| import OAI.Geometry.EinsteinFour.SphericalMoments |
|
|
| namespace OAI |
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|
| /-! Angular averages of two quadratic forms, from the precise fourth |
| spherical moment. This is used for the two Weyl blocks in the normal Jacobi |
| matrix; no curvature average is an input. -/ |
|
|
| namespace PositiveEinsteinFour.Angular |
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|
| open MeasureTheory Matrix |
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|
| noncomputable section |
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|
| abbrev Mat4 := Geometry.Mat4 |
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|
| def quadratic (C : Mat4) (q : Directions) : ℝ := |
| ∑ i, ∑ j, C i j * (q.val i * q.val j) |
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|
| theorem continuous_quadratic (C : Mat4) : Continuous (quadratic C) := by |
| unfold quadratic |
| fun_prop |
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|
| theorem integral_quadratic_mul (h : FollandMoments) (C D : Mat4) : |
| (∫ q, quadratic C q * quadratic D q ∂sphereMeasure) = |
| ∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l * momentFour i j k l := by |
| have hexp : (fun q => quadratic C q * quadratic D q) = |
| (fun q => ∑ i, ∑ j, ∑ k, ∑ l, |
| C i j * D k l * (q.val i * q.val j * q.val k * q.val l)) := by |
| funext q |
| 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 [hexp, 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.fourth] |
|
|
| /-- The finite contraction is stated on functions so that no matrix |
| coercion is unfolded inside the simplifier. -/ |
| theorem fourth_sum_formula (C D : Fin 4 → Fin 4 → ℝ) : |
| (∑ i, ∑ j, ∑ k, ∑ l, C i j * D k l * momentFour i j k l) = |
| ((∑ i, C i i) * (∑ j, D j j) + (∑ i, ∑ j, C i j * D i j) + |
| (∑ i, ∑ j, C i j * D j i)) / 24 := by |
| have hm (i j k l : Fin 4) : C i j * D k l * momentFour i j k l = |
| (C i j * D k l * delta i j * delta k l + |
| C i j * D k l * delta i k * delta j l + |
| C i j * D k l * delta i l * delta j k) / 24 := by |
| unfold momentFour |
| ring |
| simp_rw [hm, ← Finset.sum_div, Finset.sum_add_distrib] |
| congr 1 |
| simp [delta, mul_ite, Finset.sum_mul_sum] |
|
|
| /-- Fourth moments give the complete bilinear quadratic-form formula. -/ |
| theorem integral_quadratic_mul_formula (h : FollandMoments) (C D : Mat4) : |
| (∫ q, quadratic C q * quadratic D q ∂sphereMeasure) = |
| (C.trace * D.trace + (∑ i, ∑ j, C i j * D i j) + |
| (∑ i, ∑ j, C i j * D j i)) / 24 := by |
| rw [integral_quadratic_mul h] |
| simpa only [Matrix.trace, Matrix.diag] using! fourth_sum_formula C D |
|
|
| theorem integral_quadratic_mul_traceFree (h : FollandMoments) (C D : Mat4) |
| (hC : C.trace = 0) (hD : D.trace = 0) (hDs : Dᵀ = D) : |
| (∫ q, quadratic C q * quadratic D q ∂sphereMeasure) = |
| (∑ i, ∑ j, C i j * D i j) / 12 := by |
| rw [integral_quadratic_mul_formula h, hC, hD] |
| have hd (i j : Fin 4) : D j i = D i j := congrFun (congrFun hDs i) j |
| simp_rw [hd] |
| ring |
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|
| end |
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| end PositiveEinsteinFour.Angular |
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| end OAI |
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|
