Gravity Ricci Tensor Einstein Flat

The flat spacetime check

In general relativity, the Einstein tensor G_μν = R_μν − (1/2)R g_μν packages the curvature of spacetime into the left-hand side of the field equations. It is built from the Ricci tensor R_μν, which itself is a contraction of the full Riemann curvature tensor, and from the scalar curvature R, the trace of the Ricci tensor with the inverse metric. The Einstein tensor is the object that relates geometry to matter: in the sourced field equations, G_μν + Λ g_μν = κ T_μν, where T_μν is the stress-energy tensor describing the distribution of mass and energy.

Flat spacetime, the arena of special relativity, is the simplest possible geometry: no curvature anywhere. The declaration einstein_flat in the framework's machine-checked library of formal theorems proves that for the Minkowski metric, with all connection coefficients zero, the Einstein tensor is identically zero: G_μν = 0 for every pair of indices μ, ν. This is the expected consistency check: a spacetime with no curvature must produce no Einstein tensor, and therefore satisfies the vacuum field equations with cosmological constant Λ = 0. The same library also proves the Ricci tensor and scalar curvature vanish for this flat case, and that the Minkowski metric is a vacuum solution of the field equations.

In Recognition Science, this theorem is a small but load-bearing piece of the framework's gravity module. It is not a derivation of general relativity from the framework's recognition principles; it is a formal verification that the definitions of curvature and the Einstein tensor behave correctly in the simplest case. The declaration establishes a consistency property, not a physical law. It confirms that the framework's formalization of standard differential geometry reproduces the textbook fact that flat spacetime has vanishing Einstein tensor, which any correct treatment of gravity must satisfy.

The theorem is conditional on the specific structure of the Minkowski metric and the zero connection. It does not claim that all solutions of the vacuum field equations are flat, nor that the Einstein tensor vanishes for any curved spacetime. The symmetry of the Einstein tensor, proved separately in the library, requires the Ricci tensor to be symmetric, which holds when the connection is torsion-free; the flat-spacetime result does not extend to arbitrary connections. The declaration is a check on the definitions, not a statement about the content of gravity beyond the flat case.

/-- 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]

/-- 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]

/-- 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 declaration does not prove that all vacuum solutions of the field equations are flat. The declaration does not derive general relativity from recognition principles; it verifies a consistency property of the definitions. The symmetry theorem for the Einstein tensor does not apply to connections with torsion.

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)

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