| import OAI.Geometry.EinsteinFour.WeylNormal |
| import OAI.Geometry.EinsteinFour.FrameCurvature |
|
|
| namespace OAI |
|
|
| /-! The normal Jacobi trace follows from the classical Einstein curvature |
| decomposition and the two trace-free Weyl blocks. -/ |
|
|
| namespace PositiveEinsteinFour.Angular |
|
|
| open Matrix |
|
|
| theorem rotated_trace (U : Mat3) (q : Directions) : (rotated U q).trace = U.trace := by |
| have hO : (mixingMatrix q)ᵀ * mixingMatrix q = 1 := |
| mul_eq_one_comm.mp (mixing_orthogonal q) |
| rw [rotated, Matrix.trace_mul_cycle, hO, one_mul] |
|
|
| theorem normalDeviation_trace (T U : Mat3) (q : Directions) |
| (hT : T.trace = 0) (hU : U.trace = 0) : |
| (normalDeviation T U q).trace = 0 := by |
| simp [normalDeviation, Matrix.trace_smul, Matrix.trace_add, rotated_trace, hT, hU] |
|
|
| end PositiveEinsteinFour.Angular |
|
|
| namespace PositiveEinsteinFour.Geometry |
|
|
| open EinsteinFour Matrix Angular |
| open scoped Manifold ContDiff |
|
|
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M] |
| [IsManifold I4 ∞ M] |
|
|
| theorem normalCurvature_trace (g : SmoothMetric M) (c : M) |
| (e : OrthonormalFrame g c) (q : Directions) |
| (hdec : frameCurvature g c e = einsteinCurvatureTensor (plusWeyl g c e) |
| (minusWeyl g c e)) |
| (hT : (plusWeyl g c e).trace = 0) (hU : (minusWeyl g c e).trace = 0) : |
| (normalCurvature g c e q).trace = 3 := by |
| rw [normalCurvature_eq_of_decomposition g c e hdec q, Matrix.trace_add, |
| normalDeviation_trace _ _ _ hT hU] |
| norm_num [Matrix.trace_one] |
|
|
| theorem frameCurvatureField_add_one_trace (g : SmoothMetric M) (e : FrameBundle g) |
| (hdec : frameCurvature g e.base e.frame = |
| einsteinCurvatureTensor (plusWeyl g e.base e.frame) (minusWeyl g e.base e.frame)) |
| (hT : (plusWeyl g e.base e.frame).trace = 0) |
| (hU : (minusWeyl g e.base e.frame).trace = 0) : |
| ((1 : Mat3) + frameCurvatureField g e).trace = 3 := by |
| have he : (1 : Mat3) + (normalCurvature g e.base e.frame firstDirection - 1) = |
| normalCurvature g e.base e.frame firstDirection := by abel |
| change ((1 : Mat3) + (normalCurvature g e.base e.frame firstDirection - 1)).trace = 3 |
| rw [he] |
| exact normalCurvature_trace g e.base e.frame firstDirection hdec hT hU |
|
|
| end PositiveEinsteinFour.Geometry |
|
|
| end OAI |
|
|
