| import OAI.Geometry.EinsteinFour.ConnectionPullback |
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| namespace OAI |
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| /-! The connection coordinate change is derived from the actual metric |
| pullback, using metric compatibility and zero torsion. -/ |
|
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| namespace PositiveEinsteinFour.Geometry |
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| open EinsteinFour Matrix |
| open scoped Manifold ContDiff BigOperators Topology |
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| noncomputable section |
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| variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M] |
| [IsManifold I4 ∞ M] |
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| 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) |
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|
| 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 |
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|
| 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 |
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| end |
| end PositiveEinsteinFour.Geometry |
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| end OAI |
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