File size: 3,475 Bytes
695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
|