IndisputableMonolith.Gravity.MetricFromDefect

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Gravity.ZeroParameterGravity
   4
   5/-!
   6# Step 2: Metric Perturbation from J-Cost Defect Density
   7
   8Defines the metric perturbation h_mu_nu as a function of the J-cost defect
   9field on the lattice. In the weak-field limit, spacetime geometry emerges
  10from the distribution of ledger defects.
  11
  12## Physical Picture
  13
  14- Each voxel carries a J-cost defect: J(x) = ½(x + x⁻¹) - 1 ≥ 0
  15- A region with high defect density has more "strain" in the ledger
  16- This strain curves the emergent spacetime: g_mu_nu = eta_mu_nu + h_mu_nu
  17- The metric perturbation h is proportional to the integrated defect density
  18- The proportionality constant is kappa = 8*phi^5 (from ZeroParameterGravity)
  19
  20## Linearized Gravity Convention
  21
  22In linearized GR, the metric perturbation h_mu_nu satisfies:
  23- h is symmetric: h_mu_nu = h_nu_mu
  24- The trace-reversed perturbation: h_bar = h - (1/2) eta * trace(h)
  25- The linearized EFE in harmonic gauge: nabla^2 h_bar = -2*kappa * T
  26
  27In RS, the defect density IS the stress-energy source T^00.
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Gravity
  32namespace MetricFromDefect
  33
  34open Constants
  35
  36noncomputable section
  37
  38/-! ## Symmetric Tensor Structure -/
  39
  40/-- A symmetric 2-tensor in D dimensions, represented as a symmetric
  41    matrix (upper-triangular storage). For D=3+1 spacetime, indices
  42    run over {0,1,2,3}. For the spatial part, D=3 with {1,2,3}. -/
  43structure SymmetricTensor (D : ℕ) where
  44  components : Fin D → Fin D → ℝ
  45  symmetric : ∀ i j, components i j = components j i
  46
  47/-- The flat Minkowski metric in D spatial dimensions (spatial part only).
  48    eta_ij = delta_ij (Euclidean for spatial indices). -/
  49def flat_metric_spatial : SymmetricTensor 3 where
  50  components := fun i j => if i = j then 1 else 0
  51  symmetric := by intro i j; simp [eq_comm]
  52
  53/-- The trace of a symmetric tensor. -/
  54def trace (t : SymmetricTensor 3) : ℝ := t.components 0 0 + t.components 1 1 + t.components 2 2
  55
  56/-! ## Defect Field to Metric Perturbation -/
  57
  58/-- The J-cost defect density at a point. In RS, this is the source
  59    of spacetime curvature. A defect density of zero means flat space. -/
  60structure DefectField where
  61  density : ℝ → ℝ → ℝ → ℝ
  62  nonneg : ∀ x y z, 0 ≤ density x y z
  63
  64/-- The metric perturbation h_mu_nu induced by a defect field.
  65    In the Newtonian limit: h_00 = -2*Phi, h_ij = -2*Phi*delta_ij
  66    where Phi is the gravitational potential sourced by the defect density.
  67
  68    The coupling constant is kappa = 8*phi^5.
  69
  70    For a uniform defect density rho: Phi = -(1/2)*kappa*rho*r^2/(D=3)
  71    (Poisson equation: nabla^2 Phi = kappa * rho). -/
  72def metric_perturbation_from_defect (d : DefectField) (r : ℝ) : SymmetricTensor 3 where
  73  components := fun i j => if i = j then -ZeroParameterGravity.kappa_rs * d.density r 0 0 else 0
  74  symmetric := by intro i j; simp [eq_comm]
  75
  76/-- The metric perturbation is symmetric by construction. -/
  77theorem metric_perturbation_symmetric (d : DefectField) (r : ℝ) (i j : Fin 3) :
  78    (metric_perturbation_from_defect d r).components i j =
  79    (metric_perturbation_from_defect d r).components j i :=
  80  (metric_perturbation_from_defect d r).symmetric i j
  81
  82/-- Zero defect density gives zero metric perturbation (flat space). -/
  83theorem zero_defect_flat_space (r : ℝ) :
  84    let d : DefectField := ⟨fun _ _ _ => 0, fun _ _ _ => le_refl 0⟩
  85    (metric_perturbation_from_defect d r).components 0 0 = 0 := by
  86  simp [metric_perturbation_from_defect, ZeroParameterGravity.kappa_rs]
  87
  88/-- The metric perturbation is proportional to kappa (= 8*phi^5). -/
  89theorem perturbation_proportional_to_kappa (d : DefectField) (r : ℝ) :
  90    (metric_perturbation_from_defect d r).components 0 0 =
  91    -ZeroParameterGravity.kappa_rs * d.density r 0 0 := by
  92  simp [metric_perturbation_from_defect]
  93
  94/-! ## Weak-Field Regime -/
  95
  96/-- In the weak-field regime, |h_mu_nu| << 1. This means the defect
  97    density must be small: kappa * rho << 1. -/
  98def weak_field_condition (d : DefectField) : Prop :=
  99  ∀ x y z, |d.density x y z| < 1 / ZeroParameterGravity.kappa_rs
 100
 101/-- Under the weak-field condition, the metric perturbation is small. -/
 102theorem weak_field_small_perturbation (d : DefectField) (hd : weak_field_condition d)
 103    (r : ℝ) :
 104    |(metric_perturbation_from_defect d r).components 0 0| <  1 := by
 105  rw [perturbation_proportional_to_kappa]
 106  have hk := ZeroParameterGravity.kappa_pos
 107  have hd0 := hd r 0 0
 108  have h_eq :
 109      |-ZeroParameterGravity.kappa_rs * d.density r 0 0|
 110        = ZeroParameterGravity.kappa_rs * |d.density r 0 0| := by
 111    rw [show (-ZeroParameterGravity.kappa_rs * d.density r 0 0)
 112          = -(ZeroParameterGravity.kappa_rs * d.density r 0 0) from by ring,
 113        abs_neg, abs_mul, abs_of_pos hk]
 114  rw [h_eq]
 115  calc ZeroParameterGravity.kappa_rs * |d.density r 0 0|
 116      < ZeroParameterGravity.kappa_rs * (1 / ZeroParameterGravity.kappa_rs) :=
 117        mul_lt_mul_of_pos_left hd0 hk
 118    _ = 1 := by field_simp
 119
 120/-! ## Certificate -/
 121
 122structure MetricFromDefectCert where
 123  symmetric : ∀ d r i j,
 124    (metric_perturbation_from_defect d r).components i j =
 125    (metric_perturbation_from_defect d r).components j i
 126  proportional_to_kappa : ∀ d r,
 127    (metric_perturbation_from_defect d r).components 0 0 =
 128    -ZeroParameterGravity.kappa_rs * d.density r 0 0
 129
 130theorem metric_from_defect_cert : MetricFromDefectCert where
 131  symmetric := metric_perturbation_symmetric
 132  proportional_to_kappa := perturbation_proportional_to_kappa
 133
 134end
 135
 136end MetricFromDefect
 137end Gravity
 138end IndisputableMonolith
 139