Gravity Rsnull Field Equation

The null reduction

In general relativity, gravity is described by the Einstein field equation, which relates the curvature of spacetime to the energy and momentum contained in it. The equation is a matrix equation, meaning it holds for each of its 16 components. One standard technique for extracting information from such an equation is to contract it with a vector, which is a way of summing the components weighted by that vector. When the vector is null, meaning it points along the path of a light ray, a remarkable simplification occurs.

The simplification is this: the term in the equation that involves the metric tensor, which encodes the geometry of spacetime, vanishes entirely. This is not an approximation; it is an exact algebraic consequence of the null condition. The result is a scalar equation, a single number, that relates the contraction of the Ricci curvature tensor to the contraction of the stress-energy tensor. The framework's machine-checked library of formal theorems proves this reduction step in full.

In Recognition Science, this algebraic step is formalized as a theorem. The framework models the field equation in its matrix form, with a coupling constant that sets the strength of gravity. The theorem states that if the matrix equation holds, then for any null vector, the contracted form also holds. The proof is entirely algebraic and does not rely on any physical assumptions beyond the form of the equation and the null condition.

The framework's library also proves a limit of this reduction: the scalar equation cannot recover the full matrix equation. Adding a multiple of the metric tensor to the matrix leaves every null contraction unchanged, meaning the scalar equation is blind to that part of the original equation. This is a precise statement of what information is lost in the reduction.

The practical consequence is that the null-contracted equation is a reliable and simpler tool for studying lightlike geodesics and horizons. It is not a complete replacement for the full equation, but it is a valid and powerful restriction. The framework's contribution is to make this algebraic step fully explicit and machine-checked, so that any use of the reduced equation rests on a verified foundation.

/--
RS-normalized specialization: an independently inhabited Einstein-shaped
source with the RS coupling yields its null-contracted scalar form.
-/
theorem rs_null_scalar_of_source
    {T Ric : Matrix (Fin 4) (Fin 4) ℝ}
    (h : EinsteinShapedSource kappa_einstein T Ric) :
    ∀ k, MinkowskiNull k →
      quadContr Ric k = kappa_einstein * quadContr T k :=
  null_scalar_of_source h

/-- Every scalar Minkowski metric term vanishes on a null probe. -/
theorem quadContr_metric_term_eq_zero
    (f : ℝ) (k : Fin 4 → ℝ)
    (hk : MinkowskiNull k) :
    quadContr (f • minkowskiEta4) k = 0 := by
  rw [quadContr_smul, quadContr_minkowskiEta4]
  rw [show -(k 0) ^ 2 + (k 1) ^ 2 + (k 2) ^ 2 + (k 3) ^ 2 = 0 by
    simpa [MinkowskiNull] using hk]
  ring

/--
The null equation does not recover the scalar metric term: adding a nonzero
multiple of `η` changes the matrix while preserving every null contraction.
-/
theorem scalar_metric_term_is_null_invisible :
    ∃ D : Matrix (Fin 4) (Fin 4) ℝ,
      D ≠ 0 ∧
      ∀ k, MinkowskiNull k → quadContr D k = 0 := by
  refine ⟨minkowskiEta4, ?_, ?_⟩
  · intro h
    have h00 := congrFun (congrFun h (0 : Fin 4)) (0 : Fin 4)
    norm_num [minkowskiEta4] at h00
  · intro k hk
    simpa using quadContr_metric_term_eq_zero 1 k hk

What this page does not claim

This module does not derive the Einstein field equation from the framework's ledger. This module does not identify the null vector with a specific physical lightlike direction. This module does not construct a horizon patch or a solution to the field equation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/RSNullFieldEquation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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