| import OAI.Geometry.EinsteinFour.ChartMetricNondegeneracy |
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|
| namespace OAI |
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|
| /-! The actual metric pullback matrix in a chart. The tangent coordinates |
| at the image point use its own centered chart, so this congruence is the |
| zeroth-order part of the curvature coordinate-change calculation. -/ |
|
|
| namespace PositiveEinsteinFour.Geometry |
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|
| open EinsteinFour Matrix |
| open scoped Manifold ContDiff BigOperators |
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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 chartInverseMatrix (c : M) (z : FourSpace) : Mat4 := |
| fun i j => (show FourSpace from |
| mfderiv I4 I4 (chartAt FourSpace c).symm z (coordinateVector j)) i |
|
|
| theorem metricInChart_eq_center_congruence (g : SmoothMetric M) (c : M) |
| (z : FourSpace) : |
| metricInChart g c z = (chartInverseMatrix c z)ᵀ * |
| (chartMetric g ((chartAt FourSpace c).symm z) * chartInverseMatrix c z) := by |
| ext i j |
| symm |
| change (fun k => chartInverseMatrix c z k i) ⬝ᵥ |
| (chartMetric g ((chartAt FourSpace c).symm z) *ᵥ |
| (fun k => chartInverseMatrix c z k j)) = _ |
| rw [chartMetric_pairing] |
| rfl |
|
|
| def chartTransition (c d : M) (z : FourSpace) : FourSpace := |
| chartAt FourSpace d ((chartAt FourSpace c).symm z) |
|
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| def chartTransitionMatrix (c d : M) (z : FourSpace) : Mat4 := |
| fun i j => fderiv ℝ (chartTransition c d) z (coordinateVector j) i |
|
|
| theorem chartTransition_contDiffAt (c d : M) (z : FourSpace) |
| (hz : z ∈ (chartAt FourSpace c).target) |
| (hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source) : |
| ContDiffAt ℝ ∞ (chartTransition c d) z := by |
| have hc : ContMDiffAt I4 I4 ∞ (chartAt FourSpace c).symm z := |
| contMDiffAt_symm_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas c) hz |
| have hd : ContMDiffAt I4 I4 ∞ (chartAt FourSpace d) ((chartAt FourSpace c).symm z) := |
| contMDiffAt_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas d) hx |
| exact (hd.comp z hc).contDiffAt |
|
|
| theorem chartTransition_fderiv (c d : M) (z : FourSpace) |
| (hz : z ∈ (chartAt FourSpace c).target) |
| (hx : (chartAt FourSpace c).symm z ∈ (chartAt FourSpace d).source) : |
| fderiv ℝ (chartTransition c d) z = |
| (mfderiv I4 I4 (chartAt FourSpace d) ((chartAt FourSpace c).symm z)).comp |
| (mfderiv I4 I4 (chartAt FourSpace c).symm z) := by |
| have hc : MDifferentiableAt I4 I4 (chartAt FourSpace c).symm z := |
| (mdifferentiable_chart (I := I4) c).symm.mdifferentiableAt hz |
| have hd : MDifferentiableAt I4 I4 (chartAt FourSpace d) |
| ((chartAt FourSpace c).symm z) := |
| (mdifferentiable_chart (I := I4) d).mdifferentiableAt hx |
| convert! mfderiv_comp z hd hc using 1 |
| rw [mfderiv_eq_fderiv] |
| rfl |
|
|
| theorem metricInChart_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) : |
| metricInChart g c z = (chartTransitionMatrix c d z)ᵀ * |
| (metricInChart g d (chartTransition c d z) * chartTransitionMatrix c d z) := by |
| let A : FourSpace →L[ℝ] FourSpace := |
| mfderiv I4 I4 (chartAt FourSpace d) ((chartAt FourSpace c).symm z) |
| let B : FourSpace →L[ℝ] FourSpace := |
| mfderiv I4 I4 (chartAt FourSpace d).symm (chartTransition c d z) |
| let C : FourSpace →L[ℝ] FourSpace := mfderiv I4 I4 (chartAt FourSpace c).symm z |
| have hBA : B.comp A = ContinuousLinearMap.id ℝ FourSpace := |
| (mdifferentiable_chart (I := I4) d).symm_comp_deriv hx |
| have hF : fderiv ℝ (chartTransition c d) z = A.comp C := |
| chartTransition_fderiv c d z hz hx |
| have hc : B.comp (fderiv ℝ (chartTransition c d) z) = C := by |
| rw [hF, ← ContinuousLinearMap.comp_assoc, hBA, ContinuousLinearMap.id_comp] |
| have hv (i : Fin 4) : B (fderiv ℝ (chartTransition c d) z (coordinateVector i)) = |
| C (coordinateVector i) := congrArg (fun L : FourSpace →L[ℝ] FourSpace => |
| L (coordinateVector i)) hc |
| ext i j |
| symm |
| change (fun k => chartTransitionMatrix c d z k i) ⬝ᵥ |
| (metricInChart g d (chartTransition c d z) *ᵥ |
| (fun k => chartTransitionMatrix c d z k j)) = _ |
| rw [metricInChart_pairing] |
| change g.inner ((chartAt FourSpace d).symm (chartTransition c d z)) |
| (B (fderiv ℝ (chartTransition c d) z (coordinateVector i))) |
| (B (fderiv ℝ (chartTransition c d) z (coordinateVector j))) = _ |
| rw [hv, hv] |
| have hi : (chartAt FourSpace d).symm (chartTransition c d z) = |
| (chartAt FourSpace c).symm z := (chartAt FourSpace d).left_inv hx |
| rw [hi] |
| rfl |
|
|
| end |
| end PositiveEinsteinFour.Geometry |
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| end OAI |
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|
