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785c727 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 | import OAI.Geometry.EinsteinFour.AngularGradient
namespace OAI
/-! Sixth spherical moments for the directional derivative of normal curvature.
All moments are the explicit Folland specialization from `SphericalMoments`. -/
namespace PositiveEinsteinFour.Angular
open Matrix MeasureTheory
noncomputable section
/-- Contracting one fourth moment with a matrix. -/
theorem fourth_single_contraction (D : Fin 4 → Fin 4 → ℝ) (a b : Fin 4) :
(∑ k, ∑ l, D k l * momentFour a b k l) =
(delta a b * (∑ k, D k k) + D a b + D b a) / 24 := by
have hm (k l : Fin 4) : D k l * momentFour a b k l =
(D k l * delta a b * delta k l + D k l * delta a k * delta b l +
D k l * delta a l * delta b k) / 24 := by
unfold momentFour
ring
simp_rw [hm, ← Finset.sum_div, Finset.sum_add_distrib]
congr 1
simp [delta, mul_ite, ite_mul, Finset.mul_sum]
theorem delta_symm (a b : Fin 4) : delta a b = delta b a := by
simp [delta, eq_comm]
theorem momentFour_cycle (a b c d : Fin 4) :
momentFour a b c d = momentFour a d b c := by
unfold momentFour
rw [delta_symm d c, delta_symm d b]
ring
/-- Group the fifteen pairings by the partner of the first index. -/
theorem sixth_sum_recursion (C D : Fin 4 → Fin 4 → ℝ) (a b : Fin 4) :
(∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l * momentSix a b i j k l) =
(delta a b * (∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l * momentFour i j k l) +
(∑ j, C a j * (∑ k, ∑ l, D k l * momentFour b j k l)) +
(∑ i, C i a * (∑ k, ∑ l, D k l * momentFour b i k l)) +
(∑ l, D a l * (∑ i, ∑ j, C i j * momentFour b l i j)) +
(∑ k, D k a * (∑ i, ∑ j, C i j * momentFour b k i j))) / 8 := by
have hm (i j k l : Fin 4) : C i j * D k l * momentSix a b i j k l =
(delta a b * (C i j * D k l * momentFour i j k l) +
C i j * D k l * delta a i * momentFour b j k l +
C i j * D k l * delta a j * momentFour b i k l +
C i j * D k l * delta a k * momentFour b l i j +
C i j * D k l * delta a l * momentFour b k i j) / 8 := by
unfold momentSix
rw [momentFour_cycle b i j l, momentFour_cycle b i j k]
ring
have h0 : (∑ i, ∑ j, ∑ k, ∑ l,
delta a b * (C i j * D k l * momentFour i j k l)) =
delta a b * (∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l * momentFour i j k l) := by
simp only [Finset.mul_sum]
have h1 : (∑ i, ∑ j, ∑ k, ∑ l,
C i j * D k l * delta a i * momentFour b j k l) =
∑ j, C a j * (∑ k, ∑ l, D k l * momentFour b j k l) := by
simp [delta, mul_ite, ite_mul, Finset.sum_ite_irrel, Finset.mul_sum, mul_assoc]
have h2 : (∑ i, ∑ j, ∑ k, ∑ l,
C i j * D k l * delta a j * momentFour b i k l) =
∑ i, C i a * (∑ k, ∑ l, D k l * momentFour b i k l) := by
simp [delta, mul_ite, ite_mul, Finset.sum_ite_irrel, Finset.mul_sum, mul_assoc]
have hr (f : Fin 4 → Fin 4 → Fin 4 → ℝ) :
(∑ i, ∑ j, ∑ k, f i j k) = ∑ k, ∑ i, ∑ j, f i j k := by
calc
_ = ∑ i, ∑ k, ∑ j, f i j k := by
apply Finset.sum_congr rfl
intro i _
rw [Finset.sum_comm]
_ = _ := by rw [Finset.sum_comm]
have h3 : (∑ i, ∑ j, ∑ k, ∑ l,
C i j * D k l * delta a k * momentFour b l i j) =
∑ l, D a l * (∑ i, ∑ j, C i j * momentFour b l i j) := by
simp only [delta, mul_ite, ite_mul, mul_zero, mul_one, zero_mul,
Finset.sum_ite_irrel, Finset.sum_const_zero, Finset.sum_ite_eq,
Finset.mem_univ, ite_true]
rw [hr]
simp only [Finset.mul_sum]
apply Finset.sum_congr rfl
intro l _
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
have h4 : (∑ i, ∑ j, ∑ k, ∑ l,
C i j * D k l * delta a l * momentFour b k i j) =
∑ k, D k a * (∑ i, ∑ j, C i j * momentFour b k i j) := by
simp only [delta, mul_ite, ite_mul, mul_zero, mul_one, zero_mul,
Finset.sum_ite_eq, Finset.mem_univ, ite_true]
rw [hr]
simp only [Finset.mul_sum]
apply Finset.sum_congr rfl
intro k _
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
simp_rw [hm, ← Finset.sum_div, Finset.sum_add_distrib]
rw [h0, h1, h2, h3, h4]
theorem fourth_single_contraction_traceFree (D : Fin 4 → Fin 4 → ℝ)
(hD : ∑ k, D k k = 0) (hDs : ∀ i j, D i j = D j i) (a b : Fin 4) :
(∑ k, ∑ l, D k l * momentFour a b k l) = D a b / 12 := by
rw [fourth_single_contraction, hD, hDs b a]
ring
/-- A fourth-degree harmonic has no second-degree spherical moment. -/
theorem sixth_sum_traceFree (C D : Fin 4 → Fin 4 → ℝ)
(hC : ∑ k, C k k = 0) (hD : ∑ k, D k k = 0)
(hCs : ∀ i j, C i j = C j i) (hDs : ∀ i j, D i j = D j i)
(a b : Fin 4) :
(∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l * momentSix a b i j k l) =
delta a b * (∑ i, ∑ j, C i j * D i j) / 96 +
((∑ k, C a k * D k b) + (∑ k, D a k * C k b)) / 48 := by
rw [sixth_sum_recursion, fourth_sum_formula, hC, hD]
simp_rw [fourth_single_contraction_traceFree D hD hDs,
fourth_single_contraction_traceFree C hC hCs]
have hp : (∑ i, ∑ j, C i j * D j i) = ∑ i, ∑ j, C i j * D i j := by
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
rw [hDs j i]
rw [hp]
have h1 : (∑ j, C a j * (D b j / 12)) = (∑ j, C a j * D j b) / 12 := by
simp_rw [hDs b, ← mul_div_assoc, ← Finset.sum_div]
have h2 : (∑ i, C i a * (D b i / 12)) = (∑ i, C a i * D i b) / 12 := by
simp_rw [← hCs a, hDs b, ← mul_div_assoc, ← Finset.sum_div]
have h3 : (∑ l, D a l * (C b l / 12)) = (∑ l, D a l * C l b) / 12 := by
simp_rw [hCs b, ← mul_div_assoc, ← Finset.sum_div]
have h4 : (∑ k, D k a * (C b k / 12)) = (∑ k, D a k * C k b) / 12 := by
simp_rw [← hDs a, hCs b, ← mul_div_assoc, ← Finset.sum_div]
rw [h1, h2, h3, h4]
ring
/-- The raw weighted quadratic product is exactly a sixth-moment contraction. -/
theorem integral_weighted_quadratic_mul (h : FollandMoments)
(C D : Mat4) (a b : Fin 4) :
(∫ q, q.val a * q.val b * (quadratic C q * quadratic D q) ∂sphereMeasure) =
∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l * momentSix a b i j k l := by
have he : (fun q : Directions =>
q.val a * q.val b * (quadratic C q * quadratic D q)) =
fun q => ∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l *
(q.val a * q.val b * q.val i * q.val j * q.val k * q.val l) := by
funext q
have hp : quadratic C q * quadratic D q =
∑ i, ∑ j, ∑ k, ∑ l,
C i j * D k l * (q.val i * q.val j * q.val k * q.val l) := by
unfold quadratic
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro i _
rw [Finset.sum_mul]
apply Finset.sum_congr rfl
intro j _
simp_rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro k _
apply Finset.sum_congr rfl
intro l _
ring
rw [hp]
simp_rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
apply Finset.sum_congr rfl
intro k _
apply Finset.sum_congr rfl
intro l _
ring
rw [he, integral_sum h _ (fun _ => by fun_prop)]
apply Finset.sum_congr rfl
intro i _
rw [integral_sum h _ (fun _ => by fun_prop)]
apply Finset.sum_congr rfl
intro j _
rw [integral_sum h _ (fun _ => by fun_prop)]
apply Finset.sum_congr rfl
intro k _
rw [integral_sum h _ (fun _ => by fun_prop)]
apply Finset.sum_congr rfl
intro l _
rw [integral_const_mul, h.sixth]
/-- The useful matrix form of the sixth moment. -/
theorem integral_weighted_quadratic_mul_traceFree (h : FollandMoments)
(C D : Mat4) (hC : C.trace = 0) (hD : D.trace = 0)
(hCs : Cᵀ = C) (hDs : Dᵀ = D) (a b : Fin 4) :
(∫ q, q.val a * q.val b * (quadratic C q * quadratic D q) ∂sphereMeasure) =
delta a b * (∑ i, ∑ j, C i j * D i j) / 96 +
(C * D + D * C) a b / 48 := by
rw [integral_weighted_quadratic_mul h]
have hc : ∑ k, C k k = 0 := hC
have hd : ∑ k, D k k = 0 := hD
have hcs (i j : Fin 4) : C i j = C j i :=
(congrFun (congrFun hCs i) j).symm
have hds (i j : Fin 4) : D i j = D j i :=
(congrFun (congrFun hDs i) j).symm
simpa only [Matrix.add_apply, Matrix.mul_apply] using!
sixth_sum_traceFree C D hc hd hcs hds a b
end
end PositiveEinsteinFour.Angular
end OAI
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