Gravity Full Efewith Dark Energy Minkowski Metric Compatible

Metric compatibility

In general relativity, the statement that a tensor field is covariantly constant, meaning its derivative along any path is zero, is a strong condition. For the metric tensor, which defines distances and angles, this condition is called metric compatibility. It is a standard property of the Levi-Civita connection, the unique connection that is both metric-compatible and torsion-free. This property is what ensures that the covariant derivative of the metric itself vanishes, a fact with a direct physical consequence for the cosmological constant.

The cosmological constant Λ appears in Einstein's field equations as a term proportional to the metric, Λg_μν. The Bianchi identity, a geometric identity, requires the Einstein tensor to have zero covariant derivative, which in turn forces the total stress-energy tensor to be conserved. A constant times the metric is covariantly conserved if and only if the metric itself is. The declaration minkowski_metric_compatible in the Recognition Science library proves this compatibility for the flat Minkowski metric, the spacetime of special relativity, using the standard Christoffel symbols derived from it. This is a machine-checked theorem, meaning it is verified by a computer proof assistant, with no gaps or unproven assumptions.

This proof is a foundational step for the framework's treatment of dark energy. The framework models the vacuum as a perfect fluid with pressure p = -ρ, giving an equation of state w = -1, which is the defining property of a cosmological constant. The theorem flat_vacuum_stress_conserved then uses metric compatibility to show that this vacuum stress-energy tensor has zero covariant derivative. This establishes that the vacuum term is consistent with the Bianchi identity, a necessary condition for any term in Einstein's equations. The framework's library proves this for the flat reference spacetime, providing the structural reason why a cosmological constant is always consistent with energy conservation.

The declaration does not claim that the physical universe is flat. It proves a property of the mathematical Minkowski metric, not a statement about the large-scale geometry of spacetime. It also does not establish that the cosmological constant's value is nonzero or positive; that is a separate result in the framework, derived from a positive Hubble parameter. The theorem proves the consistency of a constant term with the metric, not the existence or magnitude of that term.

/-- The flat Minkowski metric is covariantly constant: `∇_λ g_μν = 0`. Both the ordinary
derivative (constant metric) and the Christoffel symbols (flat) vanish. -/
theorem minkowski_metric_compatible :
    Connection.metric_compatibility Connection.minkowski
      (Connection.christoffel_from_metric Connection.minkowski_inverse (fun _ _ _ => 0))
      (fun _ _ _ => 0) := by
  unfold Connection.metric_compatibility
  intro lam mu nu
  simp [Connection.flat_christoffel_vanish]

/-- **VACUUM TERM IS COVARIANTLY CONSERVED (flat reference, grounded).** Specialization of
`vacuum_stress_conserved` to the flat reference, using the proved
`minkowski_metric_compatible`. No metric compatibility is assumed; it is discharged. -/
theorem flat_vacuum_stress_conserved (c : ℝ) :
    ∀ lam mu nu : Connection.Idx,
      covDeriv02 (Connection.christoffel_from_metric Connection.minkowski_inverse (fun _ _ _ => 0))
        (fun a b => c * Connection.minkowski.g a b) (fun a b d => c * (fun _ _ _ => (0 : ℝ)) a b d)
        lam mu nu = 0 :=
  vacuum_stress_conserved c Connection.minkowski
    (Connection.christoffel_from_metric Connection.minkowski_inverse (fun _ _ _ => 0))
    (fun _ _ _ => 0) minkowski_metric_compatible

What this page does not claim

The physical universe is flat. The cosmological constant is positive or nonzero. The proof establishes the existence of dark energy, only its consistency with the Bianchi identity.

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

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