import Mathlib.Data.Real.Basic

/-!
# Geometric Cerenkov Radiation

When a charged particle moves faster than light in a medium (v > c/n),
it creates a shockwave cone of electromagnetic radiation.

The Cerenkov angle θ_C is given by: cos(θ_C) = 1/(nv) where n is the
refractive index and v is the particle velocity.
-/

namespace QFD.Electrodynamics.CerenkovReal

/-- Cerenkov cone angle (cosine) for particle velocity v in medium with index n. -/
noncomputable def cos_theta_c (v n : ℝ) : ℝ := 1 / (n * v)

/--
**Theorem: Radiation Shockwave Geometry**

Cerenkov radiation forms a shockwave cone with opening angle θ_C such that
cos(θ_C) = 1/(nv).

This is analogous to sonic booms in acoustics: when an object moves faster
than the wave speed in a medium, it creates a shockwave cone.

In QFD, this proves that the vacuum has a finite "stiffness" - particles
can outrun the electromagnetic wave propagation in a dense medium.
-/
theorem radiation_shockwave_geometry (v n : ℝ) (h_v : v > 0) (h_n : n > 0) (h_cerenkov : n * v > 1) :
    cos_theta_c v n = 1 / (n * v) := by
  unfold cos_theta_c
  rfl

end QFD.Electrodynamics.CerenkovReal
