Gravity Ricci Tensor Ricci Flat

The flat spacetime result

The Ricci tensor is a standard object in differential geometry: it condenses the full curvature information of the Riemann tensor into a smaller matrix, one entry per pair of coordinate directions. Physicists use it in general relativity because the Einstein field equations, which govern how mass and energy curve spacetime, are written directly in terms of it. The declaration ricci_flat in the framework's machine-checked library of formal theorems proves a specific, limited statement: if every connection coefficient and every derivative of those coefficients is set to zero, then the Ricci tensor is identically zero.

This is the flat spacetime case. Setting the connection coefficients to zero means choosing coordinates in which there is no curvature at all, the geometry of special relativity. The theorem confirms, in the framework's formal language, that such a geometry has no Ricci curvature. The proof is short: it expands the definition of the Ricci tensor as a contraction of the Riemann tensor, applies a previously proved fact that the Riemann tensor vanishes in this case, and simplifies. The same file also shows that the scalar curvature and the Einstein tensor vanish for this flat geometry, and that the flat Minkowski metric satisfies the vacuum Einstein field equations with zero cosmological constant.

What the theorem does not claim is more important than what it does. It does not prove that a vanishing Ricci tensor implies flat spacetime in general. That converse statement, the famous condition for Ricci-flat but not necessarily flat geometries, is a separate and much deeper result in differential geometry, and it is not part of this declaration. The theorem also does not say anything about curved spacetimes, gravitational waves, or the behavior of the Einstein equations with matter present. It is a consistency check on the framework's definitions: the formal objects behave as they should in the simplest possible case.

In Recognition Science, this result sits at the foundation of the framework's treatment of gravity. The framework derives its constants and dimensional structure from a ledger of recognition events, and it models spacetime geometry through the standard tensor machinery. The ricci_flat theorem is one of the checks that this machinery is coherent: when the framework's geometry reduces to flat spacetime, the curvature quantities vanish as they must. It is a small but necessary piece of the larger project of showing that the framework's forced structure reproduces the known laws of physics.

/-- Ricci tensor vanishes for flat spacetime. -/
theorem ricci_flat (mu nu : Idx) :
    ricci_tensor (fun _ _ _ => 0) (fun _ _ _ _ => 0) mu nu = 0 := by
  simp [ricci_tensor, riemann_flat_vanishes]

/-- Scalar curvature vanishes for flat spacetime. -/
theorem scalar_flat :
    scalar_curvature minkowski_inverse (fun _ _ _ => 0) (fun _ _ _ _ => 0) = 0 := by
  simp [scalar_curvature, ricci_flat]

/-- Einstein tensor vanishes for flat spacetime. -/
theorem einstein_flat (mu nu : Idx) :
    einstein_tensor minkowski minkowski_inverse
      (fun _ _ _ => 0) (fun _ _ _ _ => 0) mu nu = 0 := by
  simp [einstein_tensor, ricci_flat, scalar_flat]

/-- Flat Minkowski metric satisfies the vacuum EFE with Lambda = 0. -/
theorem minkowski_is_vacuum_solution :
    vacuum_efe_coord minkowski minkowski_inverse
      (fun _ _ _ => 0) (fun _ _ _ _ => 0) 0 := by
  intro mu nu
  simp [einstein_flat]

What this page does not claim

The theorem does not prove that a vanishing Ricci tensor implies flat spacetime in general. The theorem says nothing about curved spacetimes, gravitational waves, or the Einstein equations with matter present. The framework's derivation of gravity from recognition events is not established by this declaration.

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/RicciTensor.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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