Gravity Connection Christoffel Data

The connection in coordinates

In general relativity, gravity is not a force pulling things together; it is the curvature of spacetime itself. To describe that curvature, physicists use the Christoffel symbols, a set of numbers, written Γ^ρ_μν, that encode how a coordinate system twists and tilts as you move through it. They are the building blocks of the Levi-Civita connection, the standard mathematical rule for taking a derivative that respects the geometry. In a flat space with no gravity, all these numbers are zero; in a curved space, they are the quantities that let you compute geodesics, the paths that freely falling objects follow.

The definition is direct. For a metric tensor g, which measures distances and angles, the Christoffel symbols are computed from its derivatives: Γ^ρ_μν = (1/2) g^ρσ (∂_μg_νσ + ∂_νg_μσ − ∂_σg_μν). This formula was introduced by Elwin Bruno Christoffel in 1869, long before Einstein used it as the central tool of general relativity in 1915. Two properties define the Levi-Civita connection: it is torsion-free, meaning the symbols are symmetric in their lower two indices (Γ^ρ_μν = Γ^ρ_νμ), and it is metric-compatible, meaning the covariant derivative of the metric itself vanishes.

In Recognition Science, the framework's machine-checked library of formal theorems defines these objects from scratch in a coordinate patch. The declaration ChristoffelData is a structure that holds a function of three indices, representing the Christoffel symbols of the second kind. The library then proves the key classical facts as theorems: that the symbols constructed from a metric are symmetric in their lower indices, and that for the flat Minkowski metric of special relativity, all symbols vanish. These are not new physics; they are a formal re-derivation of standard results, checked line by line by a computer.

The library's contribution is precision, not discovery. It shows that the classical definition of the connection can be expressed cleanly in a machine-checked language, with no hidden assumptions about the smoothness of the metric or the choice of coordinates. The construction takes as input the metric, its inverse, and the partial derivatives of the metric, and outputs the Christoffel symbols. This makes the foundations of general relativity computable and auditable, a step toward using the framework's own derived constants, such as the gravitational constant, in a fully formal setting.

What the declaration does not claim is just as important. It does not prove that the Levi-Civita connection is unique, nor does it show that these symbols describe the actual curvature of physical spacetime. It is a local, coordinate-based definition, not a global statement about the universe. The library explicitly works in a coordinate patch, avoiding the abstract machinery of manifolds, so it says nothing about the global topology of spacetime. It also does not derive the Einstein field equations or any dynamical law; it merely sets up the kinematic language in which those laws are written.

/-- The Christoffel symbols of the second kind in local coordinates.
    Gamma^rho_{mu nu} = (1/2) g^{rho sigma} (d_mu g_{nu sigma} + d_nu g_{mu sigma} - d_sigma g_{mu nu})

    We represent these as a function of three indices.
    The partial derivatives d_mu g_{nu sigma} are provided as input
    (they depend on the coordinate system and the point). -/
structure ChristoffelData where
  gamma : Idx → Idx → Idx → ℝ

/-- 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 make any statement about the global topology of spacetime. The declaration does not derive the Einstein field equations or any dynamical law of gravity.

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.

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