Gravity Null Cone Quadratic Tensor Class Diff Scalar Eta Implies Null Quadratic

The null-cone probe

In the four-dimensional spacetime of special relativity, a Minkowski-null covector is a direction along which the spacetime interval vanishes: for a vector k with components (k0, k1, k2, k3), it satisfies -k0² + k1² + k2² + k3² = 0. These are the paths a flash of light would take. The theorem diff_scalar_eta_implies_null_quadratic_eq states a one-way algebraic fact: if two symmetric 4x4 matrices A and B differ by exactly a scalar multiple of the Minkowski metric η = diag(-1,1,1,1), then their quadratic forms agree on every Minkowski-null covector.

Written out, the condition is that for some real number λ, every entry obeys A(i,j) - B(i,j) = λ η(i,j). The conclusion is that for every null k, the contraction Σᵢⱼ A(i,j) kᵢ kⱼ equals Σᵢⱼ B(i,j) kᵢ kⱼ. This is the easy half of a larger result: the full theorem in the framework's machine-checked library of formal theorems proves the converse as well, that for symmetric matrices, agreement on all null directions forces the difference to be a scalar multiple of η. The one-way statement here is the direction that is used to show that adding a metric multiple does not change what an observer moving at light speed would measure.

The practical meaning is a uniqueness statement with a precise boundary. If two symmetric matrices produce identical readings on every lightlike direction, they are the same up to a single number λ times the metric. Any other difference, such as a component that changes the matrix in a way not proportional to η, would be detectable by some null probe. The theorem also shows that the antisymmetric part of a matrix is invisible to this kind of contraction, since symmetrizing a matrix leaves its null-quadratic values unchanged; only the symmetric part matters for what lightlike directions reveal.

In Recognition Science, this result is a Phase 5 algebraic prerequisite for building a geometric tensor from recognition ancestry. The framework models the values of a symmetric quadratic form on null covectors as the observable data. The theorem establishes that this data determines the symmetric matrix modulo a scalar multiple of η. It does not construct that tensor, identify any matrix with Ricci or stress-energy, or close the Einstein field equation. Those targets remain open. The declaration is a lemma in a larger program, not the program itself.

What a reader can now see is a clean boundary on what lightlike measurements can and cannot determine. The null cone is a powerful probe: it fixes a symmetric matrix up to one scalar. That scalar is the part of the metric that light itself cannot see, because the metric vanishes precisely on the null cone. This is why the theorem matters for any theory that hopes to reconstruct geometry from what light observes.

/-- Converse: a pure metric difference is invisible on the null cone. -/
theorem diff_scalar_eta_implies_null_quadratic_eq
    (A B : Matrix (Fin 4) (Fin 4) ℝ) (lam : ℝ)
    (hlam : ∀ i j, A i j - B i j = lam * minkowskiEta4 i j) :
    ∀ k, MinkowskiNull k → quadContr A k = quadContr B k := by
  intro k hk
  have hAB : A = B + lam • minkowskiEta4 := by
    ext i j
    have hij := hlam i j
    simp only [Matrix.add_apply, Matrix.smul_apply, smul_eq_mul]
    linarith
  have hcontr := congrArg (fun M => quadContr M k) hAB
  change quadContr A k = quadContr (B + lam • minkowskiEta4) k at hcontr
  have hadd :
      quadContr (B + lam • minkowskiEta4) k =
        quadContr B k + quadContr (lam • minkowskiEta4) k := by
    simp only [quadContr, Matrix.add_apply, add_mul, Finset.sum_add_distrib]
  rw [hadd, quadContr_smul_eta_of_null lam k hk, add_zero] at hcontr
  exact hcontr

/-- Biconditional for symmetric matrices. -/
theorem null_quadratic_eq_iff_diff_scalar_eta
    (A B : Matrix (Fin 4) (Fin 4) ℝ)
    (hA : Symmetric4 A)
    (hB : Symmetric4 B) :
    (∀ k, MinkowskiNull k → quadContr A k = quadContr B k) ↔
      ∃ lam : ℝ, ∀ i j, A i j - B i j = lam * minkowskiEta4 i j := by
  constructor
  · exact null_quadratic_eq_implies_diff_scalar_eta A B hA hB
  · intro h
    obtain ⟨lam, hlam⟩ := h
    exact diff_scalar_eta_implies_null_quadratic_eq A B lam hlam

/-- Quadratic contraction sees only the symmetric part. -/
theorem quadContr_eq_quadContr_symmetrize4
    (A : Matrix (Fin 4) (Fin 4) ℝ) (k : Fin 4 → ℝ) :
    quadContr A k = quadContr (symmetrize4 A) k := by
  simp only [quadContr, symmetrize4, sum_fin_four]
  ring

What this page does not claim

This theorem does not identify any matrix with Ricci or stress-energy. It does not construct horizons or Unruh temperature. It does not close the Einstein 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/NullConeQuadraticTensorClass.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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