`sbh_saturation_uniqueness`

proved

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module: IndisputableMonolith.Relativity.Compact.BlackHoleEntropy
domain: Relativity
line: 60 · github
papers citing: none yet

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IndisputableMonolith.Relativity.Compact.BlackHoleEntropy on GitHub at line 60.

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depends on

formal source

  57  use LedgerCapacityLimit A ell0 / (8 * tau0)
  58
  59/--- **CERT(definitional)**: Bekenstein-Hawking entropy as the unique saturation point. -/
  60theorem sbh_saturation_uniqueness (Rs : ℝ) (h_Rs : Rs > 0) :
  61    ∃! (S : ℝ), S = HorizonArea Rs / (4 * ell0^2) := by
  62  use HorizonArea Rs / (4 * ell0^2)
  63  constructor
  64  · rfl
  65  · intro S' h; exact h
  66
  67/-- The BH entropy saturation value is strictly positive for `Rs > 0`. -/
  68theorem sbh_saturation_positive (Rs : ℝ) (h_Rs : Rs > 0) :
  69    0 < HorizonArea Rs / (4 * ell0^2) := by
  70  have hA : 0 < HorizonArea Rs := horizon_area_pos Rs h_Rs
  71  have hden : 0 < 4 * ell0 ^ 2 := by
  72    nlinarith [sq_pos_of_pos ell0_pos]
  73  exact div_pos hA hden
  74
  75/-- The BH entropy saturation value is nonzero for `Rs > 0`. -/
  76theorem sbh_saturation_nonzero (Rs : ℝ) (h_Rs : Rs > 0) :
  77    HorizonArea Rs / (4 * ell0^2) ≠ 0 :=
  78  ne_of_gt (sbh_saturation_positive Rs h_Rs)
  79
  80end Compact
  81end Relativity
  82end IndisputableMonolith

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