Gravity Full Efewith Dark Energy Vacuum Stress Conserved

The conserved vacuum term

In general relativity, the Einstein field equations relate the curvature of spacetime to the energy and momentum contained in it. A cosmological constant, often written Λ, represents a uniform energy density that fills all of space. The equations require that this energy be conserved: its covariant derivative, a measure of how it changes as you move from point to point, must vanish. The framework's declaration vacuum_stress_conserved proves this conservation for a vacuum term of the form c times the metric tensor, where c is any constant. The proof uses metric compatibility, the fact that the metric's covariant derivative is zero, and linearity of the covariant derivative. It concludes that the covariant derivative of c times the metric is zero for any constant c, including the specific value used for dark energy.

The framework's dark energy module builds on this. It defines a vacuum energy density as Λ/κ, where κ is the coupling constant, and a pressure equal to minus that density. The equation of state is exactly w = −1, meaning the pressure is the negative of the energy density. The theorem vacuum_eos proves this ratio is −1 for any nonzero density. The module also shows that the extended Einstein data recovers the baseline when the Hubble parameter squared goes to zero, and that the cosmological constant is positive when H₀² is positive. These results are tagged THEOREM in the framework's machine-checked library of formal theorems, with zero axioms beyond the standard three.

What the declaration does not claim: it does not establish that the cosmological constant has any particular value, nor that dark energy exists physically. The absolute scale enters only through the input H₀², which is not derived. The dimensionless fraction Ω_Λ is forced, but the framework does not prove that this fraction matches the observed cosmological constant. The conservation theorem is structural: it holds for any constant c, not just the dark energy value. It does not prove that the vacuum term is the only possible source of dark energy, nor that the equations of motion are satisfied. The framework's library proves the conservation identity, but the physical interpretation remains an open question.

/-- **VACUUM TERM IS COVARIANTLY CONSERVED (general).** For any metric-compatible setup,
the (0,2) tensor `c · g` (constant `c` times the metric) has vanishing covariant
derivative. With `c = −Λ/κ` this is the static vacuum stress tensor `T^vac = −(Λ/κ) g`, so
`∇_λ T^vac_μν = 0` for all indices, hence `∇^μ T^vac_μν = 0`. -/
theorem vacuum_stress_conserved (c : ℝ) (met : Connection.MetricTensor)
    (ch : Connection.ChristoffelData)
    (dg : Connection.Idx → Connection.Idx → Connection.Idx → ℝ)
    (hcompat : Connection.metric_compatibility met ch dg) :
    ∀ lam mu nu : Connection.Idx,
      covDeriv02 ch (fun a b => c * met.g a b) (fun a b d => c * dg a b d) lam mu nu = 0 := by
  intro lam mu nu
  rw [covDeriv02_smul]
  rw [show covDeriv02 ch met.g dg lam mu nu = (0 : ℝ) from hcompat lam mu nu]
  ring

/-- **VACUUM EQUATION OF STATE IS `w = −1`.** The static vacuum fluid has `p/ρ = −1`. -/
theorem vacuum_eos (rho : ℝ) (h : rho ≠ 0) : vacuum_pressure rho / rho = -1 := by
  unfold vacuum_pressure
  rw [neg_div, div_self h]

/-- `Λ_RS` is positive for any positive Hubble scale (forced by `Ω_Λ > 0`). -/
theorem Lambda_RS_pos {H0sq : ℝ} (h : 0 < H0sq) : 0 < Lambda_RS H0sq := by
  unfold Lambda_RS
  exact mul_pos (mul_pos (by norm_num : (0 : ℝ) < 3) h) Omega_Lambda_RS_pos

What this page does not claim

The declaration does not prove that the cosmological constant has any particular numerical value. It does not establish that dark energy exists physically. It does not prove that the vacuum term is the only source of dark energy.

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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Derived articles

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