Gravity Ricci Tensor Scalar Flat

The flat spacetime check

In general relativity, the scalar curvature is a single number at each point of spacetime that summarizes how much the geometry bends there. For a completely empty, flat spacetime, the familiar Minkowski metric, that number is exactly zero. This is not a deep physical discovery but a consistency check: the simplest possible spacetime has no curvature, so its scalar curvature should vanish.

The standard calculation starts with the Riemann tensor, which encodes all information about curvature. Contracting it yields the Ricci tensor, and tracing that against the inverse metric gives the scalar curvature. For the Minkowski metric, every component of the Riemann tensor is zero, so the Ricci tensor and the scalar curvature follow as zero. The framework's library of machine-checked formal theorems records this as the declaration scalar_flat, which proves that the scalar curvature of the Minkowski metric is zero.

The same library also proves that the Einstein tensor, built from the Ricci tensor and scalar curvature, vanishes for flat spacetime. This means the Minkowski metric is a vacuum solution of the Einstein field equations with no cosmological constant. The declaration minkowski_is_vacuum_solution states this result directly.

In Recognition Science, this result is a boundary condition for the framework's account of gravity. The framework models spacetime geometry through the same tensors as general relativity, and its library shows that the flat, empty case behaves correctly. This is a foundational check, not a new physical prediction. The declaration does not claim that all flat spacetimes have zero scalar curvature, nor does it address curved spacetimes, black holes, or gravitational waves. It is a precise statement about one specific metric.

/-- 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 declaration does not prove that all flat spacetimes have zero scalar curvature. The declaration does not address curved spacetimes, black holes, or gravitational waves. The declaration does not derive the Einstein field equations from Recognition Science principles.

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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