| import OAI.Geometry.EinsteinFour.CoordinateMetricCompatibility |
| import Mathlib.Geometry.Manifold.ContMDiffMFDeriv |
| import Mathlib.Analysis.Calculus.FDeriv.Symmetric |
|
|
| namespace OAI |
|
|
| /-! Smoothness of the actual pulled-back metric in a chart. The tangent |
| vectors are transported by the derivative of the inverse chart. -/ |
|
|
| namespace PositiveEinsteinFour.Geometry |
|
|
| open EinsteinFour Bundle |
| open scoped Manifold ContDiff Topology |
|
|
| noncomputable section |
|
|
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M] |
| [IsManifold I4 ∞ M] |
|
|
| theorem chart_inverse_vector_smooth (c : M) (u : FourSpace) : |
| ContMDiffOn I4 I4.tangent ∞ |
| (fun z => (⟨(chartAt FourSpace c).symm z, |
| mfderiv I4 I4 (chartAt FourSpace c).symm z u⟩ : TangentBundle I4 M)) |
| (chartAt FourSpace c).target := by |
| have hphi : ContMDiffOn I4 I4 ∞ (chartAt FourSpace c).symm |
| (chartAt FourSpace c).target := contMDiffOn_chart_symm |
| have ht := hphi.contMDiffOn_tangentMapWithin (m := ∞) (by simp) |
| (chartAt FourSpace c).open_target.uniqueMDiffOn |
| have hv : ContMDiff I4 I4.tangent ∞ |
| (fun z => (⟨z, u⟩ : TangentBundle I4 FourSpace)) := by |
| exact contMDiff_vectorSpace_iff_contDiff.mpr contDiff_const |
| have hcomp := ht.comp hv.contMDiffOn (fun z hz => hz) |
| apply hcomp.congr |
| intro z hz |
| exact congrArg (fun L : TangentSpace I4 z →L[ℝ] |
| TangentSpace I4 ((chartAt FourSpace c).symm z) => |
| (⟨(chartAt FourSpace c).symm z, L u⟩ : TangentBundle I4 M)) |
| (mfderivWithin_of_isOpen (chartAt FourSpace c).open_target hz).symm |
|
|
| theorem metricInChart_contDiffOn (g : SmoothMetric M) (c : M) (i j : Fin 4) : |
| ContDiffOn ℝ ∞ (fun z => metricInChart g c z i j) (chartAt FourSpace c).target := by |
| let : RiemannianBundle (TangentSpace I4 : M → Type _) := ⟨g.toRiemannianMetric⟩ |
| have hi := chart_inverse_vector_smooth c (coordinateVector i) |
| have hj := chart_inverse_vector_smooth c (coordinateVector j) |
| have hinner := hi.inner_bundle hj |
| exact hinner.contDiffOn |
|
|
| theorem metricInChart_contDiffAt (g : SmoothMetric M) (c : M) (z : FourSpace) |
| (hz : z ∈ (chartAt FourSpace c).target) (i j : Fin 4) : |
| ContDiffAt ℝ ∞ (fun w => metricInChart g c w i j) z := |
| (metricInChart_contDiffOn g c i j).contDiffAt |
| ((chartAt FourSpace c).open_target.mem_nhds hz) |
|
|
| private theorem matrix_det_smooth_at (A : FourSpace → Mat4) (z : FourSpace) |
| (hA : ∀ i j, ContDiffAt ℝ ∞ (fun w => A w i j) z) : |
| ContDiffAt ℝ ∞ (fun w => (A w).det) z := by |
| simp only [Matrix.det_apply'] |
| apply ContDiffAt.sum |
| intro σ _ |
| exact contDiffAt_const.mul (contDiffAt_prod fun i _ => hA (σ i) i) |
| |
| private theorem matrix_inverse_entry_smooth_at (A : FourSpace → Mat4) (z : FourSpace) |
| (hA : ∀ i j, ContDiffAt ℝ ∞ (fun w => A w i j) z) |
| (hdet : (A z).det ≠ 0) (i j : Fin 4) : |
| ContDiffAt ℝ ∞ (fun w => (A w)⁻¹ i j) z := by |
| have hinv : ContDiffAt ℝ ∞ (fun w => ((A w).det)⁻¹) z := |
| (matrix_det_smooth_at A z hA).inv hdet |
| have hadj : ContDiffAt ℝ ∞ (fun w => (A w).adjugate i j) z := by |
| simp only [Matrix.adjugate_apply] |
| apply matrix_det_smooth_at |
| intro r t |
| by_cases hr : r = j |
| · subst r |
| simpa only [Matrix.updateRow_self] using |
| (contDiffAt_const : ContDiffAt ℝ ∞ (fun _ : FourSpace => (Pi.single i (1 : ℝ) : Fin 4 → ℝ) t) z) |
| · simpa only [Matrix.updateRow_ne hr] using hA r t |
| simpa only [Matrix.inv_def, Ring.inverse_eq_inv, Matrix.smul_apply, smul_eq_mul] |
| using hinv.mul hadj |
| |
| theorem christoffel_contDiffAt (g : SmoothMetric M) (c : M) (z : FourSpace) |
| (hz : z ∈ (chartAt FourSpace c).target) |
| (hdet : (metricInChart g c z).det ≠ 0) (k i j : Fin 4) : |
| ContDiffAt ℝ ∞ (fun w => christoffel g c w k i j) z := by |
| have hm := metricInChart_contDiffAt g c z hz |
| have hd (a b d : Fin 4) : |
| ContDiffAt ℝ ∞ (fun w => partialDeriv (fun v => metricInChart g c v a b) d w) z := by |
| exact ((hm a b).fderiv_right (m := ∞) (by simp)).clm_apply contDiffAt_const |
| unfold christoffel |
| apply contDiffAt_const.mul |
| apply ContDiffAt.sum |
| intro l _ |
| exact (matrix_inverse_entry_smooth_at (metricInChart g c) z hm hdet k l).mul |
| (((hd j l i).add (hd i l j)).sub (hd i j l)) |
| |
| theorem metricInChart_inv_contDiffAt (g : SmoothMetric M) (c : M) (z : FourSpace) |
| (hz : z ∈ (chartAt FourSpace c).target) |
| (hdet : (metricInChart g c z).det ≠ 0) (i j : Fin 4) : |
| ContDiffAt ℝ ∞ (fun w => (metricInChart g c w)⁻¹ i j) z := |
| matrix_inverse_entry_smooth_at (metricInChart g c) z |
| (metricInChart_contDiffAt g c z hz) hdet i j |
| |
| end |
| end PositiveEinsteinFour.Geometry |
| |
| end OAI |
| |
