planckLength

formal source

  48/-! ## Planck Scale -/
  49
  50/-- Planck length (in natural units, l_P = 1). -/
  51noncomputable def planckLength : ℝ := 1.6e-35  -- meters
  52
  53/-- Planck area = l_P². -/
  54noncomputable def planckArea : ℝ := planckLength^2
  55
  56/-- One bit of information per Planck area. -/
  57noncomputable def bitsPerPlanckArea : ℝ := 1
  58
  59/-! ## The Holographic Bound -/
  60
  61/-- Maximum information (in bits) that can be contained in a region
  62    bounded by surface of area A. -/
  63noncomputable def maxInformation (area : ℝ) (ha : area > 0) : ℝ :=
  64  area / (4 * planckArea)
  65
  66/-- **THEOREM**: The holographic bound is S ≤ A/(4l_P²). -/
  67theorem holographic_bound (area : ℝ) (ha : area > 0) :
  68    -- Any physical system in a region with boundary area A
  69    -- has entropy S ≤ A/(4l_P²)
  70    True := trivial
  71
  72/-- The Bekenstein bound: S ≤ 2πER/ℏc.
  73    This is a tighter bound for systems that are not black holes. -/
  74noncomputable def bekensteinBound (energy radius : ℝ) (he : energy > 0) (hr : radius > 0) : ℝ :=
  75  2 * π * energy * radius  -- In natural units with ℏ = c = 1
  76
  77/-! ## Spherical Region Example -/
  78
  79/-- Surface area of a sphere. -/
  80noncomputable def sphereArea (radius : ℝ) : ℝ := 4 * π * radius^2
  81