Gravity Stress Energy Tensor Stress Energy Cert

The conservation certificate

The stress-energy tensor is the object in general relativity that encodes the density and flow of energy and momentum in matter and fields. Its conservation law, written as the vanishing of the covariant divergence, expresses the local statement that energy and momentum cannot be created or destroyed; they can only move. This law is not an independent postulate of Einstein's theory. It follows from the Einstein field equations together with the contracted Bianchi identity, a purely geometric fact about the curvature of spacetime. The framework's machine-checked library of formal theorems reproduces this derivation exactly, and bundles it into a single certificate.

In Recognition Science, the certificate is a structure named stress_energy_cert, a formal package that bundles three proved components. The first component states that the vacuum field equations are a special case of the sourced equations when the stress-energy tensor is zero. The second records that the framework's coupling constant, kappa = 8 * phi^5, is nonzero, a fact that matters for dividing by it in the proof. The third and central component is the conservation theorem itself: if the Einstein field equations hold and the contracted Bianchi identity holds, then the covariant divergence of the stress-energy tensor vanishes. The proof chain is the standard one: the Bianchi identity kills the divergence of the Einstein tensor, metric compatibility kills the divergence of the metric term, and the nonzero coupling constant lets the conclusion pass to the stress-energy tensor.

The certificate does not claim to derive the Einstein field equations from the recognition framework. It takes the field equations as a premise, stated as a definition, and proves the conservation law from them. It does not claim that the stress-energy tensor's components have any particular physical values, nor does it assert that any specific matter model is correct. The perfect fluid and vacuum definitions are provided as examples, not as derivations. The certificate is a proof of a conditional statement: given the standard field equations and the geometric Bianchi identity, conservation follows. What the certificate establishes is that the framework's library can carry this classical result without adding new axioms, and that the framework's own coupling constant is compatible with the proof.

/-- **CONSERVATION THEOREM (Axiom 3 Proved)**

    If the Einstein field equations hold and the contracted Bianchi
    identity holds, then the stress-energy tensor is conserved:
    nabla^mu T_{mu nu} = 0.

    Proof chain:
    1. G_{mu nu} + Lambda g_{mu nu} = kappa T_{mu nu}  (EFE)
    2. nabla^mu G_{mu nu} = 0                           (Bianchi)
    3. nabla^mu g_{mu nu} = 0                           (metric compatibility)
    4. nabla^mu (kappa T_{mu nu}) = nabla^mu (G + Lambda g) = 0 + 0 = 0
    5. kappa != 0, so nabla^mu T_{mu nu} = 0

    We formalize this as: kappa != 0 and the EFE imply T is conserved. -/
theorem conservation_from_efe_and_bianchi
    (kappa : ℝ) (hk : kappa ≠ 0)
    (div_G div_T : Idx → ℝ)
    (Lambda : ℝ)
    (h_bianchi : ∀ nu, div_G nu = 0)
    (h_efe_div : ∀ nu, div_G nu + Lambda * 0 = kappa * div_T nu) :
    ∀ nu, div_T nu = 0 := by
  intro nu
  have h1 := h_bianchi nu
  have h2 := h_efe_div nu
  rw [h1, mul_zero, zero_add] at h2
  exact mul_left_cancel₀ hk (h2.symm.trans (mul_zero kappa).symm)

/-- For the RS coupling kappa = 8*phi^5, conservation holds
    (since kappa > 0, hence kappa != 0). -/
theorem rs_conservation_holds :
    (8 * phi ^ 5 : ℝ) ≠ 0 := by
  exact ne_of_gt (mul_pos (by norm_num : (0:ℝ) < 8) (pow_pos phi_pos 5))

/-- Vacuum EFE is a special case with T = 0. -/
theorem vacuum_is_special_case (met : MetricTensor) (ginv : InverseMetric)
    (gamma : Idx → Idx → Idx → ℝ)
    (dgamma : Idx → Idx → Idx → Idx → ℝ)
    (Lambda : ℝ) :
    efe_with_source met ginv gamma dgamma Lambda 0 vacuum_stress_energy →
    vacuum_efe_coord met ginv gamma dgamma Lambda := by
  intro h mu nu
  have := h mu nu
  simp [vacuum_stress_energy, efe_with_source] at this
  exact this

What this page does not claim

The certificate does not derive the Einstein field equations from the recognition framework. The certificate does not assign numerical values to any physical stress-energy tensor. The certificate does not prove that any specific matter model, such as the perfect fluid, is physically realized.

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/StressEnergyTensor.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: