IndisputableMonolith.Gravity.RicciTensor
The RicciTensor module contracts the Riemann curvature tensor over one index pair to define the Ricci tensor in local coordinates. It is cited by derivations of the Einstein field equations and the stress-energy conservation law. The module assembles the contraction from the imported RiemannTensor and Connection definitions with no new theorems.
claimThe Ricci tensor is the contraction $R_{\mu\nu} = \sum_\rho R^\rho{}_{\mu\rho\nu}$ of the Riemann curvature tensor $R^\rho{}_{\sigma\mu\nu}$ obtained from the Levi-Civita connection.
background
This module sits inside the coordinate formulation of general relativity. It imports the Levi-Civita connection (Christoffel symbols as functions of the metric) and the Riemann tensor (built from Christoffel symbols and their first derivatives) from the upstream modules. The Ricci tensor is obtained by the indicated index contraction.
Constants supplies the RS time quantum but plays no role in the curvature definitions. The module therefore provides the standard contraction step required before the Einstein tensor or stress-energy conservation can be stated.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the Ricci tensor to EinsteinHilbertAction (which proves the Hilbert variation that yields the Einstein tensor) and to StressEnergyTensor (which proves conservation from the contracted Bianchi identity). It therefore completes the curvature contraction step in the Recognition Science gravity chain leading to the Einstein field equations.
scope and limits
- Does not treat abstract manifolds or coordinate-free index notation.
- Does not prove the algebraic Bianchi identity or its contractions.
- Does not link the Ricci tensor to the Recognition Science forcing chain or phi-ladder.
- Does not derive the Einstein field equations themselves.