| import OAI.Geometry.EinsteinFour.Statement |
|
|
| namespace OAI |
|
|
| /-! Transport the exact metric classification across the classical |
| constant-rescaling identity for Riemannian distance. -/ |
|
|
| namespace PositiveEinsteinFour |
|
|
| open EinsteinFour |
| open scoped Manifold ContDiff ENNReal |
|
|
| noncomputable section |
|
|
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M] |
| [IsManifold I4 ∞ M] |
|
|
| theorem scaledEDistance_mul (g : SmoothMetric M) (a b : ℝ) (ha : 0 ≤ a) |
| (x y : M) : |
| scaledEDistance g (a * b) x y = |
| ENNReal.ofReal (Real.sqrt a) * scaledEDistance g b x y := by |
| simp only [scaledEDistance, Real.sqrt_mul ha, ENNReal.ofReal_mul (Real.sqrt_nonneg a), |
| mul_assoc] |
|
|
| /-- A normalization step needs only its ordinary distance-scaling law; |
| the final model and positive scale are then transported by this theorem. -/ |
| theorem classified_of_constant_scaling (g h : SmoothMetric M) (c : ℝ) (hc : 0 < c) |
| (hd : ∀ x y, scaledEDistance h 1 x y = scaledEDistance g c x y) |
| (hh : Classified h) : Classified g := by |
| obtain ⟨a, ha, hmodel⟩ := hh |
| have hscale (x y : M) : scaledEDistance h a x y = scaledEDistance g (a * c) x y := by |
| calc |
| _ = scaledEDistance h (a * 1) x y := by rw [mul_one] |
| _ = ENNReal.ofReal (Real.sqrt a) * scaledEDistance h 1 x y := |
| scaledEDistance_mul h a 1 ha.le x y |
| _ = ENNReal.ofReal (Real.sqrt a) * scaledEDistance g c x y := by rw [hd] |
| _ = _ := (scaledEDistance_mul g a c ha.le x y).symm |
| refine ⟨a * c, mul_pos ha hc, ?_⟩ |
| rcases hmodel with hs | hp | hr |
| · left |
| obtain ⟨f, hf⟩ := hs |
| exact ⟨f, fun x y => (hf x y).trans (hscale x y)⟩ |
| · right; left |
| obtain ⟨f, hf⟩ := hp |
| exact ⟨f, fun x y => (hf x y).trans (hscale x y)⟩ |
| · right; right |
| obtain ⟨f, hf⟩ := hr |
| exact ⟨f, fun x y => (hf x y).trans (hscale x y)⟩ |
|
|
| end |
| end PositiveEinsteinFour |
|
|
| end OAI |
|
|
