File size: 3,786 Bytes
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 | import OAI.Geometry.EinsteinFour.AngularQuadratic
namespace OAI
/-! The actual change of normal bases `I_s q` and `J_t q`. Its spherical
orthogonality is a consequence of the fourth monomial moment. -/
namespace PositiveEinsteinFour.Angular
open Matrix
noncomputable section
def oppositeMatrix (s : Fin 3) : Mat4 := Geometry.oppositeQuaternion s
def mixingCoefficient (r s : Fin 3) : Mat4 :=
(TraceHessian.quaternionMatrix r)ᵀ * oppositeMatrix s
def coefficientTable (r s : Fin 3) : Mat4 :=
![![![![-1, 0, 0, 0], ![0, -1, 0, 0], ![0, 0, 1, 0], ![0, 0, 0, 1]], ![![0, 0, 0, 1], ![0, 0, -1, 0], ![0, -1, 0, 0], ![1, 0, 0, 0]], ![![0, 0, -1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, -1, 0, 0]]], ![![![0, 0, 0, -1], ![0, 0, -1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]], ![![-1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, 1]], ![![0, 1, 0, 0], ![1, 0, 0, 0], ![0, 0, 0, -1], ![0, 0, -1, 0]]], ![![![0, 0, 1, 0], ![0, 0, 0, -1], ![1, 0, 0, 0], ![0, -1, 0, 0]], ![![0, -1, 0, 0], ![-1, 0, 0, 0], ![0, 0, 0, -1], ![0, 0, -1, 0]], ![![-1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, 1, 0], ![0, 0, 0, -1]]]] r s
theorem mixingCoefficient_eq_table (r s : Fin 3) :
mixingCoefficient r s = coefficientTable r s := by
ext i j
simp only [mixingCoefficient, Matrix.mul_apply, Matrix.transpose_apply]
fin_cases r <;> fin_cases s <;> fin_cases i <;> fin_cases j <;>
norm_num [TraceHessian.quaternionMatrix, oppositeMatrix,
Geometry.oppositeQuaternion, coefficientTable, Fin.sum_univ_succ]
def mixing (r s : Fin 3) (q : Directions) : ℝ :=
quadratic (mixingCoefficient r s) q
def mixingMatrix (q : Directions) : Geometry.Mat3 := fun r s => mixing r s q
theorem mixingCoefficient_symmetric (r s : Fin 3) :
(mixingCoefficient r s)ᵀ = mixingCoefficient r s := by
rw [mixingCoefficient_eq_table]
ext i j
change coefficientTable r s j i = coefficientTable r s i j
fin_cases r <;> fin_cases s <;> fin_cases i <;> fin_cases j <;>
norm_num [coefficientTable]
theorem mixingCoefficient_trace (r s : Fin 3) :
(mixingCoefficient r s).trace = 0 := by
rw [mixingCoefficient_eq_table]
fin_cases r <;> fin_cases s <;>
norm_num [coefficientTable, Matrix.trace, Matrix.diag, Fin.sum_univ_succ]
theorem mixingCoefficient_pairing (r s t u : Fin 3) :
(∑ i, ∑ j, mixingCoefficient r s i j * mixingCoefficient t u i j) =
if r = t then (if s = u then (4 : ℝ) else 0) else 0 := by
simp only [mixingCoefficient_eq_table]
fin_cases r <;> fin_cases s <;> fin_cases t <;> fin_cases u <;>
norm_num [coefficientTable, Fin.sum_univ_succ]
theorem integral_mixing_mul (h : FollandMoments) (r s t u : Fin 3) :
(∫ q, mixing r s q * mixing t u q ∂sphereMeasure) =
if r = t then (if s = u then (1 / 3 : ℝ) else 0) else 0 := by
unfold mixing
rw [integral_quadratic_mul_traceFree h _ _ (mixingCoefficient_trace r s)
(mixingCoefficient_trace t u) (mixingCoefficient_symmetric t u),
mixingCoefficient_pairing]
split_ifs <;> norm_num
theorem direction_sum_sq (q : Directions) : (∑ i, q.val i ^ 2) = 1 := by
have h := congrArg (fun x : ℝ => x ^ 2) q.property
simpa only [EuclideanSpace.real_norm_sq_eq, one_pow] using h
/-- The two triples give orthonormal normal bases for every unit direction. -/
theorem mixing_orthogonal (q : Directions) :
mixingMatrix q * (mixingMatrix q)ᵀ = 1 := by
ext i j
change (∑ s, mixing i s q * mixing j s q) = if i = j then 1 else 0
calc
_ = (∑ a, q.val a ^ 2) ^ 2 * (if i = j then 1 else 0) := by
simp only [mixing, quadratic, mixingCoefficient_eq_table]
fin_cases i <;> fin_cases j <;>
simp [coefficientTable, Fin.sum_univ_succ]
all_goals ring
_ = _ := by rw [direction_sum_sq]; simp
end
end PositiveEinsteinFour.Angular
end OAI
|