Space Time integral eq time space integral
SpaceTime.spaceTime_integral_eq_time_space_integral
Project documentation
The equivalence on of SpaceTime taking (1, 0, 0, ...) to of (c, 0, 0, ....) and keeping all other components the same. -/ def timeSpaceBasisEquiv {d : ℕ} (c : SpeedOfLight) : SpaceTime d ≃L[ℝ] SpaceTime d where toFun x := fun μ => match μ with | Sum.inl 0 => c.val * x (Sum.inl 0) | Sum.inr i => x (Sum.inr i) invFun x := fun μ => match μ with | Sum.i...
Exact Lean statement
lemma spaceTime_integral_eq_time_space_integral {M} [NormedAddCommGroup M]
[NormedSpace ℝ M] {d : ℕ} (c : SpeedOfLight)
(f : SpaceTime d → M) :
∫ x : SpaceTime d, f x ∂(volume) =
c.val • ∫ tx : Time × Space d, f ((toTimeAndSpace c).symm tx) ∂(volume.prod volume)Formal artifact
Lean source
lemma spaceTime_integral_eq_time_space_integral {M} [NormedAddCommGroup M] [NormedSpace ℝ M] {d : ℕ} (c : SpeedOfLight) (f : SpaceTime d → M) : ∫ x : SpaceTime d, f x ∂(volume) = c.val • ∫ tx : Time × Space d, f ((toTimeAndSpace c).symm tx) ∂(volume.prod volume) := by symm have h1 : ∫ tx : Time × Space d, f ((toTimeAndSpace c).symm tx) ∂(volume.prod volume) = ∫ x : SpaceTime d, f x ∂((ENNReal.ofReal (c⁻¹)) • volume) := by apply MeasureTheory.MeasurePreserving.integral_comp · exact toTimeAndSpace_symm_measurePreserving c · exact (toTimeAndSpace c).symm.toHomeomorph.measurableEmbedding rw [h1] simp- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/SpaceTime/Basic.lean:566-578
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