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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | 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
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