| import OAI.Geometry.Relativity.CKS.ComparatorDefinitions |
| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CKSAngularGeometry |
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise |
| abbrev Mat := Matrix I I ℝ |
| abbrev Jet := Mat × (I → Mat) × (I → I → Mat) |
| abbrev ScalarJet := ℝ × (I → ℝ) × Mat |
| abbrev CombinedJet := Jet × ScalarJet |
|
|
| def determinant (q : Mat) : ℝ := q 0 0 * q 1 1 - q 0 1 * q 1 0 |
|
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| def inverse (q : Mat) : Mat := |
| fun i j => (!![q 1 1, -q 0 1; -q 1 0, q 0 0] : Mat) i j / determinant q |
|
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| lemma determinant_eq (q : Mat) : determinant q = q.det := by |
| exact (Matrix.det_fin_two q).symm |
|
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| lemma inverse_eq (q : Mat) : inverse q = q⁻¹ := by |
| rw [Matrix.inv_def, Matrix.adjugate_fin_two, ← determinant_eq] |
| ext i j |
| simp [inverse, div_eq_mul_inv, mul_comm] |
|
|
| def christoffel (j : Jet) (a i k : I) : ℝ := |
| (∑ l, inverse j.1 a l * (j.2.1 i k l + j.2.1 k i l - j.2.1 l i k)) / 2 |
|
|
| def inverseDerivative (j : Jet) (s a b : I) : ℝ := |
| - ∑ i, ∑ k, inverse j.1 a i * j.2.1 s i k * inverse j.1 k b |
|
|
| def christoffelDerivative (j : Jet) (s a i k : I) : ℝ := |
| (∑ l, (inverseDerivative j s a l * (j.2.1 i k l + j.2.1 k i l - j.2.1 l i k) + |
| inverse j.1 a l * (j.2.2 s i k l + j.2.2 s k i l - j.2.2 s l i k))) / 2 |
|
|
| def ricci (j : Jet) (i k : I) : ℝ := |
| (∑ a, (christoffelDerivative j a a i k - christoffelDerivative j k a i a)) + |
| ∑ a, ∑ b, (christoffel j a a b * christoffel j b i k - |
| christoffel j a k b * christoffel j b i a) |
|
|
| def scalarCurvature (j : Jet) : ℝ := ∑ i, ∑ k, inverse j.1 i k * ricci j i k |
|
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| def laplace (j : Jet) (f : ScalarJet) : ℝ := |
| ∑ i, ∑ k, inverse j.1 i k * (f.2.2 i k - ∑ a, christoffel j a i k * f.2.1 a) |
|
|
| def lapseLaplacian (j : Jet) (z : ScalarJet) : ℝ := |
| (∑ i, ∑ k, inverse j.1 i k * |
| (2 * z.2.1 i * z.2.1 k / z.1^2 - z.2.2 i k / z.1 + |
| (∑ a, christoffel j a i k * z.2.1 a) / z.1)) |
|
|
| def operators (j : CombinedJet) : ℝ × ℝ := |
| (scalarCurvature j.1, lapseLaplacian j.1 j.2) |
|
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| def regular : Set CombinedJet := {j | determinant j.1.1 ≠ 0 ∧ j.2.1 ≠ 0} |
|
|
| lemma determinant_smooth : ContDiff ℝ ∞ determinant := by |
| unfold determinant |
| fun_prop |
|
|
| lemma inverse_component_smooth {j : CombinedJet} (hj : j ∈ regular) (i k : I) : |
| ContDiffAt ℝ ∞ (fun y : CombinedJet => inverse y.1.1 i k) j := by |
| unfold inverse |
| apply ContDiffAt.div |
| · fin_cases i <;> fin_cases k <;> dsimp <;> fun_prop |
| · exact determinant_smooth.contDiffAt.comp j (by fun_prop) |
| · exact hj.1 |
|
|
| lemma christoffel_smooth {j : CombinedJet} (hj : j ∈ regular) (a i k : I) : |
| ContDiffAt ℝ ∞ (fun y : CombinedJet => christoffel y.1 a i k) j := by |
| unfold christoffel |
| apply ContDiffAt.div_const |
| apply ContDiffAt.sum |
| intro l hl |
| exact (inverse_component_smooth hj a l).mul (by fun_prop) |
|
|
| lemma inverseDerivative_smooth {j : CombinedJet} (hj : j ∈ regular) (s a b : I) : |
| ContDiffAt ℝ ∞ (fun y : CombinedJet => inverseDerivative y.1 s a b) j := by |
| unfold inverseDerivative |
| apply ContDiffAt.neg |
| apply ContDiffAt.sum |
| intro i hi |
| apply ContDiffAt.sum |
| intro k hk |
| exact ((inverse_component_smooth hj a i).mul (by fun_prop)).mul |
| (inverse_component_smooth hj k b) |
|
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| lemma christoffelDerivative_smooth {j : CombinedJet} (hj : j ∈ regular) (s a i k : I) : |
| ContDiffAt ℝ ∞ (fun y : CombinedJet => christoffelDerivative y.1 s a i k) j := by |
| unfold christoffelDerivative |
| apply ContDiffAt.div_const |
| apply ContDiffAt.sum |
| intro l hl |
| exact ((inverseDerivative_smooth hj s a l).mul (by fun_prop)).add |
| ((inverse_component_smooth hj a l).mul (by fun_prop)) |
|
|
| lemma ricci_smooth {j : CombinedJet} (hj : j ∈ regular) (i k : I) : |
| ContDiffAt ℝ ∞ (fun y : CombinedJet => ricci y.1 i k) j := by |
| unfold ricci |
| apply ContDiffAt.add |
| · apply ContDiffAt.sum |
| intro a ha |
| exact (christoffelDerivative_smooth hj a a i k).sub |
| (christoffelDerivative_smooth hj k a i a) |
| · apply ContDiffAt.sum |
| intro a ha |
| apply ContDiffAt.sum |
| intro b hb |
| exact ((christoffel_smooth hj a a b).mul (christoffel_smooth hj b i k)).sub |
| ((christoffel_smooth hj a k b).mul (christoffel_smooth hj b i a)) |
|
|
| lemma operators_smooth {j : CombinedJet} (hj : j ∈ regular) : |
| ContDiffAt ℝ ∞ operators j := by |
| apply ContDiffAt.prodMk |
| · unfold scalarCurvature |
| apply ContDiffAt.sum |
| intro i hi |
| apply ContDiffAt.sum |
| intro k hk |
| exact (inverse_component_smooth hj i k).mul (ricci_smooth hj i k) |
| · unfold lapseLaplacian |
| apply ContDiffAt.sum |
| intro i hi |
| apply ContDiffAt.sum |
| intro k hk |
| apply (inverse_component_smooth hj i k).mul |
| apply ContDiffAt.add |
| · apply ContDiffAt.sub |
| · apply ContDiffAt.div (g := fun y : CombinedJet => y.2.1^2) |
| · fun_prop |
| · fun_prop |
| · exact pow_ne_zero 2 hj.2 |
| · apply ContDiffAt.div (g := fun y : CombinedJet => y.2.1) |
| · fun_prop |
| · fun_prop |
| · exact hj.2 |
| · apply ContDiffAt.div (g := fun y : CombinedJet => y.2.1) |
| · apply ContDiffAt.sum |
| intro a ha |
| exact (christoffel_smooth hj a i k).mul (by fun_prop) |
| · fun_prop |
| · exact hj.2 |
|
|
| lemma regular_open : IsOpen regular := by |
| apply IsOpen.inter |
| · exact isOpen_ne_fun (determinant_smooth.continuous.comp (by fun_prop)) continuous_const |
| · exact isOpen_ne_fun (by fun_prop) continuous_const |
|
|
| theorem operators_uniform_lipschitz {K : Set CombinedJet} (hK : IsCompact K) |
| (hreg : K ⊆ regular) : |
| ∃ δ : ℝ, 0 < δ ∧ ∃ C : ℝ≥0, |
| LipschitzOnWith C operators (Metric.cthickening δ K) := by |
| obtain ⟨δ,hδ,hsub⟩ := hK.exists_cthickening_subset_open regular_open hreg |
| refine ⟨δ,hδ,?_⟩ |
| apply LocallyLipschitzOn.exists_lipschitzOnWith_of_compact (hK.cthickening (r := δ)) |
| intro j hj |
| obtain ⟨C,t,ht,hCt⟩ := ((operators_smooth (hsub hj)).of_le (by simp : (1:ℕ∞ω) ≤ ∞)).exists_lipschitzOnWith |
| exact ⟨C,t,mem_nhdsWithin_of_mem_nhds ht,hCt⟩ |
|
|
| end |
| end CKSAngularGeometry |
|
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| end |
|
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| end OAI |
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