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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 | import OAI.Geometry.EinsteinFour.ConnectionPullback
namespace OAI
/-! The connection coordinate change is derived from the actual metric
pullback, using metric compatibility and zero torsion. -/
namespace PositiveEinsteinFour.Geometry
open EinsteinFour Matrix
open scoped Manifold ContDiff BigOperators Topology
noncomputable section
variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M]
[IsManifold I4 ∞ M]
def pulledChartConnection (g : SmoothMetric M) (c d : M) (z : FourSpace)
(i : Fin 4) : Mat4 :=
pulledConnection (chartTransitionMatrix c d z)
(matrixPartial (chartTransitionMatrix c d) i z)
(weightedConnection (chartTransitionMatrix c d z)
(fun a => connectionMatrix g d a (chartTransition c d z)) i)
theorem chartTransitionMatrix_det_ne_zero (g : SmoothMetric M) (c d : M)
(z : FourSpace) (hz : z ∈ (chartAt FourSpace c).target)
(hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source) :
(chartTransitionMatrix c d z).det ≠ 0 := by
intro he
have h := congrArg Matrix.det (metricInChart_transition g c d z hz hx)
simp only [Matrix.det_mul, Matrix.det_transpose, he, zero_mul] at h
exact (metricInChart_posDef g c z hz).det_pos.ne' h
theorem chartTransitionMatrix_contDiffAt (c d : M) (z : FourSpace)
(hz : z ∈ (chartAt FourSpace c).target)
(hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source)
(r s : Fin 4) :
ContDiffAt ℝ ∞ (fun w => chartTransitionMatrix c d w r s) z :=
jacobian_entry_contDiffAt (chartTransition_contDiffAt c d z hz hx) r s
theorem matrixPartial_transition_target_metric (g : SmoothMetric M) (c d : M)
(z : FourSpace) (hz : z ∈ (chartAt FourSpace c).target)
(hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source) (i : Fin 4) :
matrixPartial (fun w => metricInChart g d (chartTransition c d w)) i z =
(weightedConnection (chartTransitionMatrix c d z)
(fun a => connectionMatrix g d a (chartTransition c d z)) i)ᵀ *
metricInChart g d (chartTransition c d z) +
metricInChart g d (chartTransition c d z) *
weightedConnection (chartTransitionMatrix c d z)
(fun a => connectionMatrix g d a (chartTransition c d z)) i := by
have hf := (chartTransition_contDiffAt c d z hz hx).differentiableAt (by simp)
have htarget : chartTransition c d z ∈ (chartAt FourSpace d).target :=
(chartAt FourSpace d).map_source hx
have hG : IsUnit (metricInChart g d (chartTransition c d z)).det :=
isUnit_iff_ne_zero.mpr (metricInChart_posDef g d _ htarget).det_pos.ne'
rw [matrixPartial_comp hf (fun r s =>
(metricInChart_contDiffAt g d _ htarget r s).differentiableAt (by simp))]
simp only [matrixPartial_metric g d _ hG]
simp only [weightedConnection, Matrix.transpose_sum, Matrix.transpose_smul,
Finset.sum_mul, Finset.mul_sum, smul_add, smul_mul_assoc, mul_smul_comm,
Finset.sum_add_distrib]
rfl
theorem pulledChartConnection_metric (g : SmoothMetric M) (c d : M)
(z : FourSpace) (hz : z ∈ (chartAt FourSpace c).target)
(hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source) (i : Fin 4) :
matrixPartial (metricInChart g c) i z =
(pulledChartConnection g c d z i)ᵀ * metricInChart g c z +
metricInChart g c z * pulledChartConnection g c d z i := by
let J := chartTransitionMatrix c d
let H := fun w => metricInChart g d (chartTransition c d w)
let U := weightedConnection (J z)
(fun a => connectionMatrix g d a (chartTransition c d z)) i
have hG : metricInChart g c z = (J z)ᵀ * (H z * J z) :=
metricInChart_transition g c d z hz hx
have hcnear : ∀ᶠ w in 𝓝 z, w ∈ (chartAt FourSpace c).target :=
(chartAt FourSpace c).open_target.mem_nhds hz
have hcs : ContinuousAt (chartAt FourSpace c).symm z :=
(contMDiffAt_symm_of_mem_maximalAtlas (I := I4) (n := ∞)
(IsManifold.chart_mem_maximalAtlas c) hz).continuousAt
have hdnear : ∀ᶠ w in 𝓝 z,
(chartAt FourSpace c).symm w ∈ (chartAt FourSpace d).source :=
hcs.preimage_mem_nhds ((chartAt FourSpace d).open_source.mem_nhds hx)
have heq : metricInChart g c =ᶠ[𝓝 z] (fun w => (J w)ᵀ * (H w * J w)) := by
filter_upwards [hcnear, hdnear] with w hcw hdw
exact metricInChart_transition g c d w hcw hdw
have hj (r s : Fin 4) : DifferentiableAt ℝ (fun w => J w r s) z :=
(chartTransitionMatrix_contDiffAt c d z hz hx r s).differentiableAt (by simp)
have ht : chartTransition c d z ∈ (chartAt FourSpace d).target :=
(chartAt FourSpace d).map_source hx
have hh (r s : Fin 4) : DifferentiableAt ℝ (fun w => H w r s) z :=
(metricInChart_contDiffAt g d _ ht r s).differentiableAt (by simp) |>.comp z
((chartTransition_contDiffAt c d z hz hx).differentiableAt (by simp))
have hpartial : matrixPartial H i z = Uᵀ * H z + H z * U :=
matrixPartial_transition_target_metric g c d z hz hx i
have hunit : IsUnit (J z).det :=
isUnit_iff_ne_zero.mpr (chartTransitionMatrix_det_ne_zero g c d z hz hx)
rw [matrixPartial_congr heq, matrixPartial_congruence J H z hj hh, hpartial]
have hk := pulledConnection_metric_compatibility (H z) (J z)
(matrixPartial J i z) U hunit
rw [← Matrix.mul_assoc] at hG
rw [hG]
exact hk.symm
theorem pulledChartConnection_torsion (g : SmoothMetric M) (c d : M)
(z : FourSpace) (hz : z ∈ (chartAt FourSpace c).target)
(hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source)
(i j k : Fin 4) :
pulledChartConnection g c d z i k j = pulledChartConnection g c d z j k i := by
unfold pulledChartConnection
apply pulledConnection_torsion (chartTransitionMatrix c d z)
(fun r => matrixPartial (chartTransitionMatrix c d) r z)
(fun a => connectionMatrix g d a (chartTransition c d z))
· intro r s t
exact jacobian_partials_commute (chartTransition_contDiffAt c d z hz hx) t r s
· intro r s t
exact christoffel_comm g d (chartTransition c d z) t r s
theorem christoffel_transition (g : SmoothMetric M) (c d : M)
(z : FourSpace) (hz : z ∈ (chartAt FourSpace c).target)
(hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source)
(k i j : Fin 4) :
christoffel g c z k i j = pulledChartConnection g c d z i k j := by
have hG : IsUnit (metricInChart g c z).det :=
isUnit_iff_ne_zero.mpr (metricInChart_posDef g c z hz).det_pos.ne'
have he := christoffel_eq_of_compatible g c z hG
(fun r s t => pulledChartConnection g c d z s r t)
(fun r s t => pulledChartConnection_torsion g c d z hz hx s t r) ?_
· exact (congrFun (congrFun (congrFun he k) i) j).symm
· intro r s t
have hm := congrArg (fun A : Mat4 => A s t)
(pulledChartConnection_metric g c d z hz hx r)
simp only [matrixPartial, Matrix.add_apply, Matrix.mul_apply, Matrix.transpose_apply] at hm
rw [← Finset.sum_add_distrib] at hm
rw [hm]
apply Finset.sum_congr rfl
intro p _
rw [metricInChart_comm g c z p t]
ring
end
end PositiveEinsteinFour.Geometry
end OAI
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