| import OAI.Geometry.Relativity.CKS.CKSTensorFields |
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| namespace OAI |
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| noncomputable section |
| namespace CKSAngularGeometry |
| noncomputable section |
| open CKSCalculus Set Filter |
| open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise |
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| 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) |
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| 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) |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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] |
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| 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 |
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| end |
| end CKSAngularGeometry |
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| end |
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| end OAI |
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