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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 | import OAI.Geometry.Relativity.CKS.CKSFoliationMixed
import OAI.Geometry.Relativity.CKS.CKSLeadingBound
import OAI.Geometry.Relativity.CKS.CKSPhysicalRegular
import OAI.Geometry.Relativity.CKS.CollarLogSlices
namespace OAI
noncomputable section
namespace CKSMixedGeometry
noncomputable section
open CKSCalculus Set Filter Matrix
open CKSAngularGeometry (determinant)
open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise
def sourceCollarData (f : TensorFields) : CKSAngularSlice.LogCollarData where
sigma := f.base.sigma
metricError := logQError f.base
shift := logShift f.base
tError := fun y => logT f.base y-1
LError := fun y => logOriginalL f y-1
massError := fun y => logPhysicalMass f.base y-cksLeadingField f.base y
fstar := cksLeadingField f.base
etaDivR := fun y a => logOriginalEta f y a/Real.exp (y 0)
tauDivR2 := fun y a b => logOriginalTau f y a b/Real.exp (y 0)^2
structure TensorFields.LogComponentBound (f : TensorFields) (B : ℝ) (x : Point) : Prop where
sigma : ‖matrixThreeJets f.base.sigma x‖ ≤ B
mg : ‖matrixThreeJets f.base.mg x‖ ≤ B
eg : ‖matrixThreeJets f.base.eg x‖ ≤ B/Real.exp (x 0)^2
mK : ‖matrixScalarJets f.base.mK x‖ ≤ B
ek : ‖matrixScalarJets f.base.ek x‖ ≤ B/Real.exp (x 0)^2
b : ‖fun a => actualThreeJet (fun y => f.base.b y a) x‖ ≤ B/Real.exp (x 0)^3
mr : ‖actualScalarJet f.base.mr x‖ ≤ B
err : ‖actualScalarJet f.base.err x‖ ≤ B/Real.exp (x 0)^6
kr : ‖actualScalarJet f.kr x‖ ≤ B/Real.exp (x 0)^5
kb : ‖fun a => actualScalarJet (fun y => f.kb y a) x‖ ≤ B/Real.exp (x 0)^3
lemma sourceCollarData_regular {f : TensorFields} {x : Point}
(hf : f.RegularAt x) (hp : ∀ᶠ y in 𝓝 x, (logMetric f.base y).PosDef)
(hs : determinant (f.base.sigma x) ≠ 0) : (sourceCollarData f).RegularAt x := by
have hlead := cksLeadingField_diff hf.base hs
exact ⟨hf.base.sigma,logQError_diff hf.base,logShift_diff hf.base hp.self_of_nhds,
logTError_diff hf.base hp,logOriginalLError_diff hf hp,
(logPhysicalMass_diff hf.base hp).sub hlead,hlead,
logOriginalEtaDiv_diff hf hp,logOriginalTauDiv_diff hf hp⟩
theorem cks_sourceCollarData_bounded {K : Set MatrixThreeJet} (hK : IsCompact K)
(hreg : ∀ q ∈ K, determinant (fun i k => (q i k).1.1) ≠ 0)
{B : ℝ} (hB : 0 ≤ B) :
∃ R₀ : ℝ, 1 ≤ R₀ ∧ ∃ C : ℝ, 0 ≤ C ∧ ∀ f : TensorFields, ∀ x : Point,
R₀ ≤ Real.exp (x 0) → f.RegularAt x →
(∀ᶠ y in 𝓝 x, (logMetric f.base y).PosDef) → matrixThreeJets f.base.sigma x ∈ K →
f.LogComponentBound B x → (sourceCollarData f).BoundedAt C x := by
obtain ⟨Rl,hRl,Cl,hCl,hl⟩ := cks_physical_foliation_mixed hK hreg hB
obtain ⟨Rt,hRt,Ct,hCt,ht⟩ := cks_physical_tensor_mixed hK hreg hB
obtain ⟨Rm,hRm,Cm,hCm,hm⟩ := cks_physical_mass_mixed hK hreg hB
obtain ⟨Cf,hCf,hf⟩ := cks_leading_mixed_bounded hK hreg hB
let R := max Rl (max Rt Rm)
let C := 281*B+Cl+Ct+Cm+Cf
have hC : 0 ≤ C := by dsimp [C]; positivity
have hBC : B ≤ C := by dsimp [C]; linarith
have hqC : 280*B ≤ C := by dsimp [C]; linarith
have hlC : Cl ≤ C := by dsimp [C]; linarith
have htC : Ct ≤ C := by dsimp [C]; linarith
have hmC : Cm ≤ C := by dsimp [C]; linarith
have hfC : Cf ≤ C := by dsimp [C]; linarith
refine ⟨R,hRl.trans (le_max_left _ _),C,hC,?_⟩
intro f x hr hregular hp hs hb
have hrl : Rl ≤ Real.exp (x 0) := (le_max_left _ _).trans hr
have hrt : Rt ≤ Real.exp (x 0) := (le_max_left Rt Rm).trans ((le_max_right _ _).trans hr)
have hrm : Rm ≤ Real.exp (x 0) := (le_max_right Rt Rm).trans ((le_max_right _ _).trans hr)
have hr1 : 1 ≤ Real.exp (x 0) := hRl.trans hrl
obtain ⟨_,hT,hS,_,_⟩ := hl f.base x hrl hregular.base hp hs hb.sigma hb.mg hb.eg hb.mK hb.ek hb.b hb.mr hb.err
obtain ⟨hL,hE,hτ⟩ := ht f x hrt hregular hp hs hb.sigma hb.mg hb.eg hb.mK hb.ek hb.b hb.mr hb.err hb.kr hb.kb
have hF := hm f.base x hrm hregular.base hp hs hb.sigma hb.mg hb.eg hb.mK hb.ek hb.b hb.mr hb.err
have hfstar := hf f.base x hregular.base hs hb.sigma hb.mg hb.eg hb.mK hb.ek hb.b hb.mr hb.err
have hscale {a b : ℝ} (hab : a ≤ b) (n : ℕ) : a/Real.exp (x 0)^n ≤ b/Real.exp (x 0)^n :=
div_le_div_of_nonneg_right hab (by positivity)
refine ⟨?_,?_,?_,?_,?_,?_,?_,?_,?_⟩
· intro a b
exact ((norm_le_pi_norm _ b).trans ((norm_le_pi_norm _ a).trans hb.sigma)).trans hBC
· intro a b
exact (logQError_three_bound hB hr1 hregular.base hb.mg hb.eg a b).trans (hscale hqC 3)
· intro a; exact (hS a).trans (hscale hlC 5)
· exact hT.trans (hscale hlC 3)
· exact hL.trans (hscale htC 3)
· change ‖actualScalarJet (fun y => logPhysicalMass f.base y-cksLeadingField f.base y) x‖ ≤ _
rw [actualScalarJet_sub (logPhysicalMass_diff hregular.base hp)
(cksLeadingField_diff hregular.base (hreg _ hs))]
exact hF.trans (by simpa only [pow_one] using hscale hmC 1)
· exact hfstar.trans hfC
· intro a; exact (hE a).trans (hscale htC 3)
· intro a b; exact (hτ a b).trans (hscale htC 3)
end
end CKSMixedGeometry
end
end OAI
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