Gravity Connection Christoffel From Metric

The Christoffel formula

The Christoffel symbols are the standard set of numbers that describe how a coordinate system curves in a space or spacetime. Given a metric, which defines distances and angles, the Christoffel symbols of the second kind are computed from the metric and its first derivatives. The formula in local coordinates is Γ^ρ_μν = (1/2) g^ρσ(∂_μg_νσ + ∂_νg_μσ - ∂_σg_μν). This is a classical result from differential geometry, named after Elwin Bruno Christoffel, who introduced it in 1869.

The Recognition Science library formalizes this in a machine-checked collection of formal theorems. Its declaration christoffel_from_metric is a definition, not a theorem: it takes an inverse metric and the partial derivatives of a metric as input and constructs the Christoffel symbols according to the standard formula. In plain language, it establishes that the classical Christoffel formula can be written down in a formal system, with the metric as a symmetric 4x4 matrix at each point of a four-dimensional spacetime.

What the declaration itself proves is limited. It is a definition, so it does not by itself assert any property of the symbols. The library's theorems that build on it do: christoffel_symmetric proves that the symbols are symmetric in their lower two indices, which is the torsion-free condition, provided the metric derivatives are symmetric in the same way. flat_christoffel_vanish proves that for the flat Minkowski metric, with zero derivatives, all Christoffel symbols vanish. The combined certificate, connection_cert, packages these two properties together.

These are the two defining properties of the Levi-Civita connection: it is torsion-free and metric-compatible. The library defines metric_compatibility as the condition that the covariant derivative of the metric vanishes, a separate definition. The formal work here confirms that the Christoffel symbols built from a metric satisfy the torsion-free condition and vanish in flat space. It does not prove that the Levi-Civita connection is unique, nor does it derive the Einstein field equations.

In Recognition Science, this formalization is a building block for general relativity, not a new physical claim. It is a way to have a rigorous, machine-checked foundation for the computations that appear in the framework's account of gravity. The payoff for a reader is a clear statement of what is and is not established: the formula is standard, the symmetry property is proved, and the flat-space limit is proved, but the declaration itself is a definition and carries no physical content beyond its construction.

/-- Construct Christoffel symbols from metric, inverse metric, and metric derivatives.
    dg mu nu sigma = d_mu g_{nu sigma} (partial derivative of g_{nu sigma} w.r.t. x^mu). -/
noncomputable def christoffel_from_metric
    (ginv : InverseMetric) (dg : Idx → Idx → Idx → ℝ) : ChristoffelData where
  gamma := fun rho mu nu =>
    (1/2) * ∑ sigma : Idx,
      ginv.ginv rho sigma * (dg mu nu sigma + dg nu mu sigma - dg sigma mu nu)

/-- Christoffel symbols are symmetric in the lower two indices (torsion-free).
    This follows from the symmetry of the metric derivatives:
    dg mu nu sigma = d_mu g_{nu sigma} is symmetric in (nu, sigma) because
    g_{nu sigma} = g_{sigma nu}. -/
theorem christoffel_symmetric (ginv : InverseMetric)
    (dg : Idx → Idx → Idx → ℝ)
    (dg_metric_sym : ∀ mu nu sigma, dg mu nu sigma = dg mu sigma nu) :
    ∀ rho mu nu,
      (christoffel_from_metric ginv dg).gamma rho mu nu =
      (christoffel_from_metric ginv dg).gamma rho nu mu := by
  intro rho mu nu
  simp only [christoffel_from_metric]
  congr 1
  apply Finset.sum_congr rfl
  intro sigma _
  congr 1
  rw [dg_metric_sym mu nu sigma, dg_metric_sym nu mu sigma,
      dg_metric_sym sigma mu nu, dg_metric_sym sigma nu mu]
  ring

/-- For flat spacetime (Minkowski metric with constant components),
    all Christoffel symbols vanish. -/
theorem flat_christoffel_vanish :
    ∀ rho mu nu : Idx,
      (christoffel_from_metric minkowski_inverse (fun _ _ _ => 0)).gamma rho mu nu = 0 := by
  intro rho mu nu
  simp only [christoffel_from_metric, minkowski_inverse]
  norm_num

What this page does not claim

The declaration does not prove the uniqueness of the Levi-Civita connection. The declaration does not derive the Einstein field equations or any dynamical content of general relativity. The declaration does not establish that the physical spacetime is four-dimensional; it only works in a four-dimensional coordinate patch.

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/Connection.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.

Derived articles

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