Motion or Flow Amplitude in Temporal Physics

 

Equation of Motion for $A(x)$ (Flow Amplitude)

We start with the full action:

$$S = \int d^4x \left[ \frac{1}{2} (\partial_\mu A)^2 + \frac{1}{2} A^2 (\partial_\mu \phi + a_\mu)^2 - \lambda (A^2 - A_0^2)^2 - \frac{1}{4} f_{\mu\nu} f^{\mu\nu} \right]$$

Take variation with respect to $A(x)$:

$$\delta S = \int d^4x \left[ \delta A \left( -\partial^\mu \partial_\mu A + A (\partial_\mu \phi + a_\mu)^2 - 4\lambda A (A^2 - A_0^2) \right) \right]$$

Set $\delta S = 0$ ⇒ Euler–Lagrange equation for $A(x)$:

$$\boxed{ \partial^\mu \partial_\mu A - A (\partial_\mu \phi + a_\mu)^2 + 4\lambda A (A^2 - A_0^2) = 0 }$$

This governs how the magnitude of temporal flow evolves — driven by phase velocity $\partial_\mu \phi$, gauge field $a_\mu$, and potential.

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2. Mass of $a_\mu$ from Symmetry Breaking

From the term:

$$\frac{1}{2} A^2 (\partial_\mu \phi + a_\mu)^2,$$

When $A$ acquires a vacuum expectation value: $A(x) \to A_0 + \delta A(x)$, and in unitary gauge $\phi = 0$, this becomes:

$$\frac{1}{2} A_0^2 a_\mu a^\mu$$

⇒ This is a mass term for the gauge field:

$$\boxed{ m_a^2 = A_0^2 }$$

So temporal flow with stable amplitude leads to massive gauge modes — like a Higgs mechanism from within the temporal flow framework.

You can tune $A_0$ to explore massless (photon-like) vs. massive (Proca-like) behavior.

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 3. Coupling to Gravity (Curved Spacetime)

In the framework, the spacetime metric $g_{\mu\nu}$ emerges from correlations in temporal flow:

• So rewrite the action in curved spacetime:

$$S = \int d^4x \sqrt{-g} \left[ \frac{1}{2} g^{\mu\nu} \nabla_\mu A \nabla_\nu A + \frac{1}{2} A^2 g^{\mu\nu} (\nabla_\mu \phi + a_\mu)(\nabla_\nu \phi + a_\nu) - \lambda (A^2 - A_0^2)^2 - \frac{1}{4} g^{\mu\alpha} g^{\nu\beta} f_{\mu\nu} f_{\alpha\beta} \right]$$

Here:

• $\nabla_\mu$ is the covariant derivative,

• $g_{\mu\nu}(x)$ is the emergent metric from temporal flow correlations,

• This can vary the action to obtain the energy-momentum tensor:

$$T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu\nu}}$$

which will feed into Einstein's equations:

$$G_{\mu\nu} = 8\pi G T_{\mu\nu}$$

This connects flow-based dynamics directly to spacetime curvature.

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Motion: Reinterpreted in Flow Terms

Now to address motion in general:

• Phase gradients $\partial_i \phi$: determine local momentum.

• Temporal derivatives $\partial_t \phi$: relate to energy and charge.

• Amplitude $A(x,t)$: determines the “presence” or density of flow — thus of matter or energy.

• Gauge field $a_\mu$: encodes emergent interactions or fields (e.g., electromagnetic).

So “motion” in TFP = the structured evolution of these fields:
charge moving through space = structured phase gradients in evolving temporal flows.

 Solving the coupled equations of motion for

• $A(x)$: amplitude (flow density),

• $\phi(x)$: flow phase,

• $a_\mu(x)$: emergent gauge field,

• $g_{\mu\nu}(x)$: emergent spacetime geometry,

gives you a complete dynamic description of fields, matter, interaction, and gravity. These equations form a tightly interlocked system, and solutions in different limits will recover:

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 Coupled Equations of Motion (Summary)

From the action, the equations of motion are:

1. Amplitude Field $A(x)$:

$$\boxed{ \partial^\mu \partial_\mu A - A (\partial_\mu \phi + a_\mu)^2 + 4\lambda A (A^2 - A_0^2) = 0 }$$

2. Phase Field $\phi(x)$:

Varying $\phi$ gives:

$$\boxed{ \partial_\mu \left( A^2 (\partial^\mu \phi + a^\mu) \right) = 0 }$$

→ resembles charge conservation or a continuity equation for flow current.

3. Gauge Field $a_\mu(x)$:

Varying $a_\mu$ gives a Maxwell-Proca equation:

$$\boxed{ \partial_\nu f^{\nu\mu} + A^2 (\partial^\mu \phi + a^\mu) = 0 }$$

4. Metric $g_{\mu\nu}(x)$:

Einstein field equation with emergent stress-energy tensor:

$$\boxed{ G_{\mu\nu} = 8\pi G\, T_{\mu\nu}^{\text{TFP}} }$$

where $T_{\mu\nu}^{\text{TFP}}$ comes from the full flow-gauge action.

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Solving in Physical Scenarios

Here’s how to approach it in stages:

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A. Vacuum Solution (Flat Background, No Matter)

Set:

• $A = A_0$,

• $\phi = \omega t$,

• $a_\mu = 0$,

• $g_{\mu\nu} = \eta_{\mu\nu}$

This is a ground state solution:

• Satisfies all equations.

• Describes flat spacetime with constant temporal flow.

• Perturbations in $\phi$ become wave-like — similar to photons or EM waves.

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B. Electromagnetic Wave Solution (Linearized Fluctuations)

Perturb around vacuum:

• $\varphi =\omega t+\delta \varphi (x),$

• $a_\mu = \delta a_\mu(x)$
  

Assume $A \approx A_0$ constant.

Then:

• $\delta a_\mu$ satisfies $\partial_\nu f^{\nu\mu} = 0$ → standard Maxwell equations.

• $\delta \phi$ satisfies $\Box \delta\phi = 0$ → massless scalar wave.

• Together, they form a basis for EM radiation in vacuum.

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C. Static Charge or Mass Distribution

Introduce:

• $A(x)$ perturbed near a localized region (a “mass” or “charge”).

• $\phi(x)$ slowly varying.

• $a_0(x)$ (electric potential) sourced.

Solve:

• Poisson-like equations for $a_0(x)$,

• Gradual spatial variations in $A(x)$ and $g_{\mu\nu}(x)$,

• Result: Coulomb field or Newtonian gravity emerges.

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D. Spherically Symmetric Source (Curved Background)

Assume:

• Central concentration in $A(x)$ → acts as energy source.

• Solve Einstein equation $G_{\mu\nu} = 8\pi G T_{\mu\nu}^{\text{flow}}$ with spherically symmetric ansatz.

You can recover:

• Schwarzschild-like geometry for $g_{\mu\nu}(x)$,

• Motion of test particles as flow-guided geodesics,

• Lensing, time dilation, etc.

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E. Nontrivial Gauge Field Configuration (e.g., Magnetic Monopole)

Use topological configurations of $a_\mu$:

• $\nabla \cdot \vec{B} \ne 0$,

• or vortex-like behavior in $\phi(x)$.

This opens doors to:

• Emergent quantized charges,

• Dirac strings, magnetic monopoles,

• Gauge topologies from phase winding in time flow.

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F. Cosmology (Time-Dependent Background)

Let:

• $A = A(t)$, $\phi = \phi(t)$, $g_{\mu\nu} = \text{FRW metric}$
  

Solve:

• Friedman-like equation from $G_{00} = T_{00}$,

• See if flow dynamics give accelerated expansion (dark energy?), early inflation, or cosmic phase transitions.