TFP: Covariant Formulation

 

Covariant Formulation

We promote all derivatives to covariant derivatives with respect to the emergent metric $g_{\mu\nu}(x)$:

• $D_\mu \phi = \nabla_\mu \phi + a_\mu$

• $f_{\mu \nu }={\mathrm{\nabla }}_{\mu }{a}_{\nu }-{\mathrm{\nabla }}_{\nu }{a}_{\mathrm{\mu }}$

• Action (for fields on $\mathcal{M}, g_{\mu\nu}$):

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

This is the gravitationally-coupled scalar-vector system with a flow-induced mass scale.

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Step 2: Equations of Motion in Curved Spacetime

(a) Gauge field $a_\mu$

Varying $S$ with respect to $a_\mu$, we get:

$$\nabla^\nu f_{\nu\mu} + A^2 ( \nabla_\mu \phi + a_\mu ) = 0$$

Compare with curved-space Proca equation with a source from the scalar phase.

(b) Scalar phase $\phi$

Varying $S$ with respect to $\phi$, we find:

$$\nabla_\mu \left( A^2 ( \nabla^\mu \phi + a^\mu ) \right) = 0$$

This is a conserved current condition for the total covariant flow of temporal phase plus gauge field.

(c) Metric $g_{\mu\nu}$

To get the gravitational response, vary $S$ with respect to the metric:

$$T_{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu\nu}} = f_{\mu\alpha} f_{\nu}^{\ \alpha} - \frac{1}{4} g_{\mu\nu} f_{\alpha\beta} f^{\alpha\beta} + A^2 \left( \nabla_\mu \phi + a_\mu \right) \left( \nabla_\nu \phi + a_\nu \right) - g_{\mu\nu} \left[ \frac{1}{2} A^2 (\nabla^\alpha \phi + a^\alpha)^2 - V(A) \right]$$

This stress-energy tensor now sources the Einstein field equations (or their TFP-modified analog):

$$G_{\mu\nu} + \text{(TFP corrections)} = 8\pi G \, T_{\mu\nu}$$

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Step 3: Interpretation in TFP

In this model:

• The metric $g_{\mu\nu}(x)$ emerges from flow fluctuation correlations:

  $$g_{\mu\nu}(x) = \langle \partial_\mu \delta F(x) \partial_\nu \delta F(x) \rangle$$
  

• The gauge field $a_\mu$ arises from phase gradients and temporal alignment

• The scalar $\phi$ tracks internal flow clock time

Gravity, in this picture, is not imposed — it’s generated by the very temporal flow coherence whose misalignment gives rise to gauge interactions.

So the Einstein equations are not assumed, but arise via variation of an effective action $\Gamma[g_{\mu\nu}, \bar{F}]$, as I’ve been constructing.