| import OAI.Geometry.Relativity.CKS.ComparatorDefinitions |
| import OAI.Geometry.Relativity.CKS.AngularGeometry |
| import OAI.Geometry.Relativity.CKS.SmoothOuterDEC |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CKSAngularGeometry |
| noncomputable section |
| open CKSCalculus Set Filter |
| open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise |
| def actualJet (q : Point → Mat) (x : Point) : Jet := |
| (q x, (fun a i k => D (basis a) (fun y => q y i k) x), |
| fun a b i k => D (basis a) (D (basis b) (fun y => q y i k)) x) |
|
|
| def actualScalarJet (z : Point → ℝ) (x : Point) : ScalarJet := |
| (z x, (fun a => D (basis a) z x), fun a b => D (basis a) (D (basis b) z) x) |
|
|
| lemma component_diff {q : Point → Mat} {x : Point} (hq : ContDiffAt ℝ 2 q x) (i k : I) : |
| ContDiffAt ℝ 2 (fun y => q y i k) x := by |
| exact contDiffAt_pi.mp (contDiffAt_pi.mp hq i) k |
|
|
| lemma inverse_component_diff {q : Point → Mat} {x : Point} (hq : ContDiffAt ℝ 2 q x) |
| (h0 : determinant (q x) ≠ 0) (i k : I) : |
| DifferentiableAt ℝ (fun y => inverse (q y) i k) x := by |
| unfold inverse |
| apply CKSCalculus.diffAt_div |
| · fin_cases i <;> fin_cases k <;> dsimp <;> |
| first | exact (component_diff hq _ _).differentiableAt (by norm_num) |
| | exact ((component_diff hq _ _).differentiableAt (by norm_num)).neg |
| · exact ((determinant_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).contDiffAt.comp x hq).differentiableAt (by norm_num) |
| · exact h0 |
|
|
| lemma D_determinant {q : Point → Mat} {x : Point} (hq : ContDiffAt ℝ 2 q x) (e : Point) : |
| D e (fun y => determinant (q y)) x = |
| q x 0 0 * D e (fun y => q y 1 1) x + q x 1 1 * D e (fun y => q y 0 0) x - |
| (q x 0 1 * D e (fun y => q y 1 0) x + q x 1 0 * D e (fun y => q y 0 1) x) := by |
| have hd i k := (component_diff hq i k).differentiableAt (by norm_num) |
| unfold determinant |
| rw [D_sub e ((hd 0 0).fun_mul (hd 1 1)) ((hd 0 1).fun_mul (hd 1 0)), |
| D_mul e (hd 0 0) (hd 1 1), D_mul e (hd 0 1) (hd 1 0)] |
|
|
| theorem derivative_inverse {q : Point → Mat} {x : Point} (hq : ContDiffAt ℝ 2 q x) |
| (h0 : determinant (q x) ≠ 0) (s a b : I) : |
| D (basis s) (fun y => inverse (q y) a b) x = |
| inverseDerivative (actualJet q x) s a b := by |
| have hd i k := (component_diff hq i k).differentiableAt (by norm_num) |
| have hdet : DifferentiableAt ℝ (fun y => determinant (q y)) x := |
| ((determinant_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).contDiffAt.comp x hq).differentiableAt (by norm_num) |
| have hdneg i k : D (basis s) (fun y => -(q y i k)) x = |
| - D (basis s) (fun y => q y i k) x := by |
| unfold D |
| rw [fderiv_fun_neg] |
| rfl |
| unfold inverseDerivative |
| simp only [actualJet, Fin.sum_univ_two] |
| fin_cases a <;> fin_cases b <;> dsimp [inverse] <;> |
| rw [D_div _ (by first | exact hd _ _ | exact (hd _ _).neg) hdet h0, D_determinant hq] <;> |
| (try rw [hdneg]) <;> dsimp [inverse,determinant] <;> field_simp <;> ring |
|
|
| def connection (q : Point → Mat) (a i k : I) (x : Point) : ℝ := christoffel (actualJet q x) a i k |
|
|
| def curvatureRicci (q : Point → Mat) (x : Point) (i k : I) : ℝ := |
| (∑ a, (D (basis a) (connection q a i k) x - D (basis k) (connection q a i a) x)) + |
| ∑ a, ∑ b, (connection q a a b x * connection q b i k x - |
| connection q a k b x * connection q b i a x) |
|
|
| def physicalScalar (q : Point → Mat) (x : Point) : ℝ := |
| ∑ i, ∑ k, inverse (q x) i k * curvatureRicci q x i k |
|
|
| end |
| end CKSAngularGeometry |
|
|
| end |
|
|
| end OAI |
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|
