Einstein Field Equation understood in PUCS

 

🌌 Einstein Field Equation in SI:

$$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \mathrm{\nu }}$$

Now use our definition:

$$G = G_n \cdot \frac{c^3}{\text{Hz}_{\text{kg}}}$$

Substitute:

$$\frac{8\pi G}{c^4} = \frac{8\pi G_n \cdot \frac{c^3}{\text{Hz}_{\text{kg}}}}{c^4} = \frac{8\pi G_n}{c \cdot \text{Hz}_{\text{kg}}}$$

So the field equation becomes:

$$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \left( \frac{8\pi G_n}{c \cdot \text{Hz}_{\text{kg}}} \right) T_{\mu\nu}$$

---

🔍 Interpretation:

• The term $\frac{1}{c \cdot \text{Hz}_{\text{kg}}}$ is a unit conversion factor. It doesn't alter the physics — it rescales the units of $T_{\mu\nu}$ so they match those on the curvature side. It removes the SI unit scaling so that kg and  meter units becomes natural values.

• We are recasting gravity in a system where energy, mass, and frequency are on the same footing — by expressing them in natural units.

• What remains — $8\pi G_n$ — is the true gravitational coupling constant in the Physics unit coordinate system.

---

🧠 Deep Insight:

We're showing that:

• SI gravity is scaled and distorted by human-chosen units (kg, m, s), hiding the elegant dimensionless geometry underneath.

• Once we correct for the unit conversions via $c \cdot \text{Hz}_{\text{kg}}$, we're left with pure curvature = mass-energy in natural form.

• The "mystery" of $G$ vanishes — the value in SI units is just a scaling artifact.

• And yes: the Einstein equation is just saying curvature is proportional to frequency-scaled energy, where the proportionality constant is $8\pi G_n$