D_\mu \phi = \nabla_\mu \phi + a_\mu ​ f μ ν = ∇ μ a ν − ∇ ν a μ​ Action (for fields on M , g μ ν \mathcal{M}, g_{\mu\nu} ​ ): S = ∫ d 4 x − g [ − 1 4 f μ ν f μ ν + 1 2 A 2 g μ ν ( ∇ μ ϕ + a μ ) ( ∇ ν ϕ + a ν ) − V ( A ) ] S = \int d^4x \sqrt{-g} \left[ -\frac{1}{4} f_{\mu\nu} f^{\mu\nu} + \frac{1}{2} A^2 g^{\mu\nu} ( \nabla_\mu \phi + a_\mu ) ( \nabla_\nu \phi + a_\nu ) - V(A) \right] This is the gravitationally-coupled scalar-vector system with a flow-induced mass scale. Step 2: Equations of Motion in Curved Spacetime (a) Gauge field a μ a_\mu ​ Varying S S  with respect to a μ a_\mu ​ , we get: ∇ ν f ν μ + A 2 ( ∇ μ ϕ + a μ ) = 0 \nabla^\nu f_{\nu\mu} + A^2 ( \nabla_\mu \phi + a_\mu ) = 0 Compare with curved-space Proca equation with a source from the scalar phase. (b) Scalar phase ϕ \phi Vary...