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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Geometry.Relativity.CKS.BoostTransport
namespace OAI
noncomputable section
namespace CKSLorentz
noncomputable section
open scoped RealInnerProductSpace ContDiff Topology
open Set MeasureTheory
open CKSSphericalHarmonics (puncturedSpace surfaceMeasure smoothRestriction)
def pullbackRate (p : E) (F : E → ℝ) (z : ℝ × E) : ℝ :=
fderiv ℝ (normalizedPullback p F) z (1,0)
lemma normalizedDomain_open : IsOpen (Set.univ ×ˢ (puncturedSpace : Set E) : Set (ℝ × E)) :=
isOpen_univ.prod puncturedSpace.isOpen
lemma pullbackRate_smooth (p : E) {F : E → ℝ} (hF : ContDiffOn ℝ ∞ F puncturedSpace) :
ContDiffOn ℝ ∞ (pullbackRate p F) (Set.univ ×ˢ (puncturedSpace : Set E)) :=
((normalizedPullback_smooth p hF).fderiv_of_isOpen normalizedDomain_open (by simp)).clm_apply contDiffOn_const
lemma pullback_time_derivative (p : E) {F : E → ℝ} (hF : ContDiffOn ℝ ∞ F puncturedSpace)
(t : ℝ) (n : Sphere) :
HasDerivAt (fun t => normalizedPullback p F (t,(n:E))) (pullbackRate p F (t,(n:E))) t := by
have hg := ((normalizedPullback_smooth p hF).contDiffAt (x := (t,(n:E)))
(normalizedDomain_open.mem_nhds ⟨mem_univ t,CKSSphericalHarmonics.sphere_ne_zero n⟩)).differentiableAt (by simp)
have hline : HasDerivAt (fun t : ℝ => (t,(n:E))) (1,0) t :=
(hasDerivAt_id t).prodMk (hasDerivAt_const t (n:E))
convert! hg.hasFDerivAt.comp_hasDerivAt t hline using 1
lemma pullbackRate_transport (p : E) {F : E → ℝ} (hF : ContDiffOn ℝ ∞ F puncturedSpace)
(t : ℝ) (n : Sphere) :
pullbackRate p F (t,(n:E)) =
(flowEnergy p t)⁻¹ * (fderiv ℝ (fun x => normalizedPullback p F (t,x)) n (boostField p n) -
2*inner ℝ p (n:E)*normalizedPullback p F (t,(n:E))) :=
(pullback_time_derivative p hF t n).unique (normalizedPullback_transport p hF t n)
lemma continuous_restrict_sphere {G : ℝ × E → ℝ}
(hG : ContDiffOn ℝ ∞ G (Set.univ ×ˢ (puncturedSpace : Set E))) :
Continuous (fun z : ℝ × Sphere => G (z.1,(z.2:E))) := by
apply continuous_iff_continuousAt.mpr
intro z
have hi : Continuous (fun z : ℝ × Sphere => (z.1,(z.2:E))) :=
continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd)
have ho : ContinuousAt G (z.1,(z.2:E)) :=
(hG.contDiffAt (x := (z.1,(z.2:E))) (normalizedDomain_open.mem_nhds
⟨mem_univ _, CKSSphericalHarmonics.sphere_ne_zero z.2⟩)).continuousAt
exact ho.comp (f := fun z : ℝ × Sphere => (z.1,(z.2:E))) hi.continuousAt
lemma sphere_integrable_smooth {H : E → ℝ} (hH : ContDiffOn ℝ ∞ H puncturedSpace) :
Integrable (fun n : Sphere => H n) surfaceMeasure :=
(smoothRestriction H hH).continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
lemma boostField_smooth (p : E) : ContDiff ℝ ∞ (boostField p) :=
contDiff_const.sub ((ContDiff.inner ℝ contDiff_const contDiff_id).smul contDiff_id)
lemma pullbackRate_integral_zero (p : E) {F : E → ℝ} (hF : ContDiffOn ℝ ∞ F puncturedSpace) (t : ℝ) :
∫ n : Sphere, pullbackRate p F (t,(n:E)) ∂surfaceMeasure = 0 := by
let G : E → ℝ := fun x => normalizedPullback p F (t,x)
have hG : ContDiffOn ℝ ∞ G puncturedSpace := normalizedPullback_spatial p hF t
have hDG : ContDiffOn ℝ ∞ (fun x => fderiv ℝ G x (boostField p x)) puncturedSpace :=
(hG.fderiv_of_isOpen puncturedSpace.isOpen (by simp)).clm_apply (boostField_smooth p).contDiffOn
have hqG : ContDiffOn ℝ ∞ (fun x => inner ℝ p x * G x) puncturedSpace :=
(ContDiff.inner ℝ contDiff_const contDiff_id).contDiffOn.mul hG
have he : (fun n : Sphere => pullbackRate p F (t,(n:E))) =
fun n : Sphere => (flowEnergy p t)⁻¹ * (fderiv ℝ G n (boostField p n) - 2*(inner ℝ p (n:E)*G n)) := by
funext n
rw [pullbackRate_transport p hF]
dsimp only [G]; ring
rw [he, integral_const_mul, integral_sub (sphere_integrable_smooth hDG) ((sphere_integrable_smooth hqG).const_mul 2), integral_const_mul]
have hi := CKSSphericalHarmonics.sphereIntegral_boostField hG p
change (∫ n : Sphere, fderiv ℝ G n (boostField p n) ∂surfaceMeasure) = _ at hi
rw [hi]; ring
lemma normalizedIntegral_derivative_zero (p : E) {F : E → ℝ}
(hF : ContDiffOn ℝ ∞ F puncturedSpace) (t₀ : ℝ) :
HasDerivAt (fun t => ∫ n : Sphere, normalizedPullback p F (t,(n:E)) ∂surfaceMeasure) 0 t₀ := by
let H (t : ℝ) (n : Sphere) := normalizedPullback p F (t,(n:E))
let R (t : ℝ) (n : Sphere) := pullbackRate p F (t,(n:E))
have hH : Continuous (fun z : ℝ × Sphere => H z.1 z.2) :=
continuous_restrict_sphere (normalizedPullback_smooth p hF)
have hR : Continuous (fun z : ℝ × Sphere => R z.1 z.2) :=
continuous_restrict_sphere (pullbackRate_smooth p hF)
have hHt (t : ℝ) : Continuous (H t) := by
have hc : Continuous (fun n : Sphere => (t,n)) := continuous_const.prodMk continuous_id
exact Continuous.comp (f := fun n : Sphere => (t,n))
(g := fun z : ℝ × Sphere => H z.1 z.2) hH hc
have hRt (t : ℝ) : Continuous (R t) := by
have hc : Continuous (fun n : Sphere => (t,n)) := continuous_const.prodMk continuous_id
exact Continuous.comp (f := fun n : Sphere => (t,n))
(g := fun z : ℝ × Sphere => R z.1 z.2) hR hc
let K : Set (ℝ × Sphere) := Icc (t₀-1) (t₀+1) ×ˢ univ
have hK : IsCompact K := isCompact_Icc.prod isCompact_univ
obtain ⟨C,hC⟩ := hK.exists_bound_of_continuousOn hR.continuousOn
have h := hasDerivAt_integral_of_dominated_loc_of_deriv_le
(μ := surfaceMeasure) (x₀ := t₀) (s := Ioo (t₀-1) (t₀+1))
(F := H) (F' := R) (bound := fun _ => C)
(Ioo_mem_nhds (by linarith) (by linarith))
(Filter.Eventually.of_forall fun t => (hHt t).aestronglyMeasurable)
((hHt t₀).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
((hRt t₀).aestronglyMeasurable)
(Filter.Eventually.of_forall fun n t ht => hC (t,n) ⟨⟨ht.1.le,ht.2.le⟩, mem_univ n⟩)
(integrable_const C)
(Filter.Eventually.of_forall fun n t _ => pullback_time_derivative p hF t n)
convert! h.2 using 1
exact (pullbackRate_integral_zero p hF t₀).symm
theorem flow_area_substitution (p : E) {F : E → ℝ} (hF : ContDiffOn ℝ ∞ F puncturedSpace) (t : ℝ) :
∫ n : Sphere, (flowD p t n)⁻¹ ^2 * F (flowPhi p t n) ∂surfaceMeasure =
∫ n : Sphere, F n ∂surfaceMeasure := by
let H (t : ℝ) (n : Sphere) := normalizedPullback p F (t,(n:E))
have hd := normalizedIntegral_derivative_zero p hF
have he := is_const_of_deriv_eq_zero (fun t => (hd t).differentiableAt) (fun t => (hd t).deriv) t 0
have hzero : ∀ n : Sphere, H 0 n = F n := by
intro n
simp [H, normalizedPullback_sphere, weightedPullback, flowPhi, flowD, flowEnergy,
flowS, flowCoeff]
have ht : ∀ n : Sphere, H t n = (flowD p t n)⁻¹ ^2 * F (flowPhi p t n) := by
intro n
rw [show H t n = weightedPullback p t F n by exact normalizedPullback_sphere p F t n]
rfl
change (∫ n : Sphere, H t n ∂surfaceMeasure) = ∫ n : Sphere, H 0 n ∂surfaceMeasure at he
simpa only [hzero,ht] using he
end
end CKSLorentz
end
end OAI
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