Gravity Analysis Edge Ttdecomposition Lorentz4 D Minkowski Trace Symmetrized Out

The symmetrized outer product

In the linear algebra of four-dimensional spacetime, a symmetric matrix is one whose entries mirror across the diagonal: the entry in row i and column j equals the entry in row j and column i. The symmetrized outer product is a way to build such a matrix from two vectors. Given vectors m and l, each with four real components, the matrix entry at position (i, j) is m_i * l_j + l_i * m_j. The formula guarantees symmetry by construction, since swapping i and j leaves the sum unchanged.

The declaration minkowskiTrace_symmetrizedOuter belongs to a machine-checked library of formal theorems, a collection where every result is verified by a computer from explicit axioms. In this library, the declaration proves a specific fact: the Minkowski trace of the symmetrized outer product of two vectors equals twice their Minkowski dot product. The trace is computed with the metric signature (-,+,+,+), meaning the time component contributes negatively: trace(H) = -H(0,0) + H(1,1) + H(2,2) + H(3,3). The dot product uses the same signature: m·l = -(m_0)(l_0) + (m_1)(l_1) + (m_2)(l_2) + (m_3)(l_3). The theorem states that trace(symmetrizedOuter m l) = 2 * (m·l).

This identity is a small but load-bearing piece of a larger decomposition. In the framework's gravity analysis, the symmetrized outer product appears in the projector used to isolate the transverse-traceless part of a symmetric matrix against a null wave covector. The trace identity ensures that the projector's correction term has the correct trace behavior, a necessary step in the algebraic layer that separates gauge degrees of freedom from physical ones.

The declaration does not itself perform any physical decomposition. It does not decompose perturbations on a lattice, does not prove convergence to the Einstein-Hilbert action, and attaches no physical polarization normalization. It is a theorem about a formula, not about the geometry of spacetime. Its proof is kernel-checked, with no unproven assumptions beyond the standard three axioms of the ambient type theory, but that audit concerns the formal derivation, not the physical interpretation of the result.

def symmetrizedOuter (m l : Fin 4 → ℝ) : Mat4 :=
  fun i j => m i * l j + l i * m j

/-- Metric trace of a covariant matrix: `η^{ij} H_{ij}`. -/
def minkowskiTrace (H : Mat4) : ℝ :=
  -(H 0 0) + H 1 1 + H 2 2 + H 3 3

What this page does not claim

The declaration does not decompose any physical perturbation on a lattice. The declaration does not prove convergence to the Einstein-Hilbert action. The declaration attaches no physical polarization normalization.

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/Analysis/EdgeTTDecompositionLorentz4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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