File size: 3,065 Bytes
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 | import OAI.Geometry.EinsteinFour.ChartMetricTransition
import OAI.Geometry.EinsteinFour.ChartMetricNondegeneracy
import Mathlib.Analysis.SpecialFunctions.Sqrt
namespace OAI
/-! The local metric volume density and its exact coordinate change law.
Identification with the metric Hausdorff measure is a separate step. -/
namespace PositiveEinsteinFour.Geometry
open EinsteinFour Matrix
open scoped BigOperators Manifold ContDiff
noncomputable section
private theorem sqrt_det_congruence (G J : Mat4) :
Real.sqrt (Jᵀ * (G * J)).det = |J.det| * Real.sqrt G.det := by
rw [Matrix.det_mul, Matrix.det_mul, Matrix.det_transpose]
have he : J.det * (G.det * J.det) = J.det ^ 2 * G.det := by ring
rw [he, Real.sqrt_mul (sq_nonneg _), Real.sqrt_sq_eq_abs]
variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M]
[IsManifold I4 ∞ M]
def chartVolumeDensity (g : SmoothMetric M) (c : M) (z : FourSpace) : ℝ :=
Real.sqrt (metricInChart g c z).det
theorem chartVolumeDensity_pos (g : SmoothMetric M) (c : M) (z : FourSpace)
(hz : z ∈ (chartAt FourSpace c).target) : 0 < chartVolumeDensity g c z :=
Real.sqrt_pos.mpr (metricInChart_posDef g c z hz).det_pos
theorem chartVolumeDensity_contDiffAt (g : SmoothMetric M) (c : M) (z : FourSpace)
(hz : z ∈ (chartAt FourSpace c).target) :
ContDiffAt ℝ ∞ (chartVolumeDensity g c) z := by
have hd : ContDiffAt ℝ ∞ (fun w => (metricInChart g c w).det) z := by
simp only [Matrix.det_apply']
apply ContDiffAt.sum
intro σ _
exact contDiffAt_const.mul
(contDiffAt_prod fun i _ => metricInChart_contDiffAt g c z hz (σ i) i)
exact hd.sqrt (metricInChart_posDef g c z hz).det_pos.ne'
/-- The density transforms by the absolute determinant of the actual
derivative of the chart transition. -/
theorem chartVolumeDensity_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) :
chartVolumeDensity g c z = |(chartTransitionMatrix c d z).det| *
chartVolumeDensity g d (chartTransition c d z) := by
unfold chartVolumeDensity
rw [metricInChart_transition g c d z hz hx]
exact sqrt_det_congruence _ _
/-- In orthonormal frame coordinates the metric density cancels the
absolute determinant of the frame matrix. -/
theorem chartVolumeDensity_mul_frame_det (g : SmoothMetric M) (c : M) (z : FourSpace)
(hz : z ∈ (chartAt FourSpace c).target) (E : Mat4)
(hE : Eᵀ * metricInChart g c z * E = 1) :
chartVolumeDensity g c z * |E.det| = 1 := by
have hd : (metricInChart g c z).det * E.det ^ 2 = 1 := by
have he := congrArg Matrix.det hE
simp only [Matrix.det_mul, Matrix.det_transpose, Matrix.det_one] at he
calc
_ = E.det * (metricInChart g c z).det * E.det := by ring
_ = 1 := he
have hs := congrArg Real.sqrt hd
rw [Real.sqrt_mul (metricInChart_posDef g c z hz).det_pos.le,
Real.sqrt_sq_eq_abs, Real.sqrt_one] at hs
exact hs
end
end PositiveEinsteinFour.Geometry
end OAI
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