TFP: Temporal Dynamics

The Dynamic Scenario

We consider small, time-dependent fluctuations around a background:

• $A(x,t) = A_0 + \delta A(x,t)$
  

• $\phi(x,t) = \phi_0(x,t) + \delta\phi(x,t)$
  

• $a_\mu(x,t) = \delta a_\mu(x,t)$ (assume zero background)

• Spacetime is flat for now: $g_{\mu \nu }={\eta }_{\mu \mathrm{\nu }}$

We want to study wave-like behavior, so we'll use the linearized equations.

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Step 2: Linearized Equations of Motion

From the Lagrangian (simplified):

$$\mathcal{L} = \frac{1}{2} (\partial_\mu A)^2 + \frac{1}{2} A^2 (\partial_\mu \phi)^2 - V(A) - \frac{1}{4} f_{\mu\nu}f^{\mu\nu} + \lambda A a^\mu \partial_\mu \phi$$

Linearize around vacuum $A = A_0$, $\phi = \phi_0$, with small perturbations.

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1. Equation for $\delta A$

$$\Box \delta A - m_A^2 \delta A - A_0 (\partial_\mu \phi_0)(\partial^\mu \delta \phi) - A_0 (\partial_\mu \delta \phi)(\partial^\mu \phi_0) + \lambda a^\mu \partial_\mu \phi_0 = 0$$

This shows how the amplitude field $\delta A$ is driven by the phase gradients and the vector field.

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2. Equation for $\delta \phi$

$$A_0^2 \Box \delta \phi + 2 A_0 (\partial^\mu \delta A)(\partial_\mu \phi_0) + \lambda A_0 \partial_\mu a^\mu = 0$$

This is a wave equation for the phase, sourced by flow misalignment and divergence of the gauge field.

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3. Equation for $a^\mu$

$$\Box a^\mu - \partial^\mu (\partial_\nu a^\nu) = \lambda A_0 \partial^\mu \phi_0$$

This resembles the usual Maxwell equation with a source — here, the phase gradient is the effective charge-current:

$$\partial_\nu f^{\mu\nu} = \lambda A_0 \partial^\mu \phi_0$$

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Step 3: Analyze Oscillating Source

Assume a localized oscillating phase source:

• $\phi_0(x,t) = \omega t \cdot \Theta(R - |\mathbf{x}|)$ (oscillating inside a sphere of radius R)

• So:
  $\partial^\mu \phi_0 = (\omega, 0, 0, 0)$ inside, zero outside

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Vector Field Dynamics

Then the Maxwell-like equation becomes:

$$\Box a^\mu = \lambda A_0 \partial^\mu \phi_0 = \lambda A_0 \omega \delta^\mu_0 \cdot \Theta(R - r)$$

This is equivalent to a time-dependent electric charge density inside a region. The solution:

• Outside source:
  $a^0 \sim \frac{\lambda A_0 \omega R^3}{r} \cos(\omega t - kr)$
  

• Radiation field $\mathbf{E}(x,t) = -\nabla a^0 - \partial_t \mathbf{a}$ contains oscillating electric field

Electromagnetic radiation emerges naturally.

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Scalar Phase Radiation

The wave equation:

$$\Box \delta \phi = -\frac{2}{A_0} (\partial^\mu \delta A)(\partial_\mu \phi_0) - \frac{\lambda}{A_0} \partial_\mu a^\mu$$

Again, since $\phi_0 \sim \omega t$, the RHS sources oscillatory scalar waves.

This gives scalar radiation emitted due to coupling between amplitude and vector field fluctuations.

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Step 4: Gravitational Coupling (Optional for Later)

We could include gravitational backreaction by computing:

• Stress-energy tensor from fluctuations

• Plug into Einstein’s equation
  $G_{\mu\nu} = 8\pi G T_{\mu\nu}$

This would yield gravitational waves or metric deformations caused by oscillating temporal flows.

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Summary: What We Just Found

Field	Behavior
$a^\mu$	Radiates EM-like waves from oscillating $\phi$
$\phi$	Radiates scalar waves driven by $a^\mu$, $\delta A$
$\delta A$	Responds to motion of phase and gauge field
Gravity	Can be sourced by wave energy (if included)

We’ve essentially built a dynamical field theory with radiation, consistent with known EM theory and scalar-tensor extensions.