Drogidi Parallel Curved Spaces Theorem

This repository contains the mathematical formulation of the Drogidi Theory of Parallel Curved Spaces, a geometric framework proposing that Dark Matter is not a particle, but a geometric interaction between parallel manifolds mediated by the Synergy Tensor.

The theory extends General Relativity by introducing:

• multiple interacting manifolds $(M_a)$,
• a mapping field $(\Phi_{ab}: M_a \to M_b)$,
• a synergy action $(S_{\text{synergy}})$,
• and a resulting modification of the Einstein field equations.

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1. Drogidi Field Equation

$$G^{(a)}_{\mu\nu} + \Lambda_a g^{(a)}_{\mu\nu} = 8\pi G \left( T^{(a)}_{\mu\nu} + T^{(\text{synergy})}_{\mu\nu} \right)$$

This equation governs the geometry of each manifold $(M_a)$, with an additional contribution from the synergy energy–momentum tensor.

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2. Synergy Energy–Momentum Tensor

$$T^{(\text{synergy})}_{\mu\nu} = -\frac{2}{\sqrt{-g^{(a)}}} \frac{\delta S_{\text{synergy}}}{\delta g^{(a)\mu\nu}} = \sum_{b \neq a} K_{ab} J^{(ab)}_{\mu\nu}$$

The synergy tensor encodes the geometric influence of all other manifolds $(M_b)$ on $(M_a)$.

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3. Interaction Tensor

To ensure that all terms are rank‑2 tensors, the full interaction tensor is:

$$J^{(ab)}_{\mu\nu} = \alpha \big(R^{(a)} - \Phi_{ab}^* R^{(b)}\big) g^{(a)}_{\mu\nu} + \beta \big(R^{(a)}_{\mu\nu} - \Phi_{ab}^* R^{(b)}_{\mu\nu}\big) + \gamma \big(\theta_a - \Phi_{ab}^* \theta_b\big) g^{(a)}_{\mu\nu}$$

Where:

• $(R^{(a)})$, $(R^{(b)})$ are scalar curvatures,
• (R^{(a)}{\mu\nu}), (R^{(b)}{\mu\nu}): are Ricci tensors,
• $(\theta_a)$, $(\theta_b)$ are expansion scalars,
• $(\Phi_{ab}^*)$ denotes the pullback via the mapping field.

Thus $(J^{(ab)}_{\mu\nu})$ is a symmetric rank‑2 tensor on $(M_a)$.

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Citation

If you use these results in your research, please cite: CITATION.cff

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License

Mathematical content is licensed under:

Creative Commons BY‑NC‑ND 4.0
(Attribution – NonCommercial – NoDerivatives)

See: License: CC BY-NC-ND 4.0