Levi Civita basis repr eq zero of eq
realLorentzTensor.leviCivita_basis_repr_eq_zero_of_eq
Plain-language statement
The Levi-Civita tensor vanishes on any multi-index with a repeated value: if two distinct index positions i ≠ j carry the same basis index, the component is zero.
Exact Lean statement
lemma leviCivita_basis_repr_eq_zero_of_eq
{b : ComponentIdx (S := realLorentzTensor 3) ![Color.up, Color.up, Color.up, Color.up]}
{i j : Fin 4} (hij : i ≠ j) (h : b i = b j) :
(Tensor.basis _).repr ε4 b = 0Formal artifact
Lean source
lemma leviCivita_basis_repr_eq_zero_of_eq {b : ComponentIdx (S := realLorentzTensor 3) ![Color.up, Color.up, Color.up, Color.up]} {i j : Fin 4} (hij : i ≠ j) (h : b i = b j) : (Tensor.basis _).repr ε4 b = 0 := by rw [leviCivita_basis_repr_apply] have hdet : generalizedKroneckerDelta (fun i => finSumFinEquiv (b i)) (id : Fin 4 → Fin 4) = 0 := by rw [show generalizedKroneckerDelta (fun i => finSumFinEquiv (b i)) (id : Fin 4 → Fin 4) = Matrix.det (fun a c => ((kroneckerDelta (finSumFinEquiv (b a)) (id c) : ℕ) : ℤ)) from rfl] refine Matrix.det_zero_of_row_eq hij (funext fun c => ?_) rw [congrArg (⇑finSumFinEquiv) h] rw [hdet, Int.cast_zero]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/LeviCivita/Basic.lean:101-113
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