| import OAI.Geometry.Relativity.CKS.AngularGeometry |
| import OAI.Geometry.Relativity.CKS.SmoothOuterDEC |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CKSAngularGeometry |
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise |
|
|
| def rescale (c : ℝ) (j : Jet) : Jet := |
| (fun i k => c*j.1 i k, (fun s i k => c*j.2.1 s i k), fun s t i k => c*j.2.2 s t i k) |
|
|
| lemma inverse_rescale (c : ℝ) (j : Jet) (hc : c ≠ 0) (i k : I) : |
| inverse (rescale c j).1 i k = inverse j.1 i k / c := by |
| have heq : determinant (rescale c j).1 = c^2*determinant j.1 := by |
| dsimp [rescale,determinant] |
| ring |
| simp only [inverse,heq] |
| fin_cases i <;> fin_cases k <;> dsimp [rescale] <;> field_simp |
|
|
| lemma christoffel_rescale (c : ℝ) (j : Jet) (hc : c ≠ 0) (a i k : I) : |
| christoffel (rescale c j) a i k = christoffel j a i k := by |
| unfold christoffel |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro l hl |
| rw [inverse_rescale c j hc] |
| dsimp [rescale] |
| field_simp |
|
|
| lemma inverseDerivative_rescale (c : ℝ) (j : Jet) (hc : c ≠ 0) (s a b : I) : |
| inverseDerivative (rescale c j) s a b = inverseDerivative j s a b / c := by |
| unfold inverseDerivative |
| simp only [Finset.sum_div,neg_div] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro k hk |
| rw [inverse_rescale c j hc,inverse_rescale c j hc] |
| dsimp [rescale] |
| field_simp |
|
|
| lemma christoffelDerivative_rescale (c : ℝ) (j : Jet) (hc : c ≠ 0) (s a i k : I) : |
| christoffelDerivative (rescale c j) s a i k = christoffelDerivative j s a i k := by |
| unfold christoffelDerivative |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro l hl |
| rw [inverseDerivative_rescale c j hc,inverse_rescale c j hc] |
| dsimp [rescale] |
| field_simp |
|
|
| lemma ricci_rescale (c : ℝ) (j : Jet) (hc : c ≠ 0) (i k : I) : |
| ricci (rescale c j) i k = ricci j i k := by |
| simp only [ricci,christoffelDerivative_rescale c j hc,christoffel_rescale c j hc] |
|
|
| lemma scalar_rescale (c : ℝ) (j : Jet) (hc : c ≠ 0) : |
| scalarCurvature (rescale c j) = scalarCurvature j / c := by |
| simp only [scalarCurvature,inverse_rescale c j hc,ricci_rescale c j hc,Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro k hk |
| ring |
|
|
| lemma lapse_rescale (c : ℝ) (j : Jet) (z : ScalarJet) (hc : c ≠ 0) : |
| lapseLaplacian (rescale c j) z = lapseLaplacian j z / c := by |
| simp only [lapseLaplacian,inverse_rescale c j hc,christoffel_rescale c j hc,Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro k hk |
| ring |
|
|
| theorem weighted_curvature_lapse {K : Set CombinedJet} (hK : IsCompact K) |
| (hreg : K ⊆ regular) : |
| ∃ δ : ℝ, 0 < δ ∧ ∃ C : ℝ, 0 ≤ C ∧ |
| ∀ j j₀ : CombinedJet, j ∈ Metric.cthickening δ K → j₀ ∈ Metric.cthickening δ K → |
| ∀ r w A : ℝ, 0 < r → 0 ≤ w → 0 ≤ A → ‖j-j₀‖ ≤ A*w/r^3 → |
| |scalarCurvature (rescale (r^2) j.1)-scalarCurvature (rescale (r^2) j₀.1)| ≤ C*A*w/r^5 ∧ |
| |lapseLaplacian (rescale (r^2) j.1) j.2-lapseLaplacian (rescale (r^2) j₀.1) j₀.2| ≤ C*A*w/r^5 := by |
| obtain ⟨δ,hδ,C,hC⟩ := operators_uniform_lipschitz hK hreg |
| refine ⟨δ,hδ,C,C.coe_nonneg,?_⟩ |
| intro j j₀ hj hj₀ r w A hr hw hA hbound |
| have hop : ‖operators j-operators j₀‖ ≤ (C:ℝ)*(A*w/r^3) := by |
| have hh := hC.dist_le_mul j hj j₀ hj₀ |
| rw [dist_eq_norm,dist_eq_norm] at hh |
| exact hh.trans (mul_le_mul_of_nonneg_left hbound C.coe_nonneg) |
| have hfirst : |scalarCurvature j.1-scalarCurvature j₀.1| ≤ (C:ℝ)*(A*w/r^3) := |
| calc |
| _ ≤ ‖operators j-operators j₀‖ := by |
| exact norm_fst_le (operators j-operators j₀) |
| _ ≤ _ := hop |
| have hsecond : |lapseLaplacian j.1 j.2-lapseLaplacian j₀.1 j₀.2| ≤ (C:ℝ)*(A*w/r^3) := |
| calc |
| _ ≤ ‖operators j-operators j₀‖ := by |
| exact norm_snd_le (operators j-operators j₀) |
| _ ≤ _ := hop |
| have hr2 : r^2 ≠ 0 := pow_ne_zero 2 hr.ne' |
| rw [scalar_rescale _ _ hr2,scalar_rescale _ _ hr2,lapse_rescale _ _ _ hr2,lapse_rescale _ _ _ hr2] |
| constructor |
| · rw [← sub_div,abs_div,abs_of_pos (sq_pos_of_pos hr)] |
| exact (div_le_div_of_nonneg_right hfirst (sq_nonneg r)).trans_eq (by field_simp) |
| · rw [← sub_div,abs_div,abs_of_pos (sq_pos_of_pos hr)] |
| exact (div_le_div_of_nonneg_right hsecond (sq_nonneg r)).trans_eq (by field_simp) |
|
|
| end |
| end CKSAngularGeometry |
|
|
| end |
|
|
| end OAI |
|
|
