VolumeElement

formal source

   9/-- Re-export geometry and field types for ILG use. -/
  10abbrev Metric := Geometry.MetricTensor
  11abbrev RefreshField := Fields.ScalarField
  12abbrev VolumeElement := Fields.VolumeElement
  13
  14/-- Einstein–Hilbert action: S_EH = (M_P²/2) ∫ √(-g) R d^4x.
  15    Scaffold: returns symbolic R evaluation (integration machinery pending). -/
  16noncomputable def EHAction (g : Metric) : ℝ :=
  17  -- Placeholder integration over spacetime volume
  18  -- Should be: integral over manifold of √(-g(x)) * R(x)
  19  let x₀ : Fin 4 → ℝ := fun _ => 0  -- Sample point
  20  Geometry.ricci_scalar g x₀  -- Scaffold: use single-point value
  21
  22/-- Alias for consistency. -/
  23noncomputable def S_EH := EHAction
  24
  25/-- Default volume element for action integrals. -/
  26noncomputable def default_volume : VolumeElement :=
  27  { grid_spacing := 1.0, grid_spacing_pos := by norm_num }
  28
  29/-- ψ-sector kinetic term: (α/2) ∫ √(-g) g^{μν} (∂_μ ψ)(∂_ν ψ) d^4x.
  30    Now uses actual Fields.kinetic_action. -/
  31noncomputable def PsiKinetic (g : Metric) (ψ : RefreshField) (α : ℝ) : ℝ :=
  32  α * Fields.kinetic_action ψ g default_volume
  33
  34/-- ψ-sector mass/potential term: (C_lag/2) ∫ √(-g) ψ² d^4x.
  35    Now uses actual Fields.potential_action. -/
  36noncomputable def PsiPotential (g : Metric) (ψ : RefreshField) (C_lag : ℝ) : ℝ :=
  37  Fields.potential_action ψ (C_lag ^ 2) g default_volume
  38
  39/-- ψ-sector action placeholder parameterised by (C_lag, α): kinetic + potential. -/
  40noncomputable def PsiAction (g : Metric) (ψ : RefreshField) (C_lag α : ℝ) : ℝ :=
  41  PsiKinetic g ψ α + PsiPotential g ψ C_lag
  42