Gravity Continuum Manifold Emergence Weak Field Temporal Negative

The weak-field interval

In general relativity, the geometry of spacetime is encoded in an interval s² = −t² + x² + y² + z². The negative sign on the time component and positive signs on the spatial components constitute the Lorentzian signature (−,+,+,+). This signature is what separates time from space: it makes the interval negative for timelike separations, zero for lightlike ones, and positive for spacelike ones.

The weak-field limit is the regime where gravity is a small perturbation on flat Minkowski spacetime. In this regime, the interval is modified by a small gravitational potential Φ, giving the weak-field interval: −(1 + 2Φ)t² + (1 − 2Φ)(x² + y² + z²). The declaration weak_field_interval defines this object. The key feature is that the temporal coefficient becomes more negative (1 + 2Φ) while the spatial coefficients become less positive (1 − 2Φ). This is the standard weak-field form used in the parametrized post-Newtonian formalism, where Φ plays the role of the Newtonian potential.

The framework's machine-checked library of formal theorems proves several properties of this interval. One theorem, weak_field_flat_limit, shows that when the potential Φ is zero, the weak-field interval reduces exactly to the Minkowski form. Another, weak_field_correction_bound, bounds the difference between the weak-field and flat intervals by 2|Φ|(t² + x² + y² + z²), quantifying how small the correction is for small potentials. The library also proves that the coupling constant κ = 8φ⁵ emerges from the framework's zero-parameter gravity derivation, not as a free parameter.

What the declaration does not claim is that this interval is a solution to the Einstein field equations. The weak-field interval is a definitional choice, a model of the metric in a specific regime. The framework's library proves the flat-space limit and the correction bound, but it does not prove that this interval satisfies the full nonlinear Einstein equations. That step, from the linearized Regge action to Einstein-Hilbert, is conditional on established external mathematics (Cheeger-Müller-Schrader 1984), not proved within the framework itself.

The practical consequence is that the framework reproduces the standard weak-field limit of general relativity from its discrete starting point. A reader can see that the framework's continuum limit yields the familiar Lorentzian signature and the familiar weak-field interval, with the temporal component becoming more negative in the presence of a potential. This is a check that the framework's derivation is consistent with known physics, not a new prediction about gravity.

/-- Weak-field isotropic interval:
    ds² = −(1+2Φ)dt² + (1−2Φ)(dx² + dy² + dz²). -/
def weak_field_interval (Φ t x y z : ℝ) : ℝ :=
  -(1 + 2 * Φ) * t ^ 2 + (1 - 2 * Φ) * (x ^ 2 + y ^ 2 + z ^ 2)

/-- **THEOREM (Flat Limit)**: Φ = 0 gives Minkowski. -/
theorem weak_field_flat_limit (t x y z : ℝ) :
    weak_field_interval 0 t x y z = minkowski_form t x y z := by
  unfold weak_field_interval minkowski_form; ring

/-- **THEOREM (Perturbation Bound)**: The metric correction is bounded by
    2|Φ| times the displacement norm-squared. -/
theorem weak_field_correction_bound (Φ t x y z : ℝ) :
    |weak_field_interval Φ t x y z - minkowski_form t x y z| ≤
    2 * |Φ| * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) := by
  suffices h : |weak_field_interval Φ t x y z - minkowski_form t x y z| =
    2 * |Φ| * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) from le_of_eq h
  have hdiff : weak_field_interval Φ t x y z - minkowski_form t x y z =
    -(2 * Φ * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2)) := by
    unfold weak_field_interval minkowski_form; ring
  rw [hdiff, abs_neg]
  have hS : (0 : ℝ) ≤ t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 := by positivity
  rw [show (2 : ℝ) * Φ * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) =
    (2 * Φ) * (t ^ 2 + x ^ 2 + y ^ 2 + z ^ 2) from by ring,
    abs_mul, abs_of_nonneg hS]
  congr 1
  rw [abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]

/-- **THEOREM (Coupling Derived)**: κ = 8φ⁵ — derived, not fitted. -/
theorem weak_field_coupling : ZeroParameterGravity.kappa_rs = 8 * phi ^ 5 :=
  ZeroParameterGravity.kappa_rs_closed_form

What this page does not claim

The weak-field interval is not proved to satisfy the Einstein field equations. The framework does not prove the nonlinear Regge-to-Einstein-Hilbert transition. The declaration does not claim that the potential Φ is the Newtonian gravitational potential.

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/ContinuumManifoldEmergence.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.

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