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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Geometry.Relativity.CKS.LogRadialIdentity
import OAI.Geometry.Relativity.CKS.CKSScaledBounds
namespace OAI
noncomputable section
namespace CKSMixedGeometry
noncomputable section
open CKSCalculus Set Filter
open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise
structure SourceMassFields where
sigma : Angle → Mat
mg : Angle → Mat
mK : Angle → Mat
mr : Angle → ℝ
eg : Point → Mat
ek : Point → Mat
b : Point → Angle
err : Point → ℝ
def SourceMassFields.logFields (f : SourceMassFields) : MassFields where
sigma := angularLift f.sigma
mg := angularLift f.mg
mK := angularLift f.mK
mr := angularLift f.mr
eg := fun y => f.eg (logRadiusChart y)
ek := fun y => f.ek (logRadiusChart y)
b := fun y => f.b (logRadiusChart y)
err := fun y => f.err (logRadiusChart y)
er := 0
lemma angularProjection_logRadiusChart (x : Point) :
angularProjection (logRadiusChart x) = angularProjection x := by
ext a
simp [angularProjection,logRadiusChart]
structure SourceMassFields.RegularAt (f : SourceMassFields) (y : Point) : Prop where
sigma : ContDiffAt ℝ 3 f.sigma (angularProjection y)
mg : ContDiffAt ℝ 3 f.mg (angularProjection y)
mK : ContDiffAt ℝ 2 f.mK (angularProjection y)
mr : ContDiffAt ℝ 2 f.mr (angularProjection y)
eg : ContDiffAt ℝ 3 f.eg y
ek : ContDiffAt ℝ 2 f.ek y
b : ContDiffAt ℝ 3 f.b y
err : ContDiffAt ℝ 2 f.err y
lemma source_logFields_regular {f : SourceMassFields} {x : Point}
(hf : f.RegularAt (logRadiusChart x)) : f.logFields.RegularAt x := by
have hσ := hf.sigma
have hmg := hf.mg
have hmk := hf.mK
have hmr := hf.mr
rw [angularProjection_logRadiusChart] at hσ hmg hmk hmr
have hc3 := ((logRadiusChart_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 3)).contDiffAt (x := x))
have hc2 := ((logRadiusChart_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).contDiffAt (x := x))
exact ⟨angularLift_diff hσ,angularLift_diff hmg,hf.eg.comp x hc3,contDiffAt_const,
angularLift_diff hmk,hf.ek.comp x hc2,hf.b.comp x hc3,angularLift_diff hmr,hf.err.comp x hc2⟩
lemma source_logGammaRadial_actual {f : SourceMassFields} {x : Point}
(hf : f.RegularAt (logRadiusChart x)) :
logGammaRadial f.logFields x = (1/Real.exp (x 0)) • radialMatrixD (logGamma f.logFields) x := by
have hσ := hf.sigma
have hmg := hf.mg
rw [angularProjection_logRadiusChart] at hσ hmg
exact logGammaRadial_actual (source_logFields_regular hf)
(radialMatrixD_angularLift (hσ.of_le (by norm_num)))
(radialMatrixD_angularLift (hmg.of_le (by norm_num)))
def sourceGamma (f : SourceMassFields) : Point → Mat := fun y =>
(y 0)^2 • f.sigma (angularProjection y)+(1/(y 0)) • f.mg (angularProjection y)+f.eg y
def sourceTangentialK (f : SourceMassFields) : Point → Mat := fun y =>
(y 0)^2 • f.sigma (angularProjection y)+(1/(y 0)) • f.mK (angularProjection y)+f.ek y
def sourceRadialMetric (f : SourceMassFields) : Point → ℝ := fun y =>
1/(1+(y 0)^2)+f.mr (angularProjection y)/(y 0)^5+f.err y
lemma sourceGamma_pullback (f : SourceMassFields) :
(fun y => sourceGamma f (logRadiusChart y)) = logGamma f.logFields := by
funext y
simp [sourceGamma,logGamma,SourceMassFields.logFields,angularLift,
angularProjection_logRadiusChart,logRadiusChart]
lemma sourceTangentialK_pullback (f : SourceMassFields) :
(fun y => sourceTangentialK f (logRadiusChart y)) = logTangentialK f.logFields := by
funext y
simp [sourceTangentialK,logTangentialK,SourceMassFields.logFields,angularLift,
angularProjection_logRadiusChart,logRadiusChart]
lemma sourceRadialMetric_pullback (f : SourceMassFields) :
(fun y => sourceRadialMetric f (logRadiusChart y)) = logRadialMetric f.logFields := by
funext y
simp [sourceRadialMetric,logRadialMetric,SourceMassFields.logFields,angularLift,
angularProjection_logRadiusChart,logRadiusChart]
structure SourceMassFields.ComponentBounds (f : SourceMassFields) (B : ℝ) (y : Point) : Prop where
eg : ∀ i k, ScaledComponentBound (fun z => f.eg z i k) 3 2 B y
ek : ∀ i k, ScaledComponentBound (fun z => f.ek z i k) 2 2 B y
b : ∀ a, ScaledComponentBound (fun z => f.b z a) 3 3 B y
err : ScaledComponentBound f.err 2 6 B y
lemma source_log_remainder_bounds {f : SourceMassFields} {x : Point} {B : ℝ} (hB : 0 ≤ B)
(hf : f.RegularAt (logRadiusChart x)) (hb : f.ComponentBounds B (logRadiusChart x)) :
‖matrixThreeJets f.logFields.eg x‖ ≤ B/Real.exp (x 0)^2 ∧
‖matrixScalarJets f.logFields.ek x‖ ≤ B/Real.exp (x 0)^2 ∧
‖fun a => actualThreeJet (fun y => f.logFields.b y a) x‖ ≤ B/Real.exp (x 0)^3 ∧
‖actualScalarJet f.logFields.err x‖ ≤ B/Real.exp (x 0)^6 := by
refine ⟨source_matrix_three_bound hB hf.eg hb.eg,source_matrix_two_bound hB hf.ek hb.ek,?_,
source_two_jet_bound hB hf.err hb.err⟩
apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
intro a
exact source_three_jet_bound hB (contDiffAt_pi.mp hf.b a) (hb.b a)
end
end CKSMixedGeometry
end
end OAI
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