Gravity Metric From Defect Weak Field Small Perturbation

The weak-field theorem

In general relativity, the metric tensor describes the geometry of spacetime. In the weak-field limit, this metric is written as the flat Minkowski metric plus a small perturbation. The Recognition Science framework constructs this perturbation from a field of defects in its fundamental ledger, a discrete record of recognition events. The declaration weak_field_small_perturbation is a theorem in the framework's machine-checked library of formal theorems. It states that if the defect density is small everywhere, then the resulting metric perturbation is also small, specifically with magnitude less than one.

The theorem is a formal consequence of the framework's definitions. The metric perturbation is defined as proportional to the defect density, with a coupling constant kappa = 8*phi^5. The weak-field condition is defined as the requirement that the absolute defect density is less than 1/kappa everywhere. The theorem then follows directly: if the density is bounded by 1/kappa, then the perturbation, which is kappa times the density, is bounded by 1. This is a simple algebraic consequence, but its significance is that it establishes a consistency condition: the framework's weak-field regime is self-consistent, in that small sources produce small geometric effects.

The theorem does not claim that the metric perturbation is exactly the linearized gravity metric perturbation of general relativity. It does not derive the Einstein field equations, nor does it show that the framework's perturbation satisfies any particular gauge condition. It only establishes a bound on the magnitude of one component of the perturbation, given a bound on the defect density. The theorem is a local statement about the framework's construction, not a global statement about the equivalence of the framework's gravity to general relativity.

The practical consequence is that the framework can be used to study weak gravitational fields, such as those around planets or stars, without worrying about the breakdown of the linear approximation. The theorem provides a formal guarantee that the framework's own weak-field limit is well-defined. This is a necessary step for any further comparison with observed gravitational phenomena, though it is not by itself such a comparison.

/-- Under the weak-field condition, the metric perturbation is small. -/
theorem weak_field_small_perturbation (d : DefectField) (hd : weak_field_condition d)
    (r : ℝ) :
    |(metric_perturbation_from_defect d r).components 0 0| <  1 := by
  rw [perturbation_proportional_to_kappa]
  have hk := ZeroParameterGravity.kappa_pos
  have hd0 := hd r 0 0
  have h_eq :
      |-ZeroParameterGravity.kappa_rs * d.density r 0 0|
        = ZeroParameterGravity.kappa_rs * |d.density r 0 0| := by
    rw [show (-ZeroParameterGravity.kappa_rs * d.density r 0 0)
          = -(ZeroParameterGravity.kappa_rs * d.density r 0 0) from by ring,
        abs_neg, abs_mul, abs_of_pos hk]
  rw [h_eq]
  calc ZeroParameterGravity.kappa_rs * |d.density r 0 0|
      < ZeroParameterGravity.kappa_rs * (1 / ZeroParameterGravity.kappa_rs) :=
        mul_lt_mul_of_pos_left hd0 hk
    _ = 1 := by field_simp

/-- Under the weak-field condition, the metric perturbation is small. -/
theorem weak_field_small_perturbation (d : DefectField) (hd : weak_field_condition d)
    (r : ℝ) :
    |(metric_perturbation_from_defect d r).components 0 0| <  1 := by
  rw [perturbation_proportional_to_kappa]
  have hk := ZeroParameterGravity.kappa_pos
  have hd0 := hd r 0 0
  have h_eq :
      |-ZeroParameterGravity.kappa_rs * d.density r 0 0|
        = ZeroParameterGravity.kappa_rs * |d.density r 0 0| := by
    rw [show (-ZeroParameterGravity.kappa_rs * d.density r 0 0)
          = -(ZeroParameterGravity.kappa_rs * d.density r 0 0) from by ring,
        abs_neg, abs_mul, abs_of_pos hk]
  rw [h_eq]
  calc ZeroParameterGravity.kappa_rs * |d.density r 0 0|
      < ZeroParameterGravity.kappa_rs * (1 / ZeroParameterGravity.kappa_rs) :=
        mul_lt_mul_of_pos_left hd0 hk
    _ = 1 := by field_simp

/-- The metric perturbation h_mu_nu induced by a defect field.
    In the Newtonian limit: h_00 = -2*Phi, h_ij = -2*Phi*delta_ij
    where Phi is the gravitational potential sourced by the defect density.

    The coupling constant is kappa = 8*phi^5.

    For a uniform defect density rho: Phi = -(1/2)*kappa*rho*r^2/(D=3)
    (Poisson equation: nabla^2 Phi = kappa * rho). -/
def metric_perturbation_from_defect (d : DefectField) (r : ℝ) : SymmetricTensor 3 where
  components := fun i j => if i = j then -ZeroParameterGravity.kappa_rs * d.density r 0 0 else 0
  symmetric := by intro i j; simp [eq_comm]

What this page does not claim

The theorem does not derive the Einstein field equations. The theorem does not show that the framework's perturbation satisfies any gauge condition. The theorem does not establish the equivalence of the framework's gravity to general relativity.

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/MetricFromDefect.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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