Emergence of Gravitational Coupling from Quantum Action and Vacuum Geometry

Quantum-Geometric Action Framework and Derivation of k

We develop a local, action-based framework to relate quantum vacuum fluctuations with spacetime geometry, avoiding any prior assumption of the gravitational constant G.

Quantum action scale

We model the vacuum as a collection of harmonic modes with ground-state energy [7]:

$E_0=\frac{1}{2}\hslash \omega \text{ (1)}$

The characteristic coherence timescale is:

$\Delta t\sim \frac{1}{\omega }\text{ (2)}$

Thus, the action associated with a fundamental vacuum mode is:

$S_q\sim E_0\Delta t\sim \hslash \text{ (3)}$

This establishes ℏ as the natural scale of quantum action.

Effective geometric action of a vacuum cell

We consider a localized spacetime region (“vacuum cell”) characterized by a length scale l. The corresponding spacetime volume is estimated as:

$d^{4}x\sim {\ell }^3\Delta t\sim \frac{{\ell }^3}{\omega }\text{ (4)}$

This choice reflects a coherent spacetime cell whose temporal extent is set by the inverse frequency of the vacuum oscillation.

We define an effective geometric action for this cell in analogy with the Einstein-Hilbert form:

$S_g\sim \frac{1}{\kappa }{{\displaystyle \int }}^{\text{}}R{d}^{4}x\text{ (5)}$

We assume a characteristic local curvature scale associated with the vacuum fluctuation:

$R\sim \frac{1}{{\ell }^2}\text{ (6)}$

This represents a localized geometric response rather than global spacetime curvature.

Substituting, we obtain:

$S_g\sim \frac{1}{\kappa }\cdot \frac{1}{{\ell }^2}\cdot \frac{{\ell }^3}{\omega }=\frac{\ell }{\kappa \omega }\text{ (7)}$

Consistency condition and emergence of coupling

We impose a consistency condition that the geometric response of spacetime is of the same order as the quantum action scale:

$S_g\sim S_q\text{ (8)}$

Thus:

$\frac{\ell }{\kappa \omega }\sim \hslash \text{ (9)}$

Solving for k:

$\kappa \sim \frac{\ell }{\hslash \omega }\text{ (10)}$

Using the ground-state relation $E_0=\frac{1}{2}\hslash \omega$ , we rewrite:

To incorporate relativistic scaling and isotropic geometry, we introduce the speed of light c and a spherical normalization factor 4π, yielding:

$\kappa \approx \frac{4\pi {\ell }^2\omega }{E_{0}c}\text{ (12)}$

This expression has dimensions consistent with G/c⁴, [5,6] i.e., length per unit mass, and represents the emergent coupling between vacuum energy and spacetime geometry.

Interpretation

This result suggests that gravitational coupling emerges from the requirement that spacetime maintains quantum-scale action [8,9]. In this view:

• Gravity is not fundamental, but emergent
• The coupling constant reflects vacuum properties
• Geometry responds to maintain quantum coherence

Discussion

Unlike Planck-scale derivations, this framework does not assume the gravitational constant G a priori. Instead, k arises as a consistency parameter linking quantum energy and spacetime geometry.

The approach suggests that gravitational effects may depend on underlying quantum scales [6,10], potentially leading to scale-dependent behavior at high energies.

Limitations

This derivation is heuristic and relies on order-of-magnitude estimates. In particular:

• The curvature estimate is approximate
• The action balance condition is assumed
• No explicit field equations are derived

Conclusion

We have presented an action-based framework in which gravitational coupling emerges from quantum vacuum considerations. By requiring that spacetime action remains of order ℏ, we derive a coupling relation linking energy, frequency, and geometry.

This approach avoids circular dependence on G and offers a new perspective on the origin of gravitational interaction.