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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 | import OAI.Geometry.EinsteinFour.CoordinateChangeCalculus
import OAI.Geometry.EinsteinFour.CoordinateConnectionUniqueness
namespace OAI
/-! Metric compatibility under the coordinate change formula for a
connection. The matrices here are four-dimensional chart matrices. -/
namespace PositiveEinsteinFour.Geometry
open EinsteinFour Matrix
open scoped Manifold ContDiff BigOperators
noncomputable section
theorem pullback_metric_compatibility (H J K P U : Mat4)
(h : J * K = P + U * J) :
Kᵀ * (Jᵀ * H * J) + (Jᵀ * H * J) * K =
Pᵀ * H * J + Jᵀ * (Uᵀ * H + H * U) * J + Jᵀ * H * P := by
have ht : Kᵀ * Jᵀ = Pᵀ + Jᵀ * Uᵀ := by
simpa only [Matrix.transpose_mul, Matrix.transpose_add] using congrArg Matrix.transpose h
calc
_ = (Kᵀ * Jᵀ) * H * J + Jᵀ * H * (J * K) := by noncomm_ring
_ = (Pᵀ + Jᵀ * Uᵀ) * H * J + Jᵀ * H * (P + U * J) := by rw [ht, h]
_ = _ := by noncomm_ring
def pulledConnection (J P U : Mat4) : Mat4 := J⁻¹ * (P + U * J)
theorem pulledConnection_equation (J P U : Mat4) (hJ : IsUnit J.det) :
J * pulledConnection J P U = P + U * J := by
unfold pulledConnection
rw [← Matrix.mul_assoc, Matrix.mul_nonsing_inv J hJ, one_mul]
theorem pulledConnection_metric_compatibility (H J P U : Mat4)
(hJ : IsUnit J.det) :
(pulledConnection J P U)ᵀ * (Jᵀ * H * J) +
(Jᵀ * H * J) * pulledConnection J P U =
Pᵀ * H * J + Jᵀ * (Uᵀ * H + H * U) * J + Jᵀ * H * P :=
pullback_metric_compatibility H J _ P U (pulledConnection_equation J P U hJ)
def weightedConnection (J : Mat4) (C : Fin 4 → Mat4) (i : Fin 4) : Mat4 :=
∑ a, J a i • C a
theorem weightedConnection_torsion (J : Mat4) (C : Fin 4 → Mat4)
(hC : ∀ a b r, C a r b = C b r a) (i j r : Fin 4) :
(weightedConnection J C i * J) r j = (weightedConnection J C j * J) r i := by
have he (s t : Fin 4) : (weightedConnection J C s * J) r t =
∑ a, ∑ b, J a s * C a r b * J b t := by
simp only [weightedConnection, Matrix.mul_apply, Matrix.sum_apply,
Matrix.smul_apply, smul_eq_mul, Finset.sum_mul]
rw [Finset.sum_comm]
rw [he, he, Finset.sum_comm]
apply Finset.sum_congr rfl
intro a _
apply Finset.sum_congr rfl
intro b _
rw [hC b a r]
ring
theorem pulledConnection_torsion (J : Mat4) (P C : Fin 4 → Mat4)
(hP : ∀ i j r, P i r j = P j r i)
(hC : ∀ a b r, C a r b = C b r a) (i j k : Fin 4) :
pulledConnection J (P i) (weightedConnection J C i) k j =
pulledConnection J (P j) (weightedConnection J C j) k i := by
change (∑ r, J⁻¹ k r * (P i r j + (weightedConnection J C i * J) r j)) =
∑ r, J⁻¹ k r * (P j r i + (weightedConnection J C j * J) r i)
apply Finset.sum_congr rfl
intro r _
rw [hP i j r, weightedConnection_torsion J C hC i j r]
theorem matrixPartial_congruence (J A : FourSpace → Mat4) (z : FourSpace)
(hJ : ∀ r s, DifferentiableAt ℝ (fun w => J w r s) z)
(hA : ∀ r s, DifferentiableAt ℝ (fun w => A w r s) z) (i : Fin 4) :
matrixPartial (fun w => (J w)ᵀ * (A w * J w)) i z =
(matrixPartial J i z)ᵀ * A z * J z +
(J z)ᵀ * matrixPartial A i z * J z + (J z)ᵀ * A z * matrixPartial J i z := by
have hAJ (r s : Fin 4) :
DifferentiableAt ℝ (fun w => (A w * J w) r s) z := by
change DifferentiableAt ℝ (fun w => ∑ p, A w r p * J w p s) z
exact DifferentiableAt.fun_sum (fun p _ => (hA r p).mul (hJ p s))
rw [matrixPartial_mul (fun w => (J w)ᵀ) (fun w => A w * J w) z
(fun r s => hJ s r) hAJ i,
matrixPartial_transpose, matrixPartial_mul A J z hA hJ i]
noncomm_ring
/-- The mixed Jacobian derivative cancels from the curvature transformation.
The two last premises are exactly the differentiated connection laws. -/
theorem curvature_intertwining_of_connection
(J C₁ C₂ U₁ U₂ P₁ P₂ D₁₂ D₂₁ V₁₂ V₂₁ E : Mat4)
(h₁ : J * C₁ = P₁ + U₁ * J) (h₂ : J * C₂ = P₂ + U₂ * J)
(h₁₂ : P₁ * C₂ + J * D₁₂ = E + V₁₂ * J + U₂ * P₁)
(h₂₁ : P₂ * C₁ + J * D₂₁ = E + V₂₁ * J + U₁ * P₂) :
J * (D₁₂ - D₂₁ + C₁ * C₂ - C₂ * C₁) =
(V₁₂ - V₂₁ + U₁ * U₂ - U₂ * U₁) * J := by
have hd : J * D₁₂ - J * D₂₁ =
(V₁₂ - V₂₁) * J + U₂ * P₁ - U₁ * P₂ - P₁ * C₂ + P₂ * C₁ := by
calc
_ = (P₁ * C₂ + J * D₁₂) - (P₂ * C₁ + J * D₂₁) -
P₁ * C₂ + P₂ * C₁ := by noncomm_ring
_ = _ := by rw [h₁₂, h₂₁]; noncomm_ring
calc
_ = (J * D₁₂ - J * D₂₁) + (J * C₁) * C₂ - (J * C₂) * C₁ := by
noncomm_ring
_ = ((V₁₂ - V₂₁) * J + U₂ * P₁ - U₁ * P₂ - P₁ * C₂ + P₂ * C₁) +
(P₁ + U₁ * J) * C₂ - (P₂ + U₂ * J) * C₁ := by rw [hd, h₁, h₂]
_ = (V₁₂ - V₂₁) * J + U₂ * P₁ - U₁ * P₂ - P₁ * C₂ + P₂ * C₁ +
P₁ * C₂ + U₁ * (J * C₂) - P₂ * C₁ - U₂ * (J * C₁) := by noncomm_ring
_ = _ := by rw [h₁, h₂]; noncomm_ring
end
end PositiveEinsteinFour.Geometry
end OAI
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