| import OAI.Geometry.EinsteinFour.AngularGradient |
|
|
| namespace OAI |
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| /-! Sixth spherical moments for the directional derivative of normal curvature. |
| All moments are the explicit Folland specialization from `SphericalMoments`. -/ |
|
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| namespace PositiveEinsteinFour.Angular |
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|
| open Matrix MeasureTheory |
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| noncomputable section |
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|
| /-- 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 |
|
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| end |
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| end PositiveEinsteinFour.Angular |
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| end OAI |
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