StressEnergyTensor

formal source

  61
  62/-- **DEFINITION: Recognition Energy-Momentum Tensor**
  63    The stress-energy tensor derived from the RRF potential Ψ. -/
  64noncomputable def StressEnergyTensor (psi : RRF) (g : MetricTensor) : BilinearForm :=
  65  fun x _ low =>
  66    let μ := low 0
  67    let ν := low 1
  68    (partialDeriv_v2 psi μ x) * (partialDeriv_v2 psi ν x) -
  69    (1/2 : ℝ) * g.g x (fun _ => 0) low * (
  70      Finset.sum (Finset.univ : Finset (Fin 4)) (fun α =>
  71        Finset.sum (Finset.univ : Finset (Fin 4)) (fun β =>
  72          inverse_metric g x (fun i => if i.val = 0 then α else β) (fun _ => 0) *
  73          (partialDeriv_v2 psi α x) * (partialDeriv_v2 psi β x)
  74        )
  75      )
  76    )
  77
  78/-- **DEFINITION: Time-Translation Invariance**
  79    A property of the metric and potential where they are independent of coordinate time x₀. -/
  80def IsTimeTranslationInvariant (psi : RRF) (g : MetricTensor) : Prop :=
  81  ∀ t₁ t₂ : ℝ, ∀ ijk : Fin 3 → ℤ,
  82    let x₁ : Fin 4 → ℝ := fun i => if i.val = 0 then t₁ else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
  83    let x₂ : Fin 4 → ℝ := fun i => if i.val = 0 then t₂ else (ijk (match i.val with | 1 => 0 | 2 => 1 | _ => 2) : ℝ)
  84    psi x₁ = psi x₂ ∧ g.g x₁ = g.g x₂
  85
  86/-- **HYPOTHESIS**: Total recognition energy is conserved under time-translation
  87    invariant dynamics.
  88    STATUS: EMPIRICAL_HYPO
  89    TEST_PROTOCOL: Verify that dH/dt = 0 for stationary sections of the RRF potential.
  90    FALSIFIER: Discovery of a time-translation invariant system where TotalHamiltonian
  91    is not conserved. -/
  92def H_EnergyConservation (psi : RRF) (g : MetricTensor) : Prop :=
  93  IsTimeTranslationInvariant psi g →
  94  (∀ t, TotalHamiltonian psi g t = TotalHamiltonian psi g 0)