Gravity Connection

The gravity connection

A connection in differential geometry is a rule for comparing vectors at nearby points. On a curved surface, a vector carried from one point to another must be adjusted to stay parallel; the connection specifies that adjustment. In general relativity, the relevant connection is the Levi-Civita connection, which is uniquely determined by the metric tensor, the object that encodes distances and angles in spacetime. The connection's components are the Christoffel symbols, which are built from the metric and its first derivatives. This construction is formalized in local coordinates.

The formalization works in a coordinate patch, where the metric is a smooth 4x4 matrix function g : R^4 -> R^{4x4}. The Christoffel symbols are defined by the standard formula: Gamma^rho_{mu nu} = (1/2) g^{rho sigma} (d_mu g_{nu sigma} + d_nu g_{mu sigma} - d_sigma g_{mu nu}). Two key properties of this construction are proved. First, the Christoffel symbols are symmetric in their lower two indices, which means the connection is torsion-free. Second, the covariant derivative of the metric vanishes, which is metric compatibility. Together, these two properties define the Levi-Civita connection: the unique connection that is both torsion-free and metric-compatible.

For flat spacetime, the Minkowski metric with constant components, all Christoffel symbols vanish. This is the expected result: in the absence of curvature, there is no need to adjust vectors as they are parallel transported. These results are packaged into a certificate, a single object that bundles the two theorems. This certificate provides a machine-checked guarantee that the formalized connection has the defining properties of the Levi-Civita connection.

In Recognition Science, this work is part of a larger effort to build a machine-checked library of formal theorems. The library is a collection of definitions and theorems that have been verified by a computer. This contribution adds the foundational geometry of general relativity to that library. The framework does not derive the connection from its first principles; it formalizes the standard mathematical construction. The payoff is a rigorous foundation for further work on gravity within the framework's formal system.

/-- 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

/-- Metric compatibility: the covariant derivative of the metric vanishes.
    nabla_lambda g_{mu nu} = d_lambda g_{mu nu} - Gamma^rho_{lambda mu} g_{rho nu}
                                                  - Gamma^rho_{lambda nu} g_{mu rho} = 0

    This is the defining property of the Levi-Civita connection:
    the unique connection that is both torsion-free (symmetric Christoffel)
    and metric-compatible (nabla g = 0). -/
def metric_compatibility (met : MetricTensor) (ch : ChristoffelData)
    (dg : Idx → Idx → Idx → ℝ) : Prop :=
  ∀ lambda mu nu : Idx,
    dg lambda mu nu -
    ∑ rho : Idx, (ch.gamma rho lambda mu * met.g rho nu) -
    ∑ rho : Idx, (ch.gamma rho lambda nu * met.g mu rho) = 0

/-- 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 module does not derive the connection from Recognition Science's first principles; it formalizes the standard Levi-Civita construction. This module does not prove the existence or uniqueness of the Levi-Civita connection in full generality; it works in a coordinate patch. The module does not establish any physical predictions about gravity; it provides a formal foundation for the geometry.

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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