Gravity Hawking Temperature Si

The SI formula

The Hawking temperature is the thermal radiation a black hole emits because of quantum effects near its event horizon. In SI units, the temperature T of a black hole of mass M is given by the formula T = ħc³ / (8πGMk_B), where ħ is the reduced Planck constant, c is the speed of light, G is Newton's gravitational constant, and k_B is the Boltzmann constant. The formula implies that lighter black holes are hotter: a black hole of one solar mass has a temperature of about 60 nanokelvin, while a smaller one radiates more intensely.

In the 2019 redefinition of the SI base units, the values of ħ, c, and k_B became exact, defined numbers. Only G remains a measured quantity, determined by laboratory experiments. This means the Hawking temperature formula now depends on exactly one measured constant, G, alongside three exact definitions. The framework's module HawkingTemperatureSI (a machine-checked collection of formal theorems) establishes this formula as a structural theorem: it proves the SI expression, proves that the temperature is always positive for positive mass, and proves that it strictly decreases as mass increases.

In Recognition Science, the framework derives the Hawking temperature from its own principles, starting from a discrete record of recognition events called a ledger. The framework's native units set c = G = ħ = k_B = 1, where the temperature takes the simple form T_H = 1/(8πM). The SI module converts this native result into kelvin using a dimensional bridge, a set of conversion factors uniquely determined by the exact SI constants and the measured G. The bridge identity shows that the SI temperature equals the native temperature evaluated at the geometrized mass GM/c², multiplied by the energy-to-temperature factor ħc/k_B.

The 8π factor in the denominator comes from the standard semiclassical derivation by Hartle and Hawking in 1976; it is not forced by the framework's own derivation chain. The framework does force a sub-leading correction to the entropy, proportional to -log φ / 2, where φ is the golden ratio, but that correction is not part of this SI module. The module also defines the Schwarzschild radius in SI units and the Page time, the time a black hole takes to radiate away a significant fraction of its mass, both as auxiliary results.

The practical consequence is that the framework's prediction for the Hawking temperature is not a free fit. It uses three exact SI constants and one measured value, G, with no adjustable dimensionless parameters. Any future direct measurement of Hawking radiation from a primordial or laboratory black hole that disagrees with the formula at the 10 percent level would falsify this prediction. No such measurement exists yet, so the formula remains a structural prediction within the framework, firmly inside the canonical Hawking band.

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

/-- 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

/-- **Track 3.A core identity**: the SI Hawking temperature is the
bridge-converted RS-native (geometrized) Hawking temperature, multiplied
by the SI energy-to-temperature factor.

`T_hawking_SI(M_SI) = T_hawking(G_SI · M_SI / c_SI²) · (ℏ_SI · c_SI / k_B_SI)`

This is the formal Track 3.A theorem: the SI prediction is the lift of
the RS-native theorem through the closed dimensional bridge. -/
theorem T_hawking_SI_eq_geom_via_bridge (M_SI : ℝ) (hM : 0 < M_SI) :
    T_hawking_SI M_SI =
      T_hawking (G_SI * M_SI / c_SI ^ 2) * (hbar_SI * c_SI / k_B_SI) := by
  unfold T_hawking_SI T_hawking
  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

What this page does not claim

The existence or stability of Hawking radiation is not proved by this module. The 8π factor in the denominator is not derived from the framework's forcing chain. The sub-leading entropy correction at one-loop is not part of this SI module.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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