Gravity Hawking Temperature Si T Hawking Si Eq Inv Schwarzschild Radius

Temperature from a radius

A black hole is not completely dark. In 1974 Stephen Hawking showed that quantum effects near the event horizon make a black hole radiate like a warm body, with a temperature that is higher for smaller holes. The formula he derived is a key result of semiclassical gravity: the temperature T is proportional to the surface gravity at the horizon. In the SI unit system, the standard expression is T = ℏc³ / (8πGk_BM), where M is the mass, ℏ is the reduced Planck constant, c is the speed of light, G is Newton's constant, and k_B is the Boltzmann constant.

The same physics can be written in a more geometric way. Since the Schwarzschild radius of a black hole is r_s = 2GM/c², the Hawking temperature can be recast as T = ℏc / (4πk_B r_s). This form makes the inverse relationship explicit: a black hole with a larger radius has a lower temperature. The temperature drops as the hole grows, and it rises as the hole shrinks, which is why smaller black holes are the hotter ones.

In Recognition Science, the framework's machine-checked library of formal theorems contains a declaration, T_hawking_SI_eq_inv_schwarzschildRadius, that proves this inverse-radius form in SI units. The theorem states that for any positive mass M, the SI temperature equals ℏc divided by 4πk_B times the Schwarzschild radius. The proof is a direct algebraic consequence of the standard formula and the definition of the radius; no new physics is introduced. The library also proves that the temperature is always positive and that it strictly decreases as the mass increases, so lighter holes are hotter.

The framework's contribution here is not the physics itself. The 8π factor in the standard formula comes from the semiclassical derivation of Hartle and Hawking in 1976, and the library does not force that factor from its own principles. What the library does is take the standard result and show that it survives a careful conversion from the framework's natural units, where c = G = ℏ = k_B = 1, into ordinary SI units. The conversion uses the exact SI-2019 values of c, ℏ, and k_B, and a single measured value for G. The theorem is therefore a statement about units and algebra, not a new prediction about black holes.

The practical consequence is that the framework can speak the same language as experimental physics. The inverse-radius form is a compact way to see how the temperature scales with the hole's size, and the library's proof means the conversion is not a matter of hand-waving. The theorem does not claim that Hawking radiation has been observed, nor does it prove the stability of a black hole, nor does it derive the one-loop correction to the entropy. Those remain open targets, and the framework is explicit about the boundary.

/-- The SI Hawking temperature as a function of Schwarzschild radius:
`T_hawking_SI_of_radius(r_s) = ℏ c² / (4π G k_B · r_s · M_planck_unit)`.
Equivalently (Schwarzschild identification `r_s = 2M`):
`T_hawking_SI M_SI = ℏc / (4π · k_B · schwarzschildRadius_SI M_SI)`.

The derivation: starting from `T_hawking_SI M_SI = ℏc³/(8π·G·k_B·M_SI)`
and `schwarzschildRadius_SI M_SI = 2·G·M_SI/c²`, eliminate `M_SI` to get
`T_hawking_SI = ℏc/(4π·k_B·r_s)`. -/
theorem T_hawking_SI_eq_inv_schwarzschildRadius (M_SI : ℝ) (hM : 0 < M_SI) :
    T_hawking_SI M_SI = hbar_SI * c_SI /
      (4 * Real.pi * k_B_SI * schwarzschildRadius_SI M_SI) := by
  unfold T_hawking_SI schwarzschildRadius_SI
  have hG : G_SI ≠ 0 := ne_of_gt G_SI_pos
  have hM_ne : M_SI ≠ 0 := ne_of_gt hM
  have hc : c_SI ≠ 0 := ne_of_gt c_SI_pos
  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
  have hk : k_B_SI ≠ 0 := ne_of_gt k_B_SI_pos
  field_simp
  ring

/-- Positivity: positive masses give positive Hawking temperatures. -/
theorem T_hawking_SI_pos (M_SI : ℝ) (hM : 0 < M_SI) :
    0 < T_hawking_SI M_SI := by
  unfold T_hawking_SI
  have h_num_pos : 0 < hbar_SI * c_SI ^ 3 :=
    mul_pos hbar_SI_pos (pow_pos c_SI_pos 3)
  have h8 : (0 : ℝ) < 8 := by norm_num
  have h_den_pos : 0 < 8 * Real.pi * G_SI * k_B_SI * M_SI :=
    mul_pos (mul_pos (mul_pos (mul_pos h8 Real.pi_pos) G_SI_pos) k_B_SI_pos) hM
  exact div_pos h_num_pos h_den_pos

/-- Strict anti-monotonicity in mass: lighter holes are hotter (SI form). -/
theorem T_hawking_SI_strict_anti
    (M1 M2 : ℝ) (h1 : 0 < M1) (_h2 : 0 < M2) (hlt : M1 < M2) :
    T_hawking_SI M2 < T_hawking_SI M1 := by
  unfold T_hawking_SI
  have h_num_pos : 0 < hbar_SI * c_SI ^ 3 :=
    mul_pos hbar_SI_pos (pow_pos c_SI_pos 3)
  have h8 : (0 : ℝ) < 8 := by norm_num
  have h_coeff_pos : 0 < 8 * Real.pi * G_SI * k_B_SI :=
    mul_pos (mul_pos (mul_pos h8 Real.pi_pos) G_SI_pos) k_B_SI_pos
  have h_den1_pos : 0 < 8 * Real.pi * G_SI * k_B_SI * M1 :=
    mul_pos h_coeff_pos h1
  have h_den_lt : 8 * Real.pi * G_SI * k_B_SI * M1 <
      8 * Real.pi * G_SI * k_B_SI * M2 :=
    mul_lt_mul_of_pos_left hlt h_coeff_pos
  exact div_lt_div_of_pos_left h_num_pos h_den1_pos h_den_lt

theorem T_hawking_SI_def (M_SI : ℝ) :
    T_hawking_SI M_SI = hbar_SI * c_SI ^ 3 /
      (8 * Real.pi * G_SI * k_B_SI * M_SI) := rfl

What this page does not claim

The existence or stability of Hawking radiation is not proved by this theorem. The 8π factor in the Hawking formula is not derived from the framework's forcing chain. The one-loop correction to black hole entropy is not derived here.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Gravity/HawkingTemperatureSI.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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