Gravity Ricci Tensor Einstein Symmetric

Symmetry of the Einstein tensor

In general relativity, the Einstein tensor G_μν is the left-hand side of the field equations that relate spacetime curvature to matter and energy. It is built from the Ricci tensor R_μν, the scalar curvature R, and the metric tensor g_μν according to G_μν = R_μν − (1/2)R g_μν. A central structural property is symmetry: G_μν = G_νμ. This means the tensor is unchanged when its two indices are swapped, a fact that constrains how the equations can couple to the symmetric stress-energy tensor T_μν on the right-hand side.

The symmetry is not automatic. It follows from two ingredients: the Ricci tensor must itself be symmetric, and the metric tensor must be symmetric. The Ricci tensor is symmetric when the connection is torsion-free, which is the standard condition in Einstein's theory. The metric tensor's symmetry is part of its definition as a symmetric bilinear form. When both hold, swapping indices in the Einstein tensor leaves every term unchanged: the Ricci term by assumption, and the scalar-curvature term because it multiplies the symmetric metric.

The framework's machine-checked library of formal theorems records this fact as the declaration einstein_symmetric. It proves that, given a symmetric Ricci tensor and a symmetric metric, the Einstein tensor is symmetric. The proof is a direct algebraic expansion: it rewrites the definition, applies the assumed Ricci symmetry, and applies the metric symmetry. The declaration also carries a companion result: for flat Minkowski spacetime, the Einstein tensor vanishes, and the vacuum field equations hold with zero cosmological constant.

What the declaration does not claim is as important as what it proves. It does not assert that the Einstein tensor is divergence-free, a separate property that the docstring describes as stated but not proved in this file. It does not establish that any particular spacetime satisfies the field equations, only that the Minkowski vacuum is a solution. And it does not derive the field equations from the framework's deeper principles; it formalizes a classical structural fact within the library's gravity module.

The practical consequence is that any subsequent framework theorem that relies on the symmetry of the Einstein tensor can cite this declaration as its warrant. The symmetry is a precondition for writing the field equations in their standard form, coupling G_μν to a symmetric stress-energy tensor. Without it, the equations could not be posed as a clean equality between two symmetric objects.

/-- Einstein tensor is symmetric when the Ricci tensor is symmetric
    (which holds when the connection is torsion-free). -/
theorem einstein_symmetric
    (met : MetricTensor) (ginv : InverseMetric)
    (gamma : Idx → Idx → Idx → ℝ)
    (dgamma : Idx → Idx → Idx → Idx → ℝ)
    (h_ricci_sym : ∀ mu nu, ricci_tensor gamma dgamma mu nu =
                            ricci_tensor gamma dgamma nu mu)
    (mu nu : Idx) :
    einstein_tensor met ginv gamma dgamma mu nu =
    einstein_tensor met ginv gamma dgamma nu mu := by
  simp only [einstein_tensor]
  rw [h_ricci_sym mu nu, met.symmetric mu nu]

/-- The Einstein tensor: G_{mu nu} = R_{mu nu} - (1/2) R g_{mu nu}.
    This is the LHS of the Einstein field equations. -/
noncomputable def einstein_tensor
    (met : MetricTensor) (ginv : InverseMetric)
    (gamma : Idx → Idx → Idx → ℝ)
    (dgamma : Idx → Idx → Idx → Idx → ℝ)
    (mu nu : Idx) : ℝ :=
  ricci_tensor gamma dgamma mu nu -
  (1/2) * scalar_curvature ginv gamma dgamma * met.g mu nu

/-- Flat Minkowski metric satisfies the vacuum EFE with Lambda = 0. -/
theorem minkowski_is_vacuum_solution :
    vacuum_efe_coord minkowski minkowski_inverse
      (fun _ _ _ => 0) (fun _ _ _ _ => 0) 0 := by
  intro mu nu
  simp [einstein_flat]

What this page does not claim

The declaration does not prove that the Einstein tensor is divergence-free. The declaration does not derive the Einstein field equations from the framework's foundational principles. The declaration does not establish that any non-flat spacetime satisfies the field equations.

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

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