TFP: Vacuum Solution (Flat Spacetime, No Sources)

Vacuum Solution (Flat Spacetime, No Sources)

We assume:

• Constant amplitude: $A(x) = A_0$

• Uniform phase evolution: $\phi(x) = \omega t$
  

• No gauge field: $a_\mu(x) = 0$
  

• Flat metric: $g_{\mu\nu} = \eta_{\mu\nu}$

We now check each equation of motion.

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1. Equation for $A(x)$

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

Substitute:

• $\partial_\mu A = 0$
  

• $\partial_\mu \phi = (\omega, 0, 0, 0)$
  

• $a_\mu = 0$
  

Then:

$$0 - A_0 \omega^2 + 4\lambda A_0 (A_0^2 - A_0^2) = -A_0 \omega^2 = 0 \quad \Rightarrow \quad \omega = 0 \quad \text{or} \quad A_0 = 0$$

But we want non-zero flow. So to satisfy the equation, we must reintroduce $a_0 = -\omega$, such that:

$$\partial_0 \phi + a_0 = \omega - \omega = 0$$

This gives zero net kinetic term, as required in the vacuum. Thus, the equation is satisfied if:

$$A = A_0, \quad \phi = \omega t, \quad a_0 = -\omega$$

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2. Equation for $\phi(x)$

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

With:

• $A = A_0$ constant

• $a^0 = -\omega$
  

We get:

$$\partial_\mu \left( A_0^2 (\omega - \omega) \right) = \partial_\mu (0) = 0$$

 Satisfied.

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3. Equation for $a_\mu(x)$

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

With:

• $f^{\mu\nu} = \partial^\mu a^\nu - \partial^\nu a^\mu = 0$
  

• $\partial^\mu \phi + a^\mu = 0$
  

Then:

$$0 + A_0^2 \cdot 0 = 0$$

 Satisfied.

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4. Einstein Field Equation

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

In flat spacetime: $G_{\mu \nu }=0$

We compute $T_{\mu\nu}^{\text{flow}}$ from the Lagrangian:

$$\mathcal{L} = \frac{1}{2}(\partial_\mu A)^2 + \frac{1}{2} A^2 (\partial_\mu \phi + a_\mu)^2 - \lambda (A^2 - A_0^2)^2 - \frac{1}{4} f_{\mu\nu}f^{\mu\nu}$$

Substitute:

• ${\mathrm{\partial }}_{\mu }A=0$

• $\partial_\mu \phi + a_\mu = 0$
  

• $A = A_0$, $f_{\mu\nu} = 0$
  

Then:

$$\mathcal{L}_{\text{vac}} = 0 \quad \Rightarrow \quad T_{\mu\nu} = 0$$

 No energy-momentum → spacetime remains flat.

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 Summary: Vacuum Solution

Field	Value	Satisfies EOM?
$A(x)$	$A_0$	Yes
$\phi(x)$	$\omega t$	Yes
$a_0$	$-\omega$	Yes
$g_{\mu\nu}$	$\eta_{\mu\nu}$	Yes