Geometric Memory and the Cosmological Constant: Reconstructing Gravity with Curvature–Field Resonance

✍️ Author

Anonymous

🧠 Abstract

We present Geometric Proca Gravity (GPG), a new gravitational framework in which the cosmological constant Λ emerges not as a constant of nature nor as quantum vacuum energy, but as a residual geometric memory of past curvature. GPG introduces a dynamical vector field $A^\mu$, directly embedded in the manifold, whose nonlinear coupling to the Ricci tensor produces an effective, time-relaxing tension field:

$$\Lambda_{\text{GPG}} = \frac{\zeta}{A_0^2} R_{\mu\nu} A^\mu A^\nu$$

This term dynamically mimics Λ-like behavior without introducing arbitrary constants or dark energy. We show how GPG naturally replaces $f(R)$ and extended Palatini models, resolving their shortcomings by embedding memory in the structure of spacetime itself. Numerical simulations demonstrate that GPG can reproduce ΛCDM-like redshift-distance relations, explain void expansion, and break from semiclassical nuclear models near the neutron drip line. GPG unifies cosmic acceleration and nuclear binding under curvature–field resonance, and may represent a foundational shift in how we define energy in gravity.

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🔰 1. Introduction

The cosmological constant Λ appears in Einstein’s field equations as a uniform energy density of the vacuum. But its magnitude is observationally tiny — and wildly inconsistent with quantum field theory. Countless modified gravity models have been proposed, from $f(R)$ gravity to scalar–tensor theories and Palatini extensions, to address this “vacuum catastrophe.” Yet all suffer from a common flaw: they do not explain why Λ appears at all, nor do they offer a structural reason for its persistence.

In this work, we propose that Λ is not a parameter — it is a memory. We construct a model in which the geometry of spacetime retains residual tension from past curvature, encoded dynamically by a vector field embedded in the manifold. This framework — Geometric Proca Gravity (GPG) — recasts Λ as a local, slowly relaxing geometric tension. It requires no vacuum energy, no scalar fields, and no added dimensions.

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🧩 2. Theoretical Framework

We define the action of GPG as:

$$\mathcal{L} = R + \zeta R_{\mu\nu} A^\mu A^\nu - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + V(A_\mu A^\mu)$$

Here:

• $A^\mu$: vector field embedded in spacetime

• $R_{\mu\nu}$: Ricci tensor (geometry)

• $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$: field strength

• $V$: nonlinear potential

• $\zeta$: geometric coupling constant

Varying with respect to $A^\mu$ yields nonlinear equations involving:

• Proca mass-like term $m^2 A^\mu$

• Feedback with $R_{\mu\nu}$

• Nonlinear self-coupling $(A_\alpha A^\alpha)^2$

The effective cosmological energy density arises as:

$$\Lambda_{\text{GPG}} = \frac{\zeta}{A_0^2} R_{\mu\nu} A^\mu A^\nu$$

This is not constant — it evolves. But it can mimic a constant if the field–curvature system relaxes slowly.

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🔁 3. Comparison to Other Theories

Feature	$f(R)$ Gravity	Palatini	GPG
Origin of Λ	From scalar curvature	From connection dynamics	From curvature–field memory
Field content	Metric (implicit scalar)	Metric + connection	Metric + vector field
Tension/memory	❌ None	❌ Indirect	✅ Embedded
Modifies geometry?	✅	✅	✅
Predicts Λ?	✅ (by form)	✅ (with constraints)	✅ (dynamically, structurally)
Local relaxation?	❌	❌	✅

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🧪 4. Simulations & Results

a. Curvature Collapse to Λ Shell

A strong Ricci spike (simulating a direct collapse SMDBH) evolves into a slowly decaying Λ shell. The Milky Way sits on its edge — matching the location and scale of the Local Void.

b. Redshift–Distance Relation

Light traveling through relaxing Λ_GPG accumulates integrated tension — matching ΛCDM expansion curves without assuming a global constant Λ.

c. Nuclear Binding

We reinterpret the semi-empirical mass formula (SEMF) using GPG:

• Closed nuclear shells = minimal curvature tension

• Drip-line nuclei = curvature misalignment, high Λ_GPG

Simulated curvature tension diverges from SEMF in neutron-rich isotopes — explaining why exotic nuclei are unstable, geometrically.

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🌌 5. Interpretation

• Λ is not fundamental

• It's the relaxation residue of past curvature

• It varies in space and time

• Voids decompress because curvature is fading

• Galaxies may sit inside ancient gravitational echoes

Λ is not a force. It’s the memory of geometry being forced.

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🔭 6. Predictions & Tests

GPG predicts:

• Apparent Λ varies between cosmic regions (e.g. Local Void vs dense wall)

• Drip-line nuclei will exhibit GPG residuals not captured by SEMF

• CMB anisotropies may correlate with residual curvature wave structure

• ΛCDM behavior emerges locally — not globally

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🧠 7. Conclusion

GPG offers a physically embedded, curvature-informed explanation for the cosmological constant. It subsumes modified gravity models like $f(R)$ and Palatini by doing what they cannot: making Λ emerge from structure, not assumption. Geometry becomes dynamic not just in shape, but in memory.

We are not accelerating through empty space.
We are living inside the slow echo of an ancient collapse.