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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 | 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
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