Det time Space Basis Equiv
SpaceTime.det_timeSpaceBasisEquiv
Project documentation
The equivalence on of SpaceTime taking (1, 0, 0, ...) to of (c, 0, 0, ....) and keeping all other components the same. -/ def timeSpaceBasisEquiv {d : ℕ} (c : SpeedOfLight) : SpaceTime d ≃L[ℝ] SpaceTime d where toFun x := fun μ => match μ with | Sum.inl 0 => c.val * x (Sum.inl 0) | Sum.inr i => x (Sum.inr i) invFun x := fun μ => match μ with | Sum.i...
Exact Lean statement
lemma det_timeSpaceBasisEquiv {d : ℕ} (c : SpeedOfLight) :
(timeSpaceBasisEquiv (d := d) c).det = c.valFormal artifact
Lean source
lemma det_timeSpaceBasisEquiv {d : ℕ} (c : SpeedOfLight) : (timeSpaceBasisEquiv (d := d) c).det = c.val := by rw [@LinearEquiv.coe_det] let e := toTimeAndSpace (d := d) c trans LinearMap.det (e.toLinearMap ∘ₗ (timeSpaceBasisEquiv (d := d) c).toLinearMap ∘ₗ e.symm.toLinearMap) · simp only [ContinuousLinearEquiv.toLinearEquiv_symm, LinearMap.det_conj] have h1 : e.toLinearMap ∘ₗ (timeSpaceBasisEquiv (d := d) c).toLinearMap ∘ₗ e.symm.toLinearMap = (c.val • LinearMap.id).prodMap LinearMap.id := by ext tx <;> simp [e, timeSpaceBasisEquiv, toTimeAndSpace, space] rw [h1, LinearMap.det_prodMap] simp- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/SpaceTime/Basic.lean:488-499
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