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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 | import OAI.Geometry.Relativity.CKS.CKSTensorFields
namespace OAI
noncomputable section
namespace CKSAngularGeometry
noncomputable section
open CKSCalculus Set Filter
open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise
lemma matrixScalarJets_sub {q p : Point → Mat} {x : Point}
(hq : ContDiffAt ℝ 2 q x) (hp : ContDiffAt ℝ 2 p x) :
matrixScalarJets (fun y => q y-p y) x = matrixScalarJets q x-matrixScalarJets p x := by
funext i k
exact actualScalarJet_sub (component_diff hq i k) (component_diff hp i k)
lemma cksQField_two_realized (z : ℝ) {f : MassFields} {x : Point} (hf : f.RegularAt x) :
matrixScalarJets (cksQField z f) x = lowerMatrixJet (coefficientMetric z (massInputOf f x)) :=
congrArg lowerMatrixJet (cksQField_realized z hf)
lemma cksDField_realized (z : ℝ) {f : MassFields} {x : Point} (hf : f.RegularAt x)
(h0 : determinant (cksQField z f x) ≠ 0) :
actualScalarJet (cksDField z f) x = coefficientD z (massInputOf f x) := by
have hq := (cksQField_diff z hf).of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have hi := inverse_diff_at hq h0
have hmg := hf.mg.of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have heg := hf.eg.of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have hlie := cksLieField_diff z hf h0
have hp := (hmg.const_smul (-3:ℝ)).add (((hf.er.sub (heg.const_smul (2:ℝ))).sub hlie).const_smul z)
unfold cksDField coefficientD
rw [actualScalarJet_smul (1/4:ℝ) (traceProduct_diff hi hp),actual_traceProductJet hi hp,
actual_inverseJets hq h0,cksQField_two_realized z hf,
matrixScalarJets_add (hmg.const_smul (-3:ℝ)) (((hf.er.sub (heg.const_smul (2:ℝ))).sub hlie).const_smul z),
matrixScalarJets_smul (-3) hmg,matrixScalarJets_smul z ((hf.er.sub (heg.const_smul (2:ℝ))).sub hlie),
matrixScalarJets_sub (hf.er.sub (heg.const_smul (2:ℝ))) hlie,
matrixScalarJets_sub hf.er (heg.const_smul (2:ℝ)),matrixScalarJets_smul (2:ℝ) heg,
cksLieField_realized z hf h0]
rfl
lemma cksTField_realized (z : ℝ) {f : MassFields} {x : Point} (hf : f.RegularAt x)
(h0 : determinant (cksQField z f x) ≠ 0) :
actualScalarJet (cksTField z f) x = coefficientT z (massInputOf f x) := by
have hq := (cksQField_diff z hf).of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have hi := inverse_diff_at hq h0
have hmg := hf.mg.of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have heg := hf.eg.of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have hp := (hf.mK.sub hmg).add ((hf.ek.sub heg).const_smul z)
unfold cksTField coefficientT
rw [actualScalarJet_smul (1/2:ℝ) (traceProduct_diff hi hp),actual_traceProductJet hi hp,
actual_inverseJets hq h0,cksQField_two_realized z hf,
matrixScalarJets_add (hf.mK.sub hmg) ((hf.ek.sub heg).const_smul z),
matrixScalarJets_sub hf.mK hmg,matrixScalarJets_smul z (hf.ek.sub heg),
matrixScalarJets_sub hf.ek heg]
rfl
lemma cksBBField_realized (z : ℝ) {f : MassFields} {x : Point} (hf : f.RegularAt x)
(h0 : determinant (cksQField z f x) ≠ 0) :
actualScalarJet (cksBBField z f) x = coefficientBB z (massInputOf f x) := by
have hq := (cksQField_diff z hf).of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have hi := inverse_diff_at hq h0
have hb (i : I) := (contDiffAt_pi.mp hf.b i).of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
unfold cksBBField coefficientBB
rw [actualScalarJet_sum _ (fun i _ => ContDiffAt.sum fun k _ => (component_diff hi i k).mul ((hb i).mul (hb k)))]
apply Finset.sum_congr rfl
intro i _
rw [actualScalarJet_sum _ (fun k _ => (component_diff hi i k).mul ((hb i).mul (hb k)))]
apply Finset.sum_congr rfl
intro k _
rw [actualScalarJet_mul (component_diff hi i k) ((hb i).mul (hb k)),
actualScalarJet_mul (hb i) (hb k),actual_inverseMatrixJet hq h0]
have heq := cksQField_two_realized z hf
change productJet (inverseMatrixJet (matrixScalarJets (cksQField z f) x) i k)
(productJet (actualScalarJet (fun y => f.b y i) x) (actualScalarJet (fun y => f.b y k) x)) = _
rw [heq]
rfl
lemma cksVField_realized (z : ℝ) {f : MassFields} {x : Point} (hf : f.RegularAt x)
(h0 : determinant (cksQField z f x) ≠ 0) :
actualScalarJet (cksVField z f) x = coefficientV z (massInputOf f x) := by
have hmr := hf.mr
have herr := hf.err
have hbb := cksBBField_diff z hf h0
unfold cksVField coefficientV
simp (disch := fun_prop) only [actualScalarJet_add,actualScalarJet_sub,actualScalarJet_smul]
rw [cksBBField_realized z hf h0]
rfl
lemma cksFField_realized (z : ℝ) {f : MassFields} {x : Point} (hf : f.RegularAt x)
(h0 : determinant (cksQField z f x) ≠ 0) (hV : 1+z^3*cksVField z f x ≠ 0) :
actualScalarJet (cksFField z f) x = cksMassJet z (massInputOf f x) := by
unfold cksFField cksMassJet
rw [actual_normalizedMassJet z (cksDField_diff z hf h0) (cksTField_diff z hf h0) (cksVField_diff z hf h0) hV,
cksDField_realized z hf h0,cksTField_realized z hf h0,cksVField_realized z hf h0]
lemma cksLeadingField_realized {f : MassFields} {x : Point} (hf : f.RegularAt x)
(h0 : determinant (f.sigma x) ≠ 0) :
actualScalarJet (cksLeadingField f) x = leadingMassJet (massInputOf f x) := by
have hs := hf.sigma.of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have hi := inverse_diff_at hs h0
have hmg := hf.mg.of_le (by norm_num : (2:ℕ∞ω) ≤ 3)
have htg := traceProduct_diff hi hmg
have htk := traceProduct_diff hi hf.mK
have hmr := hf.mr
unfold cksLeadingField leadingMassJet
simp (disch := fun_prop) only [actualScalarJet_add,actualScalarJet_smul]
rw [actual_traceProductJet hi hmg,actual_traceProductJet hi hf.mK,actual_inverseJets hs h0]
rfl
end
end CKSAngularGeometry
end
end OAI
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