Gravity Riemann Tensor Riemann Cert

The Riemann certificate

The Riemann curvature tensor is the standard object in differential geometry that measures how much a space bends. In general relativity, it encodes the gravitational field: the Christoffel symbols (the connection coefficients that tell you how to parallel-transport a vector) and their derivatives combine into a single tensor R^rho_{sigma mu nu}. Written in local coordinates, the definition is R^rho_{sigma mu nu} = d_mu Gamma^rho_{nu sigma} - d_nu Gamma^rho_{mu sigma} + Gamma^rho_{mu lambda} Gamma^lambda_{nu sigma} - Gamma^rho_{nu lambda} Gamma^lambda_{mu sigma}. This is the classical formula that appears in every general relativity textbook.

Two properties of this tensor matter everywhere. First, it is antisymmetric in its last two indices: R^rho_{sigma mu nu} = -R^rho_{sigma nu mu}. Swapping mu and nu flips the sign, which follows directly from the coordinate formula. Second, in flat spacetime, where all Christoffel symbols and their derivatives vanish, the tensor is identically zero. These are not deep theorems; they are algebraic checks that any correct definition must satisfy.

In Recognition Science, the machine-checked library of formal theorems packages these two facts into a single certificate. The declaration riemann_cert (a certificate, meaning a bundled proof object) asserts both properties at once: for any choice of connection coefficients and their derivatives, antisymmetry holds, and for the flat case, the tensor vanishes. The certificate is a compact way to say "this definition of the Riemann tensor passes the two basic sanity checks." The proof is fully checked by the framework's kernel, with no gaps.

What the certificate does not claim is equally important. It does not prove that the Riemann tensor is the only curvature object, nor that it satisfies the algebraic Bianchi identity in general. The library does prove that identity separately, but it requires the additional assumption that the connection is torsion-free, meaning the Christoffel symbols are symmetric in their lower indices. The certificate itself omits that condition, so it cannot be read as a statement about the full Bianchi structure. It also says nothing about the physical content of general relativity, such as the Einstein field equations or the geodesic equation. The certificate is a narrow, precise object: two algebraic facts, bundled for reuse, with the conditions for each stated explicitly.

theorem riemann_cert : RiemannCert where
  antisymmetric := riemann_antisymmetric_last_two
  flat_vanishes := riemann_flat_vanishes

/-- R^rho_{sigma mu nu} is antisymmetric in the last two indices:
    R^rho_{sigma mu nu} = -R^rho_{sigma nu mu}.

    Proof: swapping mu <-> nu negates the d_mu Gamma - d_nu Gamma terms
    and swaps the quadratic Gamma terms. -/
theorem riemann_antisymmetric_last_two
    (gamma : Idx → Idx → Idx → ℝ)
    (dgamma : Idx → Idx → Idx → Idx → ℝ)
    (rho sigma mu nu : Idx) :
    riemann_tensor gamma dgamma rho sigma mu nu =
    -(riemann_tensor gamma dgamma rho sigma nu mu) := by
  simp only [riemann_tensor]
  ring

/-- For flat spacetime (all Gamma = 0, all dGamma = 0), the Riemann tensor vanishes. -/
theorem riemann_flat_vanishes (rho sigma mu nu : Idx) :
    riemann_tensor (fun _ _ _ => 0) (fun _ _ _ _ => 0) rho sigma mu nu = 0 := by
  simp [riemann_tensor]

What this page does not claim

The certificate does not prove the algebraic Bianchi identity without the torsion-free condition. The certificate does not derive the Einstein field equations or any dynamical content of general relativity. The certificate does not establish that the Riemann tensor is the unique curvature object.

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

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Derived articles

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