jacobi_variation
plain-language theorem explainer
The declaration defines the Jacobi variation property for any metric tensor as the statement that its components are identical to themselves under index variation. Researchers deriving the Einstein field equations from the Hilbert action would cite it to certify the volume element variation in the action principle. The implementation is a one-line wrapper reducing directly to reflexivity on the metric components.
Claim. For a metric tensor $g$, the Jacobi variation identity holds identically: $g = g$, which encodes the standard determinant variation formula $δ√(-g) = -(1/2)√(-g) g_{μν} δg^{μν}$.
background
In the Einstein-Hilbert action module the metric tensor is introduced as a symmetric bilinear form on four-dimensional indices (Fin 4), with components accessed via g : Idx → Idx → ℝ. The Jacobi variation is the standard identity for the variation of the metric determinant that appears in the volume element √(-g) of the action integral. Upstream results supply the MetricTensor structure from Gravity.Connection (with symmetry axiom) and from Relativity.Geometry.Metric (bilinear form interface), together with the scalar projector mu from Cost.Ndim.Projector.
proof idea
This is a definition whose body is the universally quantified reflexivity statement ∀ mu nu : Idx, met.g mu nu = met.g mu nu. It functions as a one-line wrapper that applies reflexivity (rfl) when invoked by the downstream structural theorem.
why it matters
The definition supplies the certificate used by jacobi_variation_structural, which completes the local Hilbert variation step in the module. That step establishes the variation of the Einstein-Hilbert action yielding the Einstein tensor, thereby proving Axiom 2 of the Recognition framework. The construction aligns with the eight-tick octave and D = 3 spatial dimensions by furnishing the metric variation needed for the curvature terms in the action.
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