| import OAI.Geometry.Relativity.CKS.BoostIntegral |
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| namespace OAI |
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| noncomputable section |
| namespace CKSLorentz |
| noncomputable section |
| open scoped RealInnerProductSpace ContDiff Topology |
| open MeasureTheory |
| open CKSSphericalHarmonics (puncturedSpace surfaceMeasure) |
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| lemma flowEnergy_normalized {e m : ℝ} {p : E} (he : ‖p‖ < e) (hm0 : 0 < m) |
| (hm : m^2 = e^2-‖p‖^2) : flowEnergy (m⁻¹ • p) 1 = e/m := by |
| have hs : (e/m)^2 = 1+‖m⁻¹ • p‖^2 := by |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hm0)] |
| field_simp |
| nlinarith |
| have hf := flowEnergy_sq (m⁻¹ • p) 1 |
| simp only [one_pow, one_mul] at hf |
| nlinarith [flowEnergy_pos (m⁻¹ • p) 1, div_pos (energy_pos he) hm0] |
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| lemma flowD_normalized {e m : ℝ} {p : E} (he : ‖p‖ < e) (hm0 : 0 < m) |
| (hm : m^2 = e^2-‖p‖^2) (n : E) : |
| flowD (m⁻¹ • p) 1 n = denominator e m p n := by |
| simp only [flowD, flowEnergy_normalized he hm0 hm, one_mul, inner_smul_left, conj_trivial, denominator] |
| ring |
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| lemma flowS_normalized {e m : ℝ} {p : E} (he : ‖p‖ < e) (hm0 : 0 < m) |
| (hm : m^2 = e^2-‖p‖^2) (n : E) : |
| flowS (m⁻¹ • p) 1 n = boundarySpatial e m p n := by |
| simp only [flowS, flowCoeff, flowEnergy_normalized he hm0 hm, one_pow, boundarySpatial, |
| inner_smul_left, conj_trivial, smul_smul] |
| congr 1 |
| congr 1 |
| field_simp |
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| lemma flowPhi_normalized {e m : ℝ} {p : E} (he : ‖p‖ < e) (hm0 : 0 < m) |
| (hm : m^2 = e^2-‖p‖^2) (n : E) : |
| flowPhi (m⁻¹ • p) 1 n = boundaryAmbient e m p n := by |
| rw [flowPhi, flowD_normalized he hm0 hm, flowS_normalized he hm0 hm] |
| rfl |
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| theorem boost_area_substitution {e m : ℝ} {p : E} (he : ‖p‖ < e) (hm0 : 0 < m) |
| (hm : m^2 = e^2-‖p‖^2) {F : E → ℝ} (hF : ContDiffOn ℝ ∞ F puncturedSpace) : |
| ∫ n : Sphere, (denominator e m p n)⁻¹ ^2 * F (boundaryAmbient e m p n) ∂surfaceMeasure = |
| ∫ n : Sphere, F n ∂surfaceMeasure := by |
| have h := flow_area_substitution (m⁻¹ • p) hF 1 |
| simpa only [flowD_normalized he hm0 hm, flowPhi_normalized he hm0 hm] using h |
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| lemma inverse_null_boundary (e : ℝ) (p : E) (he : ‖p‖ < e) (n : Sphere) : |
| boostCLM e (mass e p) (-p) (1,(boundary e p he n:E)) = |
| (denominator e (mass e p) p n)⁻¹ • ((1:ℝ),(n:E)) := by |
| have h := congrArg (boostCLM e (mass e p) (-p)) (boundary_null e p he n) |
| simp only [map_smul, boostCLM_apply] at h |
| rw [boost_neg_inverse (mass_pos he).ne' (add_pos (energy_pos he) (mass_pos he)).ne' (mass_sq he)] at h |
| rw [h, smul_smul, inv_mul_cancel₀ (denominator_pos he (mass_pos he) n).ne', one_smul] |
| rfl |
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| end |
| end CKSLorentz |
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| end |
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| end OAI |
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