The Dimensionless Form of Temporal Flow Physics

by John Gavel
Date: 5/25/2025

In the ongoing development of Temporal Flow Physics (TFP), a key step in linking the theory to observable physics—and comparing it to established models like General Relativity and Quantum Field Theory—is casting the entire action in a dimensionless form. This blog explores how the fundamental constants of nature—ℏ, $c$ , and $G$ —disappear into the fabric of the theory, and what this means for interpreting the action, metric, and dynamics of flow.

Why Make the Action Dimensionless?

Every physical theory is governed by an action $S$ , whose variation yields the field equations. But $S$ itself carries units: specifically, units of ℏ. By expressing the entire action in dimensionless form—dividing through by ℏ and rescaling all fields and coordinates—we distill the theory to its pure form, where the structure of interactions is laid bare, unclouded by unit conventions.

This process also reveals how constants like the speed of light and Planck length are encoded in the dynamics. In TFP, where flow is fundamental and space is emergent, this is especially illuminating.

Setup: Planck Units and Natural Normalizations

We use Planck units as the basis of our dimensionless formulation. These are defined via:

Planck length: $\ell_P = \sqrt{\hbar G / c^3}$
Planck time: $t_P = \sqrt{\hbar G / c^5}$
Planck mass: $m_P = \sqrt{\hbar c / G}$
Planck energy density: $\rho_P = m_P c^2 / \ell_P^3$

We define dimensionless quantities:

Coordinates: $\tilde{x}^\mu = x^\mu / \ell_P$ , $\tilde{t} = t / t_P$
Flow field: $\tilde{F}(x) = F(x) / m_P$
Metric: $\tilde{G}_{\mu\nu}(x) = G_{\mu\nu}(x) \ell_P^2$
Action: $\tilde{S} = S / \hbar$

Derivatives scale accordingly:

$\partial_t = \frac{1}{t_P} \partial_{\tilde{t}}$ , $\nabla = \frac{1}{\ell_P} \nabla_{\tilde{x}}$

This sets the stage for re-expressing the TFP Lagrangian.

The Original TFP Lagrangian

The Lagrangian density $\mathcal{L}_{\text{TFP}}$ in TFP includes:

\mathcal{L}_{\text{TFP}} = \frac{1}{2} G^{\mu\nu} (\partial_\mu F)(\partial_\nu F) - V(F) + \mathcal{L}_{\text{nonlocal}}[F] - \frac{1}{2\kappa_G} R(G)

Here:

The kinetic term governs local flow dynamics.
The potential $V(F)$ defines symmetry structure.
$\mathcal{L}_{\text{nonlocal}}$ captures nonlocal ratchet-like bias.
The final term is the Einstein-Hilbert term, with $\kappa_G = 8\pi G/c^4$ .

Transforming Each Term

1. Kinetic Term

Rescaling:

F = m_P \tilde{F},\quad G^{\mu\nu} = \frac{1}{\ell_P^2} \tilde{G}^{\mu\nu},\quad \partial_\mu F = \frac{m_P}{\ell_P} \partial_{\tilde{\mu}} \tilde{F}

So the kinetic term becomes:

\frac{1}{2} G^{\mu\nu} (\partial_\mu F)(\partial_\nu F) = \frac{1}{2} \frac{1}{\ell_P^4} m_P^2 \tilde{G}^{\mu\nu} (\partial_{\tilde{\mu}} \tilde{F})(\partial_{\tilde{\nu}} \tilde{F})

When multiplied by the volume element $\mathrm{d}^4 x = \ell_P^3 t_P \, \mathrm{d}^4 \tilde{x}$ , we get:

\tilde{S}_{\text{kinetic}} = \frac{1}{2} \frac{m_P^2 \ell_P^{-1} t_P}{\hbar} \int \mathrm{d}^4 \tilde{x} \, \tilde{G}^{\mu\nu} (\partial_{\tilde{\mu}} \tilde{F})(\partial_{\tilde{\nu}} \tilde{F})

Plugging in the Planck unit definitions, all prefactors collapse to 1:

\Rightarrow \tilde{S}_{\text{kinetic}} = \frac{1}{2} \int \mathrm{d}^4 \tilde{x} \, \tilde{G}^{\mu\nu} (\partial_{\tilde{\mu}} \tilde{F})(\partial_{\tilde{\nu}} \tilde{F})

2. Potential Term

We normalize the potential energy density by $\rho_P$ :

V(F) = V(m_P \tilde{F}) \quad \Rightarrow \quad \tilde{V}(\tilde{F}) = \frac{V(F)}{\rho_P}

So:

\tilde{S}_{\text{potential}} = - \int \mathrm{d}^4 \tilde{x} \, \tilde{V}(\tilde{F})

This ensures all potential terms (e.g., $\lambda_2 F^2, \lambda_3 F^3$ ) are redefined in terms of dimensionless couplings.

3. Nonlocal Term

Since this also carries energy/volume units, it’s normalized the same way:

\tilde{\mathcal{L}}_{\text{nonlocal}}[\tilde{F}] = \frac{1}{\rho_P} \mathcal{L}_{\text{nonlocal}}[m_P \tilde{F}]

4. Gravity Term

The Einstein-Hilbert term becomes:

\frac{1}{2\kappa_G} R(G) = \frac{c^4}{16\pi G} R \quad \Rightarrow \quad \tilde{R} = R \ell_P^2, \quad \Rightarrow \quad \tilde{S}_{\text{grav}} = -16\pi \int \mathrm{d}^4 \tilde{x} \, \tilde{R}(\tilde{G})

The Final Dimensionless Action

Putting it all together, the dimensionless action for TFP is:

\boxed{ \tilde{S}_{\text{TFP}}[\tilde{F}, \tilde{G}] = \int \mathrm{d}^4 \tilde{x} \left[ \frac{1}{2} \tilde{G}^{\mu\nu} \partial_{\mu} \tilde{F} \, \partial_{\nu} \tilde{F} - \tilde{V}(\tilde{F}) + \tilde{\mathcal{L}}_{\text{nonlocal}}[\tilde{F}] - 16\pi \tilde{R}(\tilde{G}) \right] }

This formulation encodes all dynamics using dimensionless fields, coordinates, and constants, a hallmark of a fully scale-invariant, background-independent theory. The speed of light is no longer explicit—it is hidden in the ratio of coefficients in the discrete version. The structure of interactions defines the causal speed, not the other way around.

Why This Matters for Physics

By writing the action this way:

We clarify how Planck-scale physics shapes all dynamics.
We isolate physically meaningful dimensionless coupling constants.
We prepare TFP for quantum and statistical treatments, like RG flow or effective actions.
We enable comparison with standard models that are usually defined in natural units.

This also reinforces the central tenet of TFP: space and mass are not fundamental, but rather emerge from flow interactions. The dimensionless action reflects that: the geometry $\tilde{G}_{\mu\nu}$ and field $\tilde{F}$ are defined relative to Planck units, not against any external background.

Dist Babble

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